THE LIQUID STATE' JOHN F. KINCAID
Department of Chemistry, University of Rochester, Rochester, New York AND
HENRY EYRING
Frick Chemical Laboratory, Princeton University, Princeton, New Jersey Received October 12, 1958 INTRODUCTION
The van der Waals centenary number of Physica (16) and the recent symposium of the Faraday Society on liquids (18) present many of the recent advances in this field. Since the whole thermodynamic theory of matter can be calculated from explicit partition functions, it is a problem of great interest to find such functions for liquids by every means a t our disposal. The liquid must be quite similar to the solid a t the melting point and similar to the vapor a t the critical point, so that we require a function which approximates the Debye partition function for the solid near the melting point and passes continuously over into a form which will yield the van der Waals equation of state for gases. We now consider a particular case. T E E APPLICATION O F A VAN DER WAALS TYPE OF EQUATION TO DENSE LIQUIDB
In a previous paper (14) a number of kinetic theory equations for gases were modified in such a way as to make them applicable to liquids. In this section we extend the program started there by examining the conditions necessary for an equation of the van der Waals type to be valid over the entire range of volumes from the dilute gas to the dense liquid. It is well known that the van der Waals equation itself has a range of applicability only slightly greater than that of the ideal gas law, and that it fails badly even at volumes as great as that a t the critical point. There are two principal causes for deviations from the simple van der Waals equation. The excluded volume b can be taken to be equal to four times the volume of the molecules only at low gas pressures. As the gas
* Presented a t the Symposium on Intermolecular Action, held a t BrownUniversity, Providence, Rhode Island, December 27-29,1938, under the auspices of the Division of Physical and Inorganic Chemistry of the American Chemical Society. 37
38
JOHN F. KINCAID AND HENRY EYRINQ
is compressed, overlapping becomes important and must be corrected for. This problem has been considered by a number of people, including Happel (9), Majumdar (15), and Hirshfelder, Stevenson, and Eyring (10). These authors give an equation for the free volume Vi of the form: V, = V exp[- (b/u)
- al(b/~)~ - az(b/u)* . . . . . I
(1)
where a1,az,etc. are parameters. Another cause for deviations becoming important a t volumes much smaller than the point a t which overlappiag becomes important is the repulsive force that the molecules exert on one another a s they condense to form a close-packed liquid. I n addition t2 these two effects, the temperature dependence of the effective volume of the atoms causes a significant decrease in van der Waals b as the temperature is increased. Mercury was chosen as a test liquid for an equation of state corrected for these effects, because it provides an extreme example of a close-packed liquid and because all the necessary data have already been summarized (13). I n setting up the partition function the static energy E. is defined (as before (13)) as the potential energy that the liquid would have if the atoms remained fixed in their equilibrium positions. The form that we choose for E, is: E. = -a/V + d e ~ p ( - c V l / ~ ) (2) Here a/V and d exp ( - c V " ~ ) represent the attractive and repulsive portions, respectively, of the static energy. The three parameters a, d, and c were very simply evaluated from three properties of the liquid a t the absolute zero. The ones selected were the energy of vaporization, which is equal to the value of E, at T = 0, the volume which coincides with the minimum in E., and the compressibility which is given in terms of the second derivative.2 The simultaneous equations to be solved are: a/V
- d exp(-cl."la) a/V4/3
= AE,.,.
= 63.18 X 10lOergsper mole
- (dc/3) e ~ p ( - c V ' / ~ )= 0
(3) (4)
and dV/dP
=
V4/a[4a/3V6/3-(c2d/9) exp( - C V ~ / ~ )=] - ~
-38.2 X
absolute units
(5)
These equations may be readily reduced to two linear equations and a quadratic, to yield the expression for E.:
E, = - 10.069 X 10l2/V
+ 8.560 X loz1exp( - 10.50V1/a)ergs per mole (6)
* No absolute significance is attached to the properties of the liquid at the absolute zero. The extrapolated properties at T = 0 are regarded merely as those which the liquid would have if i t continued to behave at temperatures below the melting point as it does at temperatures above it.
