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Heat of Combustion of AlanineBE 2104/30/97
Wednesday Six
Jason ChristosLatressa Fulton
Vinod MapranathMelinda Patterson
Table of Contents
1) Abstract 3
2) Background 4
3) Apparatus and Materials 8
4) Procedure 9Energy Equivalent 9Bomb Calorimetry 9
5) Results 12Overview 12Standardization wih Benzoic Acid 13Alanine 16
6) Discussion 22Overview 22Presentation and Quatitation 22
Nitric Acid Correcion 23Fuse Wire Correction 23
Precision and Accuracy 24Error Analysis 25Limitations 28Conclusion 29
7) Appendix 31
(**A complete listing of all of the tables and figures are found in the appendix, along with the calculations for the uncertainty interval.**)
8) References 51
2
Abstract
It is useful to measure the heat of combustion of alanine because it represents the
maximum energy an organism can obtain from a metabolite. With the availability of an
advanced instrument called a Bomb Calorimeter, it is possible to accurately measure the
heat combustion of a material with the appropriate standardized energy equivalent.
A crucial part of this lab is the accuracy of the given energy equivalent, W. W
describes the change in heat of the system due to a change in temperature of the water.
The bomb calorimeter is fired by causing a current to flow through a circuit. The fuse
wire heats up the benzoic acid for the calibration and the alanine for determination of its
Hc until the fuse burns off and breaks the circuit. This causes the combustion of the
material, but also causes other heat including the combustion of the fuse wire and side
reactions such as nitric acid. The heat of combustion of the fuse wire is determined by
measuring the length of the fuse wire burned and multiplying it by a constant. The
amount of nitric acid formed is found by titrating it with 0.0709N alkali. The heat of
combustion of alanine is accurately calculated from the heat transferred to the entire
system and subtracting any heat contributions from the burning of the fuse wire and side
reactions. After considering all these correction factors and the net corrected temperature
change of the water, the value obtained for the energy equivalent is 2429.9 + 51.3 cal/ °C
and alanine’s heat of combustion is 4284.8 + 113.1 cal/g. The energy equivalent deviates
from the accepted value by 0.16% and the heat of combustion deviates from the accepted
value by 0.14%. These values show that the data is extremely accurate.
3
Background
The human body is composed of numerous types of tissues, which can have
properties varying from hard and relatively brittle (e.g. bone tissue) or soft and extremely
ductile (e.g. the epidermis). However, all of the tissues in the body contain large
amounts of proteins, which are long chains of amino acids. It follows that the human
body contains enormous amounts of amino acid molecules. These amino acids are
involved in countless processes including the regulation of body functions, synthesis of
biological compounds, and the breaking down of molecules. Therefore, it is important
to understand as much about these significant molecules as possible.
The amino acid alanine, which has a methyl -R group, is used by the body for
many important functions. Alanine is used by the kidneys to form the ammonia that is
excreted in urine, and therefore aids in the maintenance of the systemic acid/base balance
in the body. It is also used as a precursor for the formation of glucose, and in some
cases, it is used to produce ATP. Alanine is possible that alanine is simultaneously
broken down and created in the kidneys for various reasons. For these reasons, it would
be beneficial to determine the energy involved in the breaking down and formation of the
alanine molecule.
The bioprocesses involved in the chemical conversion of alanine is accomplished
through a reaction chain that is thermodynamically equal to combustion. Therefore, by
finding the heat of combustion of alanine, the maximum possible energy that can be
obtained by the body during alanine metabolism can be determined, as well as the energy
involved in the formation of alanine.
The standard heat of combustion (calorific value) of a sample is a defined
thermodynamic quantity that represents the energy released during complete combustion
in oxygen at 298 K and 1 atm. Calorimetry is the science of measuring these quantities
of heat (the energy released by the fuel during combustion). In order to determine this
value, an instrument called an oxygen bomb calorimeter is used. The value determined
by a bomb calorimeter refers to the heat released by the combustion of all the carbon and
hydrogen with oxygen, in the formation of carbon dioxide and water, as well as the heat
4
released by the oxidation of other elements which may be present in the sample, such as
nitrogen.
In an over-simplified model, the heat of combustion can be determined by
assuming that all of the energy released by combustion of the compound goes into the
water, thereby allowing the heat of combustion to be calculated from the Energy Balance
Equation:
Qreaction = msample * DHcomb,sample = msample * Cp,water * Dtwater (Equation 1)
This equation, however, is inaccurate due to the fact that much of the energy goes
into heating the metal of the bomb calorimeter and creating non-standard reaction
products (such as nitric acid instead of N2 gas). At the same time, some energy is
expended during the burning of substances other than the sample (such as the fuse wire
that is used to ignite the reaction) and energy of the system may be lost through the walls
of the calorimeter (if it is non-adiabatic). Therefore, it is necessary to consider all of the
energy added to the system and all of the energy lost during the combustion. The
simplest way to accommodate for all of the influential factors is to calibrate the
calorimeter.
The calibration constant, W (cal/°C), is unique for each calorimeter and is
determined by performing the experiment using a fuel that has a known heat of
combustion (Benzoic Acid). The process of determining the calibration constant is called
a standardization procedure. Once the value of W is calculated, the heat of combustion
can easily be determined using the equation:
msample * DHcomb = W * Dtwater (Equation 2)
Because of the fact that the heat of combustion is directly related to the change in
temperature of the system, precise temperature measurements are imperative in bomb
calorimetry, with a required accuracy of 0.001 °C or better. The temperature change of
the entire system is calculated using the following equation.
5
t = tc - ta -r1(b - a) - r2(c - b) (Equation 3)
where tc - ta is the temperature change between the time of firing and the maximum
temperature. The values r1 and r2 are the rates of temperature change per minute during
the two 5 minute constant stages before firing, and after the maximum temperature is
reached. The values a, b, and c are times of firing, time at the 60% increase in
temperature, and time when the maximum temperature is reached.
Similarly, it is important to make precise and repeatable measurements when
measuring the water to be used in the calorimeter, since a difference of one gram (one
mL) of water will change the energy equivalent of the calorimeter by one calorie per
degree Celsius.
Since the combustion in the calorimeter takes place in an environment of oxygen
at high temperature and pressure, several reactions could take place that ordinarily would
not under normal atmospheric or physiological conditions. These reactions generate
considerable amounts of heat, which must be corrected in order to obtain valid results.
