THE THEORY O F T H E LIQUID ST,4TE1 HENRY EYRIKG
AND
JOSEPH HIRSCHFELDER
Frick Chemistry Laboratory, Princeton I;nioersity, Princeton, h’ew Jersey Received October lY, 1936
During the past fen- years there has been a great deal of progress made in the development of a theory of liquids. A general survey of its present status was presented at the symposium2 of the Faraday Society held in September, 1936, and the subject has been discussed by Hildebrand (7), Frenkel ( l l ) , Herzfeld aiid Goeppert->layer (12), and others. In this paper we shall treat the liquid as though it were compoFed of individual molecules, each moving in an average potential field due t o its neighbors. The partition function for the liquid can then be written:
This equation may he considered as the dehitioii of the free volume, 1;. Here AE is the energy of T-aporization; R , and TibL are the rotational and vibrational parts of the partition function of a molecule in the liquid; and m, k , h, and T are the mass of the molecule, the Boltzmaiin constant, the Planck constant, and the absolute temperature, respectively. The partition function for a molecule in the gas is:
F,
=
1‘,
(2~n2kT)i R, Vib, h3
(2)
where T‘, is the total volume of the gas divided by the number of molecules, R , and T-ib, are the rotational and vibrational parts of the partition functions of a molecule in the gas. K e shall consider first the inolecules hich are sufficiently symmetric that the partition functions for rotation and interiial vibrations are the same for the liquid as for the gas. A4tequilibrium between gas and liquid,
kT In FI - = p(V, F,
1’1)
Presented a t the Symposium on Molecular Structure, held a t Princeton L-niversity, Princeton, New Jersey, December 31, 1936 t o January 2, 1937, under the auspices of the Division of Physical and Inorganic Chemistry of the American Chemical Society. The papers presented a t this symposium are published in the Transactions of the Faraday Society, 1936. 249
250
H E S R T ETRING A S D JOSEPH HIRSCHFELDER
so that:
We now can see what is meant by VI. If a fluid which obeys the perfect gas law : p V = RT (4) expands isothermally from V j to V g ,the work done is
);(
A H = RT log
This is equation 3 and therefore T.'! is the effective volume in which a particular molecule in the liquid can move and obey the perfect gas law,
1 Substituting -
V,
= ' 2 ' into
RT
equation 3, we obtain the vapor pressure
equation:
RT
pvap= - exp f'J
AH (m)
The external pressure, p,,, acting on the liquid may be calculated from the well-known equation
Here V Lis the total volumr of the liquid divided by the number of molecules in the liquid. To go further, we require the relatioiiship between the V1 and Vf. For simplicity we shall consider cubical packing, in which one molecule oscillates about the origin and the six nearest neighbors are imagined fixed a t their mean positions along the three axes. Each molecule is thus at a distance Vi from the origin. Now if d is the incompressible diameter of each of these molecules, then 2V5 - 2d is the distance that the central molecule is free to move along each axis. The free volume is then just this quantity cubed, i.e. l', = 8 ( V ) - d ) 3 (8) (see figure 1). For other typcs of packing and for a inore rigorous derivation, we still expect the relationship
v,
=
b3(V;5 -
4 3
(9)
THEORY O F T H E LIQUID STATE
251
in which b may vary somewhat n-ith temperature and differ from 2. A critical discussion of equation 9 is given later in this paper. Using equa-
Since the kinetic energy ternis in the liquid and gas are equal (see equations l and a), it follows that AH is equal to AI?, the difference in potential energy of a molecule in the gas and in the liquid state, plus R T , the work done against the atmosphere. Kow combining equations 7 and 10 we obtain an equation of state for the liquid: [pex -
g]
Vf V j = bRT
Eliminating V, from equation 11 by means of equation 9:
FIG.1. The relationship between the free and total volume T’,
= (2vf - 2 4 3
I t is interesting to note that equation 12 is independent of the value of the “packing factor,” b. We can solve equation 11 for the free volume: (13)
This value of V, niay be substituted into equation 6 to obtain the equation for vapor pressure: P \ = ~ b-3 ~ (&)2[p.x
-
%I3
exp
(-g)
For many liquids it has been shomn that, to a good approximation, AE is a function only of the volume of the liquid. This is bh0m-n by the fact that for normal liquid.. the specific heats at constant volume of liquid and gas are nearly equal (10). We shall take a AE= -
v;
252
H E K R T EYRING A S D JOSEPH HIRSCHFELDER
where a and n are constants.
