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Potential energy curve for the interaction of two xenon atoms
Physica 29 227-233
Chakraborti, P. K 1963
POTENTIAL ENERGY CURVE FOR THE INTERACTION OF TWO XENON ATOMS by P. K. CHAKRABORTI I ndlan Association for the Cultivation of Science, Calcutta, India
Synopsis The potential energy curve for Xe-Xe interaction has been determined on a six parameter model by using the crystal properties, second vidal coefficient and molecular beam scattering data for xenon. The potential energy curve thus obtained should be more realistic than the usual two or three parameter models. I. Introduction. The semi-empirical potential energy functions involving
two or three constants have been quite successful in representing some bulk properties of molecules over limited temperature ranges. However, it is to be remembered that these potential functions though comparatively simple in form (consequently easier to handle mathematically) are at best approximations to the real potential energy functions even for the spherically symmetric molecules. Recently, this point has been emphasised by Guggenheim and McGlashan 1) and they have obtained a six-parameter potential energy function for argon by utilizing different properties to obtain different regions of the potential energy curve. The potential energy curve thus obtained was quite different from the usual potential energy curve on the Lennard-Jones (12: 6) model. Barua and Ch akr abo r ti s) have followed the procedure of Guggenheim and McGlashan 1) to determine the potential energy curvefor Kr-Kr interaction on the six-parameter model. The Lennard-jones (12:6) potential function for krypton as derived from the six-parameter potential could correlate the various transport and equilibrium properties of Kr reasonably well. This is in contradiction to the results obtained for Kr on the LennardJones (12: 6) model by Mas cn s) who utilized mainly the transport properties data. Subsequently, however, it has been pointed out by Fe nder 4) and also by Mason 5) that most probably the thermal diffusion data used by Masons) were not very accurate which was the main cause of the discrepancy observed by him. More recently Fender and Ha lsey ") have observed that low temperature second vidal coefficient data for Ar and Kr can be better represented with the six parameter model than with the Lennard- Jones (12: 6) model. -
227-
228
P. K. CHAKRABORTI
The encouraging results obtained by using the six-parameters potential for Ar and Kr have prompted us to determine the potential energy curve for xenon on the same model. The zero-point energy for xenon is quite small which is an advantage and the dearth of accurate X-ray data for the crystalline state of Xe has been removed by the recent measurements of Eatwel17). One disadvantage of Guggenheim and McGlashan's procedure was that they assumed the molecules to behave as rigid spheres in the positive part of the potential energy curve which is certainly not very realistic. In this paper we have utilized the potential energy function for Xe-Xe interaction as obtained by Amdur and Mas on.t) to draw the potential energy curve in the repulsive region. This should minimise the uncertainties in this region as existent in the calculations of Guggenheim and Mc Gl ash a.n-) and B arua and Ch a kr ab cr t i e). We believe that the potential energy curve for Xe-Xe interaction thus determined should be much closer to the actual potential energy curve than those given by the usual potential energy functions.
2. Determination of the potential energy curve. In the neighbourhood of the minimum the potential energy of interaction between two xenon atoms may be represented as 9) ep(r) =
-8
+ 1((1' -
1'm /rm)2 -
IX(1' -
1'm/1'm)3
+ p(1' -
1'm /rm)4
(1)
where - 8 is the energy minimum at r = 1'm, r being the distance between the two atoms tX:, p and K are constants. When r is very large 4>(1') may be expressed as (2) A. being another constant. The details for the calculation of the constants P, 8, rm, A and I( have been described in detail by Guggenheim and McGlashan 1) and Barua and Ch akr abo r ti s). Consequently we shall not go into the details again. However, in this paper we have made a significant change in the determination of the potential energy of interaction for l' < d, where d is the value of r for which ep(1') = O. Instead of assuming 4>(1') = 00 for r < d we have utilized the potential energy function in this region as determined by Amd ur and Mason from scattering experiments. Their result may be expressed as 8) IX,
~(1')
= 1.13
X 10- 8/1'7.97 erg
(for r