For example, when a fuel sample is burned in the oxygen environment, the molecules of
nitrogen that are sealed in the bomb become oxidized to form nitric acid. To account for
these unwanted reactions, the amount of the unwanted product must be measured in order
to calculate the amount of energy that is absorbed during their formation. This is done
by rinsing out the bomb calorimeter after the reaction has taken place with distilled water
to collect all of the acid that is formed. The solution is then titrated to determine the
number of moles of acid formed. Therefore, the amount of energy released during the
exothermic reaction can be determined and subtracted from the total energy increase
recorded. The equation for the Gross Heat of Combustion Hg is given below:
Hg = (tW - e1 - e3) / m (Equation 4)
where t is given in Equation 3, W is the energy equivalent, and e1 and e3 are the heat
contributions due to nitric acid formation and the fuse wire combustion.
Once the heat of combustion of alanine is accurately determined, assumptions or
predictions can be made as to the effects of this on the human body, as well as the
6
efficiency of the body as it continuously metabolizes and synthesizes the compound.
This may provide insight into the mechanisms by which the reactions take place and the
reasons behind the reactions themselves.
7
Apparatus and Materials
· Parr Pellet Press
· Associated components for the test, including sample cups, ignition wire, stands for bomb head and calorimeter cover, thermometer magnifier.
· Burette with 0.0709 N sodium carbonate solution and methyl red indicator (4)
· Top-loading 5kg capacity balance with a resolution of 0.1g
· Powder d-l alanine and powder Benzoic Acid
· Parr instrument Model 1108 Oxygen Combustion Bomb
· High Pressure oxygen cylinder, equipped with Model 1825 filling connection for bomb
· Parr Instrument Model 1341 Oxygen Bomb Calorimeter, including stirrer, precision thermometer, and associated components, W=2426 cal/O C (refer to diagram 1)
· Parr Model 2901 Ignition Unit (refer to diagram 1)
Diagram 1
8
Procedure
Energy Equivalent
The first part of this lab is to determine the energy equivalent of the particular
bomb calorimeter that is used for experimenting. This is the most crucial part because
this calculation is used in all other calculations in the lab for determining the heat of
combustion of the amino acid alanine. Therefore, precision and accuracy are extremely
important. In this part, the combustion of a known material, benzoic acid is performed.
This procedure is the same as that for the combustion for alanine. The combustion of
benzoic acid is performed twice for each week of experimentation to determine an
accurate energy equivalent for the bomb calorimeter.
Bomb Calorimetry
To prepare the alanine for the bomb calorimeter it is necessary to compress it
into a pellet or tablet form. To make the pellets with the Parr Pellet Press, the die is
filled, the charge is compressed, the die holder is reversed and the pellet is ejected. The
pellet is formed so that it is in the range of 0.9 to 1.25 grams.
After the sample pellet is made, it is necessary to prepare the 1108 Oxygen
Combustion Bomb. The calorimeter bucket is filled by first taring the dry bucket on a
trip balance, then adding 2000 + 0.5 grams of distilled water. The water temperature
should be approximately 1.5°C below room temperature. The amount of water used must
be duplicated for every trial.
The bomb head is set on the A38A support stand, and a 10cm length of Parr
45C10 nickel alloy fuse wire is fastened between the two electrodes. To close the bomb,
water is added to the sealing ring so that it slides freely into the cylinder. The gas release
valve is left open as the bomb head carefully slides into the bomb cylinder. Oxygen for
the bomb is drawn from a commercial standard oxygen tank. The filling connection
control valve is opened slowly until the gage indicates that the bomb pressure is 25 atm,
and then the control valve is closed.
The bucket is set in the 1341 Oxygen Bomb Calorimeter. The two ignition lead
wires are pushed into the terminal sockets on the bottom head, while being careful not to
9
remove any water from the bucket. The stirrer is then run for 5 minutes to reach
equilibrium before starting a measured trial (refer to diagram 2). At the end of this
period, the time is recorded and the temperature is read to one-tenth of the smallest scale
division. The temperature is then recorded at one minute intervals for 5 minutes and at
the start of the 6th minute, the bomb is fired.
Diagram 2
The bucket temperature starts to rise within 20 seconds of firing. This rise is
rapid during the first few minutes, but becomes slow as the temperature approaches a
stable maximum. Accurate time and temperature observations must be recorded to
identify certain points needed to calculate the calorific value of the sample.
The time required to reach 60 percent of the total rise is determined by estimating
the temperature at that point and observing the time when the rising mercury thread
reaches that level. Temperature readings at 45, 60, 75, 90 and 105 seconds after firing
are taken, and interpolations of these readings are performed in order to identify the 60%
point. After the rapid rise period, the temperatures are recorded to one-tenth of the
10
smallest scale division at one minute intervals. This is done until the difference between
successive readings has been constant for five minutes. The difference between
successive readings must be noted and the readings continue at one-minute intervals until
the rate of the temperature change becomes constant over a period of 5 minutes.
At the end of the readings, all interior surfaces of the bomb are rinsed with a jet
of distilled water and the bomb washings collected in a beaker. Also, all unburned pieces
of fuse wire are collected from the bomb electrodes. They are then straightened and
measured to determine their combined length in centimeters. This length is then
subtracted from the initial length of 10 centimeters and recorded in the lab notebook.
The bomb washings are titrated with the standard sodium carbonate solution using
methyl red indicator. A 0.0709N sodium carbonate solution is used for titration. These
measurements are used in determining the correction factors for the heat of combustion
of the material that was burned.
This procedure was repeated 15 times for the measurement of alanine’s the heat
of combustion of alanine with only one of the 15 trials not firing. Combustion of
benzoic acid was completed twice each week for three weeks for the calculation for the
energy equivalent.
11
Results
Overview
The heat of combustion of a sample of alanine is obtained by measuring the heat
of the entire system and subtracting the heat generated by the formation of nitric acid and
the heat from the ignition wire. To determine the total heat of the system, a calibration
constant or energy equivalent W, is calculated using a standardized sample, benzoic acid.
The standardization procedure is the most essential calculation in this experiment,
because the slightest variation in its value could have a tremendous effect on the
determinations of alanine’s heat of combustion. Six trials of standardization were done
to account for the small variations in laboratory conditions on different days and possible
random or human errors produced in each trial.