From the work of Hildebrand and his co-
workers (8) it is found that n is usually close to unity. Since *E
a vi
is of
the order of thousands of atmospheres, p,, may be neglected when it is of the order of 1 atmosphere. Using equation 15:
Thus :
(i:)i (-g-
= n3RTb-3V'f1 - exp
Pvap
1)
Taking n = 1 (corresponding to AE = aVi') and b = 2 (cubical packing), equation 17 becomes : Pvap
[-
RT AH RT
= 8V1 -
-
exp
1]3
(-g)
This equation for the vapor pressure gives the Hildebrand rule when it is realized that the specific volume, V l , is about the same for most normal liquids. Using V I = 82 cc. we obtain: P w p
=
[
T AH - -8 RT
11 (-E) exp
at,mospheres
(18)
Thus if liquids 1 and 2 have the same concentrations in the vapor
['
=
v-1
*
-
Pvap] RT
a t the respective temperatures TI and T,, then it follows that AH1 - AH2
TI
Tz
Trouton's rule can be obtained if we realize that most of the liquids usually considered boil in the neighborhood of 300 to 400" Absolute. Taking T = 300°K :
At the boiling temperature, Tb, i.e., when pvnp= 1 atm., then
which is the usual Troutoii rule value.
In this derivation of
AH
__ =
R Tb AH
-=
R Tb
10.4, 10.4
253
THEORY O F THE LIQCID STATE
there
a certain arbitrariness in the 1-alues chown for Vl and T , but for
ic
AH
most normal liquids the change in these quantities will not change
I by R rb more than one unit. For the molten salts where Tb is around 1400°H. AH and 1'1 is as small as 40 cc., - may become as large as 13. For gases RT,'
boiling a t lon- temperatures equation 18 yields a low value for ,
-,AH in
RTb
accord x i t h experiment. The thermal expansion can be investigated using the equation of state (equation 12). Taking:
and differentiating both sides of equation 12 with respect to T oce obtains:
[-n(n
+ l)aV,'"+2'(V, - dV!) a v1= R + [ p e X+ ~ n V ~ ( ~ + ' -) ] $dV;')] (l aT -
(20)
Solving this equation for
a=
Vl[p,,
+
+
3 R h X nVY'AE1 nV,'AE]2 2RTp,, - (3n
+
+ 1)nRTVylAE
(21)
The external pressure can be neglected when it is small compared with nV,'AE and one obtains:
(3 + (g)
a = -T 3
1 - (3n
(22)
1)
Using the Trouton approximation: LY
= 3[9.4 Tb - 4T]-'
(23)
Similarly the equation for the compressibility of the liquid may be obtained by differentiating both sides of equation 12 with respect to p,, . Solving for
P = we find:
vT1(s)T
254
HENRY ETRING AND JOSEPH HIRSCHFELDER
When it is justifiable to neglect the external pretsure,
This is a special form of the general thermodynamical relationship : c?,
a2VT
- cu = P
T o the approximation of Trouton’s rule:
Therefore a t the boiling point Trouton’s rule leads t o : (a)Tb
= 0.55 T i 1
(p)Tb
=
-7.1
x
10“vtTbl
These formulas usually give the coefficients of thermal expansion and of compressibility to within a factor of two. I n table 1, a and P are calculated for a number of liquids, all at 25°C. The experimental values of
and of
are taken from the u-ork of Hildebrarid and his coworkers (9). The values of the observed pvaZi,a and P are obtained from the Iizternational Critical Tables and from the Landolt-Bornstein Tabellen. n2 is the experimental value for 72 which i. obtained by the use of equation 2.5, i.e.
I n most cases, it nil1 be noticed that nl agrees quite well n-ith n 2 and is usually close to unity. a and P are calculated from equations 22 and 24, respectively. The values of b which are given in this table are calculated A E , n, and T’l from equation 17, in which the experimental values of pap, are used. For most of the ‘Lnormal”liquids considered, b has a value close to 2 . The free volumes are calculated from equation 13, and it is t o be noticed that they are of the order of 0.1 cc. The liquids considered here are of a complicated structure, and therefore it is surprising that we should
255
THEORY O F THE LIQUID STATE
obtain any agreement between the calculated and the observed properties. I n methyl alcohol and in mercury there must be important deviations from the properties of normal liquids, due to causes which we have not considered in this treatment. Xewton and Eyring (10) proceeding from a different point of view, have given an equation for the vapor pressure. The free volume which they have introduced differs from that used here in two particulars. First, TABLE 1 Vl
LIQLID
-_
147 IC4 97 80 89 60 73 40 14
n-Heptane Ethyl ether
cc1,
CHCls Ben7ene
CS? Ace toile CHSOH IIeicur! ~
nz
ni
_ _ _ _ _ _ _ _ I_ _ _ _ _ _
oc
cc
_ ~-__
k~;i!C (y{>T
tb
6 98 5 34 1 76 7 61 3 I 80 7 46 3 56 4 64 8 396 ~
1
1 ;seor/
0 0 0 0 0 0 0 0 0
4 6
5
1 I
5 7 9 1
a i i n 1 er I-l--
cin
2
062 2510 707 2370 3311 151 262 3660 3642 125 481 3670 302 3331 163 2940 00184 14650
1
'
1 12 1 03 1 10 1 11 1 09 0 89 0 89 0 34 0 33
I '
1 60 1 16 1 15
I 1 1 0 0
09 16 29
18 35
~ _ a
vJ
LIQLID
I
~
0 174 I 0 714 0 386 0 667 0365 0 696 0 404 0 028 0 00019
,
Benzene CS> Acetone CHSOH IIFrcuiy
1 1
~
1
1
cc
n-H ept an e E,tIiyl ethei CCl? CHCls
b
i
1 60 1 92 2 08 2 44 212 2 05 1 76 0 43 0 21
Calculat~d
x
103
1
1
Obsened
de#-1
10.
Observed
___ -___ nini -1 1 n ni 1
deg-1
0 94 1 68 1 1 58 1 14 1 23 1 31 I 1 27 1 1 2 1 1 2 4 1 86 1 22 1 60 1 49 3 33 1 19 146 0 18
x
Calculated
___ -__
1
B
'
11 2 21 2 10 7 1 10 3 8 5 15 2 1 14 9 35 3 3.3
12 10 10 9 7 9 8 0
9 5 0 3 6 2 5 39
they chose to define it in such a way that it contains all the variatioiis in the potential energy of the liquid n-ith temperature, i.e., their Boltzmann factor is exp(-E&lT-l) where EOdoes not vary with the temperature,
( i;). Secondly, they use the equation:
while we use exp - -
1
av$ -
vt aT
cy
3
256
HENRY EYRISG 4 S D JOSEPH HIRSCHFELDER
while the free volume which we have used is defined in such a manner that :
The first of these differences is purely a formal one. The second difference would be non-existent if b were taken as 1 in the present treatment. The problem of deriving the “packing constant” b from a knowledge of the mutual potential energy between pairs of molecules is analogous to the problem of calculating the characteristic frequencies of a crystal. A rigorous treatment of the problem would require the simultaneous consideration of the oscillations of the N molecules in the liquid, t o obtain the 3N normal modes of vibration, and from these it would be necessary to compute the mean distance which the molecules move from their respective equilibrium positions. The derivation of b given here is not strict, but it is similar to the treatment which Gruneisen (6) used for the solid and for which he obtained fairly good agreement with the properties calculated by the more accurate treatments of Debye and Born (see, for example. references 2 and 5 ) . Since the equations for a and for /3 depend only on the functional form for the relationship between V f and VI [not on b and d , for example], the agreement obtained between the experimental and the theoretical values is an indication that V ) is really linearly related to Vi. If b or d were temperature dependent, a , but not p, would be affected, but the agreement which we obtained for a as well as for p and the fact that nl is almost equal to n2 (see table 1) indicates that such temperature dependence is not large. The partition function which we have obtained here can be used in calculating the viscosity of a liquid. I n a previous paper by one of us (3), the viscosity of a liquid has been shown to be proportional t o the partition function for the normal state, F,, divided by that for the activated state,
Fa*, times a Boltzmann factor, exp
(3 -
,where Eois the activation energy
for the individual process. For liquids in which the slow process is for a single molecule to pass from one position of equilibrium to an adjacent one, we expect that the partition functions will only differ in an important way in this one degree of freedom. Thus the activation process may be thought of roughly as the vaporization in one degree of freedom, Le., the activation energy will be of the order of one-third the heat of vaporization. Also, using equation 17:
THEORY O F THE LIQCID STATE
257
If we had used equation 13 instead of equation 17 in obtaining equation 27, we would have a slightly more general relation. Andrade (1)compared the Boltzmann factor in viscosity with twice the van der Waals constant a divided by the volume of the liquid, i.e., 2 AH - 2 R T ; however the Boltzniann factor m-hich was required for the large number of liquids which he considered was slightly less than one-sixth of this value, or a little less than one-third of the heat of vaporization. REFERENCES (1) ANDRADE: Phil. hlag. 17, 698 (1934). (2) BORN,M.: Atomtheorie des festen Zustandes, Encyc. Math. Wiss. 6, part 3, No. 25 (1923). H.: J. Chem. Physics 4, 283 (1936); see equation 10. (3) EYRINQ, (4) FOWLER, R. H.: Statistical Mechanics, p. 45, equations 122 and 123. Cambridge Press, London (1928). ' (5) Reference 4, Chapters 4 and 10. (6) GRUSEISEN,E.: Ann. Physik 39, 245 (1912); Handbuch der Physik, Vol. X, 1 (1926). (7) HILDEBRAND, JOELH. : Solubility. American Chemical Society Monograph, No. 17. Reinhold Publishing Co., New York (1936). (8) Reference 7, p. 98. (9) Reference 7, p. 104. (10) NEWTON, R. F., A N D EYRINQ,H.: Trans. Faraday SOC.,in press. (11) FRENKEL, J.: Acta Physicochimica U.R.S.S. 3, 633, 913 (1935). (12) HERZFELD, K. F., AND GOEPPERT-MAYER, R I A R I A : J. Chem. Physics 2, 38 (1934); Phys. Rev. 46, 995 (1934).