< 3.6 A)
(3)
The potential energy curve obtained from the calculated values of the constants has been found to join smoothly with the potential energy function given by eq. (3). This procedure fixes d uniquely for any particular
POTENTIAL ENERGY CURVE FOR THE INTERACTION OF TWO
Xe
ATOMS
229
set of constants (which was considered as a variable by Guggenheim and Mc Gl as h arnx) and Barua and Ch a.kr ab or ti s]. The crystal properties which have been used for determining the potential energy curve are the following: (a) Entropy and enthalpy of the crystal relative to those at T = 0 obtained by the graphical integration of the heat capacity measured by Clusius and Ri c co b arii t''}. These values have been given in table I. (b) Molecular volume of the crystal at different temperatures and at effectively zero pressures obtained from X-ray measurements of Eatwel17). (c) The experimental second virial coefficients data of Beattie, Berriault and Brie r le y-") and of Michels e.a.1.2) and Whally, Lupien and Schneider 18 ) have been used to select the best set of constants on the six-parameter model. TABLE 1 Calculated and the experimental values of the entropy and enthalpy of Xenon SIR (OK) TOK
0 20 40 60 80 100 120 140 160
expo
I Set I 0
0 1.490 3.201 4-.451 5.334
1.420 3.222 4.432 5.311
6.099
6.068
6.723 7.200
6.684 7.123 7.527
7.590
H(T) - H(O)
calc. from
I
Set 2 0 1.458 3.330 4.427 5.331 6.062 6.690
7.128 7.540
I
Set 3 0 1.498 3.384 4.582 5.398 6.104 6.754 7.220 7.608
R 0 19,30 70,60 130,70 194,10 260,20 330,00 405,00 485,70
-HIR (OK) (' K)
expo 1927 1908 1856 1796 1733 1667 1597 1522 1441
I 0
I
calc. from Set I
1939 1887 1829 1768 1701 1632 1569 1501
I Set 2 I Set 3 1891 1839 1783 1721 1676 1611 1533 1452
1895 1843 1786 1722 1675 1609 1547 1457
As has been discussed in detail by Guggenheim and Mc Glash an-}, at present it is perhaps best to neglect anharmonicity and set fJ = 0 in all our calculations. For AIk we have assumed the values 200o K , 25s o K and3000K (giving rise to three sets of data) of which J.../k = 255°K has been obtained in quantum mechanical calculations of Margenau-s). The constants on the six-parameter model together with the corresponding Lennard-jones (12:6) parameters are given in table II. The constants obtained by Guggenheim and Mc G'lash an 15) for Xe by applying the principle of corresponding states to the constants determined for Ar, are listed as set 4 in table II. The set c data for Lennard-Jones (12: 6) parameters are those determined by Whalley and Schne ider ts) from crystal and equilibrium properties. The potential energy curves for the different sets of constants are shown in fig. J. For the sake of comparison, the LennardJones (12: 6) potential curve as given by set c of table II is also shown in this figure.
230
P. K. CHAKRABORTI TABLE II Potential parameters for Xenon
I
Set ).,Ik oK 1 2 3
4
200 255 300 288.4
I rm A 4.404 4.418 4.435 4.438
I
Six parameter model 2 4 3 alk x 10- 1 f3l k x 10- 1 K/k X to- 1 /k oK I cl in A -x OK -rc s 37.30
0
38.58 39.24 37.7
0 0 3.77
I
L-J (12:6) model Set
IE/k OK I "m A
-
-
-
a
267.0
3.940 3.944 3.960
264.4
-
c
238.4 225.0 225.3
4.417 4.436 4.568
89.5D 87.90
293.1 277.7
85.10
86.3
b
3S0r--------------------------------,
-+-300
I 1 1
I I
+200
I
I
I 1
I I
+100
0"i!! >!
So"
0
~
\~. ~ ~
tt.
\~,
-100
'I'1
1\
1~1
1,1
,I,
+200
I" II:, ,~/
\\\
\:~~}
~·o
~'8
rIM
5'.
A-+
Fig. 1. The interaction energy ~(r) plotted against the internuclear separation of a pair of xenon atoms for each of the sets 1, 2 and 3 of parameters given in table II. The full lines of each curve represent the parts calculated from the expressions 1, 2 and 3 in their respective range of validity whereas the broken lines are the free-hand drawn parts. The 12 : 6 interaction energy curve corresponding to the parameters of set C of table II is also included in the plot for the sake of comparison.
3. Comparison with Experiment. (i) Crystal properties: The experimental and the calculated values of the entropy and enthalpy of xenon at different temperatures are given in table 1. For the entropy the agreement between the experimental and the calculated values for the sets I, 2 and 3
POTENTIAL ENERGY CUR VE FOR THE INTERACTION OF TWO
Xe
ATOMS
231
38·0,'------------------, no ~ 0
:I
U~
36'0
ii
> 35-0
0
304'0 0
20
40
60 T IN
80
100
120
0l(-+-
Fig. 2. The curve represents the temperature variation of the molecular volume of solid xenon when p -+ 0, corresponding to the potential parameters of set 3 of table II. The experimental points are those obtained from the X-ray measurements of Eatwell i'20,---------------,
+
-160
-leo
........
-200·~--~--....,..,.. - - - ' - - . . . I - - - l 273'2 47J'2 673·2 873'2 973-2 TIN"/( _ _
Fig. 3. The second virial coefficient B(T) of xenon plotted as a function of temperature. The curves I, II and III are those calculated, as described in the text, for the sets 2, 3 and 1 respectively of parameters given in table II.
o e 6
Beattie, Beniault and Brierley. Michels, Wassenaar and Lauwerse Whalley, Lupien and Schneider.
232
P. K. CHAKRABORTI
is more or less the same. But for the enthalpy which very sensitive to elk, set 3 represents the data best. Set 1 is of course definitely worse than the other two. We have used set 3 only to calculate the molar volume of xenon at different temperatures. The results are shown in fig. 2. The agreement is generally better than one half percent. This agreement is significantly better than that obtained by Guggenheim and McGlashan (1960) by using the constants of set 4 in table II. The larger discrepancy at the lower temperatures may be due to the failure of the Einstei.n approximation, (ii) Second vidal coefficient: The temperature variation of the second vidal coefficient of a gas can be fitted by a wide variety of intermolecular potential energy functions and as such this equilibrium property of a gas by itself is useless as a source of information regarding the potential form. But after obtaining a number of forms of the interaction energy which fit all the experimental properties of the crystal, it is possible that the second virial data may distinguish between these. We are now going to investigate this possibility. The second virlal coefficient B(T) of a gas in terms of the interaction patential is co
=
B(T)
2nN / {I - exp( -~(r)/kT)} r 2 dr
o
(4)
The integral is split up as co
3.G
rL
1',
co
1=1+1+/+/ o 0 3.G rJ,
(5)
1'.
Betweenr = 0 and r = 3,6 A, ep(r) is very large so that exp{-~(r)/kT}=O. d is the value of r for which ep(r) = 0 and its value is obtained from the potential energy curve. Eqn. (5) now becomes rl
B(T)
=
inN [(3.6)3
+
J{1-
exp(-rP(r)lkT)}d(r S)
+
3.G
r.
+ J{I -
exp( -~(r)/kT)} d(r S)
-
k~ rm6/r33J .
(6)
rl
The experimental and the calculated values of B(T) are shown graphically in fig. 3, The experimental values of the second virial coefficient are best fitted with the parameters of set 3 given in table II. Thus we consider this set of parameters to be the most reliable one and the potential curve corresponding to this set of parameters in a close approximation to the actual curve.
POTENTIAL ENERGY CURVE FOR THE INTERACTION OF TWO Xe ATOMS
233
Acknowledgement. The author wishes to express his thanks to Prof. B. N. Srivastava, D. Sc., F.N.I., for this valuable guidance. He is also extremely thankful to Dr. A. K. Barua for suggesting the problem and for making available some of his valuable suggestions. Received 27-8-62 REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)
Guggenheim, E. A. and McGlashan, lVI. 1., Proc, roy. Soc. A255 (1960) 456. Barua, A. K. and Chakraborti, P. K., Physica 27: (1961) 753. Mason, E. A., J. Chern. Phys., 32 (1960) 1832. Fender, B, E. F., J. chem. Phys., 35(1961)2243. Mason, E, A., J. chern. Phys. 3li (1961) 2245. Fonder, B, E. F. and Halsey Jr-, G. D., 311 (1962) 188\. Eatwell, A. J. and Smith, B. L., Phil. Mag., u (1961) 461, Amdur, 1. and Mason, E. A., J, chern. Phys, 25 (1956) 624. Rice, O. K, J. Am-chern. Soc., 63 (1941) 3, Clusius, K. and Riccobani, L., Z. phys. Chem., 038 (1937) 81. Beattie, J. A., Barriault, R. J. and Brierley, J.]., J. chemv Phys., 19 (1951) 1222. Michels, A., Wassenaar, T. and Louwerse, P., Physica 20 (1954) 99. Whalley, E., Lupien, Y. and Schneider, W. G_, Canad. J. Chern., 33 (1955) 633. Margenau, H., Rev. mod. PllYS., 11 (I939) I. Guggenheim, E. A. and McGlashan, 1\'1. L" Mol. Phys.,:1 (1960) 563. Whalley, E. and Schneider, W. G., J. chem, Phys., 23 (1955) 1644.