Once the energy equivalent is determined, the heat of combustion of alanine is
calculated by measuring the temperature changes in the water, the effects of nitric acid
formation, and the heat produced from the burning of the ignition wire in fourteen trials.
12
Standardization with Benzoic Acid
A graph of temperature versus time for the first trial of Benzoic Acid Standardization is
given below in Figure 1. Graphs of each of the six trials of standardization are given in
the appendix.
0 2 4 6 8 10 12 14 16
27
28
29
30Benzoic Acid Trial 1: Temp. vs. Time
Time (minutes)
Tem
pera
ture
(deg
C)
30
26.3
28.2248tempi
17.50
7.26
timei
Figure 1The dotted line at 6 minutes is the time of firing, and the time 7.2 minutes represents the the 60% increase in temperature (value ‘b’ of Trial 1 in Table 1). There is an error interval of + 0.001 °C for the temperature and an estimated error interval of + 0.0167 minutes (+ 1 second) for the time. Therefore, due to the accuracy of the equipment, the error bars are very small in comparison to the values in the graph. As seen in the graph, there is a five minute constant interval before the firing of the bomb at time 6 minutes. After the bomb is fired, the temperature increases until it reaches the maximum temperature at time 12 minutes and 15 seconds (Trial 1, value ‘c’ in Table 1) where it remains approximately constant for 5 minutes.
From this graph, the values of a, b, c, r1, and r2 for the standardization are
determined and displayed below in Table 1. The value r1 is the rate at which the
13
temperature is rising during the 5 minute period before firing. The value r2 is the rate at
which the temperature is rising during the 5 minute period after time c. A value of zero
corresponds to a constant temperature during the 5 minute interval.
Trial a
(min)
b
(min)
c
(min)
r1
(°C /min)
r2
(°C /min)
c1
(mL)
c3
(cm)
1 6 7.2 12.25 0.012 0.002 10.3 6.1
2 6 7.3 12.25 0.0 0.002 12.0 8.4
3 6 7.3 12.25 0.0182 0.0 4.2 8.8
4 6 7.2 12.25 0.01 0.002 8.8 8.45
5 6 7.1 12.25 0.0642 0.002 7.5 8.65
6 6 7.2 12.25 0.0002 0.0 10.9 8.3
Table 1
The titration of the bomb washings was useful in calculating the heat of formation
of the nitric acid e1. The following equation is used for calculating e1:
e1 = volume (in ml) of 0.709N alkali titrated (Equation 5)
The heat of combustion of the fuse wire e3 = 2.3*c3 where c3 is the length of the
fuse wire consumed. The data for six trials of standardization and the calculated energy
equivalent W are given below in Table 2:
14
Sample mass
(g)
ta
(°C)
tc
(°C)
t
(°C)
H
(cal/g)
e1
(cal)
e3
(cal)
W
(cal/°C)
1 1.017 26.649 29.312 2.639 6311.1 10.3 14.03 2441.8
2 0.999 24.361 27.007 2.636 6311.1 12.0 19.32 2403.6
3 0.997 25.182 27.797 2.591 6311.1 4.2 20.24 2437.6
4 1.020 24.901 27.604 2.681 6311.1 8.8 19.44 2411.7
5 0.951 24.911 27.431 2.439 6311.1 10.29 19.90 2471.9
6 1.116 24.810 27.743 2.932 6311.1 10.9 19.09 2412.6
Table 2
The values for the energy equivalent in the six trials are analyzed and summarized
below in Table 3. The average of the samples, 95% confidence interval, accepted value,
and percent error are given.
W (cal/°C)
Average of Samples 2429.9
95% Confidence Interval 51.3 (2.1%)
Accepted Value 2426
Percent Error 0.16%
Table 3
Using the accurate value of the energy equivalent, the heat of combustion of
alanine is calculated with the following results.
15
Alanine
Once an accurate energy equivalent is determined, essentially the same procedure
as the standardization is followed for the alanine sample. A graph of temperature versus
time for the first sample of alanine is given below in Figure 2. Graphs of each of the
fourteen trials of alanine combustion are given in the appendix.
0 2 4 6 8 10 12 14 16 1825.5
26
26.5
27
27.5
28Alanine Sample 1: Temperature vs. Time
Time (minutes)
Tem
pera
ture
(deg
C)
26.8844tempi
7.2
timei
Figure 2The dotted lines that intersect display the time at which the temperature has reached its 60% increase (value ‘b’ for Sample 1 in Table 4 below). As before with the standardization, there is an error interval of + 0.001 °C for the temperature and an estimated error interval of + 0.0167 minutes (+ 1 second) for the time. The error bars are very small in comparison to the values in the graph. There is a five minute constant interval before the firing of the bomb at time 6 minutes. After the bomb is fired, the temperature increases until it reaches the maximum temperature at time 13 minutes and 15 seconds (Sample 1, value ‘c’ in Table 4) where it remains approximately constant for 5 minutes.
16
From this graph, the values of a, b, c, r1, and r2 are determined for the fourteen
samples and displayed below in Table 4.
Sample a
(min)
b
(min)
c
(min)
r1
(°C /min)
r2
(°C /min)
c1
(mL)
c3
(cm)
1 6 7.2 13.25 0.034 0.001 18.3 8.6
2 6 7.2 12.25 0.0122 0.0 17.0 8.7
3 6 7.2 11.25 -0.01 0.0 16.1 8.8
4 6 7.2 12.25 0.01 0.004 15.5 8.4
5 6 7.2 11.25 0.02 0.004 16.7 8.4
6 6 7.2 11.25 0.012 0.0 15.1 8.6
7 6 7.2 12.25 0.006 0.006 16.9 8.5
8 6 7.3 12.25 0.008 0.002 13.4 8.6
9 6 7.4 11.25 0.0 0.0 17.3 5.9
10 6 7.0 11.25 0.002 0.004 17.7 8.9
11 6 7.0 10.25 0.004 0.0 15.9 9.1
12 6 7.0 10.25 0.004 0.002 18.3 9.0
13 6 7.2 10.25 0.006 0.0 16.8 8.7
14 6 7.2 10.25 0.0078 0.01 15.3 8.7
Table 4
Data for the heats of combustion of the different samples of alanine immediately
follows in Table 5. The temperature effects and the energy effects of the nitric acid and
the burning of the ignition wire are also displayed below for the fourteen samples.
