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Thermodynamic properties of clathratesN.R. Grey a & L.A.K. Staveley aa Inorganic Chemistry Laboratory , OxfordPublished online: 12 Aug 2006.

To cite this article: N.R. Grey & L.A.K. Staveley (1964) Thermodynamic properties of clathrates,Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, 7:1, 83-95,DOI: 10.1080/00268976300100841

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Thermodynamic properties of clathrates IV. The heat capacity of carbon monoxide, nitrogen and

oxygen in the quinol clathrates

by N. R. G R E Y and L. A. K. S T A V E L E Y

Inorganic Chemis t ry Laboratory, Oxford

(Received 30 January 1963)

Measurements have been made from a temperature between 14 and 20~ to ~ 300~ of the heat capacity of three carbon monoxide quinol clathrates, two oxygen quinol clathrates and one nitrogen quinol clathrate. The results have been used to estimate the contribution C~ made to the heat capacity by a mole of carbon monoxide, nitrogen or oxygen in the clathrate. Analysis of the Qi values leads to the conclusion that the barrier hindering rotation of the molecules in the cavities is ~200 calmole -1 for oxygen, ~1100 cal mole -1 for nitrogen and slightly larger for carbon monoxide than for nitrogen. Possible reasons for these differences are briefly discussed. An attempt has been made to calculate the heat capacity contribution of the trapped molecules at low temperatures where a classical treatment is no longer applicable. The agreement with experiment is not good, which may be due partly to the alteration of the heat capacity of the lattice itself by the trapped molecules, and partly to the inadequacy of the potential energy function used in the calculations.

1. INTRODUCTION

The three previous papers of this series have dealt with heat capacity measure- ments of the fi-quinol clathrates of argon [1], methane [2] and krypton [3]. T h e measurements on the argon and krypton clathrates made it possible to test van der Waals ' statistical mechanical t rea tment [4] of these systems (based on the Lennard-Jones cell model) and to assign, within comparat ively narrow limits, a distance paramete r characteristic of the quinol lattice. I t was then possible to analyse the heat capacity results for the methane clathrate, and to conclude that the barr ier hindering rotation of the methane molecules is so small that above ,-~ 150~ rotation is virtually free.

In this paper we present results on the fi-quinol clathrates containing carbon monoxide, nitrogen and oxygen. We shall carry out a similar analysis of the heat capacity contr ibut ion of the t rapped molecules, with the object of investigating the extent to which their rotation is restricted. Some information on this is already available f rom entirely different experimental sources. F r o m magnetic suscep- tibility measurements [5], it has been concluded that the barrier opposing rotat ion of the oxygen molecules inside the clathrate is quite small (~, 128cal mole - l ) , a conclusion confirmed by a recent electron spin resonance s tudy [6]. On the other hand, it appears f rom an examination of the nuclear quadrupole resonance spec t rum that for the ni trogen clathrate the barrier hindering rotation is conside- rably higher [7].

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84 N.R. Grey and L. A. K. Staveley

Exper imenta l values of the heat capacity of the carbon monox ide clathrate in cal deg -1 m o l e - L 1 calorie = 4"184 abs. joules

3C6H4(OH)2 �9 0 '699 CO (0"04559 moles in the calor imeter)

T (~ C~ T (OK) C~ T (OK) C~o

17.58 18.64 21-05 24.93 29.70 35.18 41.47 49.62 57.67 64.45 69.44 74.67 80.12 85.27 90.09 95.09

5.53 6.35 8.14

11-13 14.49 18.13 21.90 26.15 29.23 31"51 33.32 34.96 36-71 38.50 40.03 41.22

99.46 104-36 113.74 118-67 123.89 128.87 133.99 144.52 149.51 154.37 159.33 164'56 169.75 175-30 180.87 186.24

42.49 43-74 46.57 47.96 49.40 50.97 51.95 55.35 56-79 58-22 59.65 61.13 62.69 64.23 66.08 67.76

192.00 198.01 204.36 210.82 217.65 232.92 240.77 248.53 256.50 265.56 274.25 282"45 290"81 299"11

69-39 71.71 73.25 75"20 77.62 82.40 84"68 87.18 89.93 92.78 95.76 98.43

101.37 103.64

3C6H4(OH)2.0"390 CO (0-04782 moles in the calorimeter)

