W. A. Oates Department of Metallurgy The University of Newcastle N. S. W., 2308 Australia
A n ideal (perfect) solution is usually defined in terms of the limiting experimental law of Raoult a* (rel. to pure A)
= p ~ / p ~=' z*
where xn is the atom or mole fraction of component A. It then follows that for a binary ideal solution
where AGM is the Gibbs free energy (or free enthalpy) of mixing. Since A G M / R T is independent of temperature for such a solution then the heat of mixing,
must be zero. It may also be shown that the volume of mixing is also zero. The equation obtained in this manner for the entropy of mixing
is identical in form to that obtained for the mixing of perfect gases by a rather different approach. It can also be obtained for the ideal solution by a simple statistical treatment. If the component atoms or molecules are distributed at random over the lattice sites, as shown in Figure 1, then the thermodynamic probability, W, is given by
where NA, and NBare the number of atoms (molecules) of components A and B. Using Stirling's approxi-
I mation and the Boltzmann equation a result for AS' in full agreement with the equation derived from the purely thermodynamic consequences of the experimental law of Raoult is obtained. If the mixing is not random, e.g., if the atoms tend to take up some ordered arrangement or tend to cluster together in groups or if there is such a disparity in size that one atom or molecule occupies more than one site, then the entropy calculated statistically does not agree with that derived from Raoult's law. It is apparent that as an alternative to the Raoult's law definition an ideal solution may be defined as one whose heat and volume of mixing is zero and whose entropy of mixing is that corresponding to a completely random mixing of the solution components. With this alternative definition Raoult's law follows as a conseqence. Since both definitions are consistent with one another it would seem unimportant as to whether Raoult's law or random mixing is used as the starting point. However, classes of solution are known in which random mixing of the constituents does not yield Raoult's law as an experimental consequence and below two examples of this lack of consistency are treated in some detail. I n view of this lack of consistency it seems preferable to define ideal solutions from the statistical standpoint. I n this way the hypothetical ideal solution a t least bears some resemblance to the real mixture in each case. Solutions Possessing Two Independent Sub-lattices
The type of solution illustrated in Figure 1is termed a substitutional solution since one type of atom can be substituted for the other type on the same crystal lattice. I n some types of solution the substitution is more restricted as illustrated schematically in Figure
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FeF3 Figure 1.
0, A
Model of a solution used in deriving Rooult'r low statistically otomr; X, B atoms.
2. Two sