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SPECIFIC HEAT MEASUREMENTS OF SOME SOLID GASES
IN A He3 CRT OS TAT
by
JOHN C. BURFORD
© John C. Burford 1967
A Thesis
submi-t-ted in partial fulfillment of the requirements for the
degree of Doctor of Philosophy in the University of Toronto.
September 1967
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V*--CONTENTS
Page
Abstract x
Chapter 1 INTRODUCTION1.1 — The Third Law of Thermodynamics-
The Residual Entropy. . ......11.2 — The Electronic Entropy of Oxygen............ .....20
Chapter 2 DESCRIPTION OF THE APPARATUS2.1 - Introduction................ 232.2 - The Main Features of the Cryostat.......... .....242 .3 - The Calorimeter............... .......... ......... 262.4 - The Heat Switch........................... ......282.5 - The Vacuum Systems....................... ..312.6 - The He 3 Systems....... 322.7 - The Gas Handling System...........................362.8 - The Electrical Systems............... 38
Chapter 3 PERFORMANCE OF THE APPARATUS ' - - —
3.1 — Preparation of the Samples....................... 463.2 — Preparations for the Specific Heat
Measurements............ . .51
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3*3 - Procedure of the Measurements «........... .58
Chapter 4 THERMOMETER CALIBRATION AND TEMPERATURE .SCALES
4*1 — Introduction......... 61*>
4.2 — Considerations of the Choice ofCalorimeter Thermometer....6l
4*3 “ Thermometer Calibration Procedure. ..... .644.4 — Discussion of Errors in the
Temperature Scales........ 694*5 — The Thermometer Calibration
Interpolation Formulas.....76
Chapter 5 THE DATA HANDLING - THE CALORIMETER AND
.. COPPER MEASUREMENTS
5.1 - Reduction of the Raw Data ........ 835*2 - The Calorimeter Heat Capacity...................85
5-3 - The Copper Measurements............ ....90
Chapter 6 THE OXYGEN AND NITROGEN RESULTS
6.1 — The Purpose of the Measurements....... 966.2 — Presentation and Discussion of the Results 98
Chapter 7 THE "CARBON MONOXIDE AND~NITRIC OXIDE RESULTS
7.1 — Presentation and Discussion of the CO results.1037.2 — Presentation and Discussion of the NO results.104
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Chapter 8 MEASUREMENTS ON THE IMPURE SAMPLES8.1 — Account of the Chronological Sequence
of the Experiments,8.2 — Discussion of the Results,..............
,106110
Chapter 9 FURTHER DISCUSSION OF THE RESULTS
9-1 — The Residual Entropy of CO and N O ..,
9-2 — The Effect of Oxygen as an Impurity.112
117
Acknowledgements
References
Appendices
1 .2.
Thermometer Calibration Data Specific Heat Results
2—A The Empty Calorimeter Results2—B The Copper Results
f
2-C The%
Oxygen Results2-D The Nitrogen Results2-E The Carbon Monoxide Results2-F The Nitric Oxide Results —2-G The CO-O2 Results2—H The ^2~^2 Results
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ABSTRACT
The specific heats of solid CO, NO, 0 , N , and some dilute mixtures of 02 in CO and N2 in the temperature range 0.6° to 4°K are reported. The measurements were made in a mechanical heat switch calorimeter in a He" cryostat to which had been added a He stage. A commercial germanium resistance thermometer was used which was calibrated against the He^ and He^ vapor pressure scales.
The specific heats of CO and NO were measured in an attempt to settle the question as to the origin of the residual entropy of these two substances. For these cases, there exists a discrepancy between the entropy calculated from spectroscopic data (Sspec) and that calculated from specific heat data (Sca^); the difference Sgpec-Sca^ being called the residual entropy. For many years, the usually accepted explanation for the appearance of a positive, finite value of the residual entropy in the cases of CO and NO has been in terms of *frozen-inT non—equilibrium states of the crystal. For these cases, it was assumed that the orientation of the molecules becomes frozen—in at a high temperature because the forces tending to produce orientational order are insufficient to overcome the high potential barriers to molecular
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rotation in the crystal. In this way, the disorder persists to
the absolute zero, resulting in the observed value of the residual entropy.
Recently, considerable doubt as to the validity of this
kind of argument has been raised, especially because molecular
rotation in solid CH^ has been demonstrated even at 1.8°K from a recent spin—lattice relaxation study. In an attempt to find a
specific heat anomaly which could remove the residual entropy,
this study of CO and NO was undertaken. No anomalous behavior in
either case was revealed down to 0.6°K. It is pointed out that
our present incomplete knowledge of molecular rotation in solids
at low temperatures needs to be improved by extensive infrared
absorption, spin—lattice relaxation, and other studies in order
to be in a better position to understand the origin of the resid
ual entropy in those few simple substances for which such an effect
persists.
During the course of this work, a sample of CO was found to
have been contaminated with CO2 and air. The specific heat
measurements on this sample revealed a rather broad anomaly, not accounted for by the Debye theory. A subsequent experiment in which more oxygen was deliberately added to the sample showed that the anomaly was caused by the oxygen impurity. A further exp
eriment in this series in which oxygen was added to a nitrogen
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host was performed and the results allowed certain conclusions
to be made regarding the origin of the anomaly. The concentrations
of oxygen were very low, generally a few tenths percent.
The results may be interpreted in terms of a model consistent
with the low-lying rotational energy levels of the oxygen molecule.
The observed anomaly was in excellent agreement with the Schottky
anomaly for a system containing two levels with a degeneracy ratio,.j—
upper to lower, of 2:1, and an energy spacing of 5.14°,
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CHAPTER 1
INTRODUCTION
1*1 — The Third Law of Thermodynamics, The Residual Entropy.
1. Historical sketch. The Third Law of Thermodynamics
has its origins in physical chemistry and it was mainly the
work of Nernst in the early 1900*s which laid the foundations
for the understanding of the Law as we know it today* Nernst
was originally interested in finding general rules of chemical
equilibria in gas reactions from the application of chemical
thermodynamics to the systems, and from these rather restricted
beginnings, the Law took shape. This section is devoted to a
brief description of the lines of thought which Nernst and
others used in developing the Law. Then the present status
of the Law will be made apparent, and finally, the case for a
re-examination of the simple gases, CO, ^ 0 and NO, which
apparently do not conform to the Law will be presented.
As a basis for predicting the conditions of chemical
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equilibrium of a gas mixture, Nernst started with the Gibbs-
Helmholtz Equation for chemical reactions at constant volume,
where F is the change in Free Energy, and U is the change in
Internal Energy.
Now if F(T) is known, then U(T) is also known, but not
vice versa, for if there is a solution of Eq. 1.1, F(T), then
any other solution of the form (F(T) + constant x T) will
also be a solution. Of course, there is only one unique F(T)
for any given system under prescribed conditions, and Nernst
felt that the most suitable point of reference for finding the
form of F(T) was at the absolute zero of temperature, since the
term (constant x T) would vanish.
is a change in the total number of degrees of freedom during
a chemical reaction. Thus, 6 f/6 T becomes infinite at the
absolute zero, and as a point of reference for finding F(T),
the absolute zero has no significance. Nernst then turned to
solids, where classically, the specific heats on both sides of
the chemical equation are equal. If this remained true down to
the absolute zero then c> U/ c> T = 0 and c> F/ c> T can remain
Eq. 1.1
In the case of gases, <b U/1) T 0 in general, since there
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finite there. Nernst then postulated that for condensed states,
both ^ F / ^ T and & U / & T become zero at the absolute zero,
I*t F/ <3 T — 6V/ T — 0 Nernst Heat TheoremT~> 0
Note that the first condition immediately gives the result that
the change in the entropy vanishes at the absolute zero, but
Nernst did not recognize this, and it was several years before
the attention of physicists was drawn to the theorem. Nernst
published this idea in 1906 and called it a Heat Theorem.
At first , the theorem was restricted to chemical reac
tions between condensed states and no account was taken of
quantum theory. Nernst assumed that the specific heats of the
condensed states remained at their classical values down to
the absolute zero. After the publication of the theorem, Nernst
set about testing it experimentally by measuring the specific
heats of various condensed substances at lower temperatures than
had been obtained previously. He developed the science of low
temperature calorimetry through his design of a hydrogen liqui-
fier and his adiabatic calorimeter. His work soon showed that
the specific heat of all substances tended to zero as the
absolute zero was approached. This was very satisfying to Nernst
because it was the simplest way of satisfying the condition
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Lt(T->0) £>U/ <^T = 0, as well as being in agreement with the
trend predicted by Einstein some years earlier using quantum
.mechanics. With this success, Nernst extended his theorem to
include systems which did not undergo chemical reactions, that
is, purely physical systems. The theorem was used to predict
that, like the specific heat, the thermal expansion tended to
zero at the absolute zero, a prediction which was later verified
by Lindemann .
There are several corollaries of the theorem which have
since found general agreement, such as the unattainability of
the absolute zero, but the statement S = 0 at the absolute zero
is of most interest here and it will be discussed in the light
of its meaning in thermodynamics and statistical mechanics.
This statement of what is now known as the Third Law is very
useful in predicting properties of the solid state in terms of
inter-molecular forces, and a good demonstration of this is
found when the law encountered its first difficulty when it was
applied to a certain class of substances.
The result S = 0 at the absolute zero found an immediate explanation in quantum mechanics, namely, that at the absolute
zero, the substance is in a state of perfect order where all
systems are in the ground state, which is non-degenerate. Now
glasses and solutions are definitely not expected to be in this
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state of perfect order at the absolute zero , and it was
interesting to Nernst to ask whether the Law could be applied
directly to such systems. Planck and Einstein stated that
such systems should have a non-zero entropy at the absolute
zero, but Nernst maintained that his Law could be applied
directly to any system, for he believed his Law to be a con
sequence of the Second Law and the vanishing specific heats,
and there was no doubt that the specific heats of all sub
stances vanished at the absolute zero, even for disordered
structures. This was one of the major difficulties with the
Law, and led Simon to propose a new formulation in 1927^ ^ stating:
At the absolute zero, the entropy differences
vanish between all those states of a system
between which reversible transitions are
possible, at least in principle.
In spite of these difficulties, it now seems certain
that the state of lowest energy at the absolute zero is one of
perfect order, in which the lattice vibrations have been frozen
out, the spins have been aligned, and so on. There has been
no general proof of this statement, except through the Third
Law, although intuitively it is satisfactory. The only
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difficulty seems to be that fox* a certain class of substances,
the perfectly ordered state is not allowed to exist in the solid
and the. solid is then in a non-equilibrium state at the absolute
zero. Glasses and solutions are in this class, and it is
generally thought that crystalline CO, N^O, and NO may also be
included. This idea will be discussed later on.
2* Verification of the Law. The most fundamental way
of verifying the Third Law is to make a comparison between the
values of the entropy of a system evaluated first by the methods
of thermodynamics and second, by the methods of statistical
mechanics. It is noted that the two approaches are fundament
ally different. On the one hand, the thermodynamic approach
describes the macroscopic variables of the system and how they
behave when the system is subjected to changes of heat and work.
On the other hand, the statistical mechanical approach is con
cerned with the microscopic picture, where the behavior of the
particles comprising the system is analysed on an atomic scale.
Now the thermodynamic state of the system is known when a small
number of macroscopic variables is specified, such as the
pressure, the volume, the temperature, the chemical composition,
and so on. However, each thermodynamic state can be realized in
many ways on an atomic scale, and one refers to the number of
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•ways of realizing the -thermodynamic state as the number of
micro—states corresponding to the given (thermodynamic) macro-
state. .If the number of different micro—states is g, then
the statistical definition of entropy is S = king, where k is
Boltzmannrs constant. Now the determination of the thermo
dynamic entropy depends on measurements of heat and temperature
(that is, on calorimetry), while the determination of the
statistical entropy depends on counting the number of micro-
states in the given macrostate. Also, the thermodynamic entropy
is obtained by accounting-for the quantities of heat which take
the system from an initial to a final thermodynamic state. Thus,
the value depends on the thermal behavior of the substance all
the, way from the initial to the final state. On the other hand,
the statistical entropy depends only on counting the number of
microstates in the macro-state at which the comparison is being
made, that is, the final thermodynamic state, and it requires
no knowledge of the existence of the solid and liquid phases if
the comparison is made in the gas phase, as is the usual practice
It is a major achievement of physics and chemistry that
the verification of the Third Law is shown by the fact that
the two approaches lead to the same result for the entropy of
most, but not all,substances. Because the Third Law is obeyed
in a great many cases, it was felt desirable to re-examine those
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few simple cases which still do not appear to conform to the
requirement S — 0 at the absolute zerOj hut before this matter»
is discussed, it is appropriate to consider in more detail
what is meant by the notion of entropy, both in thermodynamics
and in statistical mechanics, and how the different values are arrived at in practice.
of a chemically pure substance in going reversably from the
where S(0) is the entropy of the system at the absolute zero,
which, according to the Third Law, is zero for stable and metastable equilibrium states.
Experimentally, we measure the amount of heat necessary to raise the temperature of a system which comprises more than one
A'. Entropy in thermodynamics. The entropy change
T*.
A ST,
If we start at the absolute zero (Tj =0), then
T*.
0
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phase. Corrections for the higher temperature phase and the
latent heats must be made in order to obtain the specific heat
of the lower temperature phase under its saturated vapor, and
further corrections are necessary to obtain the specific heat
at constant pressure, which is required to determine the entropy
For an account of the procedure involved in making these correct
ions, one is referred to Fagerstroem*s thesis. Then the
entropy may be readily found by integrating Cp/T over the whole
experimental temperature range ^ to T2, which yields a value
of S(T2 ) «• S(Tj^). To obtain S(T^) — S(0), the specific heat is
extrapolated from T^ to the absolute zero. O f course, it is
advisable to extend the measurements to as low a ■ T^ value as
possible in order to be able to make a reliable estimate of
S(Tj^) — S(0). Then, having made our estimate of S(T^) — S(0),
we make use of the Third Law to put S(0) = 0 for all transitions
between states in stable or meta-stable equilibrium. Thus we
obtain S(T ) which may be compared with the value found from
statistical mechanics for the system in the same final thermo
dynamic state. The value of the thermodynamic entropy, S(T ),
found in this way is generally called the calorimetric entropy.
^ c a l •
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10
B. Entropy in statistical mechanics. To calculate the entropy on this basis, we must first know the detailed microscopic properties of the systems which comprise the assembly
(the terms here taking their statistical mechanical sense)* It is possible to calculate approximate values of the entropy of
any molecule theoretically by idealizing the translational,
rotational, and vibrational motions. Thus, the translational
entropy is giveli. by the Sackur-Tetrode equation,
trans Nk r^lnT - InP + In fzJTm\ 3/?k5/2 +L 2 V ix2 ) 2 J
the rotational energy is given by the expression,
Ejpob = J(J+l). J = 0,1,2,,.* the rotational21
quantum number. I is the moment
of inertia of the molecule, and the vibrational energy is given by,
Evit) = w(v + ^), v = 0,1,2,... the vibrationalquantum number. w is the vibrational frequency.
The state sum (or partition function) of the assembly is given by,
^tot ^trans* ^rot* ^vib
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where S R _3_(TlnZ)\ , where R is the gas constant.I dT /v
However, to obtain accurate values for the entropy, it
is necessary to take into account the facts that the molecule
is not rigid, and that the molecular vibrations are not harmonic
Exact entropy values may be obtained by using the exact spectro
scopically determined energy levels to find the state sum and
from that the entropy. In this way, no use is made of any
mechanical model of the molecule and the method should there
fore yield the most accurately obtainable entropy values,
provided that the degeneracies of the levels are known. However,
it has been shown that empirical formulae for expressing the
energy levels are sufficiently accurate for most purposes, a
fact which contributes a significant reduction in the work of
computation. Following the method developed by Giauque^^, the
centrifugal effects on the end-over-end rotational energies is
expressed by the following expansion formula,
Erot. = B . J (J+l) - D.J2(J+1)2 + ...
and the effect of anharmonicity on the molecular vibrational
energies is,
Evib = we-(v+2') “ xewe(v+2)2 + ...
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where the constants B,D,we,xe, ... characteristic of each
molecule are determined from the band spectrum of the gas.
Also, if the molecule is in a higher electronic state, the
rotational and vibrational constants will be altered and
account must be taken of this.
With these equations representing the more accurate
energy levels, the entropy may now be calculated. This is a
very involved process and one is referred to the work of
Giauque^^ for the details. Let it be noted here that the
essence of the method involves the calculation of the rotational
and vibrational entropies together from the combined state sums,
and then the translational entropyj which is found from the
Sackur-Tetrode equation, is added to give the total entropy of
the gas, usually evaluated at the pressure of 1 atm and at the normal boiling point. Thus, we find the statistical entropy of
the gas in the hypothetical perfect gas state. This value of
the entropy (which excludes nuclear spin) is usually called the
spectroscopic entropy. SSpec.
C. Comparison between S5pec and Scaj. At this
point, we are not quite ready to compare SSpec with Sca^ for
the gas at the pressure of 1 atm and at its normal boiling point, because Sgpec here refers to the gas in the perfect gas state,
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13
while Scal refers to the real gas. To correct for the non
ideality of the gas, it is usual practice to normalize Scal to
reduce it to the value of the gas in the perfect gas state
since this entropy value is the easier one to correct. First,
the real gas is imagined to have its pressure reduced from 1 to 0 atm where the real and perfect.gases are identical. Then,
the gas in the perfect state is imagined to be compressed back
to 1 atm. The entropy change may be readily calculated from a
knowledge of the equation of state of the real gas. With this
correction made, we are now in a position to compare Sca^ with
SSpec* Table 1.1 shows the entropy values of certain simple
molecular gases. The Sca]. values were obtained from specific
heat measurements usually down to 10 or. 15°K; the region below
these temperatures was covered by an extrapolation where the
specific heat was assumed to conform with the Debye theory.
3# The residual entropy. Reference to Table 1.1 shows
that for the last four entries, Sspec an<* ®cal no't agree. The difference between them is called either the zero—point, or the
residual entropy, Sres,
Sres = SSpec - Seal * the residual entropy.
Note that for these cases, SSpec is always greater than Sca^ .
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TABLE 1,1 COMPARISON BETWEEN Sspec AND Scal FOR SOME SIMPLE
MOLECULES
ENTROPY (E.U.)MOLECULE TEMP Sspec Scal Sres REFERENCE
02 B.Pt. 40.68 40.70 - A
N 2 B.Pt. 36.42 36.53 - B
Cl B.Pt. 51.55 51.56 - C40
HC1 B.Pt. 41.45 41.3 - D
CH. B.Pt. 36.61 36.53 ~ E4
CO B.Pt. 37-8 37-O 0.8 F
NO B.Pt. 43.75 43.03 0.72 G
N20 B.Pt. 48.50 47.36 1.14 H
H20 298°K 45.10 44.29 0.81 I
See next page for References
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REFERENCES:
A — W. F. Giauque and H. L. Johnston, J.Am.Chem.Soc. £1, 2300 (1929)
B - » and J. 0. Clayton, « » 4875 (1933)
C — ** and T. M. Powell, » n 61, 1970 (1939)
D — w and R. Wiebe, ** ** j[0, 101 (1928)
E *• A. Frank and K. Clusius, Zeit.Phys.Chem. B36. 291 (1937)F — E. K. Gill and J . A. Morrison, J.Chem.Phys. 45. 1585 (1966)
G - H. L. Johnston and W. F. Giauque, J.Am.Chem.Soc. J£l, 3194 (1929)
H - R. W. Blue and W. F. Giauque, J.Am.Chem.Soc. 991 (1935)
X - P. fic.T. <k A. T. A. JTlomSorij T,Cke*\*Phys. 33^175
NOTE: 1 E.U. (Entropy Unit) is equivalent to 1 cal/mole deg.
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Therefore, the possibility always exists that the specific
heat measurements were not extended to low enough temperatures
and that an unforeseen specific heat contribution, not accounted
for by the Debye—type extrapolation, may have been overlooked.
For example, in the case of CH^, the first specific heat measure
ments, which extended only down to the liquid hydrogen temper
ature range ^), gave no hint whatever of the low temperature
anomaly which was found in later measurements to lower temper
atures^^. The anomaly which was found completely removed the
apparent disagreement between SSpec and Scai and in order to
reach agreement it was necessary to allow for the existence of
different nuclear spin species of the molecule. This was also true for hydrogen^).
A .second.type of possibility may arise which results in a
finite value of Sres and that is by the display of what is usually
called *frozen«in* disorder, where certain internal states of
the crystal are prevented from attaining equilibrium. A typical
case is a glass, which may be thought of as a rigid super-cooled
liquid which is prevented from crystallizing by the strong internal forces. Thus, even taking specific heat measurements down to the absolute zero on such a substance would yield a
value of Scai which would be too small by an amount equal to the
degree of frozen—in disorder, provided that the disorder
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persisted down, to the absolute zero. If so^ then the substance
would not be expected to obey the Third Law which covers only
equilibrium states. This kind of explanation has been used
frequently to explain the residual entropy in CO, 3X141although there has been very little experimental evidence to
* confirm this picture in these cases. Because the types of
crystal disorder which are put forward to explain the residual
entropy in the cases of CO (and N2O) and of NO are different, they will now be discussed separately.
A. Carbon monoxide. The idea in this case is
that as the temperature of the crystal is lowered from the
freezing point, the molecules eventually cease their rotation
at a temperature where the thermal energy is of much the same
value as the height of the potential barriers which oppose
rotation. Now if the ends of the molecule are quite *similart,
then the energy of re«orientation of a molecule in the crystal
will be small and would probably be less than the height of the
potential barriers opposing rotation. In this case, since
rotation is no longer possible, the forces tending to produce
orientational order will be unsufficient to remove the disorder,
which was essentially frozen-in when the rotation died out. If
there is complete orientational disorder, then the residual entropy
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16
would be Rln2 = I .38 E.U.. The fact that the observed residual
entropy is less than Rln2 for the case of CO is rationalized
by saying that a partial ordering takes place before re
orientation becomes impossible.
