M. E. ZANDLER,J. A. WATSON,JR.,AND H. EYRING
2730
Application of Significant Structure Theory to the Correlation of Thermodynamic Properties of CO,, COS, and CS, in Terms
of the Respective Molecular Parameters by Melvin E. Zandler, Department of Chemistry, Wichita State University, Wichita, Kansas
James A. Watson, Jr., Department of Chemistry, Southeastern Louisiana College, Hamond, Louisiana
and Henry Eyring Department of Chemistry, University of Utah, Salt Lake City, Utah (Received October 20,1967)
Significant structure theory is applied to COZ,COS, and CSZover the entire temperature ranges of their liquid states. The partition function used for these compounds allows for hindered rotation in the liquid state. The molar volume, vapor pressure, entropy, free energy, enthalpy, heat capacity, and critical properties are all calculated in excellent agreement with experimental observations.
Although molecules of COz, COS, and CSzhave similar structures, as we see in Figure 1, some of the bulk thermodynamic and mechanical properties of these compounds in the liquid state are quite dissimilar (Table I). For example, the triple-point vapor pressures of COz and CSZ are 5.112 and 0.000018 atm, respectively, compared with typical values near 0.1 atm for most other small molecules. Since significant structure theory’ has been applied to many types of liquids ( i e . , normal liquids,2 molten metal^,^ quantum liquid^,^ fused saltsj5 water,6 liquid mixtures,’ and plastic crystals*) with excellent success, it was felt that the series of compounds Cot, COS, and CSZ would provide an interesting test of the theory. In this paper, we show that the significant structure theory is capable of describing quite accurately the liquid state of these compounds. Furthermore, the theory serves as a convenient framework for discussing the bulk properties of these compounds in terms of the respective molecular parameters. The Partition Function. The significant structure theory, as applied to the liquid state, is a technique, using physical models, for writing an approximate partition function of the liquid. In this theory a liquid is considered as a disordered lattice structure in which (‘fluidizedvacancies” are randomly distributed throughout the structure. These fluidized vacancies are nonstatic lattice imperfections or vacancies assumed to be of molecular size, since this expedites their transport through the liquid. The theory is based on the recognition that the following three “structures” give the most important contributions to the partition function of a liquid: (1) The Journal of Physical Chemistry
molecules with oscillational degrees of freedom, ( 2 ) the positional degeneracy of these molecules undergoing oscillatory motions, and (3) molecules with translational degrees of freedom. A brief argument given below is used to relate the relative proportions of molecules with oscillational and translational degrees of freedom to the excess volume of the liquid compared with the reference solid. &!lore explicitly, the fraction of molecules undergoing translational motions (structure 3) is taken as (V - Vo)/V, where V and V o are the molar volumes of the liquid and the reference solid, respectively. The argument is as follows. The difference between (1) H. Eyring and T. Ree, Proc. Nat. Acad. Sci. U . S., 47, 526 (1962); H. Eyring and R. P. Marchi, J . Chem. Educ., 40, 562 (1963) ; M. E. Zandler and M. S.Jhon, Ann. Rev. Phys. Chem., 17,373 (1966) ; H. Eyring, T. S.Ree, and T. Ree, Int. J. Eng. Sci., 3, 285 (1965). (2) T. R. Thomson, H. Eyring, and T. Ree, Proc. Nut. Acad. Sci. U.S., 46, 336 (1960); J. Grosh, M. S. Jhon, T. Ree, and H. Eyring, ibid., 4, 1004 (1965); T. R. Thomson, H. Eyring, and T. Ree, J . Phys. Chem., 67, 2701 (1963); M. S.Jhon, J. Grosh, T. Ree, and H. Eyring, ibid., 70, 1591 (1966); M. S.Jhon, J. Grosh, and H. Eyring, ibid., 71, 2253 (1967). (3) C. M. Carlson, H. Eyring, and T. Ree, Proc. Nat. Acad. Sci. U.S., 46, 649 (1960). (4) D. Henderson, H. Eyring, and D. Felix, i.Phys. Chem., 66, 1128 (1962). (5) C. M. Carlson, H. Eyring, and T. Ree, Proc. Nat. Acad. Sci. U . S., 46, 333 (1960); R. Vilcu, and C. Misdolea, J . Chem. Phys., 45, 3414 (1966). (6) R. P. Marchi and H. Eyring, J . Phys. Chem., 68, 221 (1964); H. Pak and S. Chang, J . Korean Chem. SOC.,8, 68, 121 (1964); M. S. Jhon, J. Grosh, T. Ree, and H. Eyring, J . Chem. Phys., 44,1465 (1966). (7) K. Liang, H. Eyring, and R. P. Marchi, Proc. Nat. Acad. Sci. U.S., 52, 1107 (1964); B. A. Miner and H. Eyring, ibid., 53, 1227 (1965); 8.Ma and H. Eyring, J. Chem. Phys., 42, 1920 (1965). (8) M. E. Zandler and T. R. Thomson, Solid State Commun., 4, 219 (1966).