T (~ C~ T (~ C~ T (~ C~

14"44 16"14 18"55 20"34 21"52 22"55 25"39 30'23 35"54 41"76 48"52 54"52 59'67 63"01 64"57 67'94 73'31 80'72

3"04 4"11 5"64 7"06 7"83 8"28

10"30

87.06 91.48 91.72 96.97

102.73 108.46 113.59

35.81 37.38 37.48 38"79 40.49 42.12 43.62

176.84 182-37 187.81 192.90 198.20 211.75 217.89

13.38 16.75 19.97 23.45* 25.54* 27.53* 28-42* 28.90* 29-90* 31.38 33.78

117.94 123-22 128.64 134.31 139.80 145.35 150.76 156.08 161-17 166.29 171.44

44.90 46.39 47.90 49-61 51.07 52.52 54.19 55.81 57.48 58.93 60.44

225.55 233.99 242.08 249.80 257.64 265.20 272.74 281.86 291.73

62.08 63.94 65.27 67.31 68.71 72.91 74.68 77.21 80.21 82.82 85.24 87.67 90.26 92-82 95.65 99.22

T (~

18-63 20.02 22.34 25.84 29.61 33.53 38.26 43-38 48.57 54"12 60.34 65.58 70"51 75-70 81-22 86.84 92.92

3C6H4(OH)~ �9 0"248 C O (0"04771 moles in the calorimeter)

C~

5'44 6"30 7"80

10'01 12'46 14'70 17'42 19"94 22"35 24"51 26"89* 28"95* 29"86* 31"34" 33'07 34"84 36'53

T (~

98.99 105.27 112.05 115.43 118-73 121-86 125.45 128.08 134.59 140.96 147.06 154.41 161.72 167,74 174-10 180.82 188.29

C~o

38"07 39"80 41 '60 42.63 43-56 44-53 45.49 46.29 48.13 50-14 51.80 53-97 56.18 58.04 59.94 61-86 64.53

T (~

195-79 202-73 209-78 217.61 225.63 230.77 233.06 237-81 244.74 251-81 259.17 267.79 276-66 283'96 291.34 298.77

C~

66-58 68.76 70.96 73'50 76.15 77.27 78.54 79.27 81.96 84.45 87.06 89.90 92.40 94.95 97.95

100.34

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Experimental values of the heat capacity of the nitrogen clathrate

3C6H4(OH)~. 0"680 N2 (0"04930 moles in the calorimeter)

T (~ C~o T (~ C~ T (~ C~

20.90 22.17 24.14 26.77 30.30 34.69 40.17 46-12 51.65 57.38 62.76 67.78 68-03 72.83 73.47 78-01 83.49

8"16 9"55

10'69 12"32 15"24 18"06 21'23 24"35 26"77* 29"12" 31"25" 32"98* 33'12 34"44 34'53 36"05 37"89

88.85 94.62

100-18 105-84 111.85 117.67 123.18 128.75 134.30 139.49 145.03 150.82 154.12 159-94 165.53 171-20 176.33

39-94 40-95 42-41 43-97 45.60 47-12 48.73 50-35 51-96 53-46 55-01 56-62 57-68 59-25 60-88 62-58 64.11

181"20 186.65 192'29 197-85 203'77 210-05 216-23 222-54 228-98 235-69 243-35 251.09 258'32 265.81 273.58 281"49 289.17 296.84

65.57 67'30 69.02 70.68 72'59 74.45 76'40 78.44 80.69 82"67 85.36 87.79 90'00 92-55 95.26 97'83

100-30 102.95

Experimental values of the heat capacity of the oxygen clathrate

3C6H4(OH)z. 0"646 02 (0-05472 moles in the calorimeter)

T (OK) C/o T (OK) Cp T (OK) C~

15-33 16-44 17.37 18-20 19.23 20.34 21-24 22.67 23.46 26.30 29.51 33.84 39.02 39.51 45.37 51.36 56.69 61-67 66.31 70.78 75-55

5.84 6.79 7.45 8.19 8.81 9.50

10.19 10.93 11-70 13.69 15.63 18.09 20.78 21.31 23.81 26.45 28.24 29.99 31.75 32-75 34.53

77.57 80.83 85.19 85.96 90.62 91.46 92.63 96.71

102.29 107.54 113.16 118.87 124.16 129'19 134.60 138.39 142.44 148-28 153.65 159.02 164.81

35-45 35-58 36-81 37-08 38-00 38-61 38-74 39-88 41-42 42-78 44-33 45-89 47-46 48-86 50.39 51-93 52.62 54-34 55-96 57-32 59-23

171-02 176-80 182.22 186.70 191.41 197.59 204-31 210-47 216.53 222-70 229.37 235.07 241.15 248.28 256.22 264.74 273.27 281.17 288-71 296.55

61.05 62.85 64.56 65.99 67.39 69'41 71'48 73.49 75.35 77.45 79-48 81.66 83.32 85"53 88.53 90-37 94.01 97.01 99.90

102"31

3C6H4(OH)2.0-240 02 (0"04745 moles in the calorimeter)

T (~

20"00 21"06 23"21 26"95 32"15 28"32 45"94 54"16 60'58 66"73 71"48 76"12 79"99 80"17 80"55 84"64 84"83

C~

7"00 7"63 9'21

11'40 14"36 17"69 21"15 24"30 26"49 28"47 29"79 31"08 32"13 32"15 32'39 34"17 33"70

T (~

86.00 90-16 96.30

102.44 108.94 115.78 t22-88 130.13 137.35 144.23 151.04 157-69 164.24 170.89 177.45 184-27 191.64

Table

C~

34.00 35.20 36.79 38-49 40.16 42.02 44-18 46.13 48.24 49.04 52.09 54.00 55.93 58.02 60.02 62-09 64.38

(continued),

T (~

198"52 204"91 212"24 220"73 229"85 233"61 238-66 243'43 250"95 258"35 266"25 274-47 282-52 290"55 298"70

C~

66"51 68"57 71 "28 73"93 77"08 78"00 79" 50 81"14 84-84 86"23 88"84 91 '76 94"41 97"55

100"20

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2. EXPERIMENTAL

As in the previous work, the clathrates were prepared by slow crystallization from hot solutions of recrystallized quinol in n-propanol in the presence of the appropriate gas under pressure. In preparing the samples with the higher per- centages of the holes filled, care was taken to ensure that crystallization did not begin until the temperature had fallen to ~ 45 ~ The oxygen clathrate, which is pale yellow, was stored in the dark as the oxygen is capable of bringing about photochemical oxidation of the lattice. Measurements were made on three carbon monoxide clathrates with values of x of 0"248, 0.390 and 0.699, where x is the number of moles of trapped gas per three moles of quinol, on two oxygen clathrates with x = 0.240 and 0-646, and on one nitrogen clathrate only with x = 0-680. Attempts to make measurements on samples with smaller values of x were not successful. An oxygen clathrate with x=0-18 lost nearly all its oxygen in the course of the heat capacity measurements, thereby generating sufficient pres- sure to damage the calorimeter, while the results for a sample of the carbon mono- xide clathrate with x=0.189 were discarded when on removing the specimen from the calorimeter there were indications that there had been some transfor- mation to the s-form. Small losses of gas occurred during the measurements on the samples for which results are recorded in this paper, corrections for which were applied as described in paper I [1]. The Cv values for which such corrections had to be made are indicated by asterisks in table 1.

3. DISCUSSION

The experimental Cp values are given in table 1. For the carbon monoxide clathrate, Cj, at all temperatures within the range studied is a linear function of x. From the isothermal plots of Cp against x, we have evaluated Coo (the contribution to the heat capacity made by a mole of carbon monoxide inside the cavities) and Cvo, the heat capacity of three moles of/3-quinol. Values of Cco and Cp q at regular temperature intervals are recorded in table 2. These values of Cv Q agree within experimental error with those derived from measurements on the two samples of the oxygen clathrate, assuming a linear relation for the latter between C• and x. Values of Co, (the heat capacity contribution per mole of oxygen) worked out on this basis are also included in table 2, together with the corresponding values of CN. These have been derived from the smoothed values for the one sample of the nitrogen clathrate on the assumption that at any given temperature the contribution made by the lattice is the mean of the two Cp• values for that temperature derived from the measurements on the carbon monoxide and oxygen clathrates.