This explanation is commonly put forward for molecules
which have ’similar* (but not identical) ends (see, for example,
pg 63 of Wilks^®^), and it is thought that there is some support for the argument by the fact that molecules with very ’different*
ends do. not display a residual entropy. However, besides the
thorny question as to the meaning of the terms ’similar* and
’different* in this context, the whole acceptance of the argument
rests on the assumption that the CO molecules may not re—orient
themselves at low temperatures. If it can be shown that some
degree of rotation is possible, then the argument cannot hold
water.
Gill and Morrison have examined the various evidence
relating to the question of orientational disorder in crystalline
CO. Unfortunately, there have been no spin—lattice relaxation
measurements on CO which could tell us directly the state of
molecular rotation at low temperatures. However, Gill and
Morrison’s own capacitance data down to 6°K indicates that there is no change in the orientational motion of the molecules doivn
to this temperature. Because of the scant information regarding
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the state of rotation in crystalline CO, it seemed worthwhile
to Gill and Morrison to extend the specific heat data of Clayton
and G i a u q u e t o lower temperatures especially because thermo
dynamic and spin-lattice relaxation measurements on crystalline
methane had shown that the usual explanation of the residual
entropy in this case, which was based on frozen—in disorder, was
incorrect. In addition, because methane was found to be able to
rotate fairly readily even down to 1.8°K, and because the
characteristic temperatures of methane and CO are comparable
(indicating a similarity in the effects of the inter-molecular
forces), then it was thought that a similar state of rotation
might appear in the two crystals.
The specific heat data of Gill and Morrison down to 2. 5°K
failed to show an anomalous behavior which would help explain
the residual entropy(9). for this reason, and also because a
third crystal structure modification was suspected to be dis
played at very low temperatures the present work, which
extended the specific heat data to still lower temperatures,
was undertaken. The results of this research are presented and
discussed in Sec. 7.1 and in Sec. 9.1.
B. Nitric oxide. One explanation of the residual
entropy of nitric oxide is as f o l l o w s ^ . Solid nitric oxide
may be considered as a solution of two polymerized isomers
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(dimers) which are present in equal amounts. Now liquid nitric
oxide is known to be highly associated into N202 dimers and it is reasonable to suppose that the dimerization persists into
the solid state. If the structure of the dimers can take one
of two different forms,
N N or, N O
0 0 O N
then during solidification, since the energy difference between
the two isomers may be assumed small, the dimers will be present
in equal amounts in the two forms. Since the higher energy
dimer is not able to convert to the other form, then the disorder
persists to the absolute zero. For complete disorder, the
residual entropy would appear to be ^Rln2 per mole of NO (or
Rln2 per mole of N2O2). This value is in very close agreement(12 )with the observed value found by Johnston and Giauque .
However, this explanation may be criticized because it assumes
that all the dimers of each kind are completely oriented, for
the entropy of orientational disorder is neglected and only the
entropy of mixing is considered.
An alternative explanation which considers only the
orientational disorder is as follows (-*-3). The nitric oxide
dimer is assumed to have only one isomeric form,
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N 00 N
where, in the crystal, the dimer can take one of two different
orientations. Complete orientational disorder is obtained when
there is no preferred orientation, and where the residual entropy
has the value ^Rln2 per mole of NO. Again, this kind of explan
ation rests on the assumption that the dimers are not able to
re-orient themselves in the lattice. As for the case of CO
(and N2O), this involves the question of molecular rotation in solids at low temperatures; a field which has been largely un
explored. However, some justification for the above viewpoint
has been provided by X-ray studies ^4)^ although this kind of
evidence may not be regarded as entirely convincing. The problem
of dimerization is one which has plagued the interpretation of
the physical behavior of solid oxygen for many years^^, and
since nitric oxide has received less attention .than oxygen in
this matter, we prefer to leave the question open and to investi
gate the behavior of the specific heat at temperatures below those
previously obtained in an attempt to discover a specific heat
anomaly which would remove the residual entropy. For this reason,
specific heat measurements on solid nitric oxide were made and
the results are presented and discussed in Sec. 7*2 and Sec. 9»1«
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20
1.2 - The Electronic Entropy of Oxygen.
In the course of this work, some of the gas samples
were found to have been contaminated with air, which led to a
study of the effects of oxygen impurity. In order that the
results from the experiments on the dilute oxygen-CO and
systems may be interpreted, we must consider the molecular
structure of the oxygen molecule - in its electronic ground state.
It is believed that the specific heat results may be interpreted
in terms of a model which contains low-lying energy levels of
the molecule, the populations of which are altered by a change
in temperature, leading to the observed specific heat bumps.3 ~The electronic ground state of the molecule is /
c(see, for example, Hersberg Chapter S) and because the
nuclei have no spin, symmetry requirements of the total wave
function impose the condition that the molecular rotation quantum
number K may take only odd integer values (see, for example, Herzberg (-*-6) page ,i30). Each rotational level is split into
three levels by the interaction of the spin S with the rotation
of the molecule (Hund*s case (b)). While these levels were
originally called F levels by Mulliken, today they are sometimes
called the Kramers—Schiapp levels.
A theoretical treatment of the Kramers-Schlapp levels was
first given by Kramers^^ who considered only the spin-spin
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interaction and showed it to be equivalent to a coupling between
the spin and the inter-molecular axis of the molecule. To improve agreement with experiment, S c h l a p p ( ! 8 ) included a term which took
into account the interaction between S and the magnetic field
produced by the molecular rotation. As a further refinement, Mizushima and Hill 9) considered the effect of centrifugal dis
tortion of the molecule and obtained still better agreement with
experiment. Thus, it is thought that the origin of the Kramers-
Schlapp levels is well understood.
The only rotational state of interest here is the ground
(K = l) state, where the total angular momentum quantum number
J ( = K + S) can take one of three values, 2, 1, or 0, correspond
ing to the three different orientations of S with respect to K.
Fig. 1.1 shows the ground state of the free O16O16 molecule.At very low temperatures, only the J = 0 state will be
occupied but as the temperature is raised the upper states will
become gradually filled. This process is accompanied by a Schottky
anomaly in the specific heat, the exact form of which may be
readily calculated from a knowledge of only the spacings of the energy levels and their degeneracies. The general expression for
the Schottky specific heat is,
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6P» 5.704° K
£ « 2.702° K
J = 0K =1
FIG. 1.1 TRIPLET SPLITTING OF THE 0,60|g ROTATIONAL GROUND STATE (K = I) DUE TO THE INTERACTION BETWEEN K AND THE SPIN S = l
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22
—^Schottky ~ iL_dT
fc=rvR S,E±Si exp(-Ei/kT)I" o ____________L*-n2 Si exp(- £±/kT)
where R is the gas constant, and gj is the degeneracy of the
i*th energy level, £ The entropy under a Schottky anomaly
may be written at once since it depends only on the degeneraciesof the levels. Following Rosenberg
for the Schottky entropy is,
(20), the general expression
SSchottky - R In 2t«0
where gc is the degeneracy of the ground state. Now the degeneracy
of the Kramers-Schlapp levels is given by (2J+1), since there are
(2J+1) distinct quantum states of the molecule corresponding to
the same J value. Thus, for the ground state triplet of the
molecule, the Schottky entropy is S = Rln9 per mole.
It must be remembered that in these discussions we have
referred to the molecule in the free state. However, when we
come to discuss the behavior of the energy levels when the molecule
is in close proximity to other molecules in the solid state, then
the energy level scheme (and hence the size and shape of the
Schottky anomaly) may have to be modified because of the influence
of the internal crystal fields. This matter is discussed in the
light of the results on the dilute oxygen mixtures in Sec. 9*2.
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23
CHAPTER 2
DESCRIPTION OF THE APPARATUS
2.1 - Introduction .Essentially, the apparatus consisted of a conventional
low temperature He^ calorimeter cryostat to which was added a He^ stage which was included in order to obtain temperatures
below those normally available in He^ cryostats. Despite its
basic simplicity, the apparatus is rather unusual because it is
thought to be the first cryostat which was designed to measure,
in the He^ temperature range, the heat capacity of substances
which are gaseous at room temperature, other than the inert gas
solids. Most other solid gas calorimeter cryostats have been
designed for use in the interval from room temperature down to
liquid hydrogen temperatures, with occasional investigations in
the He4 temperature range. However, in view of the resurgence of
interest in solid gases which display a residual entropy, it
became desirable to investigate the thermal properties of solid
gases at much lower temperatures than were previously obtained with
them.
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24
This chapter is devoted to a description of the apparatus.
First, the function of the main parts of the cryostat will be
described in relationship to each other, and then a more detailed
description of the individual components will be given.
2.2 — The main features of the cryostat.
Fig. 2.1 shows a diagram of the low temperature part of the
apparatus and Fig. 2.2 shows a photograph of the same part.
Referring to the Figures, the heart of the cryostat is the calor
imeter (see Sec. 2.3) which was suspended from its top and bottom
inside the copper shield by eight thin stranded terylene threads
(CoatsT Koban) which passed from the calorimeter to two brass
suspension rings which were mounted on four studded brass posts
fixed to the underside of the shield cap. The attitude of the
calorimeter could be adjusted by means of the pairs of nuts which
secured separately each of the four sides of the suspension rings
onto the posts.
The calorimeter was cooled through the heat switch (see Sec.
2.4) which provided a thermal link between the calorimeter and the He^ refrigerant chamber. The chamber was made of copper and was
an integral part of the shield system, since the chamber and the
shield cap were hard-soldered together. The use of hard solder
instead of the usual soft solder was made in order to reduce
temperature gradients in the shield which might have arisen if
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He^pumping tube
Thermol shunt (copper) for gas filling tube
Terminol strip for electrical leads-----------Shield strutsGas filling tube vacuum jacket
Shield cop
Gas filling tube (A)Gas filling tube (B)
Shield con (copper)
Thread tension adjustment nuts
Colorimeter suspension rings (brass)
Interspace pumping tube
— Thermal shunt (copper) for He 3 vapor pressure tube
‘"—Shield pumping tube and vapor pressure tube
dd vacuum jacket
r f ' _ Outer con capHeat switch bellows
Fusite seals
He^vaporpressuresensingtubeHe3 vapor pressure cell(copper)
refrigerantchamber(copper)
Heat switchclampingplate
Outer can (brass)ColorimeterStudded posts (bross)Terylenesuspensionthreads
FIG. 2.1 THE CRYOSTAT
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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soft solder had been used. This is so because the thermal
conductivity of soft solder is very much reduced when it becomes
superconducting, a phenomenon which does not occur in the hard
solders. Provision was made for using the chamber for a magnetic
thermometer where the susceptibility of a paramagnetic salt
(which could be submerged in the He3 liquid inside the chamber)
could be measured with a set of coils mounted on the outside of
the chamber. The magnetic thermometer was not used in this work.
Mechanical support for the shield was provided by three l/8” thin- wall struts which were placed between the shield and outer can
caps. The struts helped to reduce the strain on the soldered
joints of the shield cap when the heat switch was in operation.
A manganin heater of about 500 ohms was wound non-inductively
around the shield can and a carbon radio resistor, which was used
as the shield thermometer, was placed on the shield cap with a
generous coating of vacuum grease to improve the thermal contact
between the thermometer and the shield.
Connected to the side of the He^ refrigerant chamber was
a He^ vapor pressure ceil which was also made of copper and was
hard-soldered to the chamber. The vapor pressure sensing tube,
which led from the cell, and the gas filling tube which led from
the calorimeter, were both vacuum-jacketed. Part of the way up
the vacuum jackets from the outer can cap, the tubes passed through
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separate copper thermal shunts which were included to provide
a path of escape for the heat coming down the tubes and their
vacuum jackets into the He^ bath which sat in the inner Dewar.
The brass outer can and the copper shield can were mounted
to their respective caps using indium . *0 *-rings, according to the usual practice in this laboratory* The cryostat was mounted
inside two glass Dewars, the inner one containing liquid He^ and
the outer one containing liquid nitrogen.
2.3 - The Calorimeter.
The calorimeter (Fig. 2 ,3) was made almost entirely of copper. About 25 thin copper posts were located inside the body
of the calorimeter and hard soldered to the top and bottom to
assist in the rapid attainment of thermal equilibrium during heat
capacity determinations. A manganin heater (resistance about
500 ohms) was wound non-inductively around the side and was therm ally bonded to it by the usual method of using vacuum grease and
adhesive. The four heater leads were connected in pairs (current
and potential) to each end. of the heater wire. The soldered
electrical connections, which were separately coated with nail
varnish to provide electrical insulation, were anchored to the
side of the calorimeter by another layer of nail varnish which
was coated on a thin strip of cigarette paper. The germanium
resistance thermometer was screwed into a threaded well in the
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Gas fHIing tube
Heater leods
Paper insulation
me®
Heat switch clamping plate
Heat switch plate support arm
Copper posts
Copper wall
Mangonin heater
Thermometerwell Copper posts
Suspension lugs— 8 o ff -
Germaniumthermometer
FIG. 2.3 THE CALORIMETERReproduced with permission of the copyright owner. Further reproduction prohibited without permission.
base of the calorimeter, the threads of which were coated with
vacuum grease just before the thermometer was screwed into place.
The four thermometer leads were treated in the same way as those
of the heater. This procedure of thermally anchoring the lead
connections to the calorimeter helped prevent thermo-electric
emffs from entering the circuits to spoil the electrical measure
ments. In addition, because the thermometer leads from the ger
manium element were rather thick stranded copper (32 gauge), the total amount of the calorimeter system which was considered taking
part in the heat capacity determinations was more closely defined
when the complete length of copper leads was isothermal with the
calorimeter.
The heat switch plate was a piece of copper which was gold-
plated (see Sec. 2.4)• The switch plate was soldered with a small
quantity of Wood*s metal to the copper heat switch plate supporti ■»arm, which was itself hard—soldered to the calorimeter top. The
switch plate soldering was done with the calorimeter suspension
in tight adjustment and with the switch plate held rigidly between
the jaws. In this way, there resulted a minimum degree of vib
rations when the heat switch was opened, for the plate was then
centered perfectly inside the jaws. This method proved to be
much more convenient than trying to adjust the calorimeter
position with the switch plate already soldered to its arm.
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Wood’s metal was used for the solder because of its low melting
point. It was feared that the use of a tin-lead solder might
perhaps have caused some of the other soldered joints on the cal
orimeter to break.
2".4 ~ The Heat Switch.
In a calorimeter cryostat, there must be some mechanism
which provides for the cooling of the calorimeter in a reasonable
time from room temperature, and then isolates it for the specific
heat measurements. Also, a direct thermal link between the
calorimeter and the vapor pressure cell must be established for
the purpose of the thermometer calibration. Such a mechanism
is generally called a heat switch and there are two main basic
designs in common use which satisfy these needs. One is by the
use of exchange gas (usually helium), and the other is by using
a mechanical switch which makes and breaks thermal contact between
the calorimeter and its cooling agent. The main disadvantage of
the former method is connected with the sorption effects of helium
gas, and because the undesirable features drastically increase in
importance below 1°K, it was decided to incorporate a mechanical
switch even though it too had an undesirable feature which was
important below 1° as well as posing major design difficulties.The switch is shown in Fig. 2.4 and the actual operation of
it in an experiment is described in Sec. 3*1* The construction of
the mechanism down to the jaws closely follows the design of
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Interspace pumping hole
Stainless steel bellows
Heat switch clamping plate attached to calorim eter
Interspace pumping tube itch roo guideand heat swii
Outer can cap
Rotation of rod produces vertical bellows motion
Shield cap
Jow mechanism support tube
»
Heat switch jaws
Copper braid
Braid ends soldered to He3 vapor pressure cell
FIG 2.4 THE HEAT SWITCHReproduced with permission of the copyright owner. Further reproduction prohibited without permission.
29
(3)FagGrstroem to whose thesis one is referred for a description,
of the entire switch down to this point. However, it was felt
that the method used by Fagerstroem of pressing the calorimeter
down onto the shield floor and raising it for isolation would
have introduced too much heat to the calorimeter by both friction
and vibration effects. The heating effects of friction result
when the heat switch is being opened, that is, when the touching
surfaces are made to part. The heating effects of vibration are
induced after the switch has been opened, where the calorimeter
is set into vibrational motion in its suspension. Therefore, it
was decided to keep the calorimeter firmly fixed in its suspension
and let the mechanism work movable jaws. In this way, it was hoped
to minimize the heating effects of friction which become important
below 1°K. The effort proved later to be successful ; whereas for
Fagerstroem*s switch the frictional heat imputs were of the order
1000 ergs, the switch used in this work gave heat imputs of the order 10 ergs. Also, by using a more rigid suspension system
than that of Fagerstroem, the temperature drifts due to the ext
ernal vibrations were reduced, but still remained the major factor
which limited the precision of the heat capacity measurements
below 1°K.The jaws work on the »lazy tongs * principle which has been
used before in this a p p l i c a t i o n ^2). They were opened and
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30
closed by rotating a switch knob at the cryostat head. The touching
surfaces of the jaws and clamping plate were gold-plated. In( 2 )earlier designs of the switchv indium was used in place of
gold for it seemed to offer the advantage of providing a large
area of contact because of its malleability. However, it has been
recently pointed out^22 that to obtain full advantage of this it
is necessary to use a thick indium layer, in which case the poor
thermal conductivity of the superconducting indium would defeat
t le purpose of increasing the switch conductance. Also, it has
been shown(24) that the conductance of a switch depends mainly on the pressure across the jaws and this is independent of the
materials used in its construction. The reason for gold-plating
the switch was to prevent the copper from being oxidized which
would have caused the switch performance to deteriorate because
of the steady accumulation of the poorly conducting oxide over a
period of time.
Each of the switch jaws was thermally anchored separately
to the vapor pressure cell by lengths of copper braid which were
hard-soldered at each end. In this way, no use was made of the
rather poor conductance of the switch mechanism to effect thermal
equilibrium between the calorimeter and the shield. This was of
special importance when calibrating the thermometer because of the
necessity of keeping the conductance of the thermal link between
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the calorimeter thermometer* and the vapor pressure cell as large as possible.
2.5 ~ The Vacuum Systems.
The space surrounding the calorimeter, that is, the shield
system, could be evacuated before pre-cooling the cryostat through
a pumping tube which also served as the vacuum Jacket for the He^
vapor pressure sensing tube. The shield system needed to be
pumped only to a rough vacuum before a run, because at low temp
eratures the residual air inside the system was frozen out. How
ever, the. system had., to be leak-tight to a very high degree
particularly to the interspace (the space between the shield and
the outer can) because if any exchange gas had come into contact
with the inside of the shield system at low temperatures, then
the specific heat results would have been in serious error because
of the sorption effects df the helium gas.
The interspace could also be evacuated but here the vacuum
requirements were much more strict. The need to evacuate the
interspace of helium exchange gas at 4»2°K in a reasonable time
determined the choice of a high speed pumping system. To provide
adequate evacuation, the pumping line from the interspace (which
also served as the guide for the heat switch rod) was a 3/8,r thin-
wall monel tube which was connected to a 1” streamline copper tube
through a coupling at the cryostat head. From there, the ln line
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went to a valve with a large (about 1«) flow-through diameter.The other side of the valve was connected to a 2^”-throat oil
diffusion pvunp (Edwards Type 203) by a short length of 2" stream
line pipe. The total pumping length from the diffusion pump to
the cryostat head was about four feet, while the length of 3/8” tube inside the cryostat was about two feet. However, with a
considerable part of the latter tube being at a low temperature
throughout a run, the pumping speed of the system was not impaired
very much by its relative narrowness.
A cold cathode vacuum gauge was mounted in the pumping line
ahead of the valve. The gauge was used to monitor the exchange
gas pressure in the interspace. Provision was made for admitting
the exchange gas from the laboratory helium return line through
a by-pass line which was fitted with a needle valve.
2,6 — The Helium Three Systems.
1. The vapor pressure He3 system. The vapor pressure
system is rather simple . The vapor pressure cell was packed
with copper lathe turnings which helped improve the thermal
equilibrium inside it (see Sec. 4»4)« It is estimated that about
one half of the volume inside the cell was available for occup
ation by the He^ liquid, that is, about % ml. The quantity of
He^ inside the system was such that at the lowest temperature,
where the amount of liquid inside the cell was greatest, about
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\ ml of liquid was formed. The pressure sensing tube leading
from the cell was l/l6” thin-wall stainless steel and it passed through the thermal shunt on its way up through the cryostat head
and on to the supply tank and manometer, from which was obtainede
the vapor pressure reading.
The manometer limbs were totally enclosed in a wooden case
which served to avoid sudden changes of temperature of the mercury
in the limbs* There was a perspex window in the front and a
fluorescent lamp in the back of the case. Mounted immediately
behind the limbs was a strip of fly-screening (with a mesh of
about 7 lines per cm) which helped to delineate more clearly the mercury meniscus when it was viewed through the cathetometer
telescope. A ground glass screen was placed in front of the
lamp to provide an even illumination of the meniscus over the whole
pressure range. To obtain a reliable reference pressure in one
of the limbs, a mercury diffusion pump with a cold trap was
connected to it. The reference pressure was then taken as zero
and the vapor pressure was given by the difference in height of
the two mercury levels, which were measured with a cathetometer
manufactured by Dumoulin-Froment, Paris.
In order to obtain accurate pressure readings, the density,
and hence the temperature of the mercury in the limbs had to be
known. For this purpose, a mercury—in—glass thermometer was
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placed in a test-tube of mercury which was situated inside the
manometer case* The test-tube was made from the same piece of
glass from which the manometer limbs were taken* The limbs were
made from one length of 18 mm pyrex which w;as specially selected from the stock because it had the most perfect bore of all the
pieces. Precautions to find a ,good tube were necessary for two
reasons. First, the pressure readings can be spoiled by optical
refraction effects which occur in tubes which do not have uniform
wall thickness* Second, if the inner diameter is not.uniform
along the tube length, or if the tube is not perfectly round
everywhere inside, then the meniscus depression due to surface
tension may not be the same in either limb. Again, this may
result in errors in the measured pressure since the meniscus
depression was assumed to be the same in both limbs* To test
the bore roundness, several readings of the inner diameter at
each end were taken with a travelling microscope, and to test
the uniformity of the bore along the length of the tube, the
inner diameter readings from each end were compared.