17
Sample mass
(g)
ta
(°C)
tc
(°C)
t
(°C)
t*W
(cal)
e1
(cal)
e3
(cal)
H
(cal/g)
1 1.255 25.573 27.867 2.244 5424.3 18.3 19.78 4314.1
2 1.093 24.541 26.510 1.954 4748.9 17.0 20.01 4310.9
3 1.066 24.511 26.411 1.912 4645.9 16.1 20.24 4324.2
4 0.777 25.222 26.659 1.405 3413.5 15.5 19.32 4348.4
5 1.206 25.403 27.573 2.130 5175.2 16.7 19.32 4261.3
6 0.948 26.343 28.030 1.673 4064.2 15.1 19.78 4250.4
7 1.149 24.750 26.808 2.021 4910.0 16.9 19.55 4241.2
8 0.987 25.644 27.531 1.767 4292.9 13.4 19.78 4315.8
9 1.099 27.153 29.070 1.917 4658.1 17.3 13.57 4210.4
10 1.122 25.031 27.062 2.012 4889.0 17.7 20.47 4323.3
11 1.010 24.700 26.471 1.767 4294.0 15.9 20.93 4214.6
12 1.104 24.581 26.609 2.018 4902.3 18.3 20.70 4405.1
13 1.034 25.280 27.141 1.854 4505.0 16.8 20.01 4320.8
14 1.060 24.530 26.431 1.861 4522.3 15.3 20.01 4233.0
Table 5
Taking data of the heats of combustion of alanine in the last column of Table 5,
the averages, 95% confidence intervals, accepted value, and percent error of the fourteen
trials are all given below in Table 6.
Hg (Cal / g)
Average of Samples 4291.0
95% Confidence Interval 113.1 (2.6%)
Accepted Value 4284.8
Percent Error 0.14 %
Table 6
18
The accepted value of alanine’s heat of combustion in Table 6 was obtained from
the 75th Edition of the CRC Handbook of Chemistry and Physics. The value is the
average of d-alanine and l-alanine given in the handbook (4342.1 Cal/g and 4227.4 Cal/g
respectively).
It is now possible to look at how different variables change as the mass of an
alanine sample increases. The variables temperature (t), total energy of the system
(t*W), and heat from nitric acid formation (e1) are graphed as a function of mass.
In Figure 3, the corrected change in temperature t is graphed as a function of
mass for each of the fourteen samples. As seen by the graph, there is a positive linear
correlation between the mass and the change in temperature. The points are represented
by a best fit line and equation.
ALANINE: Change in Temperature vs. Mass
y = 1.7538x + 0.0275
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Mass (g)
Tem
pera
ture
Cha
nge
(deg
C)
Y
Predicted Y
Linear(Predicted Y)
Figure 3
The error bars are not visible in this graph because they are very small in comparison to the values in the graph.
19
Similarly, the total change in energy of the system t*W is graphed versus mass in
Figure 4, and it is also shown that there is a linear relationship. Again, a best fit line is
drawn to approximate the linear relationship of the points.
t*W (Total Change in Energy of the System) vs. Mass
y = 4261.4x + 66.738
0
1000
2000
3000
4000
5000
6000
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Mass (g)
t*W
(cal
) Y
Predicted Y
Linear (Predicted Y)
Figure 4
As for the previous graphs, the error bars are not visible in this graph because they are very small in comparison to the values in the graph.
20
The heat from the formation of nitric acid as a function of mass is graphed below
in Figure 5. There is no relationship between the size of the sample and the heat from
the formation of nitric acid.
Heat from Nitric Acid Formation (e1) vs. Mass
12
13
14
15
16
17
18
19
0.7 0.8 0.9 1 1.1 1.2 1.3
Mass (g)
e1 (C
al)
Figure 5
These data and results are further analyzed in the Discussion Section. The
accuracy and error of the experiment is also determined and explained there. The
explanation of the relationships with increases in mass are also discussed.
21
Discussion
OverviewAlanine is used by the body for many important functions. It is involved in
numerous biological reactions which produce energy for the body. In some cases,
alanine undergoes a reaction that converts it to glucose. In the glucose-alanine cycle,
alanine is converted into pyruvate, which in turn is converted into glucose which is used
by muscle tissues. By other mechanisms, alanine can be converted into ATP, which is
used by all cells to provide the energy needed to sustain life. Finally, in the kidney
tubules, alanine is continuously being metabolized and synthesized in reactions to
maintain a constant physiological pH of 7.4. To quantitatively measure the energy
associated with these reactions, a bomb calorimeter is used. The use of a bomb
calorimeter allows for the heat of combustion of compounds to be accurately determined.
The energy released by the sample is calculated using the temperature change of the
calorimeter apparatus from combustion. There are also other factors that influence the
temperature change which must be accounted for, such as formation of side products and
the burning of the fuse wire. Before the experiment can be performed, the energy
equivalent of the calorimeter, W, must be determined. From the 6 trials from this
experiment, W is determined to be 2429.9 ± 51.3 cal/°C, with an accepted value of 2426
cal/°C. In the experiments using alanine, the energy released by the compound is
calculated. From the 14 trials of this experiment, the heat of combustion of alanine
(C3H7NO2) is calculated to be 4291.0 ± 113.1 cal/g, with an accepted value of 4284.8
cal/g.
Presentation and Quantitation
To determine the heat of combustion, several other values must first be
calculated, such as c1, c3, e1, e3, etc.. The values of e1 and e2 are the correction factors
that account for exothermic reactions that take place during the combustion other than
the burning of alanine (the formations of nitric acid and sulfuric acid, respectively). In
22
this lab, since the compound involved contains no sulfur and the amount of sulfur in the
atmosphere is virtually zero, c2 is assumed to be zero; therefore, e2 is zero.
The basic premise of this lab is that there is some total heat produced in the
system of the bomb calorimeter. The desired heat calculation is the heat of combustion
of the amino acid, alanine. This is not equal to the total heat of the system, because there
are other sources that contribute to the heat of the system. These other sources of heat
are the heat of formation of nitric acid, and the heat of combustion of the fuse wire,
which are described by e1 and e3. These sources of heat are explained in detail below.
