SIGNIFICANT STRUCTURE THEORY APPLIEDTO
THE
HYDRIDES OF GROUP V ELEMENTS
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The Significant Structure Theory Applied to the Hydrides of Elements of the Fifth Group
by Mu Shik Jhon, Joe Grosh, Taikyue Ree, and Henry Eyring Department of Chemistry, University of Utah, Salt Lake City, Utah (Received December 1 , 1966)
The significant structure theory has been applied to the liquid state of PH3, AsH3, and SbH3. The molar volume, vapor pressure, specific heat, entropy, and critical properties for the liquids are all calculated. Surface tensions of the liquids are calculated by a modification of earlier techniques. Good agreement with experimental observations is obtained.
Introduction The significant structure theory of liquids has been developed182 and successfully applied to a number of have applied significant liquids. Lately, Lee, et structure theory to liquid NH3 and calculated all thermodynamic properties and the surface tension. Good agreement with observed values was found. This paper is one of a series of studies applying this theory to various common and uncommon liquids2 and is the second of a sequence dealing with hydrides belonging to one family.
The Partition Function According to the significant structure theory of liquid, the partition function of a liquid is
Here, fa and f, are the partition functions of “solidlike” and “gas-like” structure; respectively, V is the molar volume of the liquid, V. is the molar volume of the solid at the melting temperature, and N is Avogadro’s number. Except for liquid ammonia, each of the hydrides of the nitrogen family has a normal entropy of fusion and a solid state transition.’ This indicates that these compounds probably rotate in the solid state at the melting point. Therefore, the usual rotation term is included in both the gas and solid terms. For simplicity, the system was treated as an Einstein solid. Accordingly, the partition function takes the following form.
{
(2rmk T )*/’eV 8a2(8x3ABC)‘/‘(lc T )‘I2 X ha N 3h3
Here, E, and 8 are the energy of sublimation and the Einstein characteristic temperature, respectively. a is a dimensionless constant which, for simple liquids is calculable from the model. A , B, and C are the three principal moment of inertia, vi is the internal molecular vibration frequency, m is the molecular mass, X = V/V,, N is Avogadro’s number, and n is the number of nearest neighbors around a molecule of the liquid at the melting point. ~
~~
(1) (a) H. Eyring and T. Ree, Proc. Natl. A d . Sci. U.S.,47, 526 (1961); (b) H. Eyring and R. P. Marchi, J . Chem. Educ., 40, 562 (1963). (2) D. Henderson, H. Eyring, and D. Felix, J . Phys. Chem., 66, 1128 (1962); T. R. Thomson, H. Eyring, and T. Ree, ibid., 67, 2701 (1963); T. S. Ree, T. Ree, and H. Eyring, ibid., 68, 1163, 3262 (1964); J . Chem. Phys., 41,524 (1964); R. P. Marchi and H. Eyring, J . Phys. Chem., 68, 221 (1964); 9. H. Lin, H. Eyring, and J. Davis, Jr., ibid., 68, 3617 (1964); Y. L. Wang, T. Ree, T. S. Ree, and H. Eyring, J . Chem. Phys., 42, 1926 (1965); 5. M. Ma and H. Eyring, ibid., 42, 1920 (1965). (See also the series of papers appearing in Proc. Natl. Acad. Sci. U.S. from 1958 to the present.) (3) H. Lee, M. 8. Jhon, and 5. Chang, J. Korean C h m . SOC.,8, 85 (1964). (4) Landolt-Bornstein, “Zahlen Werte und Funktion aus Physik, Chemie, Astronomie, Geophisik und Technik,” Vol. 11, Part 3, Springer-Verlag, Berlin, 1956.
Volume 70, Number 6 M a y 1066
M. JHON, J. GROSH,T. RHEE,AND H. EYRING
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The parameters n, a, 6, E,, and V , are calculated in a manner described by Chang, et aZ.,6,6and the values are given in Table I. The calculations were made using an IBM 7040 computer.
Results ( a ) Volume and Pressure. The Helmholtz free energy A is given by
A
= -kTInf
(2)
’TTP
Table I : Parametric Values
L PHa
n a
11.45 62.67 41.44 0.1257 X
E., cd/mole
3964
e, O K
v.,cc
10-8
ASHI
11.514 39.63 42.13 0.9950 X 10-8
4528
SbHi
11.58 33.50 49.75 0.7491 X lo-’ 5565
T z TTP I
I
4
9
I
Volume
“g
Figure 1. The schematic representation of Helmholtz free energy us. volume Tc and TTPare the temperatures at the critical point and triple point, respectively.
A is plotted against V for a constant T, and a common tangent to the points corresponding to the liquid and vapor phases is drawn. The vapor pressure is given by the slope of the common tangent, and the abscissas of the two points of the common tangent indicate the volume of the liquid and vapor (Figure 1). By using eq 1, calculations of the molar volumes and vapor pressures at various temperatures were made. The calculated results are plotted in Figures 2 and 3 and show close agrecment with experimental value^.^-^^ (b) Entropy. Entropies of the liquid St are calculated from eq l and 2 using the relation
In Figure 4, the calculated entropies of liquids are compared with the observed values‘ of PHs and AsH8. There is excellent agreement between theory and experiment. (c) The Critical Points Properties. The critical point was found by setting the first and second derivatives of pressure with respect to volume at constant temperature equal to zero, and finding the values of P, V , and T satisfying these two conditions by a trial and error technique using an IBM 7040 computer. Since this calculation requires the second and third derivatives of the partition function, the critical point constitutes a fairly severe test of the theory. The calculated results are listed in Table 11. The observed data’711 available at present show satisfactory agreement with the calculated values. The Journal of Physical Chemdstry
Figure 2. Vapor pressures of the liquids of PHs, AsHs, and SbHa us. temperature. The observed values for PHs (A), ASH, (0),and SbHs ( 0 )are from ref 7, 8, and 9, respectively. The solid lines represent calculated values.
( d ) Heat Capacity. Differentiation of the expression for entropy with respect to temperature at con(5) 8. Chang, H. Park, W. K. Paik, 5. H. Park, M. 8. &on, and W. 6. A h , J. Korean Chem. SOC.,8, 33 (1964); M. S. Jhon and 9. Chang, ibid., 8 , 8 5 (1964); W.Paik and S.Chang, ibid., 8,29 (1964); W.5. Ahn and 8.Chang, ibid., 8,125 (1964). (6) M. S. Jhon, J. Grosh, T. Ree, and H. Eyring, J. Chem. Phys., 44, 1466 (1966). (7) “Lange’s Handbook of Chemistry,” 10th ed, McGraw-Hill Book Co., Ino., New York, N. Y.,1961. (8)R. H. Sherman and W. F. Giauque, J . Am. Chem. SOC.,77,2154 (1955). (9) A. A. Durrant and T. G. Pearson, J. Chem. SOC.,731 (1934). (10) D. MacIntosh and B. D. Steele, Z. Physik. Chem., 55, 129 (1806). (11) 8. Skinner, Proc. Roy. SOC.(London), 42, 283 (1887).
SIGNIFICANT STRUCTURE THEORY APPLIEDTO
THE
HYDRIDES OF GROUPV ELEMENTS
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I
I
55
I
I
1593
I
I
40
4 /
c
s
c
> 50
20
0
0 0
45 01
I
I
I
I50
20 0
250
T (OK ) I
I
200
150
Figure 6. The surface tensions of the liquids of PHs, AsH3 and SbH3. The observed values (A, 0, 0 ) are from ref 9; the solid lines represent calculated values.
I
250
T('K 1 Figure 3. Molar volumes of the liquids of PHa, AsHa, and SbHa us. temperature: The observed values ( A ) are from ref 10, while the 0 and 0 are from ref 8 and 9. The solid lines represent calculated values.
~~~~
~~~~
~~
Table I1 : Calculated and Observed Critical Constants" To
vc
Po > atm
I
OK
I
OD
PHa Calcd Obsd A,
%
350.72 327 15 7.20
75.82 64.50 17.55
137.17 155.95 12.04
AsHa Calcd
420.7
Calcd
505.15
139.3
89.39
SbHa
91.05
164.45
a The observed critical data for the liquids of AsH3 and SbHa are not available.
I
I
I
I
I50
200
250
T (OK)
I
Figure 4. Entropies of the liquids of PHa, AsHs, and SbHa us. temperature. The experimental data ( A , 0) are from ref 4; the solid lines represent calculated values.
150
I
I
200
250
T(OK)
Figure 5. Heat capacities of the liquids of PHs, &Ha, and SbHa us. temperature. The experimental data ( A , 0) are from ref 4; the solid lines represent calculated values.
I
stant volume, multiplied by T, yields Cv. This, in turn, can be converted to C p by the use of the compressibility P arid the coefficient of expansion CY, both of which can be calculated from the partition function as shown in the equation.
Figure 5 shows the results obtained. The calculated results give good agreement with observed value^.^ Thus, our model gives good results even for the heat capacity which involves the second derivative of the partition function. ( e ) Surface Tension. The method12 developed here for the calculation of surface tension is an improvement of the method used by Chang, et aZ.13 The calculated (12) J. Grosh, M. S. Jhon, T. Ree, and H. Eyring, to be published. (13) S. Chang, T. Ree, H. Eyring, and I. Matzer in "Progress in
International Research on Thermodynamics and Transport Phenomena," Academic Press, New York, N. Y., 1962, p 88.
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results are shown in Figure 6. Again, the calculated results are in good agreement with the observed valuesg
Discussion We next discuss the parameters found in these calculations. We assumed n = 12V,/V, for a close packed structure, V , and V , being the solid volume and the liquid volume at the melting point, respectively. The values of e are expected to be a little less than those in the solid. Even if the force constants k , were about equal for the hydrides, the e values would decrease from PH3 down to SbH3, since e = (h/27rk)(kr/m)’”. This tendency is observed in our values for 8. The values for E, are expected to increase from PH3 to SbH3 as is observed. It is interesting to see that the parametric E , values are nearly equal to the sum (in calorie~/mole)~of AH (fusion) and AH (vaporization) : 4359 to 3767 cal for PH3and 5098 to 4293 cal for AsH3; the datum for AHr is not available for SbH3. The E, value3 for liquid ammonia is exceptionally
The Journal of Physical Chemistry
M. JHON, J. GROSH,T. RHEE,AND H. EYRING
high. This is due to the hydrogen-bond formation in liquid ammonia. The values found for a are close to those which would be obtained from the formula14 n
a=--
- 11 2
( V , - V,)z 2 V,V,
where Z is taken to be 12. The usual calculation3of surface tension of liquid ammonia deviates from experiment. This was improved by considering the molecular orientation of surface molecules. However, for the nonpolar molecules such as PHI, AsH3, and SbH3, the simple model for the surface tension give excellent agreement with the observed values (Figure 6). Acknowledgments. The authors express their thanks to the National Science Foundation under Grant GP-3698 and the Atomic Energy Commission under Contract No. AT(11-1)-1144 for supporting this work. (14) T. S. Ree, T. Ree, H. Eyring, and R. Perkins, J. Phy8. Chem. 69, 3322 (1965).