The mercury used in the manometer was obtained from a
commercial source. When received, every bottle shotted a surface
scum despite the label *triple distilledT. To obtain a sample
which was suitable for manometer use, a quantity of the stock
was triple distilled again twice more, and kept under its own
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
35
vapor until the manometer was ready for filling. To maintain
this high level of cleanliness, the manometer limbs were
carefully cleaned by the usual technique using chromic acid, methyl hydrate, and distilled water.
2. The refrigerant He3 system. The refrigerant He3
system is a little more complex since the liquid He3 inside the chamber was pumped upon during a run. The 1” pumping line was
made from thin—wall monel and led up from the outer can to the
cryostat head where it was coupled by a short length of 2” streamline pipe to an oil diffusion pump (Edwards Type 203). The diff
usion pump was backed by a rotary roughing pump which was sub
merged in an oil bath in order to prevent leakage of the precious
He3 gas through the seals in the drive wheel shaft of the pump.The exhaust of the rotary pump went to aluminum storage tanks
fitted with valves which were closed to conserve the gas when the
system was being leak-tested. The rotary pump exhaust was fitted
with a dial pressure gauge which was used to give- an indication
of the amount of liquid He3 remaining in the refrigerant chamber during a run^ as well as to indicate leaks in the system between
runs.
The pumping tube was fitted with two radiation traps, one
at each of the can caps. The traps prevented room temperature
radiation from reaching the shield. If this precaution were not
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taken, then a considerable heating of the low temperature
section might have occurred*
An attempt was made to isolate the cryostat from the
effects of the vibrations of the rotary pump. For this purpose,
the pump was mounted on its own stand separate from the apparatus
frame and some Sylphon bellows were placed in the intake and
exhaust lines. With these precautions the vibrations reaching
the cryostat head were found (by touch of the hand) to be quite
small. However, they were important enough to produce a notice
able heating of the calorimeter system during a run.
2.7 “ The Gas Handling System.
The sample gas handling system is almost identical to that
described by Fagerstroem in his Ph.D. Thesis from this laboratory
The central part of the system, namely, the calibrated volume
gas reservoirj was also part of this apparatus and will not be
described. The only addition was a dial pressure gauge which
indicated the vapor pressure of the condensed sample in the
calorimeter when the supply of gas from the calibrated tank was
shut off. It is noted here that the sample gas was cooled and
condensed in the calorimeter by providing a supply of liquid
nitrogen inside the inner I3ewar (a discussion of the experimental procedure is made in Sec. 3*1)* The gas filling tube leading
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37
down from the cryostat head was made up from two tubes of
different diameters (A and B, Fig. 2,1), Initially, tube A
was a length of 0.040" 0D, 0,004" wall and tube B was a length
of 0.022" OD, 0,003" wall, both copper-nickel capillary. When
it was found that these tubes were too narrow to allow the sample
gas to collect as condensate in the calorimeter in reasonable
times, they were changed in favor of wider tubes. The new tubes
are: A - 1/8" 0D, 0,005" wall stainless steelj B - 0,040" OD,
0.004" wall copper-nickel. In addition, the new tube B (total
length, three feet) was coiled in order to increase the path
length for the heat being conducted down it to the calorimeter^
and was heated by a manganin heater (about 10 ohms) which was wound around the tube along its entire length. The heater was
used to accelerate the collection of condensed gas in the calor*?
imeter as well as to reduce the amount of condensate left in the
tube just before cooling the apparatus in preparation for an
experiment. Of course, with these wider tubes the heat leak due
to conduction down them was increased, This is always a consider
ation which must be weighed against the advantages when designing
calorimetric apparatus.
The thermal shunt through which the gas filling tube passed,
and the filling tube vacuum jacket above this point, were provided
with a manganin heater which was used to avoid the premature
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
condensation of the gas in the upper regions of the filling
tube. Such an occurrence would have plugged up the tube and
prevented any further filling of the calorimeter. Furthermore,
a quantity of condensed gas inside the filling tube would have
resulted in an over-estimate of the quantity actually inside the
calorimeter. While an accumulation of liquid here was not too
serious, it was found in the experiments on NO (which has itsto
normal freezing point above the liquid nitrogen boiling point),
that an accumulation of solid anywhere in the tube had to be
avoided because of the difficulty of unblocking the tube contain
ing solid. The temperature of the shunt was monitored with a
copper-eonstantan thermocouple.
2,8 — The Electrical Systems.
1. The germanium thermometer resistance measuring apparatus.
The resistance of the germanium thermometer was measured with an
isolating potential comparator, the design of which was first
developed by Dauphinee. The comparator used in this work closely
follows his design and to his original paper one is referred for
a full analysis of the p r i n c i p l e ^ . The outline of the basic
principle is as follows, reference being made to Fig. 2,5«The mechanical chopper C first connects the condenser A to
emf 1, where it is charged to the voltage The chopper contacts
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C mechanical chopper A condenser G galvanometer
- (a) BASIC COMPAR ATOR -PRINCIPLE
FIG 2 .5 THE ISOLATING COMPARATOR CIRCUIT
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45 volt *4r dry cell
Ammeter
w£z5
M |
C |
c2M, , M p , M s , double-^pole, double-throw mechanical
choppers driven at 3 5 c p s .
C|* C2» high quality polystyrene condensers.W standard resistance box S thermometer
(b) BASIC MEASURING CIRCUIT
FIG. 2;5 THE ISOLATING COMPARATORCIRCUIT
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
open and then close to the other side connecting the condenser
(with the same polarity) to emf 2 through the galvanometer G.
For e^ equal to e2, the total emf round the lopp is zero and
no current flows. If, however, e^ and e2 are not equal, thecondenser charges and discharges through the galvanometer since
the voltage across the condenser alternates between ej and e2.
This process repeats many times a second, since the chopper is
driven at a frequency of 35 cps, and if the galvanometer has asufficiently long time constant, there will appear on it a steady
deflection, indicating an out-of-balance condition. The emf*s
are never directly connected to each other} they are truly isolated *
and a considerable potential difference (V) may exist between
them without difficulty. At balance, there is no current flowing
in the potential leads and their resistance does not affect the
balance condition.
In the actual instrument used here, the voltages to be
compared were those across the thermometer and a reference stand
ard decade resistance box. The chopper chops the current
through the decade box and thermometer into a square wave AC
current. The choppers, and are connected to W and S as
shownj the condensers Cj and C2 bridging their contacts. Choppers
and M2 are driven synchronously with and are phased in such
a way that when the current through W and S is flowing in one
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
direction* condenser is connected to resistor W and condenser
C2 to S. When the current is reversed, the connections are
reversed also.
If we consider the action of the chopper M. (forgetting
for a moment the condenser C^), while is connected to W, it
charges to the voltage V .. On the next half of the cycle, the
chopper transfers to S, the current meanwhile being reversed.
When connected to S, C-j_ charges or discharges through the galvan
ometer G until it has the voltage -Vg. As this process is repeated
35 times per second, the difference between Vw and -Vs results in a pulsing current i^ through the galvanometer. Similarly, the
operation of M 2 results in a corresponding current i2 if Vw isnot equal to If we assume that for the moment there aresno thermal emf*s or other accidental voltages present in Vg or Vw ,
then any difference AVi = V - V giving rise to current i-, isW 6accompanied by a corresponding difference A V 2 = “*(VW - Vg) which in turn gives rise to the current i2 «
The pulsing currents flow in opposite directions and since
they have a phase difference of 180° in the chopper cycle, they add up to an alternating current i^ + 1 2 -*-n "the potential leads,
which is not blocked by the condenser The desired balance,
where A v ± = A v £ = 0, and hence = ±2 = 0, is indicated by a
null reading on the galvanometer. If now we allow thermal emfTs
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in Vw or Vg, the balance condition is not affected; such emf *s
being essentially DC send direct current through C^, which can
only persist until is charged.
The instrument is now seen to combine the advantages of
AC and DC operation, for thermal emf*s are automatically balanced-
out, and on the other hand, it uses a DC galvanometer as null
indicator and the lead resistance does not affect the balance, both
features being characteristic of true potentiometric (DC) methods.
Also, because the voltages are compared under equilibrium cond
itions there is no reactive balancing and so only one reading
(from the resistance box) is required. For these reasons, it
was decided to use this instrument in place of the potentiometer
which has been used almost exclusively in this laboratory for the
purpose of accurate resistance thermometry. In view of its con
siderable advantages, it is surprising that the use of the instru
ment is not reported more often in the literature of low temperature
calorimetry. As included in the advantages mentioned above, the
fact that only one reading was necessary for each determination of
resistance led to a considerable reduction of work during a run,
end as such, represented one of the main factors which influenced
the choice of it for the thermometer measuring system.
The resistance decade box used for the standard reference in the comparator was a General Radio Type 1432-x with six decade
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ranges from 0.1.Q- to 10K.fi-. To allow for the sensitivity ( IdR \V RdT )
of the germanium thermometer, it was necessary to read resistance
steps smaller than 0.1 ohm. For this reason, as well as to
provide a permanent continuous record of the thermometer resist
ance at all times during a run, the out of balance voltage across
the galvanometer was fed to an amplifier, the output of which
was taken to a Leeds and Northrup chart recorder. A rough balance
was first made by adjusting the dials on the decade box, and a
small correction to it was obtained from the trace on the chart
recorder in the usual way.
There was AC pick-up (mostly 60 Hz) in the circuit and this
was shown by a wavy trace on the chart. It was found necessary to
shield the current and potential leads in their separate pairs in
order to reduce the pick-up to an acceptable level. The lead pairs
were twisted together and taken through two l/l6” thin-wall tuhes leading from just inside the cryostat head down to the terminal
strip mounted just above the outer can. There was no attempt to
shield the leads from the terminal strip down to the thermometer.
However, the leads from the cryostat head (which passed through
9~pin kovar seals there) to the terminal strip outside the cryostat
were shielded. The leads f ro m this terminal strip to the comparator
were shielded cable. The shields w ere all inter-connected with a
copper w ir e which then passed to g ro u n d . In this w ay, by g ro u n d in g
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the shields at only one common point, the possibility of ground
loops was prevented.
The lower limit of detection by the instrument was deter
mined by the noise ievel as displayed on the chart trace (most of
it actually originated in the comparator amplifier). Under normal
operating conditions, it was found, using a measuring current of
1.5 micro-amps, that at4«2°K, where the thermometer resistance was of the order 100 ohms, the noise level was 15 nanovolts. At
0.7°K, where the thermometer resistance was of the order 1000 ohms, the noise level was found to be 150 nanovolts. These levels
correspond to resistance fluctuations of the order 0.01# which correspond in turn to fluctuations in temperature, limiting the
precision of the estimation of the temperature drifts (see Sec. 3*3)*
2. The heater current supply. The specimen heater current
supply is shown in Fig. 2.6. Essentially, it consisted of a source
of emf (a 6-volt car battery) which fed a DC current through the voltage dividers V. By means of the switch S-j_, the current range
was selected, the continuous current adjustment within each range
being made by the potentiometers P. The switch S2 allowed current to flow through either the specimen heater or a dummy heater which
was mounted outside the cryostat. When the specimen had been
heated for a specific heat determination, the current was switched through the dummy heater. This procedure helped stablize the
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o o o o
LsJ o
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FIG
.2.6
HEAT
ER
CURR
ENT
SUPP
LY
current for the following specimen heating cycle. The switch S~w
allowed the electric clock C (Standard Electric Time Co., Type S10)
to be turned on and off synchronously with the heater current. The
potential difference across the heater was measured with a Tinsley
microvolt potentiometer. The heater resistance was obtained by
comparing the potential difference across the standard 500 ohm Kelvin wire-xyound resistor ¥ with that across the heater, the two
resistors being connected in series. The standard resistor was
stabilized by immersing it in an oil bath.
The leads from the specimen heater to the- terminal strip
above the outer can were lead—plated manganin. For the section
of the leads inside the shield, each lead was made up from about
three feet of wire and was coiled in the usual way. This kind of
wire was used because it had small thermal conductance and also
because the lead coating prevented any joule heating in the heater
current leads which might have spoiled the specific heat measure
ments. The thermometer leads were also made of the same kind of
wire and were coiled in the same way.
3. Other circuits. The shield carbon thermometer current
and potential leads were of un-plated manganin and led up from the
thermometer through the pusite seals in the outer can cap and on
to the terminal strip where they joined other manganin wires lead
ing outside the cryostat to the external circuit. The thermometer
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circuit was very straight-forward, for it consisted essentially
of an emf source (a 6-volt car battery) connected to the series combination of the thermometer, a standard resistor^ and a voltage
divider (potentiometer). The resistance of the thermometer was
obtained by noting the potential differences across the standard
resistor and the thermometer.
The gas filling tube shunt heater was powered with AC current
from the mains, the voltage adjustment being made with a Variac.
Because the leads to the heater carried large currents (about 1
amp), they were made of copper. The lower tube heater was connected
to two thicker manganin wires which led up through the pusite seals
in both can caps to the terminal strip above the outer can. The
length of these leads was kept as small as possible to prevent
them from burning out. The leads from the terminal strip were of
copper and led out of the cryostat to a 6-volt car battery fund rheostat.
The filling tube thermocouple leads were taken up through
the cryostat to an ice bath and from there to the potentiometer,
where the thermocouple emf was read.The maximum error in the quantity of electrical heat Q which
was added to the calorimeter system for a heat capacity determin
ation (see Sec. 3.3 and Sec. 5.1) Is estimated to be +0.5%•
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46
CHAPTER 3
PERFORMANCE OF THE APPARATUS
3.1 - Preparation of -the Samples.
The system used for obtaining condensed gas samples in jthe calorimeter* ready for measurements at low temperature was
essentially the same as that used by Fagerstroemvoy for his
measurements on condensed gases in and above the liquid He4
temperature range, with the exception that there was, in the
present apparatus, no low temperature filling tube valve which
had to be warmed while condensing the gas* Distillation effects,
which were reduced in Fagerstroem*s apparatus by positioning this
valve as close to the calorimeter as possible, was not a consider
ation in the present investigation since the vapor pressures of all
the solid gases used were entirely negligible in the experimental
range of temperatures. This enabled an important reduction in the
complexity of the gas handling system to be made. Because the
sample preparation procedure follows closely that of Fagerstroem,
only an outline of it will be described.
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47
I After* a thorough leak-test, the entire gas handling system
I was evacuated, using a rotary pump, for periodSwhich varied from
run to run from a few days to 24 hours. Then the entire system
was twice flushed with small amounts (a few cm Hg pressure) of
the gas and then re—evacuated for at least one day* The pumps1Ff were then shut off and the system was filled with sample gas
through rubber hose leading from the steel cylinder containing the
gas supply. The gas pressure and the temperature of the reservoir
water bath were read after they were allowed to stabilize. A
mercury-in-glass thermometer was used to measure the water temper-
ture and a Wallace and Tieman mercury manometer (with vernier
divisions of 0.1 mm Hg) indicated the gas pressure. To within the
limits of experimental accuracy, the perfect gas law was sufficient
to give the molar quantity of gas inside the system from a knowledge
of the gas temperature, the volume of the system, and the gas
pressure. The gas filling pressures were always adjusted to be
as close to the maximum reading on the manometer (800 mm Hg) as
possible, since the greater the filling, pressure, the greater the
amount of condensed sample that could be collected in the calori
meter.At this point, with the heatsvitch closed, some liquid
nitrogen was placed in both the outer and inner B^wars, While the
amount in the outer Bev/ar was not important, the level in the inner
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IJcwsr wss kept below the level of the outer cun cap bscEuss it
wes found, that the rate of condensation could be better controlled
in this way., rather than having the level above the cap closer
to the filling tube thermal shunt. While the apparatus was
cooling, the filling tube heaters were switched on, the air in the
interspace was replaced by a few microns pressure of helium exchange
gas, and the shield system was evacuated and then sealed by closing
a valve. After a few hours (typically, three), the calorimeter
was at the temperature of the normal boiling point of liquid nitro
gen with a snail quantity of condensate in it at the vapor pressure
corresponding to that temperature.
In all cases except that of NO, the calorimeter temperature
was further reduced in order to collect more condensate in the
calorimeter. The reduction of temperature was accomplished by
pumping on the liquid nitrogen bath in the inner Dewar. In the
case of NO, where its normal boiling point (121°K) and normal
melting point (110°K) lie considerably above the liquid nitrogen
boiling point (77°K), it was not necessary to pump on the nitrogen
bath since the vapor pressure of NO at the nitrogen boiling point
was sufficiently low to provide, ideally, a sample large enough
for good measurements. Unfortunately, because of the high melting
point of NO, it was found impossible to collect a very large
sample since the filling tube became blocked easily by the solid,
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presumably in the regions of the tube which lay between successive
heater windings on Tube B. At one point, liquid air was tried in
place of nitrogen because . f its slightly higher boiling point,
but this did not result in an easier condensation. In practice,
the filling of the calorimeter NO was controlled in the easiest
way by simply adjusting the inner Dewar nitrogen level so as to
just touch the bottom of the outer can and simultaneously adjusting
the tube heater currents until the tube became unblocked, being
careful not to overheat parts of the tube which could cause solder
joints to break. The most vulnerable joint was that connecting
the two filling tubes A and B; actually this joint was overheated
during one run so as to melt the solder and cause a leak. The
unblocking of the tube could be seen quite easily in most cases
by the change on the manometer from a steady pressure to a slow
decrease. Sometimes, the tube constrictions were not completely
removed by heating the tube and the decrease in pressure took place
very slowly. The condensation of NO was the worst in this respect
and turned out to be the most difficult and tedious of all the
samples condensed, mainly because of the need to keep all parts of
the filling tube and calorimeter at temperatures in tha liquid NO
range, which only extends over about 12 degrees. The difficulties
were so formidable that it took about 50 hours of almost continual adjustment of the filling tube heater currents and nitrogen levels
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50
to collect 0.2 moles (representing about 6 ml of liquid).The condensation of the other gases was much easier, with
the collection of 0.5 moles taking about 12 hours. The filling
tube heater currents and nitrogen levels could be set once and
for all, thereby avoiding the need to supervise the condensation
continually. When enough condensate had been collected and with
the valve in the gas filling line closed, the pressure of the gas
remaining in the glass reservoir and the temperature of its water
bath were read. The perfect gas law was again used to evaluate
the molar quantity of gas inside the reservoir and a subtraction
of the amounts before and after the condensation gave the amount of
condensate inside the calorimeter, assuming that there was no
appreciable quantity remaining in the filling tube, an assumption
which is discussed in Sec. 6.2 in the light of its effect on the error in the measured heat capacities.
With a sufficient quantity inside the calorimeter and the
gas supply to it shut off, the nitrogen bath in the inner Dewar
was exchanged for liquid helium. Either the nitrogen was boiled
off under reduced pressure (in the case of NO), or the Dewars
were taken down and the liquid simply poured out. It was essent
ial that this operation was done as rapidly as possible in order that the liquid helium might be transferred quickly to the inner Dowar so as to prevent the calorimeter contents from evaporating.
During the period that the nitrogen was being exchanged by the
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51
helium bciths, the pressure on the dial gauge connected to the
calorimeter system was carefully watched in an attempt to detect
a rapid pressure increase which would have indicated evaporation
of the calorimeter contents.
During the first stages of the liquid helium transfer, the
tube heaters were left on in order to avoid distillation of the
condensate from the calorimeter into the regions of the filling
tube which were cooling faster than the calorimeter. When liquid
helium was starting to collect in the inner Dewar, the tube heaters
were switched off and the pressure on the dial gauge was observed
to fall very rapidly * The time taken for the transfer varied
somewhat depending on the sample heat capacity, but on the averageo„about four hours were needed to cool the calorimeter down to 4*2 K.
The transfer techniques used were the standard ones used in this
laboratory and they will not be described.
The sample was then in readiness, with the heat switch closed,
to be cooled to lower temperatures in preparation for the specific
heat measurements and this procedure is described in the following
section.
3-2 — Preparations for the Specific Heat Measurements.
With the solid gas inside the calorimeter at the temperature
of 4■2°K, it was then cooled in two stages to temperatures below 1 in the following way. First, the He^ bath temperature was
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reduced by pumping on it with the laboratory Kinney pump 'to a
temperature of about 1*2 , then the He^ stage was brought in to provide further cooling to below 1° by pumping on the liquid He3 in the refrigerant chamber. A description of this procedure is
described in this section.
With the heat switch still closed, the exchange gas was
removed from the interspace using the diffusion pump. Because of
sorption effects, the removal of He gas becomes more and more
difficult as the temperature is lowered and so the interspace
evacuation was done at 4.2°, even though its removal resulted in a reduction in the calorimeter cooling rate during the subsequent
cooling stage. This choice was made because it was considered
more important to avoid the presence of helium gas when trying to
cool the calorimeter below 1° with the He3 stage. For the initial
runs, a mass spectrometer leak detector, which collected the
diffusion pump exhaust, monitored the rate of removal of the inter
space gas. It was found that only after a period of about two
hours did the leak detector indicate a sufficiently low helium
level, even though the cold cathode gauge indicated a pressure of
less than 10~^ torr only a few moments after the diffusion pump started to work. For the later runs, the leak detector was not
used and a period of four hours was employed for the interspace
evacuation before proceeding with any further cooling.
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After this time, with the interspace pumps still working,
the he^ gas was allowed to enter the refrigerant chamber from
the storage tanks and then the temperature of the liquid He^
bath was lowered. The temperatures of the shield and calorimeter
were observed to fall very slowly, the heat from them being taken
to the He^ bath mainly by conduction through the He^ gas in the
pumping tube. When the He^ became cold enough, it started to
condense on the parts of the pumping tube which were in directX Acontact with the liquid He . As the liquid He*3 formed, it trickled
down to the warm refrigerant chamber where it was immediately
evaporated, thereby cooling the shield and calorimeter by taking
up its latent heat of evaporation. Eventually, after a period of
about two hours, all parts of the cryostat were at 1 .2° with sufficient liquid He in the chamber to provide further cooling
below 1°. It was estimated that the chamber contained, at most,
only about 1 ml liquid, but this was sufficent to cool all the samples without needing to re—cycle the gas, a technique that has
been developed for use i t» very long experiments with large heat
capacity samples.At this point, the heat switch was opened and then re—closed
with a lighter pressure on the jaws, because it was found that if the switch was opened only at the lowest temperature, prior to
taking measurements, then a very large amount of heat would be
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developed, presumably because the strains, which were induced in
the cryostat by cooling from room temperature (where the switch
was originally closed), had altered the proper alignment of the
heat switch components. Xn addition, the heat switch was closed
rather tightly before cooling in order to accelerate the calorim
eter cooling rate, and undoubtedly this would have led to a large
frictional heating effect on opening the switch.