AN APPLICATION OF THE SIGNIFICANT STRUCTURE THEORY
273 i
Table I : Observed ProperOies’ of COZ,COS, and CSZ Property
Triple point,” “K Vapor pressure,’ atm Molar volume, cc/mol Entropy, cal/deg mol Viscosity, cP Volume change of fusion, cc/mol Entropy of fusion, cal/deg mol Boiling point, OK Entropy of vaporization, cal/deg mol Critical temperature;
“K Pressure, atm Volume, cc/mol
COiJ
cos0
cad
216.55 5.112
134.31 0.00065
161.22 0,000018
37.36 27.76 0.19
45.36 23.95
52.34 25.02 2.1
8.28 9.66 (184.69)’
.-.. . I .
3.34
8.42 222.87
6.51 319.38
( ~ 2 1 . 6 ) ’ 19.91 304.20 72.85 94.06
375 I4 58.6 138
b
20.03 552 78 170
J. Timmermans, “Physico-Chemical Constants of Pure Organic Compounds,” Vol. I, 11, Elsevier Publishing Co., New York, N. Y., 1950, 1965; H. H. Landolt and R. Bornstein, “Zahlenwerte und Function aus Physik, Chemie, Astronomie, Geophysik, und Technik,” Vol. 11, Part 3, Springer-Verlag, Berlin, 1956; E. W. Washburn, Ed., “International Critical Tables,” McGrawHill Book Co., Inc., New York, N. Y., 1926. ’F. Din, Ed., “Thermodynamic Functions of Gases,” Butterworth and Co. Ltd., London, 1956. J. R. Partington and H. H. Neville, J . Phys. Colloid Chem., 55, 1550 (1951); J. D. Kemp and W. F. Giaugue, J. Amer. Chem. Soc., 59,79 (1937). 0. L. I. Brown and G. G. Manow, ibid., 59, 501 (1937); G. Waddington, J. C . Smith, K. D. Williamson, and D. W. Scott, J. Phys. Chem., 66, 1074 (1962); Technical Booklet, Stauffer Chemical Co., New York, N. Y., 1964. D. R. Lovejoy, Nature, 197,353 (1963). D. R. Stull, Ind. Eng. Chem., 39, 517 (1947). Extrapolated. K. A. Kobe and R. E. Lynn, Chem. Rev., 52,117 (1953).
’
Figure 1.
sumed to behave as solidlike molecules; i.e., they possess oscillatory degrees .of freedom. Based on the‘ above arguments, the partition function of a liquid is written as a product of the partition function for each of the structures, weighted by the number of molecules in each structure (fsfdegen)
the molar volume of the liquid and the molar volume of the reference solid is attributed to molecular-sized fluidized vacancies. The number of vacancies per mole of liquid is then No(V - Vo)/Vo, where Nois Avogadro’s number. A vacancy completely surrounded by molecules is assumed to confer translational degrees of freedom on one molecule of the liquid, since one of the neighboring molecules may move into the hole. However, a vacancy surrounded by vacancies is not able to confer “gaslike” properties on any molecule. Thus the probability that a vacancy gives rise to a gaslike molecule depends upon the fraction of neighboring sites that are filled. The probability that a lattice site contains a rnolecule is Vo/V. Hence the fraction of neighboring sites that are filled is Vo/V. For simplicity a linear relation is assumed so that the probability that a vacancy is able to confer gaslike properties on some molecule is taken to be Vo/V. Thus the number of effective vacancies (gaslike molecules) per mole of liquid is Vo/V times the number of vacancies, No(V VO)/VO,per mole of liquid, which equals Na(V Vo)/V. The remaining molecules, NoV0/V, are as-
z1 =
vo’v(fg)
(N(V
N(V-
vo)/v
- Vo)/V)!