Chakraborti, P. K 1963
POTENTIAL ENERGY CURVE FOR THE INTERACTION OF TWO XENON ATOMS by P. K. CHAKRABORTI I ndlan Association for the Cultivation of Science, Calcutta, India
Synopsis The potential energy curve for Xe-Xe interaction has been determined on a six parameter model by using the crystal properties, second vidal coefficient and molecular beam scattering data for xenon. The potential energy curve thus obtained should be more realistic than the usual two or three parameter models. I. Introduction. The semi-empirical potential energy functions involving
two or three constants have been quite successful in representing some bulk properties of molecules over limited temperature ranges. However, it is to be remembered that these potential functions though comparatively simple in form (consequently easier to handle mathematically) are at best approximations to the real potential energy functions even for the spherically symmetric molecules. Recently, this point has been emphasised by Guggenheim and McGlashan 1) and they have obtained a six-parameter potential energy function for argon by utilizing different properties to obtain different regions of the potential energy curve. The potential energy curve thus obtained was quite different from the usual potential energy curve on the Lennard-Jones (12: 6) model. Barua and Ch akr abo r ti s) have followed the procedure of Guggenheim and McGlashan 1) to determine the potential energy curvefor Kr-Kr interaction on the six-parameter model. The Lennard-jones (12:6) potential function for krypton as derived from the six-parameter potential could correlate the various transport and equilibrium properties of Kr reasonably well. This is in contradiction to the results obtained for Kr on the LennardJones (12: 6) model by Mas cn s) who utilized mainly the transport properties data. Subsequently, however, it has been pointed out by Fe nder 4) and also by Mason 5) that most probably the thermal diffusion data used by Masons) were not very accurate which was the main cause of the discrepancy observed by him. More recently Fender and Ha lsey ") have observed that low temperature second vidal coefficient data for Ar and Kr can be better represented with the six parameter model than with the Lennard- Jones (12: 6) model. -
227-
228
P. K. CHAKRABORTI
The encouraging results obtained by using the six-parameters potential for Ar and Kr have prompted us to determine the potential energy curve for xenon on the same model. The zero-point energy for xenon is quite small which is an advantage and the dearth of accurate X-ray data for the crystalline state of Xe has been removed by the recent measurements of Eatwel17). One disadvantage of Guggenheim and McGlashan's procedure was that they assumed the molecules to behave as rigid spheres in the positive part of the potential energy curve which is certainly not very realistic. In this paper we have utilized the potential energy function for Xe-Xe interaction as obtained by Amdur and Mas on.t) to draw the potential energy curve in the repulsive region. This should minimise the uncertainties in this region as existent in the calculations of Guggenheim and Mc Gl ash a.n-) and B arua and Ch a kr ab cr t i e). We believe that the potential energy curve for Xe-Xe interaction thus determined should be much closer to the actual potential energy curve than those given by the usual potential energy functions.
2. Determination of the potential energy curve. In the neighbourhood of the minimum the potential energy of interaction between two xenon atoms may be represented as 9) ep(r) =
-8
+ 1((1' -
1'm /rm)2 -
IX(1' -
1'm/1'm)3
+ p(1' -
1'm /rm)4
(1)
where - 8 is the energy minimum at r = 1'm, r being the distance between the two atoms tX:, p and K are constants. When r is very large 4>(1') may be expressed as (2) A. being another constant. The details for the calculation of the constants P, 8, rm, A and I( have been described in detail by Guggenheim and McGlashan 1) and Barua and Ch akr abo r ti s). Consequently we shall not go into the details again. However, in this paper we have made a significant change in the determination of the potential energy of interaction for l' < d, where d is the value of r for which ep(1') = O. Instead of assuming 4>(1') = 00 for r < d we have utilized the potential energy function in this region as determined by Amd ur and Mason from scattering experiments. Their result may be expressed as 8) IX,
~(1')
= 1.13
X 10- 8/1'7.97 erg
(for r
< 3.6 A)
(3)