While the switch was open, it was discovered that the
calorimeter temperature was rising quite rapidly and continued to
rise for a considerable time. At first, this temperature drift
was something of a puzzle, since the drift caused by vibrational
heating^which was always present and which had been investigated
beforehand, was much too_ small to account for it. The first
observation of it appeared with CO and was initially thought to
be connected with a phase transition which was suspected to be
displayed by this solid gas. However, when the effect was found
with the other solid gases (N£> 02, NO), another explanation was
sought. It now seems most likely that the effect was caused by
the existence of solid gas crystals which were not in good thermal
contact with any part of the calorimeter and hence did not cool as
rapidly as the others. In support of this viewpoint, it is known
that many molecular solids, including those investigated, possess
one or more crystal structure modifications between which there may
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be large changes in the density. If the temperature is lowered
quickly through such a phase change, then considerable strains
are set up throughout the solid and the result is the formationt
of a fsnow* of small crystallites. This explains why it is yery
difficult, experimentally, to obtain large single crystals in a
low temperature phase of these materials. It is then reasonable
to suppose there is poor thermal contact between adjacent
crystallites; therefore, the cooling of a snow would appear to
be very sluggish and thus the observed calorimeter warming effect
is thought to be the result of the system attaining thermal equil
ibrium. . Now this lack of thermal equilibrium did not necessarily
exist when making the specific heat measurements because all
regions were at low temperature. On the contrary, because the
specific heat becomes very small at low temperatures, any local
heating^which tends to establish unwelcome temperature gradients}
will be rapidly dissipated through the snow. Confirmation of
this was found from two sources; from the reaction of the calor
imeter thermometer to changes in the input heating rates, and
from the behavior of the measured specific heat to changes in the
deliberate heating rate. No effects were observed which could be
ascribed to a non—equilibrium condition of the calorimeter system
Generally, the heat switch was kept closed for about an hour in
order to let the calorimeter system attain thermal equilibrium
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with the He^ bath at 1.2°K.
Then, with the interspace evacuated and liquid He^ inside the refrigerant chamber, the He^ roughing pump was switched on
and the valve connecting the chamber to the pump was opened slowly
so as to prevent a surge of gas in the pumping system. After a
few moments, the valve was fully open and the calorimeter temper
ature was observed to fall steadily. After a period of about 20 minutes the calorimeter was at the temperature of about 0.7° and because further cooling below this temperature was much slower, an
initial series of specific heat measurements was made, leaving
the temperature range below 0,7° to be covered in a subsequent series.
The heat switch was then very slowly opened by rotating the
switch rod knob at the cryostat head. Great pains were taken to
avoid sudden switch movements, since these were found to produce
large amounts of heat in the calorimeter system. With care, the
heat generated could be reduced to as little as 5 ergs in some cases, although a heating of 20 ergs was more normal. This amount
was estimated from the observed resistance change before and after
opening the switch, the thermometer, sensitivity (to obtain the
corresponding temperature rise), and the heat capacity as measured
in the experiment. In practice however, the temperature drift
after opening the switch, which was caused by vibrational heating,
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. was so large as "to overwhelm the temperature rise mentioned above.
Typically, this vibrational heating rate was of the order 100 ergs per minute at the lowest temperature. Therefore, no attempt was
made to adjust the shield temperature during the course of the
run as is required by the methods of adiabatic calorimetry. Thus,
as the calorimeter temperature was rising during the run, the
heat leak the calorimeter, which was conducted along the
electrical leads and suspension threads from the shield, increased
in magnitude as the temperature gradient along the thermal path
increased.
Eventually, the warming trend was observed to change to
one of cooling, where the state of zero temperature drift corres
ponded to the condition of balance between the heat generated by
the vibrations and the heat taken away by conduction along the
wires. The calorimeter temperature at which this condition existed
was found, as expected, to depend mainly on the magnitude of the
calorimeter system heat capacity, since the vibrational heating
rate was least for the massive samples which were rigidly supported
and greatest for the light samples for which the amplitude of
vibration was quite considerable. The temperature at which there
was heat leak balance varied from about for the light samples
(for example, the empty calorimeter), to about 3- for the heavy
samples.
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5 8
Then, with the heat switch open, the heat capacity measure-
ments were ready to be made and the procedure is described in the following section.
3.3 - Procedure of the Measurements.
With the heat switch open, the out-of-balance signal from
the thermometer comparator was displayed on the chart recorder
and the temperature drift of the calorimeter was observed for a
period of about two minutes. The comparator resistance box setting
was changed in the meantime in order to determine the chart"
sensitivity (the measure of the change in thermometer resistance
per unit division of length across the chart paper). The amplif
ication of the output signal from the comparator was adjusted such
that the recorder pen traversed the full width of the chart in
little over two minutes, provided that the noise level of the
trace was sufficiently low. With the comparator adjusted and
with the potentiometer (for the heater voltage measurements) balanced
against the standard cell, the heater current and timer were
switched on together. During the heating period, which was usually
30 seconds long, the voltage across the heater was noted, being
careful to make sure the voltage remained steady during the entire
period. If this were not the case, then that specific heat deter
mination was discarded.
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59
At the end. of the heating period, the thermometer resistance
was again charted. on the recorder for about two minutes and
another sensitivity check was made. The resistance box settings
were written on the chart alongside the corresponding trace and
the temperature increment for the heating cycle was obtained from
the chart in the manner described in Sec. 5.1* This procedure
was repeated to obtain each specific heat pointy generally up to
a temperature of about 1.3°. In all, the first series of measure
ments took about an hour to complete. The heater voltage was
increased periodically to obtain progressively larger temperature
increments in a larger heat capacity sample.
Although the He^ roughing pump had been working all the time,
there was sufficient liquid He remaining to re-cool the cryostat O oto below 1 * The heat switch was then closed, and the He'3 diffusion
pump switched on. After about an hour, with the calorimeter at a
temperature near 0,5°>the heat switch was re—opened and a new
series of measurements was begun. Because the operation of the
He roughing pump contributed to the vibrational heating of thei
calorimeter,, the He^ pumps were switched off for this series.
During the course of the measurements the shield temperature
gradually increased because of the heat leaks reaching the shield by conduction along the tubes from the outer can. Thus, as the
experiment progressed, the heat leak to the calorimeter by conduc-
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tion from the shield decreased until the shield was steady at the
-temperature of the He^ bath. This, coupled with the reduction
in the vibrational heat leak, had the beneficial effect of reduc
ing the calorimeter temperature drift during this series.
On the average, about 70 specific heat determinations were
made over the temperature range from 0,6° up to about 4°*
Host of the points were taken in the region below 1.5° since the
accuracy of the measurements was not as good in this region as it owas above 1,5 > especially for the pure solid gas samples which
had small specific heats.
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61
CHAPTER 4
THERMOMETER CALIBRATION AND TEMPERATURE SCALES
4.1 - Introduction.
The calorimeter thermometer was calibrated against then 3 4He0 vapor, pressure scale. The advantages of He over He^ in
vapor pressure thermometry, have been substantiated in recent
years because it has been found that the film reflux effects
in He-II are very difficult to take into account. Another
consideration which favors the choice of He^ over He^ is the
fact that the He^ vapor pressure is very much greater than that
of He^ at the same temperature. This is especially important
below 2°K since the film flow corrections for He-II may represent
a considerable (uncertain) fraction of the total pressure. In
addition, the greater He^ vapor pressure may be read on the mano
meter with greater precision than that of He^,
4.2 - Considerations of the Choice of Calorimeter Thermometer.
Tlie sample thermometer was chosen very carefully, since
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upon it depended very much the reliability of the specific heat
measurements* Some of the features of a thermometer which were
considered necessary are; fast response to changes in calorimeter
temperature, small mass, good sensitivity^ and good reproducibility
between temperature cyclings from room temperature to low temper
atures. For these reasons, resistance thermometers have been
used almost exclusively and while carbon radio resistors have
enjoyed almost a monopoly in low temperature calorimetry in the
past, the use of commercial germanium resistance thermometers is
growing rapidly, especially for the He^ temperature range.
The main advantage of germanium over carbon thermometers is
that their resistance-temperature characteristics appear to be
much more reproducible between temperature cyclings, although not
every thermometer from all manufactuers is satisfactory in this
respect It, seems that the better reproducibility of germanium
is connected with the fact that these thermometers use one single
crystal as the temperature sensing element, while the carbon radio
resistors are made from many small grains. The temperature
cyclings probably cause the grains to change their detailed state
of contact with each other through thermal expansion effects, thus
altering the bulk resistance of the resistor. Therefore, with
carbon thermometers, new calibrations are required every time they
are cooled down to low temperatures, while for reproducible
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63
germanium thermometers, the calibration need only be done once and for all.
The main disadvantage of germanium thermometers is that
there is no simple interpolation formula which represents
satisfactorily the thermometer resistance as a function of!
temperature. Such is not the case for carbon thermometers however, I
for the well-known semi-empirical formula of Clement and Quinnell^2?) !3
works well for most thermometers. ]
After weighing these considerations, it was decided to usei\a commercially made germanium resistance thermometer. Actually, ]
several thermometers from different manufacturers were tried before
making the final choice, but few of them were suitable for calor-
imetry in the He*5 temperature range, for their resistance rose too
rapidly as the temperature was lowered. The need to keep the joule
heating of the thermometer as low as possible, consistent with the
sensitivity requirements of the resistance measuring apparatus,
places an upper limit on the maximum acceptable resistance. Also,
since the calibration of resistance thermometers depends to some
extent on the magnitude of the measuring current, it is advisable
to use a thermometer which does not have a very large change of
resistance over the range of temperatures used in the experiments,
In. this way, it becomes necessary to use only one measuring current
ln the entire work, both for the calibration and the specific heat
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measurements. This avoids the complication of having to split
the calibration up into several temperature ranges, one for each measuring current.
The final choice was a Cryocal. Inc. germanium resistor
(serial #117), with a 4»2°K resistance of 130 ohms and a 1.0°K
resistance of 620 ohms, A resistance measuring current of 1.5
microamps was used throughout the work, thus satisfying the
requirements of keeping the joule heating low (a maximum of 5.10~9
watts), as well as being sufficient current to produce an accept
able sensitivity in the measuring apparatus throughout the whole
temperature range * When the thermometer was ordered from the
manufacturer, the copper case containing the germanium crystal
unit was specified not to contain helium exchange gas, which is
usually used. It was feared that the gas might have given a
spurious contribution to the measured specific heat because of
sorption effects.
4*3 - Thermometer Calibration Procedure.
With the heat switch tightly closed, the cryostat was cooled
m preparation for the calibration in exactly the same way as it was
for the specific heat measurements. Indeed, some calibration points
were taken at the completion of the measurements in some of the runs.
The experimental procedure for performing the calibration below
1.2°K (the lowest He^ bath temperature) was different from that used
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65
above this temperature because of experimental difficulties. The
calibration procedures in the two regions will now be described
separately.
1. Calibration procedure below 1.2°K. With the He^ bath
at the temperature of 1*2° throughout and the interspace evacuated
of its helium exchange gas, the He** stage was put into operation.
It was found that the shield temperature, indicated by the shield
carbon thermometer, could be very easily controlled by making a
rough adjustment first of the He^ pumping valve and then balancing
the heat removed, by virtue of the evaporation of the He3, by the
addition of joule heating from the shield heater. In this way, a
very fast temperature response was obtained to changes in the heater
current which was adjusted to produce a steady shield temperature.
It was observed that the calorimeter temperature came to
equilibrium only a few moments after the shield temperature had
done so. However, no thermometer resistance reading was taken until
five or ten minutes had elapsed after the resistance had become
stable, in order to make .sure that the initial steady indication
was not spurious. When the resistance had become steady, the heights
of the two mercury columns of the He^ vapor pressure manometer were
read and the mercury temperature was noted. At the end of the
five or ten minute equilibration time, the vapor pressure was again
measured, making sure that the calorimeter theivnometer resistance
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had not changed noticeably during this period. For the purpose
of the calibration, the arithmetic mean of the two pressure read
ings was used. The thermometer resistance was displayed on the
chart recorder in the usual way, while a measuring current of
1.5 micro amps was used throughout.
Then the pumping valve and the shield heater current were
adjusted to produce a slightly higher shield temperature and
another calibration point was taken. This procedure was repeated
to obtain several calibration points up to 1.2°. After this
temperature range had been covered the calibration above 1.2°K was
performed.
2. Calibration procedure above 1.2°. For this temperature
range, it was necessary to allow for the possibility of having
cold spots in the vapor pressure sensing tube. A cold spot is
simply a region of the tube which lies at a lower temperature than
that of the vapor pressure cell. They must be avoided at all costs
since the He^ rising from the liquid in the cell may re—condense
on the sides of the tube at a lower temperature than that of the
cell. Thus the pressure read on the manometer may then be too
low. Two alternative calibration procedures were considered to
avoid this effect.
F i r s t , i t was s u g g e s te d t h a t th e same te c h n iq u e as t h a t used
below 1 .2 ° m ig h t be u se d h e r e , w i t h th e He^ b a th m a in ta in e d s te a d y
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67
at, say, 4.2°K. However, it was found, in attempting to control
the shield temperature (which was relatively easy below 1°), that
it was impossible to maintain the calorimeter temperature steady
for a sufficiently long period. Apparently this was a result of
the increasing time constant for temperature fluctuations in the
shield, as well as the need to adjust the He^ pumping valve in the
almost-closed position where the pumping speed was very sensitive
to fine adjustments.
The second alternative was to not use the He^ stage at all
and to let exchange gas into the interspace. Thus the shield
would be maintained isothermal with the He^ bath whose vapor
pressure could then be quite easily maintained steady for each
calibration point using the standard laboratory pumping line
equipment. This alternative was finally - adopted despite the
possible difficulties arising from the hydrostatic head pressure( 28)effect . This effect, which results in an increasing temper
ature with depth in a liquid, could possibly cause the formation
of cold spots in the vapor pressure sensing tube, since the He4
bath surface temperature may be lower than that of the He^ cell.
To avoid this possibility, the calibration points were taken only
when the liquid He^ bath surface had fallen to about the same
level as that of the outer can cap. In this way, the vapor pressure
sensing tube thermal shunt was not immersed in the bath liquid
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during the calibration and it was expected that the heat' conducted
down the vapor pressure tube vacuum jacket from the cryostat head
would serve to raise the shunt temperature above that of the bath.
If this were so, then cold spots were unlikely to form anywhere
in the pressure sensing tube.
As a further precaution, the bath heater was used continuously
during the measurements in an attempt to stir the liquid and hence
reduce the residual temperature gradients in the bath. About 60
milliwatts was dissipated in the heater which was situated in the
bottom of the inner dewar. This was sufficient power to cause the
formation of tiny gas -bubbles which were seen to rise and pass
around the outer can. After making these adjustments to the helium
bath level and bath heater, the calibration was undertaken. The
manner in which the measurements were made was identical with that
used in the region below 1.2°.
Because the bath level was so low, it was feared that the
beat conducted down the various tubes from the cryostat head would
produce a considerable temperature rise in the shield. But with
beliu_m exchange gas in the interspace throughout the calibration
procedure, the calorimeter temperature seemed to agree with that of
the bath at all times. Evidently, the thermal coupling between the
shield and the bath through the interspace exchange gas was suffic
ient to prevent any undesirable heating of the shield from this
source.
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4.4 - Discussion of Errors in the Temperature Scales.
1. Corrections to.the measured He* vapor pressure.
A. Thermomolecular pressure corrections. If two
regions of a closed gas system which are connected by a narrow tube
lie at different temperatures, then the condition for equilibrium
at high gas pressures is that the pressures in the two regions
shall be equal. However, as the pressures are lowered to a degree
where the mean free path of the gas molecules is of the same order
as that of the tube diameter, then the condition for equilibrium is
no longer that the two pressures shall be equal. Because a temper
ature gradient exists in the tube, there will be thermal transpir
ation effects which result in a mass flow from the cold to the warm
regions. This flow is balanced, in equilibrium, by an equal reverse
flow down the pressure gradient in the tube. Thus, at low pressures
and in equilibrium, there is a difference in pressure in the two
regions, the pressure in the cold region being lower than that in
the hot region. Although the pressure differences for perfect
gases may be calculated from a knowledge of the geometry of the gas
system, such is not the case for He^ (and He^) which behave in a
non-classical manner at the very low temperatures encountered in
the. present work.(29 QO )Experiments have shown 3 that the thermomolecular press—
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ure ratio, which is defined as the ratio of the gas pressure in the
cold region (pc) to that in the warm region (pw ), may be expressed
as a function only of the product of the tube radius R and pw . In
the present application, pc is the vapor pressure just above the
surface of the liquid (the desired quantity), and pw is that meas
ured on the manometer. Unfortunately, the magnitude of the ratios
depends somewhat on the material of the tube; for example, on
whether glass^0) or stainless steel(31) is chosen, there being a
1 Q% difference (approximately) between them for the same R.pw value.
For the present work, the tables of Roberts and S y d o r i a k ^ 2 ) were
used (based on measurements using a vacuum—jacketed Inconel pressure
sensing tube), making linear interpolations between their tabulated
values of R.pw . For the size of tube used in this investigation,
the maximum correction to the cell temperature was 8 mdeg at the
lowest temperature, about 0.6°K,
B. Hydrostatic pressure head corrections. In a liquid,,
there tend to be set up temperature gradients due to the hydro
static head pressure effect. In the absence of convection, the
equilibrium boiling temperature increases with depth because the
hydrostatic pressure of the liquid must be added to that at the
liquid surface to obtain the equilibrium pressure. This correction
be easily calculated from the values of the liquid density.
However, in practice, convection currents are set up which tend
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to reduce the temperature gradients and hence the magnitude of
the necessary temperature correction. Because it is difficult
to know the degree to which the gradients are altered by convec
tion, it is preferable to encourage the formation of convection
currents as much as possible to the point where corrections become unnecessary.
Because liquid He^ does not have a strong density-temperature dependence (compared with liquid He-I for example), there is
little natural stimulation for the formation of convection currents
in the presence of a temperature gradient. Thus, an attempt was
made in this work to reduce the temperature gradients in the liquid'iHe by allowing the liquid to come into contact with copper turn
ings inside the cell, as well as with the copper cell walls.
Although it has been found^33) -that using a copper cell alone the
hydrostatic head corrections may be reduced to a negligible amount
(less than 1 mdeg), it was decided to take the added precaution of
placing copper turnings inside the cell.
Table 4-1 shows the thermomolecular and hydrostatic head
temperature corrections which may be neccesary for the particular geometry used. The former corrections shown were actual]y employed
to compute the final corrected temperatures (see Appendix l), while
the latter ones were not used. The hydrostatic head corrections shown refer to the maximum possible (no convection) temperature
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TABLE 4.1 CORRECTIONS TO THE MEASURED He3 VAPOR PRESSURES
TEMPERATURE CORRECTIONS (mdee)T°K P(He^) nun Hg HYDROSTATIC HEAD THERMOMOLECULAR
0.6 0.54 9 8
0.8 2.89 3 2
1.0 8,84 1.5
1.2 20.16 1 *■«
1.4 38.52 1
NOTE: The thermomolecular corrections must be subtracted
from -the cell temperature that is inferred from the
manometer readings.
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difference between the surface of the liquid and the base of the
cell.
C. The effect of He^ impurity in the He3. The
effect of He 4 impurity is to overestimate the correct temperature
of the cell liquid from the measurements of the vapor pressure.
The necessary temperature correction, of course, depends on the
ratio of He4 to He^j for example, for a 0 ,0 5 % He4 content, a
correction of 0,3 mdeg at 3°K is necessary^34)4 temperature
is lowered, the correction falls to 0.1 mdeg at 1°K.
The gas used in this research was obtained from the Monsanto
Mound Laboratories and was their ’’Vapor Pressure Grade” for which
the supplier claimed the following analysis:
SupplierTs Analysis of the Vapor Pressure Grade He^ Gas
Greater than 99*98$ He^ in total helium
Greater than 99*9 % total helium
Less than 2,10~^% tritium
It is seen that the He 4 content is claimed to be less than 0,02$ in He^ giving a maximum temperature correction of 0.1 mdeg at 3°K.
This small correction was not applied to the present measurements. The 0.1% non-helium component undoubtedly gave a negligible contribution to the total vapor pressure and no temperature corrections
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73
were necessary from this source.
Other corrections. Corrections to the measured pressures were necessary to reduce them to values in terms of
standard gravity (980.665 cm/sec ) and the density of mercury at 0°C, since the vapor pressure tables refer to these standard
conditions. The corrections for local gravity were not made because
they represented differences of only 0.002% to the pressures. The
mercury density corrections were made, however, because they repres
ented more significant pressure variations. The density values
were taken from the International Critical Tables and were plotted
on a graph as a function of temperature. For each thermometer
calibration point, the density at the measured mercury temperature
was read from the graph.
Calculations of the hydrostatic head effect in the gas column
in the pressure sensing tube were made and it was found that the
maximum temperature corrections were negligible,
E. Precision of the Calibration temperatures.
1 . He^ v a p o r p re s s u re m ea su re m en ts . A t lo w te m p e r
atures, where th e He^ v a p o r p re s s u re i s v e ry s m a l l , th e c a th e to m e te r
reading e r r o r re p re s e n ts th e g r e a te s t c o n t r ib u t io n t o th e u n c e r ta in t y
°f the te m p e ra tu re m easurem ents. A t w o rs t , a t 0.6°K, th e re a d in g
er ro r , w h ich i s e s t im a te d t o be + 0 .0 5 mm f o r th e com bined re a d in g s
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on both limbs, represents +10 mdeg, and at 1.0°K, the reading
error represents +1.2 mdeg. The other major source of uncertainty
is the thermomolecular correction, but it is thoughtthat this
correction is correctly estimated to better than +1 mdeg. Above
1°K, the reading error produces small uncertainties in the temper
atures, since the slope (dP/dT.) of the vapor pressure curve
increases rapidly as the temperature is increased. Thus, the
cathetometer reading errors produce less than jKL mdeg uncertainty
in the temperatures. It is estimated that the temperatures above
1° are known to an overall accuracy of better than +1 mdeg. The
1962 He^ s c a l e d 34) was used to obtain temperatures from the meas-• • -
ured vapor pressure readings.