The values of C Q in table 2 are somewhat less than those given in paper III in this series from experiments on the krypton clathrate. Between the latter experi- ments and those described in the present paper some modifications were made to the calorimeter which have improved its performance at higher temperatures, and this may account for some of the difference in C~oQ values near room temper- ature, which is about 0.6 per cent. At lower temperatures it is possible that part of the difference in the C~Q values (which amounts to as much as 2 per cent) may be due to different degrees of interaction of the trapped molecule with the surrounding lattice. As suggested in paper III, such interactions may appreciably affect the contributions to the thermodynamic properties made by vibrations of low frequency:

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Thermodynamic properties of clathrates 87

Exper imenta l values of CCO ' the mola r cont r ibut ion of carbon monox ide to the heat capacity of the carbon monox ide clathrate in cal deg -~ m o l e - <

T (~

15 20 25 30 35 40 45 50 55 60 65 70

CCO

1"17 2"24 3.10 4.36 5-32 6-02 6.65 7"06 7.25 7.54 7"75 8.15

T (~

8O 90

100 110 120 130 140 150 160 170 180 190

Cco

8-52 8.60 9.26 9.58 9.58 9.42 9.51 9.48 9-38 9.15 9.03 9.23

T (~

200 210 220 230 240 250 260 270 280 290 298"16 300

C c o

9.20 9.05 9.18 9.06 8.96 9.03 8.63 8.55 8.60 8.59 8.55 8.50

Exper imenta l values of CN2 , the molar con t r ibu t ion of n i t rogen to the heat capacity of the n i t rogen clathrate.

T (~

20 25 30 35 40 45 50 55 60 65 70 75

CN 2

2.50 3.60 4.51 5.30 6.41 6.92 7.40 7.84 8-30 8-58 8.87 8-78

T (~

80 90

100 110 120 130 140 150 160 170 180 190

C ~ T (~

9-25 200 9.30 210 9.34 220 9.37 230 9.29 240 9.23 250 9.32 260 9-31 270 9.05 280 8.67 290 8.70 298.16 8.58 300

CN 2

8.60 8.57 8"60 8.40 8.25 8"12 8.10 8-10 7.92 7.95 8-00 8.01

Exper imenta l values of C02 ' the molar cont r ibu t ion of oxygen to the heat capacity of the oxygen clathrate.

T (~

15 2O 25 30 35 4O 45 5O 55 6O 65 7O

Co~

4.01 5.69 6.18 6.28

T (~

80 90

100 110

6.65 120 6.95 130 6.96 140 7.32 150 7.44 160 7.53 170 7.31 180 7.35 190

CO 2

7"46 7"49 7"39 7"25 7"27 7"28 7"47 7"38 6"97 7"04 7"21 7"23

T (~

200 210 220 230 240 250 260 270 280 290 298.16 300

CO,~

7.00 6.83 7.14 6-90 7-00 6.86 7.03 7.21 6.96 7-15 7.10 7.07

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Cco, CN~ and Cos are plotted against temperature in figures 1, 2 and 3, together with the values of Cvib, which is the contribution f rom the ' r a t t l i ng ' of the molecules in the holes calculated from van der Waals' theory. T o estimate Cvib it is necessary to have numerical values of the parameters in the Lennard- Jones potential for the interaction of a pair of identical diatomic molecules. Th e following values were chosen: for CO -CO , ~/k=llO.7 ~ and a = 3 . 7 9 s for N2-N2, e/k=95"05 ~ and ~ = 3 " 7 0 ~ ; for O2-O2, Elk= 118"1 ~ and (r=3.47&. T h e values for oxygen and nitrogen were recommended by Dr. G. Saville of this

......9 o E

m

_J d

o" t .9

6

4 ~ 5 0 0

~ b l O 0

. . . . . . s2o

I I I I 0 0 2 O0 3 0 0

T ~

Figure 1. Heat capacity-temperature plots for the carbon monoxide-quinol clathrate. Full circles, Coo = total heat capacity contribution per mole of CO in cal deg -1 mole-'. Open circles, calculated values of Cvib. Half-shaded circles, derived values of Cro t. Dotted curves, calculated values of Cro t for hindering energy barriers of 500, 1100 and 1500 cal mole -1.

laboratory who has recently carried out a detailed analysis of all the available relevant virial coefficient and solid-state data. I t may be mentioned that this analysis has shown that there can be considerable variation in the ' bes t ' values for e and ~, according to whether virial coefficient data at low temperatures are included in the analysis or not. The calculated thermodynamic properties of the substance t rapped in the clathrate are particularly sensitive to the value chosen for (r, and uncertainty in this quanti ty necessarily leads to uncertainty in the calculated