The potential energy curve obtained from the calculated values of the constants has been found to join smoothly with the potential energy function given by eq. (3). This procedure fixes d uniquely for any particular
POTENTIAL ENERGY CURVE FOR THE INTERACTION OF TWO
Xe
ATOMS
229
set of constants (which was considered as a variable by Guggenheim and Mc Gl as h arnx) and Barua and Ch a.kr ab or ti s]. The crystal properties which have been used for determining the potential energy curve are the following: (a) Entropy and enthalpy of the crystal relative to those at T = 0 obtained by the graphical integration of the heat capacity measured by Clusius and Ri c co b arii t''}. These values have been given in table I. (b) Molecular volume of the crystal at different temperatures and at effectively zero pressures obtained from X-ray measurements of Eatwel17). (c) The experimental second virial coefficients data of Beattie, Berriault and Brie r le y-") and of Michels e.a.1.2) and Whally, Lupien and Schneider 18 ) have been used to select the best set of constants on the six-parameter model. TABLE 1 Calculated and the experimental values of the entropy and enthalpy of Xenon SIR (OK) TOK
0 20 40 60 80 100 120 140 160
expo
I Set I 0
0 1.490 3.201 4-.451 5.334
1.420 3.222 4.432 5.311
6.099
6.068
6.723 7.200
6.684 7.123 7.527
7.590
H(T) - H(O)
calc. from
I
Set 2 0 1.458 3.330 4.427 5.331 6.062 6.690
7.128 7.540
I
Set 3 0 1.498 3.384 4.582 5.398 6.104 6.754 7.220 7.608
R 0 19,30 70,60 130,70 194,10 260,20 330,00 405,00 485,70
-HIR (OK) (' K)
expo 1927 1908 1856 1796 1733 1667 1597 1522 1441
I 0
I
calc. from Set I
1939 1887 1829 1768 1701 1632 1569 1501
I Set 2 I Set 3 1891 1839 1783 1721 1676 1611 1533 1452
1895 1843 1786 1722 1675 1609 1547 1457
As has been discussed in detail by Guggenheim and Mc Glash an-}, at present it is perhaps best to neglect anharmonicity and set fJ = 0 in all our calculations. For AIk we have assumed the values 200o K , 25s o K and3000K (giving rise to three sets of data) of which J.../k = 255°K has been obtained in quantum mechanical calculations of Margenau-s). The constants on the six-parameter model together with the corresponding Lennard-jones (12:6) parameters are given in table II. The constants obtained by Guggenheim and Mc G'lash an 15) for Xe by applying the principle of corresponding states to the constants determined for Ar, are listed as set 4 in table II. The set c data for Lennard-Jones (12: 6) parameters are those determined by Whalley and Schne ider ts) from crystal and equilibrium properties. The potential energy curves for the different sets of constants are shown in fig. J. For the sake of comparison, the LennardJones (12: 6) potential curve as given by set c of table II is also shown in this figure.
230
P. K. CHAKRABORTI TABLE II Potential parameters for Xenon
I
Set ).,Ik oK 1 2 3
4
200 255 300 288.4
I rm A 4.404 4.418 4.435 4.438
I
Six parameter model 2 4 3 alk x 10- 1 f3l k x 10- 1 K/k X to- 1 /k oK I cl in A -x OK -rc s 37.30
0
38.58 39.24 37.7
0 0 3.77
I
L-J (12:6) model Set
IE/k OK I "m A
-
-
-
a
267.0
3.940 3.944 3.960
264.4
-
c
238.4 225.0 225.3
4.417 4.436 4.568
89.5D 87.90
293.1 277.7
85.10
86.3
b
3S0r--------------------------------,
-+-300
I 1 1
I I
+200
I
I
I 1
I I
+100
0"i!! >!
So"
0
~
\~. ~ ~
tt.
\~,
-100
'I'1
1\
1~1
1,1
,I,
+200
I" II:, ,~/
\\\
\:~~}
~·o
~'8
rIM
5'.
A-+
Fig. 1. The interaction energy ~(r) plotted against the internuclear separation of a pair of xenon atoms for each of the sets 1, 2 and 3 of parameters given in table II. The full lines of each curve represent the parts calculated from the expressions 1, 2 and 3 in their respective range of validity whereas the broken lines are the free-hand drawn parts. The 12 : 6 interaction energy curve corresponding to the parameters of set C of table II is also included in the plot for the sake of comparison.
3. Comparison with Experiment. (i) Crystal properties: The experimental and the calculated values of the entropy and enthalpy of xenon at different temperatures are given in table 1. For the entropy the agreement between the experimental and the calculated values for the sets I, 2 and 3
POTENTIAL ENERGY CUR VE FOR THE INTERACTION OF TWO
Xe
ATOMS
231
38·0,'------------------, no ~ 0
:I
U~
36'0
ii
> 35-0
0
304'0 0
20
40
60 T IN
80
100
120
0l(-+-
Fig. 2. The curve represents the temperature variation of the molecular volume of solid xenon when p -+ 0, corresponding to the potential parameters of set 3 of table II. The experimental points are those obtained from the X-ray measurements of Eatwell i'20,---------------,
+
-160
-leo
........