THE LIQUID STATE
39
The volume at T E 0 was taken as 14.05 cc. The particular vdue of 63.18 X 101O ergs per mole for AEvsp.was chosen at T = 0, because it gives the correct entropy of the liquid at its melting point and thus the correct entropy of fusion. This value for AEvap, ( T = 0) is very close to what one obtains by extrapolation of the best thermal measurements (12). If the equation is to be employed at temperatures other than T = 0, some form must be chosen for the free volume. Although the form that has been previously employed by us (13) is satisfactory for the closepacked liquid, it fails to extrapolate smoothly to yield the ideal gas exvession at large volumes. Equation 1, an expansion in powers of (b/V), appears promising, but it is a rather clumsy expression with which to work A much simpler expression, which we have found to be more satisfac-
tory, is V, = V exp[- (b/V)"]
(7)
in which the sum of powers of b/V has been replaced by a single term raised to an arbitrary power. Using equation 7 the partition function becomes j = [(2~rnkT)~/~/h~]V exp[- (b/V)"] exp( -E,/RT)
(8)
and the equation of state follows from equation 8 by the conventional method.
P = RT (blnf/bV)T = -a/V*
+ (dc/3W3) e x p ( - ~ V ~ /+~ )
Since the parameters in E, have already been determined, the unknown in equation 9 at any particular temperature is the quantity n(b/V)". This was determined at the melting point and a t the boiling point from the experimental vapor pressures. This left as the only unknown the value of n, which was found to be 1.30 from equation 9, utilizing the experimental volume of the liquid a t its melting point. These data give b equal to 60.67 and 50.07 cc. at the melting point and at the boiling point, respectively. Values of b at other temperatures were computed from the relation: b
=
(119.05
- 10.70 In T ) CC.
(10)
The agreement of the calculations with the observed data is shown in table 1. The six observations that were used to determine the six constants of the partition function are given in parentheses. The compressibilities quoted are from Hubbard and Loomis ( l l ) , and the other data are the same as those previously employed (13). I n comparing the calculations with the observations it should be recalled that it has been our aim throughout to make the partition function as simple as possible,
40
JOHN F. KINCAID AND HENRY EYRING
IN
THE LIQUID STATE
41
and to evaluate the parameters in the easiest possible manner. By employing a more laborious method of securing the parameters it would no doubt be possible to obtain a better partition function. Inspection of equation 9 reveals that a t large volumes it reduces to the ideal gas law, PV = RT. Whether it gives satisfactory results a t intermediate volumes is, however, still to be determined. Although no extensive PV data are available for liquid mercury except a t temperatures below the boiling point, the critical temperature and pressure have been determined by Birch (2), who has also estimated the critical volume from the data of Bender (1; see also 20). From the conditions that ( b P / b V ) T and (bzP/bV2)Tequal 0 a t the critical point we have the following simultaneous equations to be solved for T,, V,, and P,:
P
+ a / V 2 = ( R T / V ) [ l+ n(b/V)"]
(1 1)
+ (n + nz) (b/V)"I 6a/V2 = ( R T / V ) [ 2 + (2n + 3n2 + n3))(b/V)"] 2a/V2 = ( R T / V ) [ l
The repulsive contribution to E, has been omitted from these equations, since it can be neglected a t such large volumes. The calculated values are compared with those experimentally determined in table 2. The calculated pressure is given only approximately, because it is extremely sensitive to the extrapolated value of b. The success of the partition function in predicting approximately the right critical constants is of considerable interest, since the bonding in metallic solids and liquids is usually thought of as qyite different from the van der Waals forces acting between pairs of molecules. Unfortunately, mercury is the only liquid metal for which the data afe available for setting up a partition function of this kind. We now proceed to certain general considerations.
Partition junctions For temperatures several times the Debyg: characteristic temperatures, e(= hv'lk), the Debye solid partition function approaches the Einstein form:
At still higher temperatures K takes the classical form:
Here N, Y ' , Y , and E are the number of molecules, the maximum frequency, the mean frequency (v = 3/4 VI), and the energy of vaporization
42
JOHN F. KINCAID AND HENRY EYRING
for N molecules at the absolute zero, respectively. E is usually expressible as a function of volume alone, unless important structural changes occur; k , h, and T a r e the Boltzmann constant, Planck’s constant, and the absolute temperature, respectively. fr and fi are the partition functions for the rotational and internal degrees of freedom of a molecule. The partition function which leads to van der Waals equation is:
TEMPEKATURE
Observed ‘R.
-1730 -
j -~
Cal,td;ted
I
Obaervsd cc.