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Thermodynamic properties of clathrates 89

values of Cvib. For the intermolecular distance parameter ~ characteristic of the quinol lattice we have taken the value 2-95 ~,, since we are primarily concerned here with an analysis of the heat capacity of clathrates in the region from ~ 100~ to room temperature in which the classical statistical mechanical treatment is appli- cable, and this value of ~ gave the best agreement with the experimental heat "capacities in this temperature range for the argon and krypton clathrates. In view of the approximate nature of the Lennard-Jones potential and the variability in the estimated values of the parameters depending on the nature of the experimental data from which they are derived, it does not follow that the same value of ~ would be that best suited to a calculation of say the heat capacity at much lower tempera- tures or indeed to that of a different thermodynamic property at any temperature.

_ll

i -O

~ I r ; , , _ _ -.o... , , . . . . "....~

.9O0

~LI - _ _ ~ o o

I I I I00 2 0 0 300

T ~

Figure 2. Heat capacity-temperature plots for the nitrogen-quinol clathrate. Full circles, CN~, = total heat capacity contribution per mole of N2 in cal deg -1 mole -1. Open circles, calculated values of Cvib. Half-shaded circles, derived values of Cro ~- Dotted curves, calculated values of (Trot for hindering energy barriers of 500, 900 and 1500 cal mole 1

Also plotted in figures 1, 2 and 3 are the experimental values of Crot, i.e. Cco (or C~2 or Co~ ) less C~i b. (The further reduction which should be made for the contribution from the internal vibrational degree of freedom is usually negligible ; even for oxygen which has the lowest vibrational frequency of the three molecules it amounts to only 0"06cal mole at 300~

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90 : N, R. Grey and L. A. K. Staveley

The study of the argon and krypton clathrates showed that van der Waals' equations were applicable above ~ 120 ~ It will be seen that above this tempera- ture Crot for oxygen is less than Crot for nitrogen or carbon monoxide, and at room temperature only exceeds by about 0.35 cal/mole the value for two-dimen- sional free rotation. The calorimetric evidence therefore indicates that the barrier hindering rotation is much less for oxygen than for the other two molecules. O'Brien et al. [5] analysed the magnetic susceptibility results from the oxygen elathrate on the assumption that the rotation of the oxygen molecules is hindered

81 E

- - b

~ 4 ~9

" ' . 300 ~. (.3 .200 "-. "-

_

I I ! I 0 0 2 0 0 3 0 0

T ~

Figure 3. Heat capacity-temperature curves for the oxygen-quinol clathrate. Full circles, Co, , = total heat capacity contribution per mole of oxygen in cal deg -1 mole -1. Open circles, calculated values of Cvi b. Half-shaded circles, derived values of Cro t- Dotted curves, calculated values of Cro t for hindering energy barriers of 150, 200, 300 and 500 cal mole -1.

by a potential of the form V= V 0 (1 -cos20), where 0 is the angle between the molecular axis and the c-axis of the crystal. For the height of the barrier opposing rotation (2V0) they obtained the value 12Seal/mole. In figure 3 are shown the calculated Crot curves for values of 2V 0 of 150, 300 and 500 cal/mole. It would appear that the heat capacity results require a rather higher potential barrier than 128 cal/mole. This was also implied by the electron spin resonance study of the oxygen clathrate by Foner et al. [6]. This investigation showed, moreover, that the barrier height increases according to whether none, one or

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Thermodynamic properties of clathr ates 91

both of the two cavities nearest to the oxygen molecule are occupied, the barrier being least when these two cavities are empty. It appears that the barrier height with both nearest-neighbour cavities filled may be as high as 250 cal mole -1.

Bearing in mind that the the experimental values of Crot are subject to both the experimental error in the original Cp determinations and to the uncertainty already referred to in the calculated values of Cvm, it will be seen that the h e a t capacity results are consistent with a barrier height of 200-250 cal mole -1. The barrier height will in principle be a function of temperature. As far as the expan- sion of the lattice is concerned, it will presumably decrease with rising temperature, at least for a molecule situated at the centre of the cavity. But as van der Waals has pointed out [8], it is perhaps not altogether justifiable to regard the rotational and vibrational movements of a trapped molecule as completely independent. The hindrance to rotation experienced by a molecule might depend on the vibra- tional level it happens to occupy. The barrier heights estimated by Meyer and his co-workers depend on measurements made at quite low temperatures, whereas the values reached from heat capacity measurements extending up to room temperature must be regarded as average values for a considerable temperature range.

Turning to the nitrogen and carbon monoxide clathrates, figures 1 and 2 include calculated heat capacity curves for hindered rotators of various barrier heights. The nitrogen results indicate a barrier of ~ 1100cal mole -1, while for carbon monoxide the figure would appear to be slightly higher. From their study of the quadrupole resonance spectrum of the nitrogen clathrate [7], Meyer and Scott obtained a value of 1.55 • 0.10 x 1012sec -1 for the frequency v 0 with which the nitrogen molecules oscillate about the orientation of minimum potential energy. By applying a relation between v 0 and Vc, valid for the small amplitudes of vibration which will prevail at very low temperatures, it can be shown that this value of v 0 corresponds to a barrier height (2V0) of 94-0 • 100 cal mole 1. This is in reasonable agreement with our heat capacity estimate, as is the value of 818 cal mole -1 obtained by Coulter from an independent analysis of his own heat capacity results [9].

Large differences between the carbon monoxide and nitrogen clathrates would not be expected but, since above 120~ Crot for carbon monoxide is greater than that for nitrogen, the carbon monoxide molecules presumably experience the greater hindrance to rotation. This difference cannot be attributed to the dipole moment of carbon monoxide which is so small as to have a negligible orientating influence, but it may arise from the larger quadrupole moment of this molecule as compared with nitrogen. The transition at 61"6~ in solid carbon monoxide at which the molecules appear to gain orientational freedom occurs about 26 ~ higher than the corresponding transition in nitrogen. This difference has been attributed to the higher quadrupole moment of carbon monoxide [10] and has in fact been used to calculate values for the quadrupole moment of the two molecules which agree quite well with the estimates made from microwave collision broadening effects of 1-27 x 10 -26 e.s.u, for nitrogen and 1.71 x 10 z6 e.s.u, for carbon monoxide. By contrast, the quadrupole moment of oxygen appears to be altogether smaller, less than one-third of that of nitrogen. So it is possible that the hindrance to rotation of these three diatomic molecules in their cavities in the quinol lattice arises primarily from the interaction of the molecular quadrupole with the polar walls of the cavity. (The polarity of the cavity walls

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arises from the fact that the oxygen atoms in them are grouped round the poles (in the geographical sense)while the carbon atoms are grouped round the equator.) If the oxygen atoms confer an effective negative charge on the top and bottom of the cavity, the preferred alignment of the quadrupole should be that in which the axis of the diatomic molecule coincides with the c-axis of the crystal. The work of Meyer and Scott [7] has shown that the nitrogen molecule is in fact so aligned in the cavity at low temperatures.) Alternatively, the change in potential energy of the trapped molecule as it rotates may be due at least partly to the changing degree of overlap of its electronic system with those of the atoms forming the cavity wall. Although the bond length in the oxygen molecule is greater than that in nitrogen or carbon monoxide, the critical volume of oxygen is appreciably less than that of the other two substances and it may be that from the point of view of intermolecular interaction the oxygen molecule is effectively the smallest of the three.

As already mentioned, the previous work on the argon and krypton clathrates showed that the theoretical heat capacity equations of van der Waals' treatment, which is based on classical statistics, only become valid at about 120~ It would clearly be of interest to calculate the thermodynamic functions at lower tempera- tures at which quantization is important. The results for the argon and krypton clathrates would then permit a more extensive test of the cell model, and moreover if Cvib could be reliably estimated at low temperatures then from the corresponding derived values of Crot it should be possible to obtain more precise estimates of the heights of the energy barriers restricting the rotation of diatomic and poly- atomic molecules.