Here N is the number of molecules being considered, and the factorial term takes into account the communal sharing of the free volume ( N / N o )(V - Vo)by the gaslike molecules. For fs, f,, and fdegen are used, respectively, a n Einstein perfect-crystal partition function, a partition function for an ideal gas in the free volume ( N / N o ) ( V - VO),and the expression 1 nhe -e’kT, where nh is the number of alternate positions available to a solid like molecule at the expense of the energy E . Using Stirling’s approximation for the factorial term, the partition function for a monatomic liquid such as argon becomes
+
[( 2 ~ r n k T ) ~eV]N‘v-vo’/V //” -
h3
No
Here, w = UE,,I,~V~/(V - Vo)RT, where Esubl is the sublimation energy, e is the Einstein characterVolume 78, Number 8 August 1968
M. E. ZANDLER,J. A. WATSON, JR.,AND H. EYRING
2732 istic temperature, and n and a are parameters which may be calculated from the mode1.g In the case of polyatomic molecules, additional degrees of freedom not possessed by monatomic molecules must be taken into account. The internal vibrations present no difficulty, as they remain practically unchanged in any phase change. On the other hand, rotational degrees of freedom of polyatomic molecules in the gaseous state may change to vibrations or librations in the solid state. In the usual treatment of polyatomic molecules by the significant structure theory, the gaslike molecules are considered to rotate freely, while for the solidlike molecules a choice is made between free rotation or completely hindered rotation (oscillational degrees of freedom), depending on the shape of the molecule. Since neither extreme seems to be applicable to most molecules, we use here a partition function for rotation that allows partial hindering. The following approximate partition function for hindered rotation was suggested by McLaughlin and Eyring.lo As a model of the rotational degrees of freedom of a solidlike molecule, suppose that for all energy states less than some value B , the molecule behaves as an oscillator, while for higher energies it rotates freely. According to this model, the partition function for one-dimensional hindered rotation is approximately fhr
= foe,
+ e-B/RT
(frot
Table I1 : Observed Molecular Constants of Cot, COS, and CS2 Property
Molecular weight” (W,g h o l Moment of inertiab ( X lO-4oI), g/cm* Vibrational frequencies,b om-l Ut
wa
Bond length,’
c-0
C-S
- fosc)
Here, the term e-B/RTfoso subtracts out the oscillational states which have been replaced by rotational levels. For linear molecules, this expression becomes
The barrier height constant B is a characteristic constant for each substance which depends mainly on the shape of the molecule and the strength of the intermolecular attractions. Thus for a nearly spherical molecule, B is small and fhr E frat, while for an elongated molecule in which the intermolecular forces are large (high critical point), B is large and fhr ‘v foe,. The above hindered-rotation partition function is used for the solidlike molecules in a liquid, while a free-rotation partition function is used for the gaslike molecules. The complete partition function for linear molecules such as COZ, COS, and CS2 becomes eq. 1. Inserting values of the physical constants,11 the molecular properties (Table 11), and the parameters E s u b l , 0, Vo,n, a, and B (Table 111) into eq 1 yields 2 = Z ( N , V , 7’). Using the formula A = -kT In 2 and other standard formulas of statistical thermodyThe Journal of Physical Chemistry
namics, the thermodynamic properties of GOz, COS, and CSZwere calculated on an IBM 7044 digital computer.
d
cos
css
44.010
60.075
76.139
70.6
137.9
264
667.3 1388.3 2349.3
520.6 857.7 2063.5
396.8 658.0 1532,5
1,159
1.164 1.559
1.553
COa
..,
...
“Based on carbon-12. * H . H. Landolt and R. Bornstein, “Zahlenwerte und Function aus Physik, Chemie, Astronomie, Geophysik, und Technik,” Vol. 11, Part 3, Springer-Verlag, Berlin, 1956; J. D. Kemp and W. F. Giaugue, J . Amer. Chem. SOC, 69, 79 (1937); “JANAF Thermochemical Tables,” The Dow Chemical Co., Midland, Mich.; P. C. Cross, J . Chem. Phys., 3, 825 (1935); G. Hertsberg, “Infrared and Raman Spectra of Polyatomic Molecules,” D. Van Nostrand Co., Inc., Princeton, N. J., 1945. ’ L. Pauling, “The Nature of the Chemical Bond,” 3rd ed, Cornell University Press, Ithaca, N. Y . ,1960.
