The STRUCTURE of LIQUIDS JOSEPH 0 . HIRSCHFELDER University of Wisconsin.
The Eyring theory of liquids i s deueloped i n terms of elementary thermodynamics and a simple picture. A n equation of state i s derived which i s shown to be the limitingform of the wan der Waals equation for close packing. The Hildebrand and Trouton rules for vapor pressure, the entropy change on melting, and the temperature coeficient of wiscosity are explained.
Madison. Wisconsin
liquids. But in the meantime the rough theory of liquids developed here is useful, since it predicts properties which agree surprisingly well with the experimental facts. We think of a liquid as a very dense gas with the molecules squeezed together by their own forces of mutual attraction. Each molecule dashes from side to side, knocking back any neighbor which invades its portion of the total volume of the liquid. If V is the molal volume of the liquid and N is Avogadro's number,
M
ANY of the properties of liquids can be explained in terms of a simple picture. Only a rudimentary knowledge of thermodynamics is required to understand the origin of the equation of state, the Hildebrand and Trouton rules for the vapor pressure, the entropy change on melting, the temperature coefficient of viscosity, and so forth. Our presentation follows the work of Henry Eyringl and his collaborators. The novel ideas which they developed have been solidly grounded to the experimental data of Hildebrand, Debye, Gibson, Eucken, and other^,^ and their viewpoint has formed a basis upon which LennardJonesa and others are building a more permanent theoretical structure. The Eyring theory of liquids corresponds to the Einstein or to the Griineisen approximations for the solid state. All of the molecules except one are held in their mean positions, and the properties of the whole system are estimated from the amount of freedom which the one remaining molecule possesses. In the solid, this freedom is expressed in terms of a vibration frequency or characteristic temperature, while in the liquid the concept of "freevolume" is introduced for the same purpose. In order to obtain an adequate description of the solid state it is necessary to use the elaborate Debye theory. An analog of the Debye approach must be developed for
mGmE I.-THE FREE~VOLUME, a,, IS THAT SPACE IN WHICHTHE CENTER OF A MOLECULE MAY MOVE m o m COLLIDINGWITH ITS NEIGHBORS. THE FREE-VOLUME DIFFERS FROM THE SPECIFIC VOLUME, W, WAICH IS THE AVERAGE AMOUNTOF SPACEAVAILABLEFOR
MOLECULE EACH
the saecific volume of each molecule in the liauid. v,. is eq;al to V / N . But it is the free-volume, ;e,r rather (b) NEWONAND EmNG, Trans. FaraA~ Sot., 33973 (1936). than the specific which plays the important r61e (c) EYRINOAND H~SCHPELDER, J. Phys. Chem., 41, 249 . (1937). in determining the properties of liquids. The freeAND EYRING, J . Chem. Phys., volume is .that SDace in which the center of the molecule (d)~mscnaE~D~R, STEVENSON, 5,896 (1937). can move and siill not collide (on the average) with the (e) ~ N C A I DAND EYRING, ibid., 5,587 (1937). ( ) KINCAID AND EYRING,,~~~., 6,620 (1938). neighboring molecules. In this effective volume the &) EWELL AND EYRINO. ~b~d., 5.726 (1937). molecules behave like a perfect gas. Figure 1shows the (h) HIRSCHPELDBR. EWELL.AND ROEBUCK.ibid., 6, 205 difference between the specific and the free-volumes. (1938). (i) HIRSC~ELDER AND ROSEVEARE, 3. Phys. Chem.. 43, 15 For all practical purposes, the attractive forces in the (1939). liquid are sufficiently long-ranged so that the energy of (j)Krncam AND EYRING, ibid.. 43,37 (1939). Cf. HILDEBRAND. "Solubilitv." .. 2nd ed.. Reinhold publishinz the system is the same. It is immaterial in which Dart ~orp:. New York -City.~lQ$. ~, ~ ., ~,.,.--\ of t h i free-volume the center of a molecule happens to .. * (a] LENNARD-JONES, mtZySSCa, 1". 3 4 1 ( L Y J I J . be located. The repulsive forces are short-ranged and (b) L~NN-. AND D~~~N ~ - , proc. Roy, sot. (London), A163,53 (1937); A165.1 (1938): A169,698 (1939). only comeinto play when two molecules actually collide. (a) EYRING. 3. Ckm. Phys., 4,283 (1936).