The possibility of treating the ' rattling ' motion at very low temperatures as a simple harmonic oscillation has already been considered for the argon clath- rate [2]. Using the values of the intermolecular energy parameters characteristic of the quinol lattice which fit the experimental heat capacity results in the classical region, it was found that at low temperatures the calculated values exceeded the experimental figures, sometimes by as much as 1 cal deg -1 mole-1. This calcu- lation involves expanding the potential energy of the molecule in its cell as a series of even power of r, where r is the distance of the centre of the molecule from the centre of the cell. For the simple harmonic oscillator approximation only the first term in the expansion is retained. Unpublished calculations by Dr. N. G. Parsonage have shown that if this calculation is carried to the next stage of approximation by including the term in r 4 in the expanded expression for the potential energy, the corresponding values of the heat capacity are still larger than the experimental figures between 10 and 30~ but the difference between the two for the argon clathrate is reduced to not more than 0-5 cal deg-lmole -1. On the other hand, for the krypton clathrate he found little difference between the heat capacity values for the simple harmonic oscillator and those based on the next approximation. Both calculated values were about 1 caldeg-lmole -1 higher than the experimental figures between 20 and 60~ We have therefore thought it worth while to attempt to estimate Crib at lower temperatures for the krypton clathrate by the more rigorous method of evaluating the energy levels for the Lennard-Jones potential, using a computer. A brief account of the procedure adopted is given in the Appendix. The values chosen for the intermolecular energy parameters characteristic of a pair of krypton atoms were E/k = 171.0 ~ and a-- 3.60~, while for the quinol lattice 5 was taken to be 2-95 ~ and the quantity

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Thermodynamic properties of clathrates 93

Z~/(~/k) as 283. The values so obtained for Cx, , the contribution to the heat capacity per mole of krypton, are as follows (the experimental values being given in brackets) :

T

Cxr (cal deg - i mole -i)

T ( O K )

Cr,.r (cal deg - i mole -i)

5 I0 15 20

0.56 2.91 4.10 4.57

- - - - (1 .87) (2 .85)

25 45 70 100

4.72 4.86 5.20 5.10

(3.25) (4.00) (4.44) (4.82)

It will be seen that there is still considerable discrepancy between theory and experiment. There is more than one possible reason for this. First, the effect

4 0 E

~c~ 3

d o 2

0 L

" CO

--ol /%2 ~ "

I . , ~ s o " - - / ' e I

/ i i J

50 I00 150

T ~

Figure 4. Estimated values of Cro t for the oxygen and carbon monoxide clathrates, assuming (1) that Cvib is the same for the 03, N2 and CO clathrates, (2) that the hindering energy barrier for N2 molecules is 940 cal mole -1. (Full curve, oxygen ; broken curve, carbon monoxide.) The dotted curves show calculated Cro t curves for barrier heights of 150 and 300 cal mole-: (for the oxygen molecule) and 1500 cal mole -1 (for the carbon monoxide molecule).

of the trapped molecules on the low-frequency vibrations of the quinol lattice may falsify the experimental values of Cxr , particularly at low temperatures. Secondly, the Lennard-Jones potential may be an inadequate representation of intermole- cular interaction over an extended range of conditions. As already mentioned, it cannot with two fixed parameters quantitatively account for the interaction of two gas molecules over a range of temperature. Moreover, the field of force within the cavity is not really spherically symmetrical.

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94 N . R . Grey and L. A. K., Staveley

As we cannot reliably calculate Crib at low temperatures, there remains the possibility of allowing for it empirically for one clathrate by comparison with another clathrate containing similar molecules. Since the oxygen, nitrogen and carbon monoxide molecules are roughly the same size and exert intermolecular forces of comparable strength, it is probable that Crib at low temperatures does not differ much for the clathrates containing these three molecules. If we assume that for the nitrogen clathrate the barrier opposing rotation has the value of 940 cal mole -1 given by Meyer and Scott, we may calculate Crot from Pitzer's tables [11] and by subtraction from the experimental C~ values obtain Crib. If these values of Crib are Subtracted from the experimental values of Co~ it is found that for the oxygen clathrate Crot reaches its maximum at about 21~ (figure 4). This is where the maximum should fall for a barrier of height .~ 200 cal mole 2, which lies between the estimates from the magnetic susceptibility and electron spin resonance results.

