ORDER AND DISORDER I N LIQUID SOLUTIONS1 JOHN G. KIRKWOOD Department of Chemistry, Cornel1 University, Zthaca, New Yozk Received October 1 , 1938
I A large number of non-polar liquid mixtures conform approximately to the laws of the regular solution, originally formulated by Hildebrand (3) on the basis of thermodynamic considerations. The regular solution has been treated from a molecular point of view by Scatchard (7), by Rildebrand and Wood (4), and by Guggenheim (2): A regular solution is primarily characterized by the fact that it possesses the entropy of mixing of an ideal solution. This implies the existence of a random molecular distribution in the solution. By random distribution we mean that the neighbors of each molecule are, on the average, distributed among the various molecular species of the mixture in the proportion of their mole fractions, the average local composition in the vicinity of the molecule being identical with the bulk composition of the solution. In real solutions with nonvanishing heats of mixing, random distribution can scarcely provide more than an approximate description of the actual situation. In seeking a n explanation for the departure of actual solutions from regular behavior, it is therefore of importance to study the influence of deviations from random distribution on the thermodynamic functions of the system. The present investigation is concerned with this problem. The average distribution of the neighbors of a molecule in solution, among the various molecular species present, is determined by two opposing influences,-the disordering effect of thermal motion and the ordering effect of intermolecular forces. For example, in a binary solution in which the intermolecular attraction between unlike molecules is greater than that between like molecules, each molecule will exert an ordering influence in its vicinity resulting in a local composition richer in molecules of the opposite species than the solution in bulk. On the other hand, if the attraction between like molecules is greater than between unlike, a local composition in the vicinity of each molecule, richer in molecules of the same species, will result. The extent to which local segregation of this 1 Presentedat the Symposiumon Intermolecular Action, held at Brown University, Providence, Rhode Island, December 27-29, 1938, under the auspices of the Division of Physical and Inorganic Chemistry of the American Chemical Society. 97
98
JOHN 0. KIRKWOOD
sort can be established depends upon the violence of thermal motion and will be greater the lower the temperature. At sufticiently low temperatures it may manifest itself on a macroscopic scale by causing the solution to separate into two phases. The problem of local order or order of neighbors was discussed by Bethe (1) from a quantitative point of view in his theory of superlattices in solid solutions. Into this problem there entered another type of order, longrange order, relating to the segregation of the components on interpenetrating lattices in the crystal. Since there can be no question of the establishment of long-range order in liquid solutions, we need only concern ourselves with local order. An alternative method of treating order and disorder in solid .;elutions, developed by the writer (5), is particularly well adapted to the investigation of local order in liquid solutions. Although, in the meantime, the problem has been treated by Rushbrooke (6), using Bethe’s method, it seems worthwhile to discuss the question from the standpoint of our method if only because of its simplicity and directness.
I1 We consider a non-polar binary liquid solution composed of N 1 molecules of type 1 and Na molecules of type 2, the total number of molecules N 1 + N z being designated by N . Following the general lines of Guggenheim’s method (2), we span the volume u occupied by the solution by a virtual lattice, dividing it into N cells of equal size. Neglecting configurations in which two or more molecules occupy a single cell, we may express the partition function of the system as follows2
B = l/kT where the sum extends over all configurations, c , of the system, the term “configuration” being employed in a special sense here to designate a specific distribution of the cells among the N molecule^.^ E(c) is the energy of the system in the given configuration when each molecule is situated at the origin of the cell which it occupies, and Q(c) is a vibrational partition function appropriate to the given configuration, analogous to the lattice 2 The formulation, equation 1, is really of value only for spherically symmetric molecules of equal size. However, if formally applied to non-spherical molecules, the factors, Q ( c ) , are understood to include rotation81 contributions to the partition function in the given configuration, and the free volume factors, Y, in equation 2 should properly include a factor in rotational configuration space. a By considering the distribution of cells among molecules rather than molecules among cells, we @voidthe necessity of dividing by Ni 1 N z ! ,since we make no distinction between configurations differing only in the permutation of identical molecules.