2. He^ vapor pressure measurements. The reading
error here is larger than that of the He° cathetometer and it is
estimated to be +0.1 mm. However, because the manometer was used
to read large vapor pressures, the resulting uncertainty of the
bath temperatures is thought to be quite small. At the lowest
temperature (2.6°K), the temperature uncertainty from this source
is estimated to be +0.5 mdeg, falling to ,+0.2 mdeg at 4*2°. No
other sources of error are thought to be important. The 1958 He^ s c a l e (37) was used to obtain temperatures from the measured vapor
pressure readings.
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2. The reproducibility of the thermometer.
The germanium thermometer was relied upon to demonstrate
reproducible R,T characteristics from run to run. Some of the
evidence for the good reproducibility came from two sources:
1. During the course of every specific heat run,
the thermometer resistance at 4.2° (the temperature of the He^
bath under the pressure of the laboratory gas holder), and at
1.2° (the lowest temperature reached by pumping on the bath with
the laboratory Kinney pump) were noted. These temperatures were chosen for comparison because they could be obtained with very
little variation from run to run. In every run, the respective
resistances were found to agree with each other within the error
of estimating the bath temperature,
2, The entire set of calibration points was taken
in five different runs, the apparatus being warmed to room temper
ature between runs. The points fitted all the attempted calib
ration formulae (see Sec. 4.5) either with small random deviations
or with somewhat larger smooth systematic deviations. If the
thermometer were not reproducible, there would have been large
random deviations from the formulae.Of course, the critical test for the reproducibility of the
thermometer lies in its ability to give the specific heat of a standard substance (in this case, copper) in good agreement with
the accepted values, and to give identical results for the same
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substance in . d i f f e r e n t r u n s . I n t h i s c o n n e c t io n , s p e c i f i c h e a t
measurements on c o p p e r w ere made (se e S ec, 5 * 3 ) , and tw o e x p e r im e n ts
were p e rfo rm e d on n i t r o g e n (see S ec. 6 . 2 ) ,
4,5 - The T he rm om ete r C a l ib r a t io n I n t e r p o l a t i o n F o rm u la e .
In the course of this work, approximately 2,000 temperature
values were needed for the specific heat data and it was decided
from the start to make use of the computer facilities in the
University for as much of the computation as possible. In order
to minimize the labour in computing temperatures from values of
the thermometer resistance, an analytic formula f(R,T) was sought
from which the data could be reduced with the aid of the computer,
instead of the much more tedious method of plotting the R, T points
on a large graph and reducing the data by hand. Essentially, the
method employed was to fit the thermometer calibration points,
by a least squares procedure, to various forms of. f(R, T) until a
satisfactory fit was found. Since the calibration points were
relatively widely separated, the interpolation formula (which must
give accurate R,T values in between the calibration points) had to
he chosen with care. The two criteria which were used for choosing
a satisfactory formula are;
1 . The d e v ia t io n s o f th e c a l i b r a t i o n p o in t s fro m th e v a lu e s
c a lc u la te d fro m th e fo rm u la had t o be s m a l l ,
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77
2. The specific heat of a 1standard* substance had to beaccurately reproduced.
-forUnfortunately,/the purpose of a rigorous choice of a formula,
the first condition is difficult to apply (unless the deviations
are not random), while difficulties are encountered with the
second when choosing a standard substance* Possibly the easiest
metal to obtain in a high state of purity (the condition which
most often affects the specific heat) is copper, but when examin
ing the literature of the specific heat of this metal, considerable( 3 )disagreement is found even for quite recent work. For this
reason, it was proposed at the I964 Calorimetry Conference that a batch of uniformly high purity copper should be prepared and
samples from it distributed to various laboratories in order to
determine if a consistent specific heat value could be ascribed
to all the samples and if so, to publish this value as the
accepted copper specific heat. Such a batch of copper samples
was distributed^ unfortunately, when the sample received by this
laboratory was analysed it was found to contain important quantities1Iof Mn and Fe, both of which have disastrous effects on the specific
heat of copper, especially at low temperature. Actually, in prel
iminary experiments to test the cryostat, this sample and a diff
erent calorimeter from that used for the solid gas experiments did
show a low temperature specific heat characteristic of a sample
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containing ferromagnetic impurities, although the beryllium-copper
alloy, which contains iron and which was used in the calorimeter
construction, may have been responsible. Because very precise
measurements were needed, it was decided not to use this copper
sample or the calorimeter, but to use instead a very high purity
copper sample which was very kindly furnished by Dr. D. L. Martin
of the National Research Council, Ottawa. This sample was attached
to the solid gas calorimeter for the specific heat measurements,
which are described in Sec. 5-3*
Several different forms of f(R,T) were tried* Originally,
the polynomials,
were tried, with n taking values, in turn, from 2 to 6.. None of
them gave a satisfactory fit over the entire temperature range,
although the fit at high temperatures (above 2°JC) was fairly good for them all. Therefore, it was decided to search for another
Because the Clement-Quiunell formula applicable to carbon
resistors is based on the low temperature variation of resistance
of semi-conductors, l/T InR, it was felt that a formula which also
uses this as a basis should represent fairly well the R,T relation
for germanium. The basic Clement-Quinnell equation, with various
formula which would give a good fit at low temperatures also.
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extra terms was then tried;
JL = bc + b^lnR + b2
Clement—Quinne11 __formula
This equation, with n taking values, in -turn, from 2 to 4 was
tried, but again, none of them gave satisfactory fits over the
entire temperature range. The fit at low temperatures was good
for them all, but the high termperature fit was unsatisfactory.
Instead of searching for a polynomial containing more terms
(the approach which several workers have resorted to^^^), it was
decided to split the temperature range into two parts using^ in
each sub-range, a different calibration formula for the purpose
of computing the specific heats. For the low temperature region
(below 1.3°), the modified Clement-Quinnell formula was used,
while for the high temperature region (above 1.3° ) 9 a polynomial in lnR was used:
t-"n^bidnR)-11=3-
extra terms
i - a + a lnR + a_(lnR)2 + a.(lnR)3 + a. (lnR)4 — calibration- A T 3 4for T > 1.3°K
1 = b + b lnR + b + b . + b , + b ,T 1 2 3 „ —5. . c a l i b r a t i o n - B
lnR (lnR)2 (lnR)J (lnR)4for T < 1.3°K
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A least squares analysis was performed separately on each
of the two formulae. For Calibration-A* input data from 1.27°
to 4.2° was employed (total of 22 points)* while for Calibration-B input data from the entire temperature range* 0.6° to 4.2° was used (total of 29 points). The deviation curve for the combined
calibration is shown in Fig. 4*1* The deviations were evaluated
in the computer program by obtaining from the least squares analysis
the best values of the constants. Then the calibration resistance
values were inserted back into the formula to obtain Tca^c. Tmeas
is the measured calibration temperature corresponding to that re
sistance value used to evaluate ^ca^c» Table 4*2 shows the constant
obtained from the least squares analyses* where eight figure accuracy
was used.oReferring to Fig. 4*1* "the deviations below 2 are seen to
be quite random* therefore* the specific heat errors from this
source should be random also. However* the deviations above 2
have a strong systematic trend which would likely result i.n system
atic errors in the specific heat in this region. If necessary* the
deviation curve may be used to adjust the specific heat values
computed from the calibration formula as follows.
The deviation between 3*2° and 3*6°>for example, changes by
about 4 mdeg* corresponding to a rate of change of 10 mdeg per degree 1/3- The slope of the deviation c u rv e is negative* hence theor
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73CD■o-5oQ .CoCDQ .
■oCD
C/) (f)
Oo■ O
3.CD
CD■o-5oQ .Cao■oo
CDQ .
■oCD
C/)C/)
£FHCD•O'
0a
b ?
1oCDE
H
zo!§>L?JO
FROM 1962 He3 VAPOR PRESSURE SCALE meas FROM 1958 H VAPOR PRESSURE SCALEmeas
20
4 RANGE IN WHICH CAUB. (B) WAS USED TO COMPUTE SPECIFIC HEATS
• CALIBRATION (B) ▲ CALIBRATION (A)
RANGE IN WHIGH CALIB.(A) WAS USED TO COMPUTE SPECIFIC HEATS
(see text)
FIG 4.1 CHARACTERISTICS OF THE THERMOMETER CALIBRATION FORMULAE
TABLE 4.2 THE CONSTANTS APPEARING IN THE THERMOMETER CALIBRATIONFORMULAE
CALIBRATION ~ 6(low temperatures)
bQ = 0.268391 X 103
bx = - 0.839558 x 101
b2 = - 0.318890 x 104
b3 = O.I83888 x 105
b4 = ~ 0.523626 x 105
b5 = 0.593660 x 105
input d at a f r o m the entire t e m p e r a t u r e r a n g e
CALIBRATION ~ A (HIGH TEMPERATURES)
aQ = 0.406713 x 101
■= - 0.159592 x 101
a2 = 0.548157 x 10-1
a = 0.364853 x lO"13
a . = - 0.279514 x 10~24
INPUT DATA FROM 1.27° UP
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81
specific heat values evaluated, from the calibration equation should be 1% too small in this region*- Thus, when the deviation
curve has a negative slope, the specific heat values are too
small, while for a positive slope, they are too large* Comparing
Figures 4-1 5*2,it is seen that the trend of the specific heat
is in rough agreement with this prediction, although the magnitude of the deviations is somewhat different from that expected on this
basis alone.When different calibration equations are used to compute
specific heats in contiguous temperature regions, the question arises as to how well the two sets of specific heat values (computed from each of the two calibrations) compare in the region where the calibrations meet.- Great care must be taken to. ensure
that there are no sharp discontinuities at the point where one calibration takes over from the other, in this case, at 1*3°»Fig. 4*2 shows the total heat capacity of the copper sample and
calorimeter computed from the two calibrations in the region around
!• 3°. The data are plotted as C/T against in order that the
differences may be seen more clearly than from a plot of C against T.
It is seen that the differences change sign when passing
through the temperature region around 1.3^* this region, the
differences represent only a few tenths percent (or less), which
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Reproduced
with perm
ission of the
copyright ow
ner. Further
reproduction prohibited
without
permission.
3.0
2.
C/y (mJ/deg)
2.6
24
© CALIBRATION (A )-H IG H TEMPERATURE & CALIBRATION (B)-LO W TEMPERATURE
•mm m ■1 %1
-! i
THE LINE REPRESENTS THE EQUATION
—■ M8 + c*T2) . + tg + « T 2)_T Color 0 Copperwith 8 , oc values from the least squares analyses (see text).
T*J.3*K
1.0 2.0 T 2 (°K) 3.0
FIG.4.2 HEAT CAPACITY OF THE COPPER SAMPLE WITH CALORIMETER EVALUATED FROM EACH CALIBRATION FORMULA
is g e n e ra lly le s s th a n th e s c a t te r o f I n d iv id u a l p o in ts fro m th
le a s t squares f i t t o a s t r a ig h t l i n e . Thus, th e p ro ce d u re o f
d iv id in g th e e n t i r e te m p e ra tu re ra n g e in t o tw o d i f f e r e n t c a l ib -
ra t io n re g io n s i s fo u n d t o be s a t is f a c t o r y .
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83
CHAPTER 5
THE DATA HANDLING-— THE CALORIMETER AND COPPER MEASUREMENTS
5,1 - R e d u c tio n o f t h e Raw D a ta .
I n o r d e r t o be i n a p o s i t i o n t o com pute th e s p e c i f i c h e a t
o f th e sam p les ( i n c lu d in g th e c a lo r im e t e r ) , i t was n e c e s s a ry t o
c a lc u la te , f ro m th e ra w d a ta , th e e l e c t r i c a l e n e rg y added t o th e
system and th e r e s u l t i n g te m p e ra tu re r i s e f o r e a ch h e a t in g c y c le .
The e l e c t r i c a l e n e rg y was o b ta in e d fro m th e m e a su re m e n ts o f th e
hea te r v o l ta g e , t h e h e a te r r e s is ta n c e , R ^ , and th e h e a t in g
p e r io d , t , and t h e te m p e ra tu re r i s e was o b ta in e d fro m th e th e rm o
meter r e s is ta n c e r e a d in g s . The r e d u c t io n o f th e d a ta a lo n g th e s e
lin e s i s now d e s c r ib e d .
S in ce th e te m p e ra tu re o f th e c a lo r im e t e r s ys te m was v e r y
ra re ly c o n s ta n t i n t im e , i t was n e c e s s a ry t o a l lo w f o r th e d r i f t
(dR /d t) o f th e th e rm o m e te r r e s is ta n c e t r a c e a c ro s s th e c h a r t r e
corder p a p e r . I n o r d e r t o o b ta in th e t r u e te m p e ra tu re r i s e f o r
the pu rpose o f c o m p u tin g th e h e a t c a p a c i t y , b o th th e r e s is ta n c e
d r i f t s b e fo re and a f t e r th e h e a t in g p e r io d w ere e x t r a p o la te d i n t o
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84
th is re g io n . Because th e d r i f t s were a lw ays found to be co n s ta n t
in tim e between th e h e a t in g p e r io d s , s t r a ig h t l in e s were drawn
through th e d r i f t s and extended to th e m id -p o in t o f th e h e a tin g
pe riod . From th e re s is ta n c e v a lu e s fou nd a t t h i s p o in t (Rj-,
denoting th e b e fo re -h e a t in g , and Ra th e a f te r - h e a t in g v a lu e s ) , th e
corresponding te m p e ra tu re s , T^ and Ta were computed from th e
thermometer c a l ib r a t io n fo rm u la e and th e n th e d i f fe r e n c e , Ta-T^
gave the t r u e te m p e ra tu re r is e f o r t h a t p a r t ic u la r h e a t in g c y c le .
For t h is and a l l subsequen t c a lc u la t io n s , th e com puter was
employed in th e fo l lo w in g way.
From th e f i v e measurements f o r each h e a t in g c y c le , Rb, RQ,
Vh, Rj^, and t , th e h e a t c a p a c ity was com puted. Thus, a l l i n th e
same computer p ro g ra m , th e te m p e ra tu re r is e Ta-T j) , and th e mean
tem perature, Tm = (Ta + T ^ ) /2 were c a lc u la te d from R^ and R^, th e
e le c t r ic a l energy Q = . t /R ^ was computed, and th e h e a t c a p a c ity
c = f i / ( Ta~Tb^ o b ta in e d . T h is p ro ce ss was re p e a te d f o r each one
of the s e t o f h e a t in g c y c le s , th u s g iv in g a s e t o f h e a t c a p a c ity
po in ts e x te n d in g o v e r th e e x p e r im e n ta l te m p e ra tu re ra n g e . For
the purpose o f th e c a lc u la t io n s in v o lv in g th e te m p e ra tu re v a r ia t io n
of the hea t c a p a c ity , th e mean te m p e ra tu re T^ was used to in d ic a te
the tem pera tu re c o rre s p o n d in g to th e h e a t c a p a c ity p o in t found f o r
the h e a tin g c y c le . F o r th e s m a ll te m p e ra tu re in c re m e n ts used in
th is rese a rch , t h i s ass ignm en t was s u f f i c ie n t l y a c c u ra te and no
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8 5
c o rre c tio n s w ere n e c e s s a ry t o a l lo w f o r th e n o n - l in e a r v a r ia t io n
of the h e a t c a p a c ity w i th te m p e ra tu re .
5.2 - The C a lo r im e te r H e a t C a p a c ity .
In o rd e r t o d e te rm in e th e s p e c i f ic h e a t o f th e copper and
s o lid gas sam p les , i t was n e c e s s a ry t o s u b t r a c t th e empty c a lo r«
im eter h e a t c a p a c ity fro m th e measured t o t a l . T h e re fo re , th e
empty c a lo r im e te r h e a t c a p a c ity was measured in a se p a ra te ru n .
The p rocedure and th e r e s u l t s a re d e s c r ib e d b e lo w .
A f te r e v a c u a t in g th e c a lo r im e te r system f o r ab ou t a day,
the c ry o s ta t was c o o le d i n th e manner d e s c r ib e d i n Sec, 3 .2 , and
the measurements made i n th e manner d e s c r ib e d in Sec. 3 .3 .
P a r t ic u la r a t t e n t io n was p a id t o th e m easurem ents be low 1 .3 ° K
because th e c a lo r im e te r h e a t c a p a c ity was v e ry s m a ll i n t h i s
reg ion . C o n s id e ra b le s c a t t e r i n th e m easurem ents was expected
because o f u n c e r t a in t ie s i n th e stray h e a t le a k s t o th e c a lo r im e te r ,
which co u ld re p re s e n t a c o n s id e ra b le f r a c t io n o f th e t o t a l h e a t in g
ra te d u r in g th e h e a t in g p e r io d s . F o r t h i s re a s o n , ab ou t 40 p o in ts
were taken be low 1 .3 ° , w h i le o n ly 20 p o in ts were ta k e n above t h i s
tem pera tu re .
The r e s u l t s a re shown in g r a p h ic a l fo rm in F ig . 5 -1 and th e
table o f r e s u l t s i s g iv e n i n A p p e nd ix 2 -A . I n F ig . 5.1> th e low
temperature r e s u l t s a re shown as a p lo t o f C /T against arid th e
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0.8
0.7
•LOW TEMPERATURE- DATA (BELOW) C3
©o DEVIATION*
Gmeo5-teT+~T3)X 100%
fv-®-'meos
©o
T°K», FROM LEAST SQUARES
FIT TO THE ENTIRE SET OF DATA
THE HIGH TEMPERATURE RESULTS
C/T (m«J/deg2)
„L.
5%~ r
00
0THE LINE REPRESENTS THE LEAST SQUARES FIT TO C /T = (0 .6 6 2 + 0 .0 875 T 2 ) m J/deg2
( see text)
THE LOW TEMPERATURE RESULTS
0 1.0 j2 (°K)2 2.0FIG. 5.1 THE EMPTY CALORIMETER HEAT CAPACITY
high temperature results are shown in the form of a deviation plot.
The deviation is defined as the difference between the individual
specific heat values, C(Tm ), and the expression C = ^ T + ocj3
evaluated at each of the temperatures Tm belonging to the heating
cycles. The parameters and oc were obtained from a least squares
fit of the data to the above expression, the analysis being per
formed on the computer. It is seen that there is generally more
scatter in the low temperature points than in those at higher
temperatures for the reason mentioned above. It is noted that at
the lowest temperatures, the stray heat leak represented about 40$ of the total heating rate during the heating periods. However, as
the temperature increased during the course of the run, both the
magnitude of the stray heat leakis and its percentage contribution
to the total heating rate diminished together. Thus, the errors
in the estimation of the stray heating rate became less significant
with respect to the precision of the heat capacity measurements
and this is shown by the fact that there is less scatter in the
points as the temperature is increased. At high temperatures, the
deviations appear to be systematic rather than random and this is
thought to be a result of the dominant effect of the systematic
errors in the thermometer calibration formulae.
The heat capacity of the calorimeter was assumed to have a
temperature dependence of the form:
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where the term ft T represents the heat capacity of the electrons3and the term oc T . that of the phonons. It was assumed that there
were no other contributions over the experimental range of temper
atures. To test the validity of using Eq. 5.1 to represent the
specific heat of the calorimeter, two separate least squares
analyses were performed, one oh the data from the entire temper-
ature range, 0.6° to 2.6°, and the other on the high temperature data, 1.3° to 2.6°. The results of the two analyses are shown in
Table 5 -1.
It is seen that the two linear terms agree very well and
although there is not such good agreement between the two cubic
terms (a disagreement of 1/0 , the agreement is thought to be
satisfactory in view of the 3% standard error in each of the two values.
Although the magnitude of the linear term may be accounted
for by the electronic specific heat corresponding to the amount
of copper (61 gm) contained in the calorimeter, such is not the
case for the cubic term, where the magnitude is about twice that
expected from the copper. Evidently, some other component of the
calorimeter makes a significant contribution to the heat capacity
and this contribution has a T3 temperature dependence. The
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TABLE 5.1 RESULTS OF THE LEAST SQUARES ANALYSES ON THE EMPTYCALORIMETER DATA
■-
RESULTS OF THE TWO
ANALYSES
0 .6° - 2.6° 1.2° - 2.6'
g (mj/deg2) 0*662 0.662
OC (mj/deg^) 0.0875 0.0884
Standard error in ^ (%) + 1.0 + 1.3
Standard error' in OC (/£) + 3.1 + 2.6
RMS Deviation of the points from Eq. _5.1 (mj/deg)
0.032 0.017
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possibility" of trapped gas inside the calorimeter being responsible (from incomplete evacuation^ for example), can be ruled out at
once* because about § litre STP gas would be required to account for the magnitude of the extra contribution. The materials used
in the calorimeter construction which could possibly account for
it are:
1. The vacuum grease (Apiezon T) used to anchor the
heater wire thermally to the side of the calorimeter, as well as
that used around the germanium resistance thermometer case, or
2, The Teflon covering of the thermometer leads
extending from the thermometer case to the soldered lead connect
ions which were anchored to the side of the calorimeter.
The specific heat of Apiezon T grease has been measured r e c e n t l y ( 3 8 ) ancj it has been found that below 4*2°K, the tempera
ture dependence of the specific heat has the following form:
C = 2.034T + 5.065T3 + 0.3547T5 - 0.0101T7 x 106cal/deg gm
Besides th e s p e c i f i c h e a t p o s s e s s in g s ig n i f i c a n t te rm s in powers
o f T o th e r th a n c u b ic , a b o u t 1 .5 gm w ou ld be re q u ir e d t o a cco u n t
fo r the m ag n itu de o f th e e x t r a c a lo r im e te r s p e c i f ic h e a t , w h ic h ,
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89
in view of the rather low density of the grease, represents a
much larger volume than the estimated quantity which was used.