APPENDIX

To evaluate the energy levels for the rattling of a molecule in a cavity we have assumed that the molecule is monatomic and the cavity spherical. The wave equation can then be written as the product of a radial component R(r*) and an angular component Y(O, 4), where the latter is governed by the well-known equation in 0 and ~b, and R(r*) obeys the equation :

1 d f r , 2dR~+B~(V(r , )_E)_ C l _ o (1) r*2 dr * \ dr*/ ~ ~-K21 - " In this equation r*= r/5 where r is the distance of the molecule from the centre of the cavity and 8 is the Lennard-Jones distance characteristic of the cavity wall. V(r*) is given by the expression:

V(r* )= A*[ (V-~o) "I(Y)- 2(V*" m(y)J,n

where l(y) = (1 + 12y + 25.2y~ + 12ya +y4)(1 _ y ) - l _ 1,

m(y)=( l+y) (1-y) 4_ 1,

y = r2/a 2, where a = the diameter of the cavity,

A* =Ze*, V* = (or*) 3 and V0= (1/~/2) a 3.

e* is the depth of the potential energy minimum for the molecule-cavity wall interaction, (r* is the distance for which the energy of this interaction is zero, and Zis the number of ' elements' of the cavity wall. Writing R(r*) = x(r*)/r* and c=l(l= 1), where l=O, 1, 2, 3, etc., equation (1) may be written as:

d2x +[B(E- •'(r*))-l(lr+l)]x=O. (2) dr*2 The required eigenvalues are obtained from equation (2) using the computer, making the usual assumption that the value of r never exceeds a/2, i.e. that R = 0 when r* = a/25. It is also a necessary condition that R = 0 when a = 0.

We are very grateful to Dr. D. E. Mayers for his advice in adapting a computer programme to the solution of equation (2). The method used was essentially that described by Hartree [12]. The Lennard-Jones parameters for the inter- action of a pair of trapped molecules were those applicable to krypton, namely e/k = 171.0 ~ and r = 3.60 ~, this choice being simply determined by the fact that the experimental values of the heat capacity contribution are known with greater

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Thermodynamic properties of clathrates 95

certainty for the krypton clathrate than for the argon clathrate. 5 was taken to be 2.95 A.

Two hundred energy levels were computed, all values of v up to 12 being considered for the two lowest values of l; the highest value of l considered was 28. Some further energy levels can be obtained by extrapolation. (Very close agreement of the calculated heat capacity with the limiting classical values could only have been obtained by evaluating considerably more levels.) We are reaso- nably confident that our calculations have been extensive enough to deal adequately with the low temperature range in which we are particularly interested, since all levels up to approximately Elk= 1050 ~ have been evaluated. If the energy of the molecule in the centre of the cell is taken as zero the partition function is given by the expression:

Q = ~ , v(21 + 1 )exp ( - E / k T). Finally the standard relation between Q and Crib leads to the equation

N ~l~,(21+l)E2exp(-E/kT) [~lv(2l+l)E2exp(-E/kT)~27

We are grateful to the British Oxygen Company for a Fellowship to one of us (N. R. G.), and we also wish to thank Imperial Chemical Industries Ltd. for financial help.

REFERENCES

[1] PARSONAGE, N. G., and STAVELEY, L. A. K., 1959, Mol. Phys., 2, 212. [2] PARSONAGE, N. G., and STAVELEY, L. A. K., 1960, 3/lol. Phys., 3, 59. [3] GREY, N. R., PARSONAGE, N. G., and STAVELEY, L. A. K., 1961, 2VIol Phys., 4, 153. [4] WAALS, J. H. VAN DER, 1956, Trans. Faraday Soc., 52, 184. [5] MEYER, H., O'BRIEN, ~V[. C. M., and VAN VLECK, J. H., 1957, Proc. roy. Soc. A,

243, 414. [6] FONER, S., MEYER, H., and KLEINER, W. H., 1961, J. Phys. Chem. Solids, 18, 273. [7] MEYER, H., and SCOTT, T. A., 1959, J. Phys. Chem. Solids, 11, 215. [8] WAALS, J. H. VAN DEll, 1961, J. Phys. Chem. Solids, 18, 82. [9] COULTER, L. V., ROPER, G. C., and STEPAKOEE, G., reported at the XVIIIth Internatio-

nal Congress of Pure and Applied Chemistry, Montreal, 1961. [10] DE WETTE, F. W., 1956, Physica, 22, 644. [11] PITZER, K. S., and GWINN, W. D., 1942, J. chem. Phys., 10, 428. [12] HARTREE, D. R., The Calculation of Atomic Structure, 1957 (New York : Wiley), p. 77.

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