ORDER AND DISORDER IN LIQUID SOLUTIONS
99
vibrational partition function of a crystal. In the theory of Guggenheim &(c) is assumed independent of configuration and composition and is expressed in the following form where ml and rn are the masses of the two molecular types and u is the free volume of a molecule in any celL4 On the basis of equation 2 and the assumption that the energy E(c) is independent of configuration, Guggenheim deduces the laws of the regular solution of Hildebrand. We shall employ the first assumption, embodied in equation 2, but not the second. By taking account of the fluctuations in energy among the various configurations, we are able to investigate the ordering effect of a given molecule on its neighbors and the resulting deviations from the laws of the regular solution. A configuration of the system may be uniquely specified by a set of numbers ql. . . . qN, stating the numbers of molecules of type 1 in each of the N cells. Each variable q. may assume one of two values, zero or unity. Equally suitable for the purpose is the set p 1 . . . . P N , stating the number of molecules of type 2 in each cell. Obviously these sets are not independent, but for each cell a the relation, qa + pa = 1, holds. Nevertheless, for the sake of symmetry in notation, we shall find it convenient to employ both sets of variables. The following sum relations, satisfied by the q’s and the p’s, are important.
where the sums extend over all cells. In calculating the energy, E(c), of a given configuration, we shall neglect the interaction of each molecule with all but its z nearest neighbors, an approximation which is fairly good, since we are concerned with non-polar molecules coupled by short-range intermolecular forces. If there is no volume change on mixing the pure liquid constituents 1 and 2, the energy then has the form E = Eo + NVop where EOis the energy, El + Ez, of the two pure liquids before mixing, Vll, VB,and V n are the mutual energies of the indicated types of nearest
‘
The free volume u differs from that of Eyring and Hirschfelder, since the latter authors give equal weight to configurations in which two or more molecules are in II single cell. (See Rice: J. Chem. Phys. 6, 476 (1938).) The distinction is of no practical importance in the present discussion, since the free volume does not appear in the free energy of mixing.
100
JOHN G. KIRKWOOD
neighbor pair, and p is a variable depending upon configuration in the following manner
where h a b is unity if the cells a and b are neighbors, and zero otherwise. Evidently we may write N
b-1
hab
=z
for any cell a. . The Gibbs free energy of mixing, AF, equal to AA, the work content change, if there is no volume change on mixing @),is related to the partition functions of the solution and the pure liquids before mixing, f, fl, and j z , respectively, in the following manner
Equations 1, 2, 4, and 7 then allow
and f is given by equations 1 and 2. us to write e-OAF
&)e--
=;
9
a = V&T
(8)
where the sum extends-over all values of p consistent with condition 3, and w ( p ) is the number of configurations corresponding to a given value of p . Evidently we have
where the binomial coefficient
(E)
is the total number of configurations
of the system, equal to the number of ways in which the N cells may be distributed among Nl molecules of type 1 and Nz molecules of type 2. We , to unity by the relation define a distribution function, ( ~ ( p )normalized
ORDER AND DISORDER IN LIQUID SOLUTIONS
101
The free energy of mixing, AF, may then be written
where the factorials of large numbers in
j : (
have been evaluated by
Stirling's formula and 21 and 9 are the mole fractions N l / N and Nz/N of the two components. The extreme difficulty of determining the distribution function, ~ ( p ) , prevents the direct evaluation of the sum u. However, its logarithm may be expanded in a power series in a, the coefficients of which involve the moments of ~ ( p ) .
where the quantities, in,are the semi-invariants of Thiele,' which are related to the moments M. of q ( p ) by the following set of linear equations
Mn
C P"&J)
Solution of equations 13 yields for the first few semi-invariants
MI
XI
- Mi X, = M I - 3M1Mz + 2M: Xa =
Ma
(14)
FOPthe calculations of the moments, M , (equation 13), it is convenient to introduce the variables 71. . . . q as~ indices of summation, since p depends upon these variables (equation 5). From the definitions of &)I and ~ ( p we ) may write
We note that if
?n
is a positive exponent la"
5
= 7.;
pa'" =
PO
(16)
See A. Fisher: Mathematical Theory of Probabilities, 2nd edition. The Mac-
millan Co., New York (1926).