Nitric Acid Correction
Since nitrogen exists in the alanine structure, and nitrogen from the atmosphere is
trapped in the oxygen chamber, a significant amount of nitrogen reacts to create a
significant e1 value. The total energy released by the nitric acid formation is described
by the amount of nitric acid remaining in the calorimeter. Therefore, the energy released
by the nitric acid is directly proportional to the number of ions in one mL. This
corresponds with one calorie when 0.0709N alkali solution is used. Thus, by simply
titrating the nitric acid solution obtained from rinsing the calorimeter with the 0.0709N
alkali solution, the calories released in the nitric acid formation is determined.
Since nitrogen is a component of alanine, the energy involved in the chemical
conversion of the nitrogen/nitric acid should be accounted for in the final determination
of alanine’s heat of combustion. The relationship between the nitric acid formation and
the mass of the alanine sample was examined in Figure 5, and it was determined that no
relationship exists. Therefore, no values can be determined for the energy involved in
the conversion of the nitrogen portion of the alanine molecule.
Fuse Wire Correction
There is also some heat released during the combustion of the fuse wires. Once
the bomb is fired, there is an electrical current transmitted and, because of low resistance,
the fuse wire becomes extremely hot and transfers this heat to the alanine pellet. After
the temperature has reached a certain level, the fuse breaks, and there is an open circuit.
Therefore, no current can be further transmitted, and there is no further heat transfer.
23
The fuse wires that are burned also contribute to the total heat of the system. This heat
must also be corrected for when calculating the heat of combustion of alanine. The heat
of combustion of the fuse wires varies linearly with the length of the wires. Thus, using
the given fact that 2.3 calories are released during the combustion of one centimeter of
the Parr 45C10 nickel chromium fuse wire, the value of e3 is determined by measuring
the length of the wire burned.
Precision and AccuracyThe precision is important because it describes how different trials in our data
compare to each other, and how our data compares to the accepted value. A measure of
the precision is the 95% confidence interval. This measures how the data varies from the
values in the trials. The plus or minus error intervals show that the data is subject to
some random errors that ‘could’ fall within a certain interval. Therefore, 95% of the
trials to determine value of W will fall within 2429.9 ± 51.3 cal/°C, and 95% of the trials
to determine the heat of combustion of alanine will fall within 4291.0 ± 113.1 cal/g.
Causes of error in this section are due to human error and inconsistency in the taking of
data.
However, the precision does not measure how close the data is to the actual and
accepted values, which is referred to as the accuracy. The accuracy is different because it
compares how close the data obtained is to the actual or accepted values. Errors in
accuracy are due to systematic errors in the equipment that is used. The accuracy is
determined by finding the % errors. These values are found to be 0.16% and 0.14% for
the energy equivalent, W, and the heat of combustion of alanine, respectively. These
percentages, since they are relatively small, indicate extreme accuracy. Therefore, the
experimental values are close to the accepted values. If both the precision and the
accuracy are examined, it is noted that the accepted value falls within the 95%
confidence interval of the experimental data. Thus, it is concluded that the experimental
values agree with the accepted values. The accepted value for alanine was determined by
averaging the accepted heats of combustion of d-alanine and l-alanine, as given in the
75th ED. of the CRC Chemistry and Physics Handbook (4342.1 cal/g and 4227.4 cal/g,
24
respectively). It should be noted that, although no specifications were given on the
container as to the ratio of d-alanine to l-alanine, the distribution company was contacted
and they reported that the alanine powder was 50% d-alanine and 50% l-alanine (a 1:1
ratio).
Error Analysis
This entire lab is based on finding the precision of our oxygen bomb calorimeter
by first using a standard substance (Benzoic Acid) whose heat of combustion is known
and using that along with the values measured of time and temperature to find the energy
equivalent or heat capacity for the system. Then, the specific heat that was found for the
apparatus is tested to see how it conforms to determining the heat of combustion of other
substances (alanine). In this determination, the same corrections of the nitric acid
formation and the fuse wire combustion are used.
First, there are temperature corrections which are based on the thermometer used
in the experiment. In this experiment, the thermometer was read to the nearest hundredth
of a Centigrade, and then a chart from the company was used to correct the temperature
to the nearest thousandth. Since the entire experiment took place in about a 2 degree
Celsius range, this helped the data become more accurate. The error for the temperature
is eliminated from the other calculations because the value used to calculate the heat of
combustion of the alanine (or benzoic acid) cancels due to the fact the difference in
temperature is needed, not an actual temperature.
Secondly, the calculation for the heat of combustion is corrected due to the fact of
the acid formation. The heat of formation of 0.1M Nitric Acid(HNO3) is 1cal/mol
formed. The number of moles of Nitric Acid is determined through titration with
NaCO3. Since the Molarity is known and the Volume is measured of the Sodium
Carbonate, the number of moles are easily calculated. This is then equated to the number
of moles of Nitric Acid. This is the correction for the calories of heat of formation of
HNO3. The formation of Nitric Acid is important to this lab because there is nitrogen in
the alanine. This is significant because if nitric acid is formed from part of the alanine,
this may be part of the heat of combustion of the alanine. In order to test its significance
in the amount of heat that was produced, the mass of alanine is graphed against the
25
amount of nitric acid that is formed. There is no direct correlation between them, and
since the values that are found are extremely accurate, this is determined not to be a
problem. Without any sort of graph-able relationship between the amount of alanine and
the amount of nitric acid formed, there would be no way to correct for this problem
Since the Molarity of the alkali is known, the error in the number of moles of
acid is based on the error in measuring the volume of the base from the burette. This
error is DV = 0.10 ml (1 X 10-4 L).
The final source of error in the heat of combustion calculation has to do with the
wire fuse that is burned in the experiment. It could be assumed that the entire
approximately 10cm wire were burned in the experiment; however, due to the way in
which the fuse is attached to the ignition unit, there is a small length of wire left over
after the combustion has occurred. In any case, the burning of the wire adds heat to the
system that does not occur from burning either the benzoic acid or the sucrose. The
correction for this added heat is equal to the length of the wire burned, which is the initial
length minus the unused ends of wire, all times the heat of combustion of the wire which
is 2.3 calories per centimeter. The error for this correction is equal to the error from the
measuring devices used to find the length of the wire before and after the bomb reaction
times the heat of combustion of the wire. ((DL) x (2.3 cal/cm) = error). For the device
used in this experiment, DL = 0.0010 m (1.0 mm).