-200·~--~--....,..,.. - - - ' - - . . . I - - - l 273'2 47J'2 673·2 873'2 973-2 TIN"/( _ _
Fig. 3. The second virial coefficient B(T) of xenon plotted as a function of temperature. The curves I, II and III are those calculated, as described in the text, for the sets 2, 3 and 1 respectively of parameters given in table II.
o e 6
Beattie, Beniault and Brierley. Michels, Wassenaar and Lauwerse Whalley, Lupien and Schneider.
232
P. K. CHAKRABORTI
is more or less the same. But for the enthalpy which very sensitive to elk, set 3 represents the data best. Set 1 is of course definitely worse than the other two. We have used set 3 only to calculate the molar volume of xenon at different temperatures. The results are shown in fig. 2. The agreement is generally better than one half percent. This agreement is significantly better than that obtained by Guggenheim and McGlashan (1960) by using the constants of set 4 in table II. The larger discrepancy at the lower temperatures may be due to the failure of the Einstei.n approximation, (ii) Second vidal coefficient: The temperature variation of the second vidal coefficient of a gas can be fitted by a wide variety of intermolecular potential energy functions and as such this equilibrium property of a gas by itself is useless as a source of information regarding the potential form. But after obtaining a number of forms of the interaction energy which fit all the experimental properties of the crystal, it is possible that the second virial data may distinguish between these. We are now going to investigate this possibility. The second virlal coefficient B(T) of a gas in terms of the interaction patential is co
=
B(T)
2nN / {I - exp( -~(r)/kT)} r 2 dr
o
(4)
The integral is split up as co
3.G
rL
1',
co
1=1+1+/+/ o 0 3.G rJ,
(5)
1'.
Betweenr = 0 and r = 3,6 A, ep(r) is very large so that exp{-~(r)/kT}=O. d is the value of r for which ep(r) = 0 and its value is obtained from the potential energy curve. Eqn. (5) now becomes rl
B(T)
=
inN [(3.6)3
+
J{1-
exp(-rP(r)lkT)}d(r S)
+
3.G
r.
+ J{I -
exp( -~(r)/kT)} d(r S)
-
k~ rm6/r33J .
(6)
rl
The experimental and the calculated values of B(T) are shown graphically in fig. 3, The experimental values of the second virial coefficient are best fitted with the parameters of set 3 given in table II. Thus we consider this set of parameters to be the most reliable one and the potential curve corresponding to this set of parameters in a close approximation to the actual curve.
POTENTIAL ENERGY CURVE FOR THE INTERACTION OF TWO Xe ATOMS
233
Acknowledgement. The author wishes to express his thanks to Prof. B. N. Srivastava, D. Sc., F.N.I., for this valuable guidance. He is also extremely thankful to Dr. A. K. Barua for suggesting the problem and for making available some of his valuable suggestions. Received 27-8-62 REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)
Guggenheim, E. A. and McGlashan, lVI. 1., Proc, roy. Soc. A255 (1960) 456. Barua, A. K. and Chakraborti, P. K., Physica 27: (1961) 753. Mason, E. A., J. Chern. Phys., 32 (1960) 1832. Fender, B, E. F., J. chem. Phys., 35(1961)2243. Mason, E, A., J. chern. Phys. 3li (1961) 2245. Fonder, B, E. F. and Halsey Jr-, G. D., 311 (1962) 188\. Eatwell, A. J. and Smith, B. L., Phil. Mag., u (1961) 461, Amdur, 1. and Mason, E. A., J, chern. Phys, 25 (1956) 624. Rice, O. K, J. Am-chern. Soc., 63 (1941) 3, Clusius, K. and Riccobani, L., Z. phys. Chem., 038 (1937) 81. Beattie, J. A., Barriault, R. J. and Brierley, J.]., J. chemv Phys., 19 (1951) 1222. Michels, A., Wassenaar, T. and Louwerse, P., Physica 20 (1954) 99. Whalley, E., Lupien, Y. and Schneider, W. G_, Canad. J. Chern., 33 (1955) 633. Margenau, H., Rev. mod. PllYS., 11 (I939) I. Guggenheim, E. A. and McGlashan, 1\'1. L" Mol. Phys.,:1 (1960) 563. Whalley, E. and Schneider, W. G., J. chem, Phys., 23 (1955) 1644.