.-
~~~
.--..
-
For simplicity in thesederivations, we think of the molecules as being rigid and elastic (like billiard balls), but this restriction can easily be removed and must be corrected before we can compute the correct specific heats. The centers of the molecules then move in a force-free box of volume, up It is force-free because attractions between any one molecule and all of the others are effectively balanced. The walls of the box represent the geometrical constraints imposed on the motion of a molecule by its neighbors. We can easily calculate this free-volume from the geometry of the
from the vapor to the liquid is immaterial, as is also the fact that in the liquid the pressure acting on the molecule arises from the forces of mutual attraction of the molecules themselves. Now the work done when a perfect gas is compressed isothermally from a volume v. to a volume q is kT log (v,/v,) where k is the gas constant per molecule. We identify this work with the latent beat of vaporization per molecule, A. Thus = wo
A = kT log (v./uJ) or
exp (- AIkT)
(3)
Since the molecules in the vapor obey the perfect gas equation, P.& = kT (4 where the vapor pressure is
We shall put this equation for vapor pressure into a more useful form after obtaining the equation of state. First let us examine the free energy. We know from simple statistical mechanics that the Gibbs free energy for the vapor, F,, is F,
liquid packing. For example, suppose that the molecules are arranged roughly in a simple cubic lattice. When all of the molecules except one are held in their mean positions, this one molecule can move a maximum distance of 2(v,"" d ) in each dimension if d is the diameter of the molecules. This situation is illustrated in Figure 2. The distance between the two outside molecules in this figure is just twice the cube root of the specific volume. The center of the molecule we are interested in (the black one) is then confined to a cubical box each of whose sides is 2(w,'/' - d). The free-volume is then the cube of this distance or VJ
= 8(V/h"/z
-
(1)
Usually it is convenient to express the diameter of the molecules in terms of the ordinary van der Waals constant, 6.
All of the thermodynamical properties of a liquid may be expressed in terms of the free-volume. Vapor pressure (or more accurately, the fugacity) is our first example. For this purpose we think of a vapor phase in equilibrium with the liquid. This vapor obeys the perfect gas laws if it is dilute and has a specific volume of u, per molecule. When a molecule is taken from the vapor phase and condensed into the liquid, it is effectively compressed from moving in a volume v. to moving in a volume v,. In both the initial and the final states it obeys the perfect gas laws, so that we can use thermodynamics to calculate the work required for this process. The path by which the molecule is taken
=
-NkTlogv.
- Q(T)
(6)
Here (2~mkT/h?)~'' is part of the translational partition function, fds, fmr, fin, are the partition functions for the vibrational, rotational, and internal degrees of freedom, respectively. The important property of these functions from our standpoint is that they do not change appreciably with volume unless the molecules are asymmetrical and are so tightly squeezed together that their rotational motions are hindered. We shall assume for the time being that the rotations are not hindered so that Q is the same for the liquid as for the vapor. If the vapor phase is in equilibrium with the liquid, the free energy for the liquid, 8,is equal to that for the vapor F.
=
F,
(8)
Thus the right-hand side of equation (6) is equal to
F,, and using equation (3) to express the left-hand side in terms of v, instead of vg FZ = -NkTlog (qezp (AIRT)) - Q(T)
(9)
The thermodynamical equation of state follows immediately from the free energy
In this equation it makes little difference whether we use the Gibbs free energy or the Helmholtz maximum work function since pV for a liquid is small (under ah ordinary conditions). Here - hV T = Pi, the \-.,internal pressure. Now Professor Joel Hildebrand4 has studied the internal pressure experimentally and compared i t with the heat of vaporization for a number ~ L D E E R A N D op. , Cit., P.99.