All of these corrections and their errors are brought together to determine the heat
of combustion of the alanine in the bomb. The equation:
Hg = (tW - e1 - e2 - e3) / m
is used to incorporate all of the corrections discussed above. The variables are that t is
equal to the net corrected temperature rise. The value W is the specific heat or energy
equivalent that was calculated in the standardization of the system with the benzoic acid.
The e1 is equal to the correction for the nitric acid, while the e3 is the correction
determined for the burning of the fuse wire. The value of e2 is assumed to be zero since
the % content of sulfur in the atmosphere is virtually zero, and sulfur does not exist in the
26
compounds being combusted. Finally, the m in the denominator of the equation is the
mass of the pellet used in the bomb.
The total error for the determination of the energy equivalent is based on the
equation used above and differentiated with respect to the uncertainty intervals. The
equation used can be found in the appendix.
e1 (acid
correction)
e3 (fuse wire
correction)
Temp mass
uncertainty
interval
De1 = 1.41 cal De3 = 0.0023cal DT = 0.002*C Dm = 0.001g
The error in the energy equivalent is determined to be 4.743 cal/°C.(=DW) This
error is due to the accumulation of the error of the measuring of the mass of benzoic acid,
the error in measuring the temperature from the thermometer, and the error in the heat of
combustion of the fuse wires and heat of formation of the nitric acid.
This error and the ones listed before are then used in the equation to calculate the
error in the heat of combustion of alanine. That was determined to be 18.356 cal/g.
(=DHc) This uncertainty interval is much larger than that for W because it is dependent
on the DW value. This shows the importance of precision and accuracy in the
measurements taken, especially for that of the energy equivalent, since its errors are
multiplied in the error for the heat of combustion. The calculations for both of these are
located in the appendix.
The relative value is calculated by plugging the values in the equation for a first
time. Then, all of the variables are kept constant except for increasing the correction
being tested by one unit. For instance, when finding the relativity of the error for e3,
everything is held constant from the first to the second calculation, except that the value
for the number of centimeters of fuse wire is increased by one. The values for Hc are
then subtracted from each other, thus finding the relative error.
27
Relative Error of Corrections:
e1 e3 temp mass
Correction 1 cal/mol 2.3cal/cm These two combine together to calculate the
Hg since it is dependent on temp per mass.
Relative Value
(cal per one unit)
1 cal 2.3cal 2426cal (= W)
2429.9 + 2.1% cal
e1 e3 temp mass
Correction 1 cal/mol 2.3cal/cm These two combine together to calculate the
Hg since it is dependent on temp per mass.
Relative Value
(cal per one unit)
1 cal 2.3cal 4248.8 (= Hc (alanine))
4291.0 + 2.6%
Limitations
There are several limitations involved in determining alanine’s heat of
combustion. It is important to understand some of these limitations in order to design
new methods of increasing the experiment’s precision and accuracy. These limitations
include the inconsistency of lab conditions, the inaccuracies of titration and fuse wire
measurements, and undetectable non-combusted alanine in the calorimeter.
It is imperative to repeat the experiment under the same conditions for each
experiment to achieve the best results. This limitation is more a restriction on the
precision of the data. Small variations in lab conditions including temperature cause the
data to deviate from day to day. This will cause the data to be less precise because the
values are not as close to each other as under uniform experimental conditions.
28
When measuring the volume of 0.0709 N alkali solution titrated, there may have
been small errors when reading the burette. When measuring the length of the fuse wire
burned, there was also some inaccuracy in measuring the length. These two
contributions make the data less accurate rather than less precise due to the fact that these
inconsistencies were constant throughout each trial. These values deviate from the
accepted value in each of these trials and they will cause the calculation of the heat of
combustion to be less accurate.
In each trial, there is a possibility that not all of the alanine sample was
combusted. Correct experimental procedure tells us that if the sample is not fully
combusted, then the data must be discarded. A limitation in the accuracy of this
experiment occurs when a small portion of the alanine sample does not combust, but is
undetectable. This limitation contributes to both the inaccuracy and imprecision of the
data.
In summary, there were many limitations in the experiment that were not
accounted for as correction factors. In future experiments, other techniques may be
devised to make the measurements of the fuse wire burned and titration volume more
accurate. By keeping constant temperature or conditions in the lab, higher precision in
the data can be maintained. Overall, the data’s accuracy and precision can be improved.
ConclusionIn this eight week project, the goal of the experimentation and research was to
determine alanine’s heat of combustion and relate it to biological processes.
Through research, it was found that alanine is a non-essential amino acid that can
be manufactured by the body from other sources as needed. It is involved in the energy-
producing breakdown of glucose. Alanine plays a major role in the Glocose/Alanine
Cycle (Diagram 3). Through the mechanisms of this process, alanine aids the body in
getting pyruvate, an end product resulting from the breakdown of glucose, into the liver
cytosol. In conditions of sudden anaerobic energy need, when proteins are broken down
for energy, alanine acts as a carrier molecule to take the nitrogen containing group to the
liver and kidneys. There it is converted into the less toxic urea and excreted in the urine.
As a result of this conversion, the build up toxic products in the muscle cells is
29
prevented. Through a similar process, alanine is used by the kidneys to form the
ammonia that is excreted by the body. In this way, alanine helps the body to maintain
the physiological pH of 7.4. During this process, it is necessary for the kidneys to
constantly be synthesizing and metabolizing alanine. Alanine is also an important
sources of energy for the brain and central nervous system and is known to strengthen the
immune system by producing antibodies.
Because of the numerous processes in the human body that involve either the
formation or breaking-down of alanine, the determination of the molecule’s heat of
combustion is of great concern. This value allows for quantitative analysis of the
processes involved, and provides a better understanding of the complex chain reactions
that are necessary to sustain life.
Diagram 3
30
Appendix
Qreaction = msample * DHcomb,sample = msample * Cp,water * Dtwater (Equation 1)
msample * DHcomb = W * Dtwater (Equation 2)
t = tc - ta -r1(b - a) - r2(c - b) (Equation 3)
Hg = (tW - e1 - e3) / m (Equation 4)
e1 = volume of 0.709N alkali titrated (Equation 5)
31
W Hgm e1 e3t
(Equation 6)
W Hgmt
e1t
e3t
(Equation 7)
D W .Hgt
D m .1t
D e1 .1t
D e3 .Hgm e1 e3
t2D t (Equation 8)
The values for Dm, De1, De3, and Dt are listed in the first table in the error analysis section. The values for the rest of the variables are:
Hg = 6311.1 cal/gt = 2.653 degreesm = 1.017 ge1 = 8.95 cale3 = 18.668 cal.
DW is calculated to be 4.743 cal/ °C. It is then used in the calculating the uncertainty interval for the heat of combustion of alanine.
Hc.t W e1 e3
m(Equation 9)
Hc .t Wm
e1m
e3m
(Equation 10)
D Hc .Wm
D t .1m
D e1 .1m
D e3 .( ).t W e1 e3
m2D m .t
mD W (Equation 11)
All of the uncertainty intervals used in this experiment are listed in the table in the error analysis section, except for DW, which is listed above. The rest of the constants are listed below:
W = 2429.9 cal/°C t = 1.895 degrees m = 1.065 g e1 = 16.45 cal
32
e3 = 19.533 cal
33
Trial a
(min)
b
(min)
c
(min)
r1
(°C /min)
r2
(°C /min)
c1
(mL)
c3
(cm)
1 6 7.2 12.25 0.012 0.002 10.3 6.1
2 6 7.3 12.25 0.0 0.002 12.0 8.4
3 6 7.3 12.25 0.0182 0.0 4.2 8.8
4 6 7.2 12.25 0.01 0.002 8.8 8.45
5 6 7.1 12.25 0.0642 0.002 7.5 8.65
6 6 7.2 12.25 0.0002 0.0 10.9 8.3
Table 1
Sample mass
(g)
ta
(°C)
tc
(°C)
t
(°C)
H
(cal/g)
e1
(cal)
e3
(cal)
W
(cal/°C)
1 1.017 26.649 29.312 2.639 6311.1 10.3 14.03 2441.8
2 0.999 24.361 27.007 2.636 6311.1 12.0 19.32 2403.6
3 0.997 25.182 27.797 2.591 6311.1 4.2 20.24 2437.6
4 1.020 24.901 27.604 2.681 6311.1 8.8 19.44 2411.7
5 0.951 24.911 27.431 2.439 6311.1 10.29 19.90 2471.9
6 1.116 24.810 27.743 2.932 6311.1 10.9 19.09 2412.6
Table 2
W (cal/°C)
Average of Samples 2429.9
95% Confidence Interval 51.3 (2.1%)
Accepted Value 2426
Percent Error 0.16%
Table 3
Sample a b c r1 r2 c1 c3
34
(min) (min) (min) (°C /min) (°C /min)(mL)
(cm)
1 6 7.2 13.25 0.034 0.001 18.3 8.6
2 6 7.2 12.25 0.0122 0.0 17.0 8.7
3 6 7.2 11.25 -0.01 0.0 16.1 8.8
4 6 7.2 12.25 0.01 0.004 15.5 8.4
5 6 7.2 11.25 0.02 0.004 16.7 8.4
6 6 7.2 11.25 0.012 0.0 15.1 8.6
7 6 7.2 12.25 0.006 0.006 16.9 8.5
8 6 7.3 12.25 0.008 0.002 13.4 8.6
9 6 7.4 11.25 0.0 0.0 17.3 5.9
10 6 7.0 11.25 0.002 0.004 17.7 8.9
11 6 7.0 10.25 0.004 0.0 15.9 9.1
12 6 7.0 10.25 0.004 0.002 18.3 9.0
13 6 7.2 10.25 0.006 0.0 16.8 8.7
14 6 7.2 10.25 0.0078 0.01 15.3 8.7
Table 4
35
Sample mass
(g)
ta
(°C)
tc
(°C)
t
(°C)
t*W
(cal)
e1
(cal)
e3
(cal)
H
(cal/g)
1 1.255 25.573 27.867 2.244 5424.3 18.3 19.78 4314.1
2 1.093 24.541 26.510 1.954 4748.9 17.0 20.01 4310.9
3 1.066 24.511 26.411 1.912 4645.9 16.1 20.24 4324.2
4 0.777 25.222 26.659 1.405 3413.5 15.5 19.32 4348.4
5 1.206 25.403 27.573 2.130 5175.2 16.7 19.32 4261.3
6 0.948 26.343 28.030 1.673 4064.2 15.1 19.78 4250.4
7 1.149 24.750 26.808 2.021 4910.0 16.9 19.55 4241.2
8 0.987 25.644 27.531 1.767 4292.9 13.4 19.78 4315.8
9 1.099 27.153 29.070 1.917 4658.1 17.3 13.57 4210.4
10 1.122 25.031 27.062 2.012 4889.0 17.7 20.47 4323.3
11 1.010 24.700 26.471 1.767 4294.0 15.9 20.93 4214.6
12 1.104 24.581 26.609 2.018 4902.3 18.3 20.70 4405.1
13 1.034 25.280 27.141 1.854 4505.0 16.8 20.01 4320.8
14 1.060 24.530 26.431 1.861 4522.3 15.3 20.01 4233.0
Table 5
36
Hg (Cal / g)
Average of Samples 4291.0
95% Confidence Interval 113.1 (2.6%)
Accepted Value 4284.8
Percent Error 0.14 %
Table 6
e1 (acid
correction)
e3 (fuse wire
correction)
Temp mass
uncertainty
interval
De1 = 1.41 cal De3 = 0.0023cal DT = 0.002*C Dm = 0.001g
Table 7
e1 e3 temp mass
Correction 1 cal/mol 2.3cal/cm These two combine together to calculate the
Hg since it is dependent on temp per mass.
Relative Value
(cal per one unit)
1 cal 2.3cal 2426cal (= W)
2429.9 + 2.1% cal
Table 8
e1 e3 temp mass
Correction 1 cal/mol 2.3cal/cm These two combine together to calculate the
Hg since it is dependent on temp per mass.