The specific heats of some polymers (Teflon, Kel-F nylon,
and some polyethylene samples) have been measured between 1° and 4 50 (39) and the results show that in all cases the specific heat
Ois proportional to TJ, as is required to explain the present
calorimeter results. The result for Teflon is as follows:
C = 0.045 T^ mj/deg gm
Taking the density as 2.2 gm/cm^, then about 0.5 cm^ would
he required to account for the magnitude of the extra calorimeter\
heat capacity which is just about the estimated quantity which
covers the thermometer leads* Thus, the anomalously high coeffic
ient of T^ in the calorimeter heat capacity may be explained by the
contribution of the Teflon.
Unfortunately, the calorimeter measurements do not extend up
to the temperatures obtained with the copper and solid gas samples,
that is, about 4°K. For the purpose of subtracting the calorim
eter heat capacity from the measurements on the copper and solid gas
samples, Eq. 5«1 was used to represent the calorimeter heat capacity
an the entire range of temperatures up to 4°K. However, in view
°f the fact that the measured calorimeter heat capacity below 2.6 may he accounted for by the contributions of the copper and Teflon
wsfpiReproduced with permission of the copyright owner. Further reproduction prohibited without permission.
alone (which together contain only terms in T and T3), then the
method of extrapolating Eq. 5.1 out of the range of the calorimeter
measurements up to 4°K is thought to be satisfactory, for other
contributions are not expected to be important in the region of
extrapolation.
To subtract the calorimeter heat capacity from the measure
ments on the copper and solid gas samples, Eq* 5*1 was evaluated
at each of the temperatures Tm belonging to the corresponding heat
ing cycles of the sample measurements and then this calculated
value was subtracted from the total heat capacity found for the
set of heating cycles. The calculations were performed on the
computer in the same program as that used to evaluate the total
heat capacity.
5.3 - The Copper Measurements.
The specific heat of copper was measured for three main
reasons. They are:
1. To test the general operation of the cryostat. Among
the information required here was the heat switch performance
(sample cooling times, frictional heating on opening the switch),3he stage performance (minimum accessible temperature), operation
of the electrical circuits (no short circuits), and degree of
vibrational heating of the calorimeter system.
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91
2. To test the degree of thermal isolation of the'calorimeter. During the measurements, the calorimeter was not perfectly isol
ated from its surroundings, there being the electrical leads and
the gas filling tube which provided a direct thermal link between
them. Therefore, some parts of the leads and tube undoubtedly
took part in the heating process to some extent, and only a direct
experimental test could termine whether this uncertainty contrib
uted an objectionable error to the measurements. Because the
conductance of the thermal link varied with temperature, it was
decided to reduce the conductance of the thermal link as much as
possible by choosing low conductivity materials for the leads
(manganin) and the filling tube (copper-nickel). Even so, it was
felt desirable to test the degree of isolation of the calorimeter
by making measurements on copper.
3. To test the thermometry. This has been described in Sec. 4.2.
If the copper measurements were found to be satisfactory,
then this would lend weight to the validity of the solid gas
measurements, since any unforseen systematic errors of the cryostat
arising from the sources mentioned above which could spoil the solid
gas measurements, would also spoil the copper measurements.
For the measurements, a blind hole was drilled in one end
of the cylindrical copper sample to allow for the germanium
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thermometer and a small quantity of vacuum grease was smeared onto
the base of the calorimeter. The copper sample was then slung
underneath the calorimeter using fine copper wire as a suspension
cradle, with the sample in firm contact with the calorimeter base.
The excess grease which was squeezed out from the mating surfaces
was removed.
The results are shown in graphical form in Fig. 5.2 and in
tabular form in Appendix 2—B. Again, as for the display of the
empty calorimeter results, in Fig, 5«2 the results below 1.26° are
given as a plot of C/T against T^ and the results above 1.26° are
shown in the form of a deviation plot. The empty calorimeter heat
capacity was subtracted in the manner described in Sec, 5*2 and the
resulting heat capacity was then normalized to one mole of copper.
The data from the entire temperature range was fitted by a least
squares analysis to an expression of the form:
where the term linear in temperature represents the contribution of the electrons and the cubic term represents the contribution
of the phonons. It was assumed that there were no other contrib'
jj-T + 1944 j/mole deg Eq. 5.2
utions over the experimental range of temperatures. The results of the least squares analysis are shown in Table 5*2. The values
°f V" and Og are 0.6662 mj/mole deg^ and 336 respectively,
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ZJCD-o—ioQ.coCDCl
-oCD3</>C/5o'3ooo■oV<cq'
3CDcp.
CD-o—5oQ.Cao3■ooCDQ.
"OCDC/5C/5
.+2xPl - l
F 0
0 - iQ -2
■LOW TEMPERATURE* DATA (BELOW) Crooner ~ (* T+ «»T )
DEVIATION = c p p p e r----------------------X 100%'copper
3 , T(°K) 4
l i) HIGH TEMPERATURE RESULTS
ff»«*0BTAINED FROM LEAST SQUARES FIT TO THE HIGH TEMPERATURE RESULTS ONLY
G/T(m J/mole deg2)
5 %..fUi) LOW TEMPERATURE RESULTS
THE LINE REPRESENTS THE EQUATION C / T = « + « T 2WITH *» «F R 0 M LEAST SQUARES FIT TO THE HIGH TEMPERATURE RESULTS
1 0.5
FIG 5.21 I
1.0 1.5 2.0THE MOLAR SPECIFIC HEAT OF COPPER T
in very poor agreement with the commonly accepted values; the
difference between them being 4% and 3% respectively. However,
it is possible that the low temperature results are in error
because of experimental difficulties which were encountered when
preparations were being made for the copper run. Since the
preparations were rather hurried, it was not possible under the
circumstances to re-arrange the calorimeter suspension threads
so as to prevent them coming into intimate contact with the side
of the copper sample. Consequently, there was set up a thermal
path of significant conductance between the copper sample and the
shield and as a result, there was stronger than normal thermal
coupling between them. Under these conditions, it is expected
that the heat leaks to the calorimeter would be more than normally
dependent on fluctuations in the shield temperature. Thus, the
shield temperature fluctuations could contribute a greater than
normal uncertainty to the temperature rise in a heating cycle.
This effect would be most important at low temperatures where
the heat capacity of the calorimeter system is very small.
The results above 1.26° were fitted by a least squares
analysis to an expression of the form of Eq. 5»2* The results of
the analysis are shown in Table 5.2. The values of and eg are
•7039 mj/mole deg2 and 346.1° respectively, and are in excellent agreement with the results of Martin^26). Prom the lower graph in
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TABLE 5.2 THE SPECIFIC HEAT OF COPPER
( i ) RESULTS FROM THE ENTIRE TEMPERATURE RANGE
THIS WORK M A R T I N OSBORNE ET A L ^4 1 ^
^ (mj/mole deg) 0.6662 0.6961 0.6943
Oj) (°K) 336 345.6 344.5
Standard error in ^ (%) ± 1.5 + 0.78Standard error in O^(^) ± 3.5 + 1.0RMS Deviation of the points 0.052from Eq. 5.2 (mj/mole deg)
(ii) RESULTS ABOVE 1.26°K
^ (mj/mole deg) 0.7039
(°K) 346.1
Standard error in ^ (%) +0.43
Standard error in 0° (%) + 1.0RMS Deviation of -the points 0.008
from Eq. 5.2. (mj/mole deg)
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Fig. 5*2 can be seen the disagreement, below 1°, between the high
and the low temperature data. Table 5.3 gives the impurity analysisand weight of the copper sample.
It is possible that Eq. $.2 is insufficient to represent
the copper specific heat and it may be necessary to invoke the
existence of an extra term in the lattice contribution of the formBt5(40). If so, then the constant B would have to be negative
to account for the trend of the measurements at low temperatures.
According to Martin^^^, a negative B may be required to bring
his 0j) value (and also that of this work) into agreement with the ©1Op derived from the elastic constants, However, a compilation
of recent data from different workers by Osborne et al^4-*-) has
resulted in the establishment of the Copper Reference Equation,
which is a representation of the specific heat of copper from all
the reliable sources so far available:
The Copper Reference Equationul
The best values of the constants A. are as follows:
A-l = 6.9434 x 10“1 A2 = 4-7548 x 10“2 A3 = 1.6314 x 10“6
A 4 = 9.4786 x io~8 A 5 =-1.3639 X 10“10A^ = 5.3898 X 1 0 ~ 1 4
(For Cp in inj/mole deg)
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TABLE 5 . 3 THE IMPURITIES IN THE COPPER SAMPLE - THE SAMPLEWEIGHT
SEMI-QUANTITATIVE
QUANTITATIVE SPECTROGRAPHIC SPECTROGRAPHIC
ANALYSIS BY CARRIER METHOD ANALYSIS
PPM (by weight) ppM (by weight)
IMPURITY TOP BOTTOM TOP BOTTOM
... Fe 0.14 0.13
Mn not detected not detected
Mg 0.3-0.03 0.3-0.03
Si 0.1-1.0 0.1-1.0
Ag 0.3-0.03 0.1-0.01
WEIGHT OF SAMPLE 145.0 gm
ATOMIC WEIGHT OF COPPER 63.57 gm
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It is seen that A3 (that is, our B) is positive, contrary to the present requirements. Also, the magnitude of the term
XTA T is so small below 4 K, "that it could not possibly account
for the disagreement between the low and the high temperature
results of this work.
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CHAPTER 6
THE. OXYGEN AND NITROGEN RESULTS
6.1 - The Purpose of the Measurements *.
The specific heats of oxygen, and nitrogen were measured for the following reasons:
1. Oxygen. Oxygen was measured partly because the results
of Fagerstroem^} from this laboratory were in poor agreement with
the earlier results of Kostriukova and Strelkov 42)^ but mainly
because there was some interest in the behaviour of the specific
heat at temperatures below those which had been obtained previously.
When this research was started, there was some doubt as to the
magnetic behaviour of solid oxygen and there was speculation that
there might be magnetic dipole ordering below 1°K which would show UP as a specific heat anomaly. At that time, it was not definitely
established whether or not the low temperature transition OC~ p ,
which Fagerstroem had investigated, was connected with the trans
ition to the antiferromagnetic state, although most evidence
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s u p p o r t e d this view (for discussion, and references, see Jamieson
M.A. Thesis^1"^). However, the matter was settled during the course
of this research when the neutron diffraction work of Collins^43)
showed that oxygen in the OC “phase is anti ferromagnetic and while
there is some degree of short-range order in the higher temperature
p -phase, there is no long-range anti ferromagnetic alignment of
the dipoles. Thus, the cC-p transition was found to coincide with
the onset of magnetic ordering and no specific heat anomaly from
this source could be expected other than that at the oc- p transition
temperature, 23.8°K. As expected, no anomalous behavior was found
in the entire temperature range of this work.
2. Nitrogen. Nitrogen was measured simply because specific
heat measurements have not been reported before for this solid gas
in the low temperature range obtainable with the present apparatus.
The only other reported data is that of Giauque and Clayton
down to 15°K and of K« Clusius et al^'*^ down to 10°K. Because
the characteristic temperature of molecular solids is a rapidly
varying function of temperature when the temperature is increased
out of the range where the Debye approximation is valid, that is,
where the characteristic temperature takes the constant value 0 , then the estimation of 0° from measurements at high temperatures (10° or 15°K) is liable to be in serious error. Since the low
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temperature limit & is often used in theoretical discussions^
it was thought useful to obtain values of the characteristic
temperature to as low a temperature as possible. With these low
temperature results, a much more reliable estimate of G° may then
be made. In addition, accurate specific heat values of pure
nitrogen were required in order to estimate the contribution of
the oxygen impurity to the dilute oxygen-nitrogen mixture results
(see Chapter 8).
6.2 - Presentation and Discussion of the Results.
The results are shown in graphical form in Figures 6.1 and6.2 and in tabular form in Appendices 2—C and 2—D for oxygen and nitrogen respectively. The results shown in the Figures are plots
of the characteristic temperature which was obtained from the
measurements as follows. The heat capacity of the calorimeter
was subtracted from that of the complete system in the manner
described in Sec. 5.2 and the resulting heat capacity of the solid
gas was normalized to one mole, where the mole was considered to
contain 2N atoms, N representing Avogadrofs Number. The molar A Aspecific heat was then equated to the expression 1944 (T/o)^
J/mole deg in accordance with the Debye theory• The specific heat
values and then the 9 values upwards from 2.4°K were corrected for tlie systematic deviations in the thermometer calibration formula-A
as described in Sec. 4.3.
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ission of the
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(T)
l!0 -
• O
LEGEND: • PRESENT RESEARCH* FAGERSTROEM (1965)
ioo
• - © 9• • © 0
• \A »« • a
6:
©o
©
. ® o ©A © a a ® e o
a
o i.oi
FIG. 6.1
2.0 3.0T(°K)
THE CHARACTERISTIC TEMPERATURE ®(T) OF OXYGEN
4.(
4
4
4
&a> v>o
I
o o E E <j- in o> m sj- too o
I3
to jn UJ ustr a:UJ UJ</» <n • <4
uioUJ
©94<*«
9S
4 <*•%
©
49 • 499
9 •«• 4©
4© ' ©9 49©
4,4
4 "4 4 4
44
UJoo££
0 La- rO ®
p©
v UJr - cc JL^ 31 i—r- <
tr iu o.
_ 2 O UJ<\i H
oh-co ccUJ ho < cc < X
oUJXh
OJCO©Ll
<rf-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Implicit in this procedure is the assumption that the
diatomic molecules do not rotate and where only the molecule as a whole takes part in the lattice vibrations, there being no other
contribution to the specific heat. This assumption may not be
correct, but in the absence of any detailed knowledge of molecular
rotation in solids, especially for the diatomic molecular solids,
it is considered preferable to consider only the motion of the
centre of gravity of the molecules as taking part in the lattice
vibrations. It is pointed out that in the solid state and as
the temperature is lowered, free molecular rotation is suppressed
and gives way to libration (torsional oscillations about equilib
rium positions), but in the temperature range covered in this
research, both the librational and inter-molecular vibrational
frequencies were thought to be too high for these modes to
contribute significantly to the specific heat. This expectation
was borne out by the results, since the characteristic temper
atures for both oxygen and nitrogen appeared to be fairly constant
over the experimental, temperature range»
The calculation of the amount of condensed gas inside the
caloi’imeter from the calibrated gas reservoir pressures and temp
eratures may have given an overestimate if some of the condensate
had been situated in the filling tube where it may not have taken
part in the heating process from which was determined the specific
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
heat. Since there was no way of directly knowing the amount
contained in the tube, the resulting errors in the solid gas
specific heat are difficult to estimate. However, an upper limit
may be placed on this error since the maximum error would be
obtained when the whole of the filling tube B (the lower tube)
contained condensed gas, making the reasonable assumption that
the upper tube A did not contain an appreciable amount. With this
assumption, the maximum overestimate of the amount of solid gas
inside the calorimeter is about 1%, However, because the tubes
were heated during the condensation process, it \As= most unlikely
that an error of this magnitude could occur. Thus, the resulting
errors in the characteristic temperature values from this source
are thought to be very small, less than 0 .2$, for both the oxygen and nitrogen runs.
1. Oxygen. Reference to Fig. 6.1 shows that at low temper
atures there is considerable scatter to the measurements. At
higher temperatures (above 1.8°K) the scatter is very small, being
less than 0.5$. The low temperature scatter is a result of having
to subtract two quantities of similar magnitude — the calorimeter
and total specific heats — where the errors in the two values
combine to produce a considerable uncertainty in the solid gas
specific heat. As the temperature is increased, the solid gas
specific heat quickly outgrowsthat of the calorimeter and the
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resulting errors in the solid gas specific heat rapidly diminish*
The results of Fagerstroem are also shown in the Figure
and a comparison between the two sets of results shows that the
agreement is satisfactory above 2 K. Below 2 K a comparison may
not be profitably made in view of the rather large scatter in both
sets of results. The estimates of the low temperature limit of
the characteristic temperature from the data above about 2°K are
as follows:
KostriukovaThis Work Fagerstroem^ 3 ) and Strelkov(4-2)
©°(°K) 104.5 + 1.0 104 ± 2.0 100
Comparing the results* it is thought that the Russian value is in error.
2. Nitrogen. Reference to Fig, 6.2 shows that the low
temperature points have considerable scatter* but those above 1 .8° are seen to have a slight systematic trend even after correcting
for the deviations in the thermometer calibration formula. Note that the scale in this Figure is very much larger than that used
to display the oxygen results and the deviations show up more clearly. The Figure shows results from two separate runs using
different amounts of solid nitrogen in each run.
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The low temperature results of the two runs seem to have
opposite trends. Thus, most of the points of Series~I turn upwards
while most of those of Series~II turn downwards. The reason for
this divergent behavior is not at all clear. Fig, 6.3 shows the total heat capacity (including the calorimeter) from both
runs at low temperatures, plotted as C/T against T2. Because the
divergent, trend is seen to show up in the total heat capacity also,
then it cannot be the result of faulty subtraction of the calorim
eter heat capacity or of errors in the estimation of the quantity
of solid nitrogen in the calorimeter, where these two errors are
though to be the main sources of uncertainty in the solid gas
specific heat. In addition, it is unlikely that the behavior is
a result of poor thermometer calibration, since this would affect
the results from both runs equally.
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© SERIES I “ 0 .4 9 4 moles a SERIES I I “ 0. 6 5 5 moles
THE LINES REPRESENT THE BEST STRAIGHT LINES THROUGH THE HIGH TEMPERATURE POINTS WHERE THE, LINES ARE FORCED TO PASS THROUGH THE POINT ( arc0|» 0 )
A A
FIG 6.3 TOTAL HEAT CAPACITY C/Tvs T2 FOR THE TWO NITROGEN RUNS
1 0 3
CHAPTER 7
THE CARBON MONOXIDE AND N ITRIC OXIDE RESULTS
7,1 - P re s e n ta t io n and D is c u s s io n o f th e Carbon M onoxide R e s u lts .
The CO results are shown in graphical form in Pig. 7»1
and the table of values is given in Appendix 2~E. The results
shown in the Figure are plots of the characteristic temperature
which was computed from the specific heat data using the same
method as that used for the N£ and O2 results (see Sec. 6.2). Corrections for the thermometer calibration deviations were
applied to the specific heat data above 2.4 K. in the manner des<~ cribed in Sec. 4.3. The maximum error in the specific heat res
ulting from an overestimate of the quantity of solid gas inside
the calorimeter (see Sec. 6.2) is estimated to be, at most, 0,5%>
The r e s u l t s o f G i l l and Morrison(9 ) a re a ls o shown i n F ig .
7.1, where t h e i r ta b u la te d s p e c i f i c h e a t v a lu e s were reduced to
values o f th e c h a r a c t e r i s t i c te m p e ra tu re u s in g th e same assu m ptio ns
as those used i n t h i s w o rk . The maximum d is a g re e m e n t, w h ich o c cu rs
at about 3.5°K, i s e q u x v a le n t t o a 5% d i f f e r e n c e in th e s p e c i f ic
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® Present research$ ° I FGFNfV, uctttwu. A GILL and MORRISON (1966)
1 0 5 •” •. .
(T) • ’ •
100
95
* A
Characteristic Temperature ©,given by Cps I944.( ,g ’ J/mole deg
1_____________L_2 3 4
T(°K)FIG 7.1 THE CHARACTERISTIC TEMPERATURE © (T)OF CARBON MONOXIDE
1 0 4
heat, which is equal to the sum of the maximum errors in* both sets of results.
It is seen from Fig. 7.1 that no anomalous specific heat behavior is apparent down to the lowest temperature, 0.7°K. This
result is discussed in the light of the application of the Third
Law to CO in Chapter 9 *
7.2 - Presentation and Discussion of the Nitric Oxide Results
The NO results are shown in graphical form in Fig. 7.2
and the table of values is given in the Appendix 2-F. Again,
the Figure shows the characteristic temperature plotted as a
function of temperature, with corrections for the thermometer
calibration deviations having been made. The maximum error in
the specific heat resulting from an over-estimate of the quantity
of solid gas inside the calorimeter is rather larger here than
for the other specimens. Because of experimental difficulties,
only a small amount of condensed NO could be collected. Hence,
a given amount remaining in the filling tube represents a much
larger fraction of the total quantity of condensate in the NO run.
It is estimated that the maximum error in the specific heat from
this source is about 1,5%<Reference to Fig, 7*2 shows that, as for the other solid gas
samples, there is considerable scatter in the data below 1.8 K
tut above this temperature the characteristic temperature is seen
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1 I
o
o*•oo
>»JQca>o*
©
©
' 0 o
• © • 9
• • • ©^ #
<£) o> a>g *o£ ™O oo ECl n .
E ^ I— roo H ©
"cn " t " 'C ^o 0) a —O o JC clO O
oto
oCJ
ro
OJ
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
FIG. 7
.2 TH
E CH
ARAC
TERI
STIC
TE
MPER
ATUR
E ©
(T)
OF NI
TRIC
OX
IDE
to r is e w ith in c r e a s in g te m p e ra tu re and then rem ains s teady up to
4°K. The amount o f th e r is e i s e q u iv a le n t to a re d u c t io n o f about
10# in th e s p e c i f ic h e a t . T h is i s v e ry unusual b e h a v io r , s in ce
the c h a r a c te r is t ic te m p e ra tu re f a l l s from i t s lo w te m p e ra tu re l i m i t
fo r most m o le c u la r s o l id s . There i s th e p o s s ib i l i t y t h a t th e re i s
a low te m p e ra tu re c o n t r ib u t io n t o th e s p e c if ic h e a t o th e r th a n
tha t from th e l a t t i c e v ib r a t io n s w h ich , because i t was n o t taken
in to account when com p u ting th e c h a r a c te r is t ic te m p e ra tu re , would
produce th e obse rved b e h a v io r . I n any case, even i f t h i s were so,
then th e e n tro p y a s s o c ia te d w i th th e anomaly w ould be in s u f f i c ie n t
to account f o r th e r e s id u a l e n tro p y .