1500
I
VOLUME
about 40
PRESSUHE
Calculated
Observed
cc.
dynes per cm.1
37.5
1 1600 X 10’
I-
Calculated dynes per cm.1
about loo0 X lob ~
V f is now the free volume for N molecules, and the explicit forms to be taken by V f ,E, and f, are questions to be answered in a theory of liquids. Melting and the communal entropy At the melting point we have the relation where the subscript 1 indicates the liquid and s indicates the solid. Also
THE LIQUID STATE
43
and A , = -kT In K (13). From thermodynamics the pressure satisfies the well-known equation
P =-
(g),
+
+
If Ai pVl > A , pV, the solid is stable, while for the reverse inequality the liquid is stable. If, for a given temperature, A , be plotted as ordinate with the volume as abscissa, then using equation 8 we see that the intersection of the tangent to the curve with the ordinate V = 0 gives the Gibbs free energy for the solid at the temperature and volume corresponding to the point of tangency, the pressure of course being the negative of the slope. Plotting the Helmholtz free energy for the liquid and gas the same procedure gives their Gibbs free energy as a function of
A, d
Y FIG.1. The Helmholtz free energy A for liquid, gas, and solid a t some temperature T is plotted against the volume as abscissa. The intercept a gives the Gibbs free energy for liquid and solid in equilibrium a t the pressure given by the negative of the slope of the line ab. The corresponding volumes for solid and liquid are the abscissas of the points b and c, respectively. The line def has a corresponding significance for the liquid-vapor equilibrium
V , T,and P . The common tangent to the A curves of two phases fixes the pressure and the two volumes at which the two phases are in equilibrium at that particular temperature. For the triple point there is of course a common tangent to all three phases. This procedure is discussed at length in treatments like the one by Kohnstamm (17) in the Handbuch der Physik. The situation for a solid and liquid whicb expands upon melting is indicated in figure 1. Clearly an explicit forni for the partition functions for the solid and liquid phases constitutes a theory of melting. Only those points on the A curves are thermodynamically stable for which the tangent fails to intersect any other section of these curves. The A curve for the solid differs from that for the liquid-vapor because of some change in structure, having as a result the instability of volumes
44
JOHN F. EINCAID AND HENRY EYRING
intermediate between those for the points b and c. Since the potential energy a t such intermolecular distances, for molecules held by van der Waals forces, almost certainly increases with distance in spite of any structural change, the stabilization of the liquid arises from a sudden increase of entropy with volume. For complex molecules there will be increases in the entropy of rotation when neighbors get farther apart during melting, but this is only incidental to the increase in entropy associated with translational degrees of freedom. Most monatomic molecules melt with an increase of about two entropy units at atmospheric pressure (IO). This is the value to be expected if the communal entropy is attained, principally during the process of melting. Rice (17) argues that, because the entropy of melting for monatomic molecules such as argon drops with the pressure below two units, probably the communal entropy is of minor importance on melting. This is of course evidence that the entire two units do not necessarily come in a t the melting point, but this was already clear for another reason, Le., the values for sodium and potassium are slightly less than two units even a t atmospheric pressure. We have an analogous case in the process of vaporization. Thus the term p ( V , - V , ) contributes RT to the energy and R to the entropy of vaporization a t low pressures, but both values drop to zero a t the critical point. V,, and V , are the molal volumes of liquid and vapor, respectively. Gurney and Mott (8) have given the curve for the communal entropy its a function of the number of molecules, n, in each of N / n groups which communally share volume, it being understood that the N / n groups do not share communally with each other. The corresponding Helmholtz free energy is:
and the communal entropy per mole is:
where S(1) = 0; 5(5) = 0.66R; X(27) = 0.9R; X ( m ) = R. Now the condition that there shall be local communal sharing of free volume must be that the local density drop sufficiently to permit the structure to collapse freely in the various directions. If the molecules were points we would of course'have communal sharing of volume. Thus the lack of communal sharing arises from the finite dimensions of the molecules, i.e., from their interlocking. Tonks (19), of course, finds communal sharing among molecules arranged in a linear filament. Thus if molecules were arranged in a simple cubic lattice, each of the linear
45
THE LIQUID STATE
filaments parallel to any one of the three axes would show communal sharing, so that from equation 20 we see that the system would possess practically the full entropy R. However, even in the liquid state the system has not expanded to the point where filaments slide by other filaments with ease, as is shown by the considerable activation energy for viscous flow. Viscosity of course measures the ease with which layers of molecules slide by each other (4, 5). Thus communal sharing (which measures the ease of collapse of the structure) and fluidity or diffusion (which presumably measure the ease of rotation of double molecules (10)) are only slightly different aspects of the same problem of the degree of correlation between the motion of ncighbors, and we regard the great change of fluidity upon melting as compelling evidence for a corresponding great change in the communal entropy. The correlation between the motion of neighbors is of course a strong function of the volume, and this can be expressed in some such form as the following expression for A : E VI .' A = -kTln{((y'! frft exp (21) ( E ( n 1) - E(n) . exp 1; - S(n) n-1 kT Here S(n) is defined by equation 20 and E,, is the energy required to expand N / n regions, each containing n molecules, from the average volume up to that critical volume. a t which communal sharing begins. We have not yet tested explicit forms of equation 21 in efforts to explain the change of the melting point with pressure, but a knowledge of the activation energy for viscous flow should be of assistance in estimating E(n) and in checking intuitive estimates of what the critical volumes for sharing should be. We now consider briefly methods of estimating the free volume. The free volume In a recent paper (14) we have considered a t length possible ways of
2
p+
calculating the free volume.