(-)
of so-called "normal" liquids. He found that for most liquids A = a(T)/NV
and Pi
=
a(T)/P =
+ NkT
(11)
-
(12)
Here we recognize that a ( T ) is the van der Waals constant, and we are not surprised to find that it should have slightly different values a t differenttemperatures. It can be shown from theoretical considerations that the validity of equation ( 1 1 ) corresponds to the approximation that as the liquid is compressed, the frequency but not the character of the collisions between neighboring molecules is changed. Now we are ready to express the equation of state in a simple form. From,.equations ( 1 ) and (2) we obtain the expression for . the
(*)b V
.
T
illusory. A rather sensitive test for the equation of state is given by a comparison of the observed with the calculated coefficients of compressibility and thermal expansion. Table 1 shows a sample of the agreement which is obtained. TABLE 1 TABCOBPPIEIBNZS 09 COYPRBSSIBILN AND
09
TABPXAL
EXP~SIO FOR N
LlQOlDSk
Now let us examine our equation (5) for vapor pressure, P,, = kTvl-1 exp ( - X / k T ) . We wish to express the free-volume in terms of the heat of vaporiza-
and making use of equation (12) for PI
it foliows that
( P 4- a ( T ) / V 2 ) ( v- 0.7816b'liV2/~)= NkT (13) If we had considered the molecules to be packed in a rough body-centered cubic lattice, the numerical constant would have been 0.7163 instead of 0.7816; facecentered cubical packing would have led to the constant 0.6962. These constants are pure numbers determined mathematically, not empirically, from the geometry of the free-volume. This equation is exactly like the van der Waals equation except that now the excluded volume varies as the two-thirds power of the volume instead of being a constant. Actually this equation is the limiting form of the van der Waals equation when the latter is corrected for the overlapping of hard spheres. Both van der Waals and Boltzmann realized that such corrections were necessary. They wrote the equation of state as an infinite series of the form p
Nk T + a l p = -(1 v + b / V + 0.625bVV2 + 0.2869ba/Va+
FIGURE POTENTIAL ENERGY FOR A MOLECULE.A IS IN TAE SIMPLEST VERSION OF TFIE LIQUID. B ISI N THE SOLID. C IS THE POTENTIAL FOR A MOLECULE I N THE LIQUIDSWHICA L E A D S 1 0 THE CORRECTSPECIFICHEATS
0.1928b'/V4) (14)
Boltzmann and Jaeger computed the term 0.625b2/P by considering the simultaneous collisions of three molecules. Majumdar and Happel6 introduced the term 0.2869ba/Vaby considering the simultaneous collisions of four molecules. Now in order to make equation (14) agree with our equation (13) throughout the range of densities characteristic of liquids, i t is only necessary to lump together all of the further corrections into the term 0.1928b4/V4. Since this term is of a reasonable magnitude compared to the other terms, our equation of state seems to be the logical extension of van der Waals for the region of close packing. It is interertin~that this eauation iust involves the pas constants a and b. ~ a L l e scan-be made up showing excellent numerical agreements between these constants letermined from the liquid and from the gas, but the differentmethods for determining a and b from the gas data give such varied values that such agreement is
-
-
(a) HAPPEL, Ann. Physik, 4, 21, 342 (1906). (6) MAJUMDAR, BuU. Calcutta Math. Soc., 21, 107 (1929).
tion. Combining the equation of state (13) with the expression ( 1 ) for the free-volume, it follows that This expression can be further simplified by neglecting the external pressure, p, in comparison with the internal pressure, a / V Z . Then, from equation (ll), X = aN-'V-' NkT. And so
+
P,,
=
7 N k T v - ' [ ( LkT )
- 11' erp ( - x / k T )
(16)
Recently Professor Joseph Mayer has shown that for a great many substances, in accord with this equation, P,,V/kT is a universal function of X/kT. The Hildebrand rnle for vapor pressure states that the concentration of the vapor, c, = P..,/kT, is a universal function of X/kT. According to our equation (16) this is true except for the term V-' whose variation from compound
to compound is unimportant when compared to the strong temperature dependence of the factor ex$ (-X/kT). Trouton'smle is obtained by approximating NkT - by a universal number. For example, if we set
".
11 T/
T = 300°K. and V = 82 cc. P,, = 35((A/kT) - 1)' czp (-X/kT) atmospheres
(17)
the solid. Thus the improved free-volume is formed by splitting the potential for the solid (b) along the dotted line in the middle and inserting a force-free region as shown in drawing (c). In this manner, Kincaid and Eyringl were able to calculate the specific heats and most of the other thermodynamical properties of liquid mercury to within the experimental errors. To be sure, the few constants occurring in their equations were
At the normal boiling point Ts (where P,, = 1 atm.) we can solve this equation for the heat of vaporization and obtain: XlkTs = 10.4
(18)
Here we have obtained Trouton's rule with the very best value of the Trouton constant. This constant is quite insensitive to the number which is taken for NkT/8V and changes by only a few per cent. when NkT/RV is changed by a factor of two. However, for the extremely low boiling points and small molal volFIGURE5.-IN
SOLID,
THE BUT NOT I N THE THE MOLECULES ARE GEOMETRICALLY FROM EACH OTHER'S
LIQUID
PREVENTED SHARING FREE-VOLTMES. THEIRGEOMETRICAL CONSTRAINT IS PORTRAYED BY PARTITIONS AND THE DIFFERENCE IN ENTROPY OP THE TWOSITUAT~ONS GIVES THE ENTROPYOF FusroN
Pochiny in Sold
FIG-
Pocking in Ligufd
O F PACKING O F MOLESOLID AND I N THE LIQUID AT T D MErrrNa POINT T ~ DOTTED E LINE SAOWSTHE
TIGHTNESS
CULES IN THE
A
BODY-CENTEREDLATTICE
nmes of the noble gases, equation (16) gives for X/kTs a value around 12 which is still in accord with experiment. The simple theory of liquids requires further corrections before we can calculate the specific heats. Up to this point we have assumed that the potential field in which the molecules move was a box similar to that shown in drawing (a) of Figure 3. The walls were perpendicular which would correspond to the molecules being completely rigid on collision. But we know that in a solid where the molecules are almost always in a state of collision with their neighbors, the potential field is nearly parabolic (as shown in the drawing (b)) and has a curvature related to the characteristic temperature. In improving the potential field for the molecules in the liquid we assume (1) In between collisions the molecules move in a force-free space as before. (2) During collisions the molecules are repelled by their neighbors with exactly the same forces as in the solid. And so the walls of the potential well for the liquid have the same curvature as for the potential for
determined from experimental data, but the agreements establish the theoretical relationships as a t least good empirical interpolation formulas. The process of melting is one of the most interesting features of this simple theory. Melting occurs when the molecules are no longer geometrically trapped in their unit cells and can just squeeze through the holes in their lattices. For example, if the molecules are arranged in body-centered packing (as shown in Figure 4) this occurs when the lattice is expanded twenty per cent. over its value a t the absolute zero. Grlineisen showed that the universality of the Lindemann relationship for the characteristic temperature depends on such an expansion factor heing the same for most substances. In any case, melting is accompanied by a transition from an essentially ordered array to one which is essentially chaotic. We believe that the entropy of melting is just the entropy difference between the ordered and the unordered states. This entropy is easy to estimate. In the solid, if each molecule has a free-volume,,,!I it utilizes this free-volume independent of its neighbors. In the solid (but not in the liquid) a surplus volume in one part of the system is geometrically prevented from being utilized in turn by the molecules throughout the system. This situation is portrayed in the left-hand side of Figure 5. In a liquid, the molecules have a communal sharing of their free-volumes. This means that sometimes a group of the molecules will he crowded into a small space and a t other times they will he spread apart. The prevalence of such large fluctuations is equivalent (from the statistical mechanical standpoint) to a condition where all of the N molecules move in a