102
JOHN 0. KIRKWOOD
since qo and pa are restricted to the values zero and unity. This fact, together with equation 5, leads to the following expression for the moments, n
Mn = (l/Nz)" r-1
n
C v:t)yra 8-1
where v:t) is the number of terms in
involving r + s distinct indices (cells) al, . . . a,, bl, . . . b,. The validity of equation 17 of course assumes that y,, is independent of the particular set of r + s cells involved. This we shall show to be the case. The product, V a l . . . . qar p a l . . . . pa, is unity if r specified cells a l . . . a, are occupied by molecules of type 1 and s specified cells bl . . . b, are occupied by molecules of type 2. Otherwise it is zero, since it vanishes if any of the r + s factors is zero. Thus the sum on the right-hand side of equation 17 is exactly equal to the number of ways in which N-r-s specified cells may be distributed among Nl-r molecules of type 1 and NZ- s molecules of type 2, equal to
(Nn;ls'),multiplied by the number of ways r specified
cells may be distributed among r molecules of type i and s specified cells among s molecules of type 2, equal to unity. Thus we have the following expression for yra:
a result which is independent of the particular set of r + s specified cells involved. We shall not attempt to give a general expression for the coefficients v::), merely calculating them for certain specific values of n. The first moment, Ml, is easily calculated, vi:) being equal to Nz, the total number of terms in the sum (equation 5 ) . Thus we have by Pauations 17 and 18,
103
ORDER AND DISORDER IN LIQUID SOLUTIONS
where 0 ( 1 / N 2 ) denotes terms bearing a ratio or order 1 / N 2 to the initial term. For the calculation of the second moment M2, we find by inspection of the second power of the sum in equation 5 v::)
= NZ
(2) (2) ~ 1 2= ~ 2 1=
&)
Nz(z-1)
= N2z2- 4Nz2
We note that the sum of the (Nz)", in the sum,
(20)
+ 2Nz
is less than the total number of terms,
Y!:'
since any term in which an index a is equal to an index b vanishes by virtue of the relation to pa = 1 , requiring qapo always to vanish. By equations 17, 18, and 20 we obtain, after some algebraic reductions,
+
The moment M smay be calculated by a similar procedure. Finally we obtain with the aid of equation 14 the following expressions for the semiinvariants. A1
= XlQ
A2
= ~x:x:/Nz
(22)
As = - ~ X ~ X : ( X ~ - ~ ) ~ / " ? +
In the calculation of As, the details of which have not been given, it is necessary to retain terms of order N-' in M I and M z . The method just described for the calculation of the moments is similar in principle to that employed by Van Vleck in his treatment of the Heisenberg theory of ferromagnetism. An alternative method has been described by the writer in an earlier article which involves the use of interpenetrating lattices. It may equally well be used in the present calculations. Equations 1 1 , 12, and 22 lead to the following expression for the free energy of mixing of the solution: AF/NkT
=
+
~ 1 1 0 g ~ lQ
logs
+ as122 - (cY'/z)z:z: - (~cY~/~z~)x:xa :) (' x ~ (23)
Equation 23 is valid to terms in the fourth and higher powers of means of the thermodynamic formulas,
E = B(F/T)/B(l/T)
a.
By
104
JOHN
Q.
KIRKWOOD
and
s = - aF/aT expressions for the average energy and entropy of mixing may be obtained.
AE/NkT = azlq - ( 2 a 2 / z > z ~2 ~(2aP/z2)s:2:(~1 - a)' (24) AS/Nk = - 21log 21 - log - (CY~/Z)Z;Z: - (4as/3Z2)Z:2:(21- sei>2 The chemical potentials of the two components may be obtained by differentiationof AF with respect to N I and Na, respectively, a t constant temperature, pressure, and number of moles of the other component.
- p:)/RT = log Zi + ad - (rU2/Z)Zi2:(3Zs - 1) + . . . . + - (a2/2)2~S(321- 1) + . . . , (p2 - $)/RT = log
(pi
LIZ:
}
(25)
where p i and p i are the chemical potentials of the pure liquid components. The retention of terms in the first power of a alone in equations 23, 24, and 26 leads to the laws of the regular solution. The t e r n the higher powers of a represent the ordering effect of each molecule on its neighbors and the resulting deviation from random distribution. The constant VO,equal to d T , may be computed from the heat of solution a t any temperature by means of equation 24. Since there is no volume change on mixing, A33 is equal to AH,the negative of the integral heat of solution. If we designate by L the negative of the integral heat of solution per mole of an equimolal mixture of the components and retain only terms in a2 in equation 24, we obtain a=
2
11
- (1 - 8L/zRT)"*]
(26)
In the greater number of non-polar liquid mixtures L is positive, corm sponding to a negative heat of solution, and a and V Oare also positive. When a is positive there exists a critical solution temperature below which the solution separates into two phases. Although in most nonpolar liquid mixtures this temperature lies so low that the solution is unstable with respect to solid phases before it is reached, a discussion of the question is not without theoretical interest. For the coexistence of two liquid phases y and e, the conditions of heterogeneous equilibrium require
where p i y ) , p p ) , etc., are the chemical potentials of the componenta in the respective phases. From the symmetry of equations 26 remark that in either phase
ORDER AND DISORDER IN LIQUID SOLUTIONS
105
Thus equations 27 reduce to
2jf)
+ 2:')
P
1
Equations 29 are satisfied if I(
(T) l + 6 &)
= Ir (l+)
= 2 2( 0 E
2i7)
+ 24''
(30)
.&)
=:
1
where 6 is the difference, z:') - 2:') or 21') - zi'), in composition of the two phases. Equations 28 and 30 lead to the following condition on 6: 6 = tan h (B6)
B = $ [ a - (a*/z)(l- 6 9 1 (31) For two phases to coexist, equation 31 must have a real solution differing from zero, the vanishing solution corresponding to the trivial case in which the phases are identical. A non-vanishing real solution exists only if B is greater than unity. Thus there exists a critical value a. which must be exceeded for two phases to coexist, satisfying the equation CY:
cyc
- 2za0 + 42 = 0 = Z [ 1- (1- 4/~)"']
(32)
and a critical solution temperature, T,,below which two phases can coexist.