(m)v)
+
-)>
In this paper we have seen that V exp -
(:>"
gives a satisfactory expression for this quantity for mercury, and we now wish to discuss briefly the velocity of sound method (14). The basic idea is that the time used by the sound wave in travelling through the atoms themselves is negligible compared with that used in traversing the free volume, where the wave is carried by the atoms moving with the velocity of sound in the gas. This yields the equation
46
JOHN F. KINCAID AND HENRY EYFUNG
where U I , up, m, V , and V , are the velocity of sound in the liquid, the velocity in the gas, the weight of a molecule, the molal volume, and the molal free volume of the liquid, respectively. If for the moment we neglect communal sharing, me obtain for the partition function for one translational degree of freedom in the liquid:
Now at high temperatures compared with the characteristic temperature
e
=hv‘ = h-
k
(-)9N
k 4TV
uR
the Debye theory (7) for solids yields the limiting partition function:
Here udis the velocity of sound in the solid. Thus we see that our very simple model for the free volume of a liquid keds to formally the same translational partition function as the Debye theory for the solid. The free volume calculated from equation 22 is in satisfactory agreement with that found by other methods (14). However, it is possible that improvements in our model and examination of additional data will lead to the adoption of a proportionality constant in equation 23 more nearly equal to that in equation 24. Brillouin (3) has treated liquids in a manner analogous to Debye’s solid treatment and has obtained interesting results. When the activation energy for viscous flow disappears, as it does in the neighborhood of the critical point,, the liquid loses its ability to store the potential energy resulting from shearing stresses. Thus transverse sound waves only involve kinetic energy. I n this way Brillouin explains why the translational specific heat drops from about 6 cal. near the melting point to about 4 a t the critical point. Our velocity of sound method provides an alternative approach to these questions. REFERENCES (1) BENDER: Physik. Z. 16,246 (1915); 19,410 (1918). (2) BIRCH,FRANCIS: Phys. Rev. 41, 641 (1932). (3) BRILLOUIN: Trans. Faraday SOC.33, 54 (1937). (4) EWELLAND EYRING:J. Chem. Phys. I,726 (1937). (5) EYRING:J. Chem. Phys. 4, 283 (1936). (6) EYRING AND HIRSCHFELDER: J. Phys. Chern. 41,249 (1937). (7) FOWLER: Statistical Mechanics, p. 127. University Press, Cambridge (1936).
THE LIQUID STATE
(8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20)
47
GURNEY AND MOTT:J. Cheni. Phys. 6, 222 (1938). H.: Ann. Physik 4, 21, 312 (1906). HAPPEL, HIRSCHFELDER, STEVENSON, . ~ K DEYRINO: J. Chem. Phys. 6,896 (1937). HUBBAED AND LOOMIS: Phil. Mag. [7] 6, 1177 (1928). KELLEY, K. K.: U. S. Bur. Mines, Bull. 393 (1936). AND EYRING: J. Chem. Phys. 6, 587 (1937). KINCAID A N I EYRING: J. Chem. Phys. 6,620 (1938). KINCAID MAJUMDAR, R.: Bull. CalcuttsJIath. SOC.21, 107 (1929). Physica 4, No. 10 (1937). RICE,0. K.: J. Chem. Phys. 6,476 (1938). ON LIamDs: Trans. Faraday SOC.33, 1-282 (1937). SYMPOSIUM TONKS,LEWI:Phys. Rev. SO, 955 (1936). WEBER:Commun. Kamerlingh Onnes Lab. Univ. Leiden, Suppl. KO. 4.9 to Nos. 146-166, p. 23 (1920).