volume N times the free-volume, i. e., utilize their total free-volume in a gas-like manner (as is illustrated in the right-hand drawing in Figure 5 ) . Simple thermodynamics sufficesto show that if N molecules of perfect gas move in a volume NQ they have an eutropy of R entropy units more than they have for the situation where the N molecules are separated and each one is constrained to move in a volume of q. Here R is the mold pas constant equal to 2.0 E.U. We expect, then, that tlhe entropy of-fusion of monatomic substances should he roughly R = 2.0 E.W. and it is an experimental fact6that the average entropy of fusion of a large number of monatomic substances is 2.2 E.W. The entropy of fusion of polyatomic molecules is further complicated by vibrations in the solid phase being converted into rotations in the liquid phase. We can estimate the eutropy of vibration, So,if we know the vibration frequencies from Raman spectra or other data. Here for the purpose of numerical calculations we use 50 cm.-' in accordance with Pauling's' obsemations. From the moments of inertia, we can compute the entropy of rotation, S,. The entropy of fusion, SJ, should then be SJ = R S, S, (19)
cannot flow past one another. To make this flow possible, it is necessary to have a surplus volume. To make a hole the size of one molecule in the liquid it requires work equal to the heat of vaporization. For this reason, the exponential temperature dependence seems to show that the unit process for viscous flow requires the formation of a hole approximately a third or fourth the size of a molecule. If the flow takes place by the collision of two molecules followed by a rotation of the pair
+ +
Table 2 shows the sort of agreement which is obtained. TABLE 2
Srbsrnnrr Clr
Bn I,
CSz N10
I
13.6
16.1 18.3 14.5
13.4
S* '
S
t
8.8 9.9
8.9
7.2 7.7
7.7
10.7
6.8 9.6 9.3
.
(.SJ)&
6.8
9.5 9.5 8.4 8.6
In general the eutropy of fusion of polyatomic molecules should be roughly S., = 2 3 0 E.W., where D is the number of degrees of vibration which are converted into rotations during the melting process. Viscous flow is one of the most characteristic properties of liquids. Andrade8 found that the temperature coefficientfor the viscosity, 7, of hundreds of substances could be represented in the form n = no exp (X/3.5kT) (20)
+
.-----
FOLLOWED BY A ROTATION OP THE PAIR. TRB AMOUNT0s VOLUMEREQUIRED FOR THISMOTION ACCOUNTS POR TEE TEMPERATURE COEPPICrENT OF
~
~~
~
VISCOSITY
through approximately 90°, we can show geometrically that the additional volume required is in agreement with this interpretation. Metals have a much smaller temperature coefficient for viscous flow since in this case the atoms can move without their outer shells and a much smaller surplus volume is required. At the present time, attempts are being made to place the theory of liquids on a more rigorous basis. ?he rough t h e o j here seems to-be adequate where qo is temperature independent. The constant for treating most-of the qualitative aspects of the 3.5 in this equation variesbetween 3 and 4 for different physical properties. This viewpoint is being extended substances. ~h~ reason for this regularity may be to a consideration of the properties of solutions. The seen from a simple picture (pigure 6). ~h~ molecules calculation of activity coefficients should prove parin a liquid are ordinarily packed so tightly that they ticularly useful since they are one of the factors which determine the rate of chemical reactions in solutions. 0 Cj. Tanon, editor, "A treatise an physical chemistry," 2nd The author wishes to thank the Wisconsin Alumni ed., D. VanNostrand Co., Inc., NewYork City, 1931. Research Foundation for financial support throughout ' PAULrNG, Phys. Reu., 36,430 (1930). the course of this work. Phil. Mag., 17, 497,698 (1934). ANDRADE,