Parameters. In the case of monatomic liquids, solid data may be used for Esubl, 8,and Vo. That is, the solidlike molecules in liquid argon can be described by nearly the same parameters as molecules in the actual solid. For more complicated molecules, however, there may be changes in structure upon melting that require the use of different parameters to describe the solidlike molecules in the liquid and molecules in the actual solid. For example, a phase transition occurs in some solids below the melting point, while no transition occurs in the solid state in other similar compounds. This transition has been interpreted to be due to the onset of molecular rotation. Thus it ap(9) H. Eyring and T. Ree, New Mer. Acad. Sci. Bull., 6 , 6 (1966). (10) D. R. McLaughlin and H. Eyring, Proc. Nat. Acad. 8ci. U.S., 55, 1031 (1966). (11) National Bureau of Standards Values for Physical Constants, J . Chem. Educ., 40,642 (1963).
AN APPLICATION OF
THE
SIGNIFICANT STRUCTURE THEORY
Property
con
cos
CSn
5153.72
6019.82
8637.80
6266.1 102.18
6545.2 67.62
9019.6 73.87
105.0
71.2
67.5
30.633
43 873
51.025
29, 0gc 12.000 0.001023 450.0
,-,*
18.000 0,000329 640.0
49.0 24.000 0.000211 770.0
Esubi (liquid)?
cal/mol (at T = O°K),bcal/mol e(liquid),“ OK 8 = S/&D (at T 16”K),*OK Vo(liquid),a cc/mol V,,bI(solid a t T Tt,),cc/mol ~ ~ ( l i q u i dcc/mol ),~ liquid)," cc/mol B(liquid),” cal/mol AHsubl’
-
I
24 was used, since each sulfur atom in the molecule was assumed to serve as a lattice site. Values of B were selected by varying the parameter from 0 (free rotation) through 200,000 (essentially libration) by appropriate increments and selecting the value of B that gave the best fit of the calculated properties. Approximate fits were obtained for both B = 0 and B = m , while for intermediate values the calculated properties varied appreciably. In the case of CS2, for example, the accuracy of the calculated properties (especially Cp) was very much better for B = 770 f 10 than for either extreme, i.e., B = 0 or B = co.
n
Table 111: Parameters for GO$, COS, and CSI
a Fitted to properties by “simplified method” of ref 13. H. H. Landolt and R. Bornstein, “Zahlenwerte und Function
aus Physik, Chemie. Astronomie, Geophysik, und Technik,” Vol. 11, Part 3, Springer-Verlag, Berlin, 1956. F. Din, Ed., “Thermodynamic Functions of Gases,” Butterworth and Go., Ltd., London, 1956.