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CHAPTER 8
MEASUREMENTS ON THE. IMPURE SAMPLES
8,1 - Account of the Chronological Sequence of the Experiments.The specific heat measurements on the first CO sample
received from the manufacturer revealed an anomaly, not accounted
for by the Debye theory, whose maximum was of the same order of
magnitude as the expected lattice contribution (see Fig. 8.2,
points labelled CO + 0.14$ ^ first, the anomaly was thought
to be connected with a change of phase to a third low temperature
crystal modification. The broadness of the anomaly suggested that
there might be a sluggish transition to the low temperature phase
similar, for example, to a martensite transition. If so, a time-
effect study of the anomaly might reveal its nature. Such a study
was undertaken and the chronological order of the experiments is as follows:
1. In an attempt to suppress the transition, the sample was cooled quickly from the freczing point to 2.5°h. and measurements
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were made up to 3.2 . If the attempt were successful, the measure
ments should have reproduced the data of Gill and Morrison^9^,
which were taken upwards from 2.5°. Then the sample was quickly
cooled to 1° and further measurements were made up to 3.2°.2. The sample was held at 4.2° for about 40 hours and the
above sequence of measurements was repeated.
3. The sample was held for about 40 hours at a high temper
ature (about 25°K), which was well above the region of the anomaly, in an attempt to anneal the sample. The sample was cooled again
and the same sequence of measurements was followed, with the excep
tion that for one series the deliberate heating rate was decreased
by a factor ten below that normally used. For some types of
transitions, the specific heat may depend on the magnitude of the
heating rate and it was hoped to observe such a dependence.
The results of these experiments are shown in Fig. S.l .
where the total measured heat capacity of the sample and calor
imeter is plotted in the form C/T against T2. The results show
that the specific heat is entirely reproducible and that no time—
effects are apparent.Soon after these experiments, the gas was sent for analysis
which revealed that it was contaminated with large quantities of
C02 and air (sec Table 8.1). At first, it was thought that the
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• •o»
111
© CO
CM
o>■
• o*S
•a CM
•• •
CMlO
""Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
FIG. 8
.1 TH
E TIM
E EF
FECT
ST
UDY
AND
THE
EFFE
CT
OF AD
DING
Cog
TO
THE
IMPU
RE
Co SA
MPL
E
TABLE 8.1 IMPURITY ANALYSIS OF THE IMPURE CARBON MONOXIDE SAMPLE
IMPURITY % (by Volume)
C02 0.108
N2 0.39
°2 0.137
DEW POINT -30°F
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CC>2 might; be responsible for the anomaly because the characteristic rotational temperature o^^the CO2 molecule was consistent with the temperature of the maximum, and in addition, the amount of CO,,
present was consistent with the entropy under the anomaly. Therefore,
the obvious experiment was to add some more to the sample. If
the above explanation were correct, then the magnitude of the
anomaly would be increased in proportion to the extra amount of C02.
About 0.57% C02 was added to the sample already containing 0.108/S C02 and the results are shown in Fig. 8.1, again as a plot of the total measured heat capacity C/T against T . The results show that
no discernable effect may be ascribed to the added C02#To continue the investigation, it was decided to study the
effect of adding more oxygen to the sample. The procedure was to
evacuate the entire gas handling system for about a day and then
admit a small amount of oxygen from the cylinder to the reservoir
where the amount was measured by noting the reservoir pressure
reading. The host gas was then admitted to the reservoir to bring
the total pressure up to about 800 mm Hg. The mole fraction of
the oxygen was equated to the ratio of the two pressure readings
before and after admitting the host gas. It is estimated that the
mole fraction was obtained with an uncertainty of no less than +5%.
An amount of 0.22% 02 was added to the CO sample already containing 0'-l37/£ 02 and the results are shown in Fig. 8.2 (points labelled
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C0+ 0.47:5 02). The anomalous specific heat, Canom, was’obtained
from the measurements in the following way. First, the calorimeter
heat capacity was., subtracted from the total and then the result
was normalized to one mole of CO. Then, .the molar specific heat
of pure CO was subtracted and the result was normalized to one
mole of 02 to obtain C^nom. All calculations were performed in
one computer program where the specific heat of pure CO was rep
resented by its characteristic temperature which was taken as a
constant of value 103° over the entire temperature range, 0.6° to 3.5°K.
The final experiment to study the effect of adding oxygen
was to add about 0.55% P2 to a pure nitrogen host. This experiment
was undertaken in order to try to find a difference in the size or
shape of the anomaly which could be ascribed to a difference in
the surroundings of the oxygen molecule. The anomalous specific
heat for this sample is shown in Fig. ,8.2, and the table of results
is given in Appendix 2-H. For the purpose of subtracting the
specific heat of pure N2, the characteristic temperature was token
as a constant of value 83.5° over the entire temperature range,°.6° to 4°K.
The temperature of the maximum of the anomaly is estimated
to be, Tmax = (1.95 ± 0,05)°K for both the N2 -I* 0.55% 0% samplethe CO + 0.14% 02 sample, and TjnaX - (1.90 + 0.05)°K for the
I- 0.47% 0 sample.
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with perm
ission of the
copyright ow
ner. Further
reproduction prohibited
without
permission.
■o-5oQ.
LEGEND: ’ Ne+0.55%0** CO+O.14% Oe• ca+O47%02
■SCHOTTKY ANOMALY FROM TWO-LEVEL, Q|/Qo"2»
6-5.14* SCHEME
o*a>T3 SCHOTTKY ANOMALY FROM FREE MOLECULE’ SCHEME • y
SCHOTTKY ANOMALY FROM-TWO LEVEL NON-DEGENERATE SCHEME
0 3
FIG 8.2 THE ANOMALOUS SPECIFIC HEAT C on^lT ) OF THE SAMPLES CONTAINING OXYGEN IMPURITY
8.2 - Discussion of the Results.
The results show quite clearly that the anomalous heat
c a p a c i t y of the original CO sample is due to the presence of
oxygsn and that the magnitude of the anomaly is linearly propor
tional to the amount of oxygen present. The last result is of
great significance because it rules out the possibility of an
interpretation based upon exchange effects between the oxygen
molecules. This matter is discussed in more detail in Sec. 9.2.
For the CO sample containing 0.47% 0^, the anomalous specific
heat is seen to fall faster on the high temperature side than the
other two results and eventually goes negative (the negative
results are not shown in Fig. 8.2). For this sample, Canom is
everywhere less than that of the other two samples. Also, the
temperature of the maximum lies below that of the other two. This
behavior suggests that the lattice heat capacity (that of pure CO)
subtracted from the total was over-estimated for this sample. To
explain the observed difference between the two CO—O2 results, it is necessary to introduce a difference of about 2% in the character
istic temperatures. If this interpretation is correct, then it
becomes necessary to explain why an impurity of only 0.5% of 0 in
CO modifies the lattice vibrational frequency spectrum to correspond
to a change in the characteristic temperature of this magnitude,
furthermore, in the case of N2 it appears that the same proportion
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of 02 (0.5%) does not modify the frequency spectrum to the samedegree. From a study of the thermodynamic properties of CO +
(o)2,6/s Gill and Morrisonw 7 concluded that the specific heat behavior at low temperatures of this mixture was no different
from that of pure CO. If so, then it is difficult to see that
the above interpretation of the present negative results is just
ified, since the perturbing effect of small additions of 0 and
N to a host lattice should be about the same because of the 2similarity in their molecular weights.
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CHAPTER 9
FURTHER DISCUSSION OF THE RESULTS
9.1 - The Residual Entropy of CO and NO,The results of this research show that both CO and NO
display no anomalous specific heat behavior down to about 0.6°Kj
therefore, the residual entropies of these substances remain.
However, the case of CO needs re-examination because of the incorrect value of the characteristic temperature used by Clayton
and Giauque(l0) to estimate the calorimetric entropy beneath the
range of their measurements, that is, below 12°K. The value used
by Clayton and Giauque was 80°, whereas the average value m the
region below 12°K found by Gill and Morrison^9 is about 100
Therefore, Clayton and GiauqueTs estimate of the entropy mregion of extrapolation is in error by a factor of about two. The
j • ~ ,-*TH-r>r>nv found by the two groups calculations of the calorimetric en py
is given belo\\r.
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113
CO
12° - O K
12 — B.Pt. (includingall latent heats)
Entropy of real gas at 1 atm, B.Pt.
ENTROPY cal/mole deg
Clayton and^10^Gill a n d ^ Giauque Morrison
0.46 0.23(extrapolated) (measured)
3 6 . 5 5
37.01
3 6 . 5 5
3 6 . 7 8
Spectroscopic Entropy of real gas at 1 atm, B.Pt. 3 7 . 8
Residual Entropy = 1.0 + 0.2
NO
(12)Residual Entropy obtained by Johnston and Giauque = 0.75cal/mole deg
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As pointed out above, if the usual explanation of the
residual entropy* which is based on the idea of Tfrozen-int
disorder of the molecules in the crystal* is correct* then it
is necessary to say that molecular rotation in the solid is
impossible. Infrared studies^^^* spin-lattice studies (4 7)
dielectric studies(48 )# antj thermodynamic studies (49) show
that* to some extent* molecular rotation is possible in many solids
even at temperatures well below the freezing point* although the
rotation may not be as free as it is in the gas. There may be
energy barriers which oppose rotation in the solid* but some
degree of molecular re-orientation is very often possible. In
an extreme case* the molecule may be regarded as performing
torsional oscillations about one of several possible equilibrium
orientations where the molecule may tunnel through the potential
barriers to switch from one orientation to another. Unfortunately*
there seems to have been no detailed study made of this problem
in the cases of CO and NO* although from our limited knowledge we
have no reason to suppose that they show any essential differences
from those solids for which molecular r e —orientation has been(50 )definitely established* for* example* CH^ .
The CO molecule has a small electric dipole moment (0.11 debye)
and it is possible that dipole ordering will occur at low temper
atures such as to remove the residual entropy. On a Curxe-Wexss
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scheme, the transition, temperature is given by:
T ^ Nyx2 3k
where N = number of molecules/cm^
= dipole moment
k = Boltzmann*s constant.
Evaluating this expression, it is found that dipole ordering
may occur at around 0.6°K. (I am grateful to Dr, J. A. Morrison
for suggesting this idea.) From a theoretical, study of the
various interactions in some molecular c r y s t a l s t h e dipole
orientation energy of CO molecules at 0°K was calculated to be
-3.4 cal/mole,* Equating this energy to kT, one finds for the
temperature at which dipole orientation becomes important a value of about 2°K. Thus, it is possible that dipole ordering occurs
below the present range of measurements where a transition to a
third.low temperature crystal modification takes place analogous,
for example, to the transition in oxygen^4J'. Some evidence
for a third low temperature phase of CO has been provided by an
X-ray study of the CO-Ar phase diagram^'1" , although it appears
that there have been no subsequent investigations which could
improve on this limited amount of information. It is interesting
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116
that nitrous oxide ^ 0, which has a linear structure NNO, also has a weak dipole moment (0.17 debye) and may therefore be discussed along similar lines.
Nitric oxide may present us with a different situation,
however. Prior to this work, there had been no specific heatomeasurements on the solid below 14 K; therefore, before any
further investigations are made, it is suggested that the temper
ature range between 4° and 14°K be examined. It is possible that
quadrupole interactions between the dimers may produce an
ordering effect similar to the oc— j-S transition in both CO and
N2 ~* t although the present uncertainty regarding the structure
of the N2O2 dimer^4) may preclude any useful analysis along these lines.
Unfortunately, from thermodynamic studies alone it is only
rarely possible to determine the mechanism responsible for any
observed anomalies. In the cases under discussion here, we would
like to have at hand extensive infrared absorption, neutron
scattering, dielectric relaxation, spin-lattice relaxation, and
perhaps further thermodynamic data - such as studies of the effects
of pressure - to very low temperatures (below 1°K) in order to be
in a better position to make a decision regarding the origin of
the re si-dual entropy.
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117
9*2 — Tlie Effect of Oxygen as an Impurity.
The results from the three dilute oxygen mixtures described
in Chapter 8 show the effect of small additions of oxygen to twohosts, CO and N • To understand the results, a satisfactory theory2must explain the following observations:
1. The temperature of the maximum of the anomaly is about
2°K and -is the same value for all of the mixtures.
2. The magnitude of the anomaly is linearly proportional
to the amount of oxygen present.
3. The size and shape of the anomaly does not differ accord
ing to whether the host is CO or N2 •4* Additions of other gases such as and N2 (see
Table 8.1) to CO is without measurable effect on the specific heat.
5. No irreversible or other non-equilibrium effects are
apparent for one of the mixtures (see Sec. 8.1 ).
From observations 1 and 2 we may at once rule out an inter
pretation based upon exchange effects between the oxygen molecule
spins. This is so for two reasons: First, if exchange forces
were operating, we would expect the magnitude of the anomaly to
depend on a higher* than linear power of the density of the oxygen
impurity because of the co-operative nature of exchange forces.
Second, the temperature of the maximum would depend on the
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118
concentration of oxygen; decreasing the concentration would shift
the maximum to lower temperatures. For example, in the case of (53 )solid air (20^ 0^ in N^) the antiferromagnetic transition is
found to be displaced to about 3°K from the OC- transition of
23-8°K in pure oxygen.
Some independent evidence from which we may conclude that
exchange effects are unimportant is that provided by a study of
the magnetic susceptibility of oxygen molecules enclosed in a
clathrate compound from room temperature down to 0.25° K ^ 4> 55) <The experiments showed that exchange effects were negligible in
the entire temperature range, for the observed variations from the
magnetic behavior expected from isolated freely-rotating molecules
could be successfully explained in terms of a hindered rotator
model, where the rotation of the molecules was perturbed at low
temperatures by the internal crystal field of the clathrate.
Whereas the average separation of the oxygen molecules in the clath-orate materials was 8 A, the average separation in the mixtures used
in the present study was never less than 4-0 X . Clearly, on this
basis it is improbable that exchange effects are important in the
present work.
It is now thought that the results may be interpreted in terms
of a model in which the oxygen molecule has a set of low-lying
molecular energy levels (see Sec. 1.2). Immediately, there is no
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119
difficulty choosing a suitable set of levels, for the3 -vimolecule has an unusual electronic ground state, , which is
split into three closely-spaced levels by the interaction of the
spin with the rotation of the molecule (see Fig. l.l). Clearly,
we can now explain observation 4 because the molecules C02, H20,3 -vand do not possess a ground state and hence cannot give
rise to a low temperature anomaly in the specific heat.
An attempt must now be made to fit a Schottky-type specific
heat anomaly corresponding to a particular set of energy levels
to the observed anomaly. When an agreement has been found between
the observed anomaly and a Schottky anomaly, we may conclude that
the energy level scheme which resulted in the agreement is that
actually in operation in the molecule. It must be remembered that
from the form of the Schottky specific heat we may obtain the
energy level spacings and degeneracies, but only as a ratio of the
degeneracy of the ground state. The first step was to evaluate
the Schottky anomaly corresponding to the ground state triplet of
the free oxygen molecule (see Fig. l.l). The result is shown in
Fig. 8,2. It is seen that the agreement with the observed anomaly
is very poor; clearly, the free molecule energy 3.evel scheme has
been modified by the surroundings of the host molecules in the solid
state. .
As a basis for obtaining an agreement with a particular energy
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level scheme, it was decided to use only those schemes for which
the corresponding anomaly had the same temperature of the maximum
as that of the observed anomaly, that is, the energy level spacings
were adjusted so as to obtain this agreement. The first proposed
scheme was a simple two-level, non-degenerate system. The energy
spacing was put at 4«66°.in order to obtain agreement with the
maximum of the observed anomaly. The result of the calculations
shown in Fig. 8.2 is in very poor agreement with the observed
anomaly. Thus, there is no two-level, non-degenerate scheme which
could fit the observed results.
The final attempt at a fit was to evaluate the Schottky anom
aly for a two-level system where the degeneracy ratio, upper to
lower, was set at 2:1. The spacing .of the levels was put at
5 * 14° order to obtain agreement with the maximum of the observed
anomaly. Figure 8.2 shows that the agreement with the experimental
results is excellent in view of the rather large uncertainty (about
5%) in the estimation of the mole fractions of oxygen in the samples.
The above energy level scheme (shown in Fig. 9*1) which is operating
in the molecule in these experiments differs considerably from the
free molecule scheme. Moreover, the observed scheme does not
differ according to whether the host molecule is either CO or N2 (observational and 3).
The fact that excellent agreement is found with a Schottky
/
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FIG 9.1
g =5.14
9 |/9 0 *2
9q
THE EXPERIMENTALLY-DETERMINED ENERGY LEVEL SCHEME OF THE OXYGEN MOLECULE IN ITS GROUND ROTATIONAL STATE
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121
expression may appear remarkable at first sight. However, it
seems that we must accept the fact that the 0^ molecules are
rotating at these low temperatures, although not necessarily as
freely as in the gas. It is thought that the rotational ground
state degeneracy is still lifted by the interaction of.the spin
with the molecular rotation, but because the rotation is most
likely hindered, the magnitude of the interaction is expected to
be different.from that in the freely-rotating molecule. This
accounts for the difference in the energy level schemes between
that observed in these experiments and that of the free molecule.
Curiously, we are brought up against the very same problem
as that encountered when trying to interpret the residual entropy
of CO and NO; that is, the question of molecular rotation at low
temperatures. As discussed in Sec. 9.1, we would like to have
some more direct evidence relating to this question in the case
of the dilute oxygen mixtures, such as infrared absorption and
spin-lattice relaxation data. Perhaps the most profitable line
of- enquiry would be through a spin-lattice relaxation study of
dilute 0^0-^-enriched oxygen mixtures.
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ACKNOWLEDGEMENTS
I wish, "to thank Dr, G. M. Graham for the supervision ofi*
this research and for his helpful suggestions concerning the
interpretation of the dilute oxygen mixture results.
My deepest gratitude is due Dr_ J. A, Morrison at whose
suggestion this research program was initiated. His kind
consideration throughout our correspondence was a constant
stimulus to me*
I wish to thank Dr. D. L. Martin who very kindly provided
the copper sample and its impurity analysis and various cryogenic
materials which were used in the apparatus construction.
Many thanks are due Dr. F. D, Manchester who very generously
loaned to me his isolating potential comparator and furnished the
vapor pressure He° gas from his own short supply.
Finally, I wish to acknowledge the generous financial support
provided by the National Research Council by whom I was awarded a
Studentship in the years 1965-1967*
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REFERENCES
I. W.Nernst, Kgl. Ges, d. Wiss. Gott., 1 (1906)■ 2. F.E.Sirnon, Z.Phys. JJL, 806 (1927)3. C.-H. Fagerstroem, Ph.D. Thesis, University of Toronto (1965)4. W.F.Giauque and R.Overstreet, J.Am.Chem.Soc. jy, 1731 (1932)
5. A.Frank and K.Clusius, Z.Phys. Chem. B36. 291 (1937)6. J.H.Colwell, E.K.Gill, and J.A.Morrison, J.Chem.Phys. 39.
635 (1963)7. R.W.Hill and B.W.Ricketson, Phil.Mag. yfj, 277 (1954)
8. J.Wilks, The Third Law of Thermodynamics. Oxford UniversityPress, (1961)
9. E.K.Gill and J .A .Morrison, J.Chem.Phys. 1585 (1966)
10. J.O.Clayton and W.F.Giauque, J.Am.Chem.Soc. jy, -2610 (1932)
II. C.S.Barrett and L.Meyer, J.Chem.Phys.' 43. 3502 (1965)12. H.L.Johnston and W.F.Giauque, J.Am.Chem.Soc. juL, 3194 (1929)
13. E.S.R.Gopal, Specific Heats at Low Temperatures. Plenum Press,New York (1966)
14. W.J.Dulmage, E,A.Meyers, and W.N.Lipscomb, Acta Cryst. .6,760 (1953)
15. H .C .Jamieson, M.A.Thesis, University of Toronto (1965)16. G.Herzberg, Molecular Spectra and Molecular Structure.
Van Nostrand, 2nd. Ed. (3.965)
17. H.A.Kramers, Z.Phys. jy, 422 (1929)18. R.Sch.lapp, Phys.Rev. j£l, 342 (1937)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
19.20.
21.22.
23.
24.
25.26.27.28.29.
30.
31.
32.
33.
34.
35.
36.
37.38.
39.
M.Mizushima and R.M.Hill, Phys.Rev. 5L3j 745 (1954)H.M.Rosenberg, Row Temperature Solid State Physics.
Oxford University Press (1963)R.G.Scurlock and E.M.Wray, J.Sci.Inst. J4J,. 421 (1965)
R.W.Hill and G.R.Pickett, Proc.Low Temp.Calorimetry Conf.,Helsinki (1966)
F.D.Manchester, Can.J.Phys. 2lL> 989 (1959)R.Berman, J.Appl, Phys. 318 (1956)
T.M.Dauphinee, Can.J.Phys. jy., 577 (1953)
D.L,Martin, Phys.Rev. 141. 576 (1966)J.R.Clement and E.H.Quinnell, Rev.Sci.Inst. 213 (1952)
F.E.Hoare and J.E.Zimmerman, Rev.Sci.Inst. I84 (1959)S.Weber et al, Leiden Comm. 246A. (1936)S.Weber and G.Schmidt, Leiden Comm. 246C (1936)
G.T.McConville, R.A.Watkins, and W.L.Taylor, Proc.Low Temp.Calorimetry Conf., Helsinki (1966)
T.R.Roberts and S.G.Sydoriak, Phys.Rev. 102. 3O4 (1956)S.G.Sydoriak and T.R.Roberts, Phys.Rev. 106, 175 (1957)
S.G.Sydoriak and R.H.Sherman, J.Res.Nat.Bur.Stand. 68A,547 (1964)
N .E .Phillips, Phys.Rev. 134. 385 (1964)
J.T.Schriempf, Cryogenics .6, 362 (1 9 6 6)H.Van Dijk et al, J.Res.Nat.Bur.Stand. 64A, 1 (i9 6 0)E.F.Westrum et al, Cryogenics _2, 43 (1967)
W,Reese and J.E.Tucker, J .Chcm.Phys. _43? 105 (1965)
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40. T.K.H,Barron and J.A.Morrison, Can,J.Phys. 799 (1957)41. D#W.Osborne, H.E.Flotow, and F.Schrein.er, Rev.Sci.Inst. 58.