Relative Value
(cal per one unit)
1 cal 2.3cal 4248.8 (=Hc (alanine))
4291.0 + 2.6%
Table 9
37
0 2 4 6 8 10 12 14 16
27
28
29
30Benzoic Acid Trial 1: Temp. vs. Time
Time (minutes)
Tem
pera
ture
(deg
C)
30
26.3
28.2248tempi
17.50
7.26
timei
Figure 1
0 2 4 6 8 10 12 14 16 1825.5
26
26.5
27
27.5
28Alanine Sample 1: Temperature vs. Time
Time (minutes)
Tem
pera
ture
(deg
C)
26.8844tempi
7.2
timei
Figure 2
38
ALANINE: Change in Temperature vs. Mass
y = 1.7538x + 0.0275
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Mass (g)
Tem
pera
ture
Cha
nge
(deg
C)
Y
Predicted Y
Linear(Predicted Y)
Figure 3
t*W (Total Change in Energy of the System) vs. Mass
y = 4261.4x + 66.738
0
1000
2000
3000
4000
5000
6000
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Mass (g)
t*W
(cal
) Y
Predicted Y
Linear (Predicted Y)
Figure 4
39
Heat from Nitric Acid Formation (e1) vs. Mass
12
13
14
15
16
17
18
19
0.7 0.8 0.9 1 1.1 1.2 1.3
Mass (g)
e1 (C
al)
Figure 5
40
Standardization with Benzoic Acid
Temp. vs. Time Sample 1
26.5
27
27.5
28
28.5
29
29.5
0 5 10 15 20
Time (minutes)
Tem
pera
ture
(deg
C)
Temp. vs. Time Sample 2
24
24.5
25
25.5
26
26.5
27
27.5
0 5 10 15 20
Time (minutes)
Tem
pera
ture
(deg
C)
41
Temp. vs. Time Sample 3
25
25.5
26
26.5
27
27.5
28
0 5 10 15 20
Time (minutes)
Tem
pera
ture
(deg
C)
Temp. vs. Time Sample 4
2425
2627
28
0 5 10 15 20
Time (minutes)
Tem
pera
ture
(deg
C)
Temp. vs. Time Sample 5
24.5
25
25.5
26
26.5
27
27.5
28
0 5 10 15 20
Time (minutes)
Tem
pera
ture
(deg
C)
42
Sample 6: Temperature vs. Time
24.5
25
25.5
26
26.5
27
27.5
28
0 5 10 15 20
Time (minutes)
Tem
pera
ture
(deg
C)
Alanine
Sample 1: Temp. vs. Time
25
25.5
26
26.5
27
27.5
28
0 2 4 6 8 10 12 14 16 18
Time (min)
Tem
p. (D
eg C
)
43
Sample 2: Temp. vs. Time
24
24.5
25
25.5
26
26.5
27
0 2 4 6 8 10 12 14 16 18
Time (min)
Tem
p. (d
eg C
)
Sample 3: Temperature vs. Time
24.4
24.624.8
25
25.2
25.425.6
25.8
26
26.226.4
26.6
0 2 4 6 8 10 12 14 16 18
Time (Min)
Tem
p. (d
eg C
)
44
Sample 4: Temperature vs. Time
25
25.2
25.4
25.6
25.8
26
26.2
26.4
26.6
26.8
0 2 4 6 8 10 12 14 16 18
Time (min)
Tem
p. (d
eg C
)
Sample 5 Temperature vs. Time
25
25.5
26
26.5
27
27.5
28
0 2 4 6 8 10 12 14 16 18 20
Time (Minutes)
Tem
pera
ture
(deg
C)
45
Sample 6: Temp. vs Time
26.2
26.426.6
26.827
27.2
27.427.6
27.828
28.2
0 5 10 15 20
Time (min)
Tem
p. (d
eg C
)
Sample 7: Temp. vs. Time
24.5
25
25.5
26
26.5
27
0 5 10 15 20
Time (Min)
Tem
p. (D
eg C
)
46
Sample 8: Temp. vs Time
25.6
25.8
26
26.2
26.4
26.6
26.8
27
27.2
27.4
27.6
0 5 10 15 20
Time (min)
Tem
p. (d
eg C
)
Sample 9: Temperature vs. Time
2727.227.427.627.8
2828.228.428.628.8
2929.2
0 2 4 6 8 10 12 14 16
Time (min)
Tem
p. (d
eg C
)
47
Sample 10: Temp. vs. Time
25
25.5
26
26.5
27
27.5
0 2 4 6 8 10 12 14 16
Time (min)
Tem
p (d
eg C
)
Sample 11: Temperature vs. Time
24.6
24.8
25
25.2
25.4
25.6
25.8
26
26.2
26.4
26.6
0 2 4 6 8 10 12 14 16
Time (min)
Tem
p. (d
eg C
)
48
Sample 12: Temp. vs. Time
24.5
25
25.5
26
26.5
27
0 2 4 6 8 10 12 14 16
Time (min)
Tem
p. (d
eg C
)
Sample 13: Temp. vs Time
25.225.425.625.8
2626.226.426.626.8
2727.2
0 2 4 6 8 10 12 14 16
Time (min)
Tem
p. (d
eg C
)
49
Sample 14: Temp. vs. Time
24
24.5
25
25.5
26
26.5
0 2 4 6 8 10 12 14 16
Time (min)
Tem
p. (d
eg C
)
50
References
BE 210 Laboratory Manual, Spring 1997.
Brodsky, Irwin G., and John Devlin. Effects of dietary protein restriction on regional amino acid metabolism in insulin-dependent diabetes mellitus. American Physiological Society, 1996. pp. E148-57.
Brot, Fred. Research Technical Service. Correspondence ([email protected])
Castellan, Gilbert W. Physical Chemistry: 3 rd Edition . The Benjamin/Cummings Publishing Company, Inc., Reading, Massachusetts, 1983, Chapter 24.
CRC Handbook of Chemistry and Physics, 75th Edition.
Fouque, Denis, Sylvie Dugelay, and Guy Martin. Alanine metabolism inisolated human kidney tubules: Use of a mathematical model. European Journal of Biochemistry, 236, Feb. 1996. pp. 128-37.
Nurjhan, N. A. Bucci, and G. Perriello. Glutamine: A Major Gluconeogenic Precursor and Vehicle for Interorgan Carbon Transport in Man. The American Society for Clinical Investigation, Vol. 95, January 1995. pp. 272-6.
Oliver, Javier, Rafael Salto, Maria Sola, and Alberto Vargas. Cell Biochemistry and Function, Vol. 12, 1994. pp. 229-35.
Perriello, G., and R. Jorde. Estimation of glucose-alanine-lactate-glutamate cycles in postabsorptive humans: role of skeletal muscle. American Physiology Society, 1996. pp. E443-9.
Stumvoll, Michael, Gabriele Perriello, Nurjahan Nurjhan. Glutamine and Alanine Metabolism in NIDDM. Diabetes, Vol. 45, July 1996. pp. 863-8.
51