pears that, in some substances, molecules may undergo rotation (probably hindered) in the solid state (plasticcrystal state12), while in other substances molecular rotation is not allowed even in the liquid state (liquid crystal). For substances that have a large entropy of fusion, the onset of hindered molecular rotation probably occurs at the melting point. Thus the solidlike molecules in the liquid state of these substances are undergoing hindered rotation, while in the actual solid they are not. The rotating molecules exhibit reduced cohesiveness, thereby justifying the use of larger Vo and 0 values and smaller values of Eeublfor the solidlike molecules compared with the actual solid (Table 111). The entropies of fusion for COz, COS, and CSz are all quite large (Table I). Several methods13 are available for evaluating the 0, Vo,n, and a. The method used parameters bere is described in detail as the “simplified method” in ref 13. In this method, values of the parameters are selected such that values calculated from eq 1 for the vapor pressure (Ptp)and molar volume (V,,) at the triple point and the boiling point (Tap) agree with the observed values of these three properties (Table I). These three experimental properties serve to fix Eeubl,0, and Vo,the defining parameters for the solidlike molecules. The parameters n and a are selected by a technique similar to the Seoul method (Jhon,
et aL8). The parameter a is a measure of the cooperative effectla or holes in melting and has relatively less influence on other properties. The parameter n is approximately equal to the number of neighboring positions around a molecule; i.e., n, = 12 for spherical molecules. In the elongated CS2 molecule, a value of
2733
=i
Results The observed values for the following properties were obtained as follows. All experimental values of the property found in the literature were plotted vs. temperature on a large-scale graph, the best curve was constructed through the points, and values were read at convenient intervals. Where the data warranted, the accuracy of the procedure was increased by plotting the difference between the experimental value and an appropriate approximate analytical function vs. temperature and interpolating at the chosen temperatures. Molar Volume and Pressure. The Helmholtz free energy A is given by
A ( N , V , T ) = -kT In Z ( N , V , T )
(2)
If A is plotted as a function of V at constant T and N and a common tangent to the points corresponding to the liquid and vapor phases is drawn, the vapor pressure is given by the negative of the slope of the comThe abscissas mon tangent, P = -@A/dV),. at the two points of tangency represent the volumes of the liquid and vapor phases in equilibrium, i.e., at equal chemical potentials. Using eq 1 and 2, calculations of the molar volumes and vapor pressures a t various temperatures between the triple point and critical point were made. The calculated and observed values are given in Tables IV-VI and are plotted in Figures 2 and 3. Taking N = No and using the molar volumes of the liquid ( V I )and vapor (V,) obtained above at various temperatures in eq 1, the following properties were calculated. Entropy. Molar entropies of the liquid calculated from eq 1 and 2, using the relation S = -(bA/bT)v, are given in Tables IV-VI and are plotted in Figure 4. Helmholtz Free Energy, Enthalpy, and Heat of Vaporization. The molar Helmholtz free energy, A , the molar enthalpy, H,and the molar heat of vaporization, AH,,, calculated using the relations A = -AT In 2, H = A TS PV, and AH,,, = H I - H,, are
+
+
(12) J. Timmermans, J . Phys. Chem. h‘oZids, IS, 1 (1961); L. A. K. Staveley, Ann. Rev. Phys. Chem. 13,351 (1962). (13) M.E.Zandler, Ph.D. Thesis, Arizona State Univeraity, Tempe, Aria., 1965.
Volume 7.9, Number 8 August 1968
M. E. ZANDLER,J. A. WATSON,JR.,AND H. EYRINQ
2734 Table I V : Calculated and Observed5 Properties of Carbon DisuE.de 8, oal deg-1 mol -1
0. 000015c 0.000015
49.00
18.54
0 .000015c 0.000015
52.34 52 34
25.05 25.04
0. 0013QC 0.00136
54.40 53.63
29.01 28.99
0.0283 0.0284
56.71 55.18
32.21 32.22
-5123 -5134
0.228 0.227
59.34 57.06
34.96 34.98
4402 -4418
1.000 1.000
62.26 59.31
37.38 37.40
3637 -3693
2.49 2.45
...
...
...
...
...
...
39.13
-3111
- 1681
...
61.40
1.22
1.15
6189
19.23
...
... -2099 ...
*..
...
2.04
3.51
v, P , cltm
Solid Obsd Calcd
161 11
Liquid Obsd Calcd
161.11
I
TtP
TtP
Obsd Calcd
200.00
Obsd Calcd
240,OO
Obsd Calcd
280.00
Obsd Calcd
319 39
Obsd Calcd
350.00
Obsd Calcd
450.00
Obsd Calcd
552.0
Calcdd
609.1
I
Tbp
...
I
20.3 18.0
103
Table I, footnotes a, d,f,and h.
...
...
...
71 58
44.20
I
78.0 62.6
TO
4,
co/mol
T,OK
...
H,
ca1/ m o1
-7609 *..
-6560
-6553 -5859
-5844
-
- 1064 ...
JO-lA, oal/mol
1O*a, OK-1
-1059 ...
I , .
*\
.
10'5,
AHYBD,
atm-1
cal/mol
....
(8803)
.*.
..*
-1059 - 1059
0.98 0.62
0.18
-1166 -1164
1.01 0.66
0.25
- 1285
1.os 0.77
0.38
1.17 0.91
0.57
1.31 1.07
0.84
- 1287 - 1419
- 1421 - 1558 -1564
...
...
... ...
...