159 (1967)42. M.O.Kostriukova and P.G.Strelkov, Dokl.Akad.Nauk.SSR, 90,
525 (1953)43. ,M.F.Collins, Proc.Roy„Soc. 8<), 415 (1966)
44. W.F.Giauque and J.O.Clayton, J .Am.Chem.Soc. Jj£j 4 875 (1933)45. K.Clusius et al, Z.Naturforsch. 14a. 793 (1959)
46. W.C.Price and G.R.Wilkinson, J.Phys.Chem.Solids ,18, 74 (1961)
47* E.R.Andrew, J.Phys.Chem.Solids ,18, 9 (1961)48. C.P.Smyth, J.Phy s. Chem. Solids lj$, 40 (I96I)49. L.A.K.Staveley, J.Phy s. Chem. Solids lj3, 46 (I96I)50. G.A.de Wit, Thesis, University of British Columbia (1966)51. L.Jansen,A.Michels, and J.M.Lupton, Physica j20, 1235 (1954)
52. M.W.Melhuish and R.L.Scott, J.Phys.Chem. 68, 2301 (1964)53* W.H.Lien and N .E .Phillips, J .Chem.Phys. 3^ , 1073 (1961)
54* A.H.Cooke et al, Proc.Roy,Soc. 225. 112 (1954)
55* H.Meyer, M.C.M.0*Brien, and J.H.Van Vleck, Proc.Roy.Soc.2Al, 414 (1957)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX 1
THERMOMETER CALIBRATION DATA
He^ vaporpressure: i mm (meas)
h mm corr for P He Po/Pw pc (corr) mm
T°K R ohms
0.71J
0.71 0.927 0.658 0.618 1436.42.20 2.19 O.984 2.15 0.757 972.06.29 6.26 0.997 6 .24 O.93O 693.6
11.54 11.48 1 11.48 1.058 573.514.73 14 • 66 1.116 534.016.48 I6.40 1.145 516.2518.93 18.84 1.181 495.5020.78 20.68 1.207 480.4021.75 21.65 1.22.0 472.8025.55 25.42 1.267 449.0028.20 28.06 1.297 437.4528.22 28.08 1.297 437.0831.87 31.72 1.335 421.3032.90 32.74 1.346 417.3139.42 39.23 I.406 395.6344.98 44.76 1.453 379.7551.43 51.18 1.503 365.2566.08 65.76 1.603 339.4365.90 65.58 I.601 339.2578.10 77.72 1.673 322.85
(Continued....)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
86.1195.19
85.69
94.73
1.716 313.261.763 304.68
NOTES:
u 4He vapor pressure:h mm (meas) T°K R ohms
67.5 2.431 217.9795.45 2.610 204.60
149.4 2.872 187.05
231.1 3.166 170.86
317.1 3.406 159.57
426.7 3.655 149.38
536.2 3.865 141.79
642.2 4.041 135.96
771.1 4.231 130.15
1. The therinomolecular pressure ratios, pc/pw were
obtained from the tables of Roberts and Sydoriak2. The tempei’atures vrere obtained from the 1962 He^
vapor pressure scale and the 1958 He^ vaporpressure scale(37).
(32).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX 2
TABLES OF SPECIFIC HEAT RESULTS
APPENDIX 2—A THE EMPTY CALORIMETER
1, Range in which Calibration-B was used to compute specific heats
T°K AT (mdeff) C (mj/dee) T°K AT (mdes) C (mj/dee:)
0.6535 23.5 0.497 0.6904 23.7 0.5120.7282 20.7 0.564 0.7699 22 .6 0.538
078132 32.1 0 583 0.8653 29.7 0.615
O .8887 29.2 0.644 0.9615 46.0 O.7420.9445 26.5 0.710 1.0502 44.3 0.770
0.9903 25.0 0.752 0.6889 42.4 0.440
1.0435 44.7 0.763 0.7944 56.0 0.595
1.1023 39.3 0.867 0.9219 47.7 0.7001.1764 40.1 0.850 I.OO48 43.7 0.7621.2717 32.7 1.04 1.0825 41.9 0.796
1.2609 32.7 1.01 1.1575 39.4 O.8460.6485 26.4 O.404 1.2199 34.6 0.965
0.7994 27.5 0.609 1.2890 56.3 1.06
0.8655 26.4 0.637 0.8476 31.3 0.596
0.9368 26.7 0.630 0.9228 28.7 0.6640.9869 24.I 0.698 1.0002 44.0 0.7881.0313 23.1 0.732 1.0745 41.9 0. 8311.0851 42.0 0.787 1.1465 38.4 0.906
(Conti nuecl. . . . )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.1382 39.8 0.830 1.2100 36.1 0.9641.1881 34.6 0.955 1.2669 36.4 0.956
Range in which Calibration-A was used to compute specific heats
. 1 ►3 O A T (mdeja?) C (mj/deg) T°K AT (rndeg) C (mj/deg)
1.2742 32.5 1.05 2.0692 83.4 2.16
1.2634 32.5 1.02 2.1330 79.6 2.261.3330 55.8 1.08 2.2109 117 2.461.4060 52.1 1.16 2.3031 111 2.61
1.4672 49.5 1.22 2.3884 103 2.82
1.5320 45.8 1.32 2.4811 131 2.98
1.9310 73.4 1.37 2.5755 121 3.23
1.6735 66.7 1.51 1.2913 56.0 1.07
.1.7387 63.5 1.59 1.3621 51.2 1.171.7960 59.6 1.69 1.4239 48.2 1.24
1.9523 58.5 1.72 1.4777 47.6 1.26
1.9205 95.7 1.88 1.2694 36.2 0.962
1.9989 88.1 2.04
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission
APPENDIX 2-B. THE COPPER RESULTS
1. Range in which Calibra-fcion-B was used to compute specific heats
T°K A T fmdee) C (mJ/des) T°K A T (mdee) C (ml/dee)
0.707 11.0 1.32 0.623 9.71 1.600.730 12.6 1.75 0.647 9.61 1.620.753 12.8 1.71 0.667 12.5 1.25
0.775 12.3 1.78 0.692 18.3 1.530.801 20.3 1.88 0.718 17.0 1.640.832 19.6 1.96 0.746 17.5 1.590.858 18.2 2.10 0.775 25.8 1.76
0.889 33.9 2.07 0.811 23.8 1.920.928 31.6
* J-
2.22 O.840 22.4- 2.03
O.964 31.0 2.26 O.867 23.0 1.98
0.997 31.2 2.25 0.900 36.4 2.121.026 29.1 2.41 0.939 35.0 2.201.061 49.1 2.60 0.976 31.8 2.43
1.103 45.3 2.81 1.007 30.5 2.53
1.142 44.6 2.86 I.050 49.9 2.581.179 42.8 2.98 1.105 46.I 2.79
1.228 75.0 3.19 1.153 44.4 2.891.290 70.6 3.38 1.198 41.1 3.13
0.567 . 53.4 1.09 1.257 70.4 3-24
0.604 7.26 1.38
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Range in. which Calibration-A was used to compute specific heatsT°IC A T (mdeg) C (mj/deg) T°K AT (mdeg) C (mj/<
1.326 65.3 3.50 2.439 163 8.471.386 62.3 3.66 2.581 148 9.30
1.441 58.4 3.91 2.736 189 10.41.518 106 4.15 2.905 172 11.51.608 97.7 4.51 3.058 157 12.61.692 90.7 4.86 3.229 211 14.01.767 85.7 5.15 3.412 187 15.7I.876 156 5.61 3.579 171 17.1
2.015 140 6.22 3.777 2 55 19.22.137 128 6.82 3.998 224 21.92.281 182 7.58
NOTES: 1, The Table shows the total heat capacity of the
copper sample and calorimeter.
2. Number of moles of copper = 2.315
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APPENDIX 2~C. THE OXYGEN RESULTS
1. Ranee in w h ich C a l ib r a t io n - B was used to compute s p e c i f ic h e a ts
MOLAR MOLAR MOLARSPECIFIC HEAT SPECIFIC HEAT SPECIFIC HEAT
T°K (m j/m o le dee) T°K (m j/m o le dee) T°K (mj/mole deg)
o. 848 0.859 1.116 1.886 1.094 1.8720.919 1.040 1.178 2.331 1.173 2.496
0.985 1.150 0.917 1.003 1.233 2.963
1.052 1.509 1.046 1.395
2 . Ranee in w h ich C a l ib r a t io n - A was used to compute 's p e c i f ic h e a ts
1.289 3.510 1.812 10.29 2.630 31.08
1.343 3.857 2.013 13.94 2.752 35.58
1.397 4.565 2.089 15.69 2.861 39.51
1.451 5.176 2.156 17.19 2.958 43.55
1.514 5.749 2.218 18.57 3.O64 48.631.569 6. 384 2.285 20.34 3.179 54.37
1.628 7.169 2.360 22.34 1.285 3.0081.690 7.707 2.439 24.84 1.336 3.738
1.744 8.888 2.524 27.35 1 .390 4.567
(Continued....)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.447 4.731 1.827 10.50 2.447 25.021.490 5.277 1.888 11,45 2.553 28.471.532 5.866 1.954 12.81 2.648 31.64
1.571 6.246 2.013 13.92 2.758 35.29
1.614 7.029 2.088 15.71 2,883 40.841.662 7.808 2.174 17.69 2.996 45.60
1.715 8.845 2.261 19.71
1.774 9.563 2.349 22.25
NOTES: 1. Number of moles of oxygen = 0.555
2. The gas was obtained from Matheson Ltd,
and was their Extra Dry Grade.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX 2-D . THE NITROGEN RESULTS - SERIES-1
1. Range in which Calibration*-!} was used to compute specific heats
oT K
MOLAR SPECIFIC HEAT (mj/mole deg) T°K
MOLAR SPECIFIC HEAT (mj/mole deg)
MOLAR SPECIFIC HEAT
T°K (mj/mole deg)
0.943 2.260 0.811 1.545 1.123 4.371
1.032 3.560 O.89I 2,147 1.168 5.295
1.116 4.338 0.949 2.613 1.214 5.857
1.175 4.9H 1.017 3.296 1.258 6.5661.236 6.310 1.078 3.998
2. Range in which 12alibration-A was used to compute specific heats
1.302 7.454 2.415 47.14 1.664 15.531.369 8.191 2.552 55.08 1.736 17.50
1.450 10.35 2.681 64.09 1.800 19.50
1.536 12.20 2.855 77.44 1.897 23.15
1.605 13.95 3.055 94.34 2,020 27.70
1.667 15.39 3.225 110.8 2.124 32.141.722 16.91 3.372 126.1 2.215 36.27
1.794 19.39 1.306 71.60 2.322 42 .20
(Continued....)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.879 22.26.1.954 25.10
2.064 29.662.192 35.50
2.294 40.69
1.360 8.471
1.419 9.676
1.481 11.09
1.543 12.34
1.599 13.79
2.442 48.5-8
2.585 . 57.69
2.745 68.762.882 79.37
3.045 93.81
1 . Range i n w h ich C a lib ra t io n - 'B was used S E R IE S -II t o compute s p e c i f ic h e a ts_____________
MOLAR MOMR MOLARSPECIFIC HEAT SPECIFIC HEAT SPECIFIC HEAT
T°K (m J/m ole deg) T°K (m j/m o le deg) T°K (m j/m o le -deg)
0.713 1.437 I.I48 5.253 0.923 2.5910.758 1.517 1.199 5.998 0.956 3.0990.812 1. 880 0.598 .8343 0.991 3.267
0.851 2.269 0.643 1.027 1.028 3.815
0.886 2.392 0,681 1.101 1.063 4.199
0.920 2.614 0.724 1.200 1.102 4.590
0.954 2. 938 0.755 I.638 1.139 5.095
0.990 3.491 0.788 1.788 1.171 5.437
1.023 3.586 0.824 2.011 1.220 6.2121.063 4.165 0.859 2.144
1.102 4.572 0.893 2.412
(Continued...)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2. Range in which Calibration.~A was used to compute specific beats
MOLAR SPECIFIC HEAT
T°K (mj/mole deg)
1.247 6.6821.303 7.583
1.368 8.792
1.423 9.8381.270 7.070
1.323 8.010I.38I 9.065
1.434 10.05
1.489 11.25
1.546 12.62
MOLAR SPECIFIC HEAT
T°K (mj/mole
1.597 13.72
1.660 15.37
1.732 17.47
1.795 19.30
1.880 22.27
1.983 25.98
2.070 29.682.412 46.55
2.509 52.54
2.649 61.79
MOLAR SPECIFIC HEAT
T°K (mj/mole deg)
2.823 74.08
2.969 86.033.096 97.773.261 113.23.454 135.43.618 158.1
3.804 I83.24.004 216.3
4.175 251.8
NOTES: 1. Number of moles in Series-I = 0.494
Number of moles in Series-II = 0.536
2. The gas was obtained from Matheson Ltd.
and was their Extra Dry Grade.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX 2-E. THE CARBON MONOXIDE RESULTS.
1, Range in which Calibration-B was used to compute specific heats
T°K
MOLAR SPECIFIC HEAT (mj/mole dee) T°K
MOLAR SPECIFIC HEAT fmJ/mole deg) 1-3 0 *
i
MOLAR SPECIFIC HEAT (mj/mole deg)
0.725 0.7H 1.094 2.301 0.923 1.560
0.774 0.666 1.138 2.417 0.959 1.301
0.818 I.032 1.191 2.897 0.991 1.7510.8 58 0.962 1.245 3.357 1.033 1.809
O.904 1.326 O.683 Q.574 1.073 2.125
O.964 1.516 0.779 0.701 1.111 2.081
1.001 1.709 0.824 O.836 1.152 2.853
1.045 1.880 0.886 1.238 1.238 3.499
2. Range in which Calibration--A was used to compute specific heats
1.296 3.705 1.620 7.763 2.686 35.24
1.350 4.257 1.683 8.456 2.799 39.29
1.410 4.882 1.751 9.666 2. 896 43.79
1.465 5.461 1.824 10.92 3.025 50.01
(Continued....)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T°K
MOLAR SPECIFIC HEAT (mj/mole deg) T°K
MOLAR SPECIFIC HEAT (mj/mole deg) 0 T K
MOLAR SPECIFIC HEAT (mj/mole deg)
1.515 6.451 1.917 12.75 3.182 58.481.555 6.645 2.027 14.88 3.326 66.731.282 3.799 2.121 17.09 3.461 74.991.325 - 4.070 2.228 19.91 3.633 88.261.377 4.796 2.344 23.20 3.831 105.41.438 5.369 2.445 26.14 4.042 124.81.553 6.797 2.559 30.18 4.263 152.9
NOTES: 1. . Number of moles = 0.536
2. The gas was obtained from Matheson Ltd,
and was their C.P.Grade.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX 2-F. THE NITRIC OXIDE RESULTS
1. Range in which Calibration-B was used to coraoute SDecific heats
MOLAR SPECIFIC HEAT
T°K (mj/mole deg) T°K
MOLAR SPECIFIC HEAT (mj/mole deg) 0 T K
MOLAR SPECIFIC HEAT (mj/mole deg)
0.768 '' 0.343 1.266 1.860 1.125 1.2960.840 0.688 0.719 0.544 1.181 1.741O.968 0.712 0.778 0.344 1.242 1.7201.160 1.226 1.012 0.667
1.211 2.091 1.079 1.106 -
2. Range in which (Salibration-A was used to compute specific heats
1.319 2.197 1.412 2.994 2.551 17.45
1.373 3.016 1.491 3.325 2 .664 19.48
1.431 3.092 1.557 4.118 2.798 22.45
1.482 3.555 1.636 4.778 2.953 26.05
1.537 3.823 1.721 4-770 3.093 29.76
1.595 4.076 1.790 6.116 3.232 33.96
1.649 4*668 1.873 7.021 3.378 38.25
1.719 5.468 1.980 8.476 3.506 43.05
(Continued....)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
MOLAR MOLAR MOLARSPECIFIC HEAT SPECIFIC HEAT SPECIFIC HEAT
T°K (mj/mole deg) T°K (mj/mole deg) T°K (mj/mole deg)1.802 6.351 2.070 9.566 3.666 48.691.876 7.240 2.186 11.25 3.847 57.44
1.304 2.213 2.322 13.24 4.OO3 65.121.353 2.884 2.436 15.08 4.143 71.65
NOTES: 1. Number of moles = 0.203
2. The gas was obtained from Matheson Ltd.
and was their regular grade.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX 2-G THE C0-02 RESULTS. - SERIES-I
1. Range in which Calibration-B was used to compute specific heats
MOLAR MOLAR MOLARSPECIFIC HEAT SPECIFIC HEAT SPECIFIC HEAT
T°K (mj/mole dee) T°K (mJ/mole dee) T°K (mj/mole dee)
0.707 1.283 1.175 8.024 0,930 4.030
0.745 1.868 1.228 8,8 55 O.963 4.360
0.777 2.234 1.275 9.625 0.995 4.8910.827 2 .616 0.588 0.433 1.025 5.580
0.871 3.099 0.632 0.925 1.055 5.977
0.909 3.527 0.681 1.015 1.084 6.329
0.955 4.567 O.718 1.474 1.123 -7.2080.994 5.089 O.76O 1.834 1.169 7.863
1.037 5.560 0.796 2.140 1.212 8.651
I.O84 6.295 0.873 3.382 1.250 9.058
1.128 7.090 0.900 3.429
2. Ranee in which Calibration-A was used to compute specific heats
1.321 10.55 1.392 11.62 2.653 39.85
1.376 11.38 1.445 12.58 2.787 44.85(Continued.,.,)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T°K
MOLAR SPECIFIC HEAT (m j/m o le de ff) T°K
MOLAR SPECIFIC HEAT (m j/m o le des:) T°K
m o la :SPECIFIC(m j/m o le
1.438 12.49 1.504 13.44 2.930 50.23
1.497 13.35 1.560 14.46 3.055 55.911.585 14.97 1.612 15.24 3.190 62.881.697 16.87 1.689 16.68 3.333 69.481.796 18.52 1.788 18.31 3.508 81.591.886 20.22 1.876 20.05 3.707 95.00
1.967 21. 88 1.981 22.07 3.877 109.32.066 23.91 2.098 24.46 4.081 129.1
2.179 26.40 2.237 27.72 4.308 155.6
1.297 10.06 2.394 31.85
1.346 10.94 2.532 35.84
S E R IE S -Ii;
1 . Range in w h ich C a l ib r a t io n - B was used t o compute s p e c i f ic h e a ts
0.700 2.463 1.095 12.20 0. 828 5.398
0.735 3.367 1.130 13.55 0.856 5.809
0.773 3.724 1.170 14.53 0.895 7.0610.804 4.788 1.216 15.81 0.921 7.364
0.833 5.541 1.258 16.99 0.952 8.374 (Continued,..)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T°K
MOLAR SPECIFIC HEAT (m j/m o le dee) T°K
MOLAR SPECIFIC HEAT (m j/m o le dee) T°K
MOLAR SPECIFIC HEAT (m j/m o le dee)
0,863 5.853 0.603 1.163 0.990 9.471
0.899 6.848 0.639 1.845 1.024 10.260.929 7.803 0.680 2.207 1.065 11.50
0.960 8.354 0.711 3.044 1.098 12.32
0.993 9.282 0.736 3.369 1.139 13.70
1.023 10.19 0.764 3.796 1.187 15.17
1.057 11.22 0.802 4.507 1.240 16.47
2 . , Ranee i n w h ich C a l ib r a t io n - A w as 'use d to compute s p e c i f ic h e a ts
1.310 1.465 2.884 51.70 1.465 22.161.364 1.531 3.023 56.15 1.531 23.69
1.4U 1.624 3.176 62,08 1.624 25.44
1.482 1.730 3.340 69.33 1.730 27.55
1.565 1.829 3.487 76.17 1.829 29.14
1,642 1.949 3.653 85.15 1.949 31.43
1.734 2.084 3.862 99.20 2.084 33.66
I .844 2.209 4.073 116.7 2.209 35.72
1.298 2.370 4.257 133.1 2.370 38.74
1.347 2.560 2.560 43.00
(C o n tin u e d , . . . )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
NOTES:
MOLAR MOLARSPECIFIC HEAT SPECIFIC HEAT
T°K (mj/mole deg) T°K (mj/mole deg)
1.403 20.92 2.730 47.17
1. Number of moles of CO in Series-I = 0.519
” »» 11 » » « Series-II = 0.513
2. Mole fraction of oxygen in Series-I = 0.14$« ' n « Series-II = 0.34^
3. The specific heat values shown in the Table
refer to one mole of CO.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX 2—H. THE N2 - 02 RESULTS.
1. Range in which Calibration-B was used to compute specific heats
MOLAR SPECIFIC HEAT
T°K (raj/mole deg)
0.700 5.115
0.748
0.767 0.788 0.811
0.831 0.856 O.883 0.908
0.940
0.977 1.011 1.050
6.528
7.3978 , 2 6 6
9.506
10.3011.82 12.86
14.36
15.9317.72
19.42
21.77
MOLAR SPECIFIC HEAT
T°K (mj/mole deg)
1.093 24.03
1.133 26.131.174 28.52
1.217 30.64
1.257 32.600.571 I.8360.603 2.684
0.635 3.452
0.660 4.1110.709 5.481
0.734 6.7020.759 7.298
0.781 8.222
MOLAR SPECIFIC HEAT
T K (mj/mole cleg)
0.808 9.705
0.840 11.03
0.869 12,46
0.903
0.937
0.967
0.995
I.O641.099
1.1311.168
1.207
1.258
14.20 15.74
- 17.58
18.8022.73
24.17 24.0228.17
30.37
32.99
( C o n t i n u e d . . . . )
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2. Range in which Calibration-^ was used to compute specific heats
MOLAR SPECIFIC HEAT
MOLAR SPECIFIC HEAT
MOLAR SPECIFIC HEAT
T°K (mj/mole deg) T°K (mj/mole deg) T°K (mj/mole deg)
1.321 36.18 1.861 58.48 3.014 118.71.376 38.73 1.937 61.46 a .166 131.31.441 41.67 2.036 65.11 3.334 147.91.515 44.98 2.156 69.76 3.514 169.3
1.583 47.94 2.268 74.41 3.720 195.6
I.648 50.95 2.438 82.65 3.943 230.5
1.709 52.81 2.656 94.51 4.133 267.11.780 55.43 2.846 106.8
NOTES: 1. Number of moles of N^ = 0.;
2. Mole fraction of oxygen = 0
566
.55%3. The specific heat values shown in the Table
refer to one mole of ^ •
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