T, = Vo/kaC (33) For a regular solution in which only the linear term in a is retained, T , is equal to 0.5 Vo/k. However, from equation 32 we calculate for aC a value, 2.34, with body-centered packing, z = 8, and a critical solution temperature 0.427 V&. This is lower than the regular solution value by 15 per cent. Thus the local order established by a molecule among its
106
JOHN 0. KIRKWOOD
neighbors opposes the tendency of the solution to separate into two phases. The latter process may be regarded as a macroscopic mechanism for establishing order, satisfying the tendency of a molecule to make its environment rich in its own species when a is positive. This tendency may be partially satisfied without separation into two phases through the microscopic ordering mechanism by means of which a molecule establishes a local composition richer in its own species than the solution in bulk. A remark about the relative magnitudes of the various terms in the free energy of mixing, equation 23, is perhaps appropriate. We shall consider only the regular solution term azlzz and the quadratic term ( ~ t ~ / z ) 5 1 ~ 2 2 ~ . The latter term bears a ratio of azlfi/z to the regular solution term. With z equal to eight in an equimolar mixture a t the critical solution temperature, this ratio is equal to 0.07. Thus the local order effect produces a rather small deviation from the regular solution, smaller, indeed, than one might surmise on qualitative grounds. Rushbrooke (6) obtains an expression for the free energy of mixing in which the local order contribution involves an exponential. When the exponential is expanded and notations are brought into correspondence, his equation and equation 23 are in agreement in the linear and quadratic terms in a. However, the cubic terms in a do not agree, and indeed Rushbrooke’s term depends upon composition in an entirely different manner from our own. Since the present treatment provides an exact method for the expansion of the free energy in powers of a, within the frame of the simplifying assumptions underlying both theories, it would appear that not much significance can be attached to the higher powers of a or 1/T in the expansion of Rushbrooke’s exponential. The conclusions based upon the two treatments are, however, essentially the same. When we come to consider the influence of the local order effect on the deviation of actual solutions from regular behavior, we find that it is generally overshadowed by other effects. An analysis of the data on a large number of non-polar liquid mixtures by Professor Scatchard (9) shows that in solutions for which AE and a are positive, the entropy of mixing, in excess of the ideal value, R [ - x1 log z1- z2 log 221, is in general positive, whereas the local order contribution, - (a2/z)s:z:, from equation 24 is always negative. We shall not speculate on the nature of the other effects at the present. However, they are doubtless concealed in the vibrational factors &(c) of the partition function (equation 1). Moreover it seems certain that the Guggenheim approximation, in which these factors are treated as independent of both composition and configuration, is far too drastic to provide an exact theory. In conclusion, we remark that although the local order effect is relatively small and generally overshadowed by other influences, a n analysis of the type which has been described seems not without value, since these conclusions could scarcely have been reached by qualitative reasoning.
ORDER AND DISORDER IN LIQUID SOLUTIONS
107
REFERENCES (1) BETHE,H.: Proc. Roy. SOC.(London) AlM), 552 (1935). (2) GUGGENHEIM, E. A.: Proc. Roy. SOC.(London) A148, 304 (1935). (3) HILDEBRAND, J. H.: Solubility, American Chemical Society Monograph, 2nd edition. The Chemical Catalog Co., Inc., New York (1936). (4) HILDEBRAND, J. H., AND WOOD,S. E.: J. Chem. Phys. 1,817 (1933). (6) KIRKWOOD, J. G.: J. Cbem. Phys. 6, 70 (1938). (6) RUSHBROOKE: Proc. Roy. Soc. (London) AlM, 296 (1938). (7) SCATCHARD, G.: Chem. Rev. 8, 321 (1931). (8) SCATCHARD, G.: Trans. Faraday SOC.33, 160 (1937). (9) SCATCHARD, G.,AND HAMER, W. J.: J. Am. Chem. SOC.67, 1805 (1935).