170 95.3
48.98
1418
-2576
5.56
21.1
173.8
52.88
3831
-2879
m
m
AHsubl.
E
Extrapolated.
13,74
..,
...
7754 7747
18 00 19.06
7406 7390
17.96 17.81
7062 7072
18 00 17.75
6750 6767
18.06 18.12
6409 6454
18.10 18.69
.
m
c1)
oal deg -1 mol -1
#
I
I
...
.
5075
21 83
0 3153
31.13
0
I
m
m
Predicted critical point.
coefficient of thermal expansion, from eq 1 and 2 using the relations
C,
CY,
were calculated
T ( b S / b T ) v = T(b2A/bT2)v
v
dV
Values of the heat capacity a t constant pressure, C,, were calculated from eq 1 and 2 using the above relations and the expression
C, = C,
7
6
I
4
3
2
I
IOOO/T(OK)
Figure 2. Logarithm of vapor pressure of liquid COa, COS, a n d CSa.
given in Tables IV-VI. The standard state for the enthalpy and free energy is the ideal gas a t 0°K. Heat Capacity, Compressibility, and Coeficient of Thermal Expansion. The heat capacity a t constant volume, C,, the isothermd compressibility, 0,and the
+ (TVa2/8)
Calculated and observed values of C,, a, and 0 are given in Table IV-VI, and C, is plotted in Figure 5. Since these quantities involve second derivatives of the logarithm of the partition function, the agreement with observed values must be considered excellent. Critical Constants. The critical temperature was found by setting
Since this calculation requires second and third derivatives of the logarithm of the partition function, pre-
AN APPLICATION OF
THE
SIGNIFICANT STRUCTURE THEORY
2735
Table V : Calculated and Observeda Properties of Carbon Oxysulfide
v, T,OK
Solid Obsd Calcd
134.31
Liquid Obsd Calcd
134.31
TtP
TtP
P, atm
cc/mol
... .
0.00064c 0.00064
..*
8, oal deg-1 mol-1
H, cal/mol
A. cal/mol
5.53
-5416
-7,502
...
... ...
... ...
-7,504 -7,535
0.86 1.07
0.32
-8,155 -8,188
1.11 1.13
0.44
.-..
CP I
...
m01-1
(6361)'
12.04
5233 5285
17.59 18.51
4983 5028
17.11 17.14
4800 4846
16.96 16'.92
4627 4669
16.95 17.01
4438 4461
17.05 17.38
...
...
45 36c 45.36
23 95 23.79
0.0142 0.0141
46.51' 46.63
26.96 26.89
-4287 -4340 -3842 -3887
0.0789 0.0794
47.64 47.76
28.96 28.89
-3500 -3547
-8,711 -8,747
1.32 1.27
0.60
0.300 0 302
49.03 49.06
30.76 30.67
-3159 -3208
-9,312 -9,343
1.54 1.43
0.81
1.ooo 1.000
50.73 50.82
32 60 32.53
-2771 -2814
- 10,006
-10,038
1.84 1.65
1.13
62.14 60.05
3.48 2.96
...
...
...
37.98
4.03
3587
20.12
0 1997
33.15
I
I
160.00
Obsd Calcd
180.00
Obsd Calcd
200.00
Obsd Calcd
222.87
Obsd Calcd
300.00
Obsd Calcd
378.00 TC
58.7 54.1
138 84.6
43.12
435
Calcdd
409.1
82.1
149.1
46.46
1743
a
os1 deg-1
0. 00064c 0.0063
Obsd Calcd
I
Tbp
AHvap,
cal/mol
12.8 12.8
Table I, footnotes a, c,f, and h.
' AHsubl.
I
...
...
- 1378
...
Extrapolated.
#
,
.
... -12,790
*..
m
-15,970
10.1
- 17,460
m
. I .
...
*.. ... . , I
m
35 m
m
m
Predicted critical point.
Table VI : Calculated and Observeda Properties of Carbon Dioxide 8,
v, T,OK
P,a t m
cc/mol
cal deg-1 mol-1
Solid Obsd Calcd
216.56
5.112 5.113
29.09
18.10
Liquid Obsd Calcd
216.56
5.112 5.113
37.37
TtP
TtP
UP 8
H,
1 0 4 ~ ~
cal/mol
a t m -1
AHv~vp,
cal/mol
oal deg-1
mol -1
-4341
...
-8,284
...
.*.
(5755y
14.4
*..
37.37
27.76 27.46
-2269 -2293
-8,285 -8,245
3.05 2.90
1.3 1.7
3663 3783
18.82 18.81
...
...
...
...
Obsd Calcd
228.16
8.224 8.221
38.79 38.72
28.71 28.46
-2043 2069
-
-8,601 -8,569
3.44 3.33
1.7 2.2
3463 3619
19.40 19.74
Obsd Calcd
248.16
16.62 16.62
41.81 41.68
30,39 30.17
- 1634 - 1653
-9,192 -9,156
4.34 4.37
3.6 3.7
3092 3280
20.59 21.92
Obsd Calcd
268.16
30.07 29.99
46.01 45.92
32.07 31 -91
-1187 -1187
-9,820 -9,779
6.11 6.21
7.6 6.9
2612 2846
22.77 25.48
Obsd Calcd
288.16
50.19 49.59
53.39 52.71
33.92 33.78
-648 -644
-10,487 -10,442
14.6 10.4
1867 2258
30.00 33 04
Obsd Calcd
304.20 TO
72.85 70.62
94.06 62.90
37.33 35.54
-95
-29
-11,551 -11,013
22.5
0 1560
53.30
Calcd'
317.8
92.50
102.11
38.44
769
-11 ,573
m
a
See Table I, footnotes a, b, e,f, and h,
m
30 16 W
52 m
0
I
m
m
* AHaubl. ' Predicted critical point.
diction of the critical point constitutes a fairly severe test of the theory. The results are given in Tables IV-VI. Predicted values of T , are about 4 4 % high.
Similar results have been obtained with most other simple molecules. It has been found1%that changing the form of the degeneracy term to Volume '78,Number 8 August 1968
M. E. ZANDLER,J. A. WATSON,JR.,AND H. EYRING
2736 1 fdepn
+ vo
E
+ ,[(V 1 - vo)/vo12)e-"
-
I
CALCULATED
0
considerably improved the accuracy of the calculated critical point of argon. Preliminary testing of this and other altered forms of fdesen for other liquids is encouraging.
A OBSERVED
Discussion The agreement between the calculated and observed properties of Cot, COS, and CS2 is quite satisfactory. Thus it may be concluded that the properties of Cog, COS, and CS2 may be adequately described within the framework of the significant strbcture theory. According to the bond distances listed in Table I and the van der Waals radii of sulfur (1.85 A) and
C, (00
CALCULATED
A
160
200
e50
Temperature
a00
S60
(OK1
Figure 3. Molar volumes of liquid COZ,COS, and CSZ.
-
too
-
A
CALCULATED
0 0
A
OBSERVED
160
250
300
550
Figure 5 . Molar heat capacities (C,) of liquid Cog, COS, and CSS.
OBSERVED (SMOOTHED)
08
100
200
HEAT CAPACITY
Temperature VK)
0 0
180
,
200
800
e50
Temperature
a60
(OK)
Figure 4. Molar entropies of liquid COZ,COS, and CSZ. The Journal of Physical Chemistry
oxygen (1.40 A), the CS2 and C02 molecules have very similar shapes (Figure 1). The easiest way for a molecule to rotate in the liquid phase is by excluding competitors from the cavity required for its rotation. To form a hole the size of a molecule requires the energy of sublimation, E s u b l . A barrier preventing rotation should thus depend on the shape of the molecule and should be some fraction, f, of E s u b l . Thus for molecules of similar shape f = B/E, should be about constant. Actually the values of f are 0.087, 0.106, and 0.089 for COZ, COS, and CSz, respectively. If one assumes that the COS barrier should be the arithmetic mean of the other two, as the intermediate structure suggests, one would get 610 for B as compared with 640 chosen to give the best fit of the specific heat. Additional cases must be examined before we can be sure that B can always be predicted as closely as this from the molecular structure and the value of E s u b l . The polar nature of COS probably contributes to f being slightly higher than for C02 and CSZ. Carbon dioxide has an abnormally high vapor pressure at the triple point. This is presumably due to the large polarity of the C-0 bond together with its regular shape which causes solid C02to pack well. The bond polarity gives rise to relatively large short-range intermolecular forces in the solid state where the molecules are not rotating (ie., the molecules are interlocked). This is shown by comparing ratios of E s u b l (solidlike molecules in liquid) to A H s u b l o (actual solid). For c02, COS, and cS2, A H s u b l o / E s u b l = 1.21, 1-09, and 1.05, respectively. For the rare gases and other substances in which no structure change occurs upon melting, the ratio A H s u b l o / E s u b l is almost unity. Thus solid cot is stable at relatively high temperatures. When the temperature is raised to the triple point, the solid
TEMPERATURE DEPENDENCE OF
THE
SOLVENT STARKEFFECT
melts, undergoing a very large volume and entropy change. In the resulting liquid, the molecules are undergoing hindered rotation and behave very much like CS2 molecules when compared a t similar vapor pressures. The observed (V - V,)/V, and calculated (V - VO)/Vofractional volume expansion at the triple point are: CS2: 0.068 (obsd), 0.034 (calcd); C02: 0.284 (obsd), 0.220 (calcd). Thus liquid C02 has an
2737
abnormally large number of vacancies at the triple point (22% compared with 6-10% for normal liquids). This is also borne out by the abnormally small viscosity of C02 a t the triple point (0.19 CPas compared to the usual 1-2 cP). Acknowledgment. The authors express their thanks to the Army Ordnance (Contract DA-31-124-ARO-D408) for supporting this work.
Temperature Dependence of the Solvent Stark Effect by G. A. Gerhold and E. Miller Department of Chemistry, University of California, Davia, California 96616
(Received November I S , 1967)
The temperature dependence of the solvent Stark effect in electronic absorption spectra which was predicted in the theory of Baur and Nicol is verified experimentally for the 22,200-cm-1 absorption band of naphthacene in 25 solvents. These solvents naturally divide into two groups, double bond containing and others, which are distinguished by their dispersion interactions with the solute. The estimate of the excited-state polarizability for naphthacene is somewhat lower than previously reported values.
Introduction The position of the electronic absorption spectrum of any molecule is affected by its environment. The interactions which cause this dependency in a solution can be conveniently classified as specific interactions, e.g., hydrogen bonds, or nonspecific interactions. Calculation of the latter type of interactions can be carried out in the point-dipole approximation by a straightforward application of the second-order perturbation theory; questions of accurate wave functions, integrals, and distribution functions can then be circumvented by the introduction of reaction-field expressions. This approach has been used by several The use of the dipole approximation leads to an expression which contains four terms, which correspond, respectively, to the permanent and induced moments of the solute interacting with the permanent and induced moments of the solvent. Only one of these terms, the solvent Stark effect term which is the induced solute moment, permanent solvent moment term, presents real difficulties, and most workers have assumed that it could be neglected. 1-3 Recently Baur and Nicol have obtained the following expression for the solvent shift of a nonpolar solute molecule (nonpolar in both the ground and excited states) in any nonspecifically interacting solvent4
The solvent Stark effect is given by the last term. In eq 1, v is the observed frequency, YO is ideally the single-molecule frequency, A is a complex function of the solvent energy levels, n is the index of refraction of the solvent, and E is the dielectric constant of the solvent. B is a characteristic of the solvent which will be discussed later. Baur and Nicol have successfully applied this expression to the data of Weigang and Wild on the absorption spectra of naphthacene in a series of solvents.6 However, all of these measurements were taken at a single temperature; this work is an experimental investigation of the predicted temperature dependence of the solvent Stark effect.
Experimental Section The solvent shift predicted by eq 1 will only apply to the case of a solute with no permanent moment in both its ground and excited states. Furthermore, the coefficient of the induced dipole-induced dipole term ( A )can usually be assumed to be a constant if the solute absorption is remote from the solvent absorption. The 22,200-cm-1 band of naphthacene is an absorption which satisfies these conditions; this same solute was (1) E.G. McRae, J.Phys. Chem., 61,662 (1957). (2) W. Liptay, Z.Naturforsch., 20a, 1441 (1966). (3) N.G.Bakhshiev, Opt. Spectrosc., 10,379 (1961). (4) M.E.Baur and M. Nicol, J. Chem. Phys., 44,3337 (1966). (6) 0.E.Weigangand D. D. Wild, {bid., 37,1180 (1962).
Volume 78, Number 8 August 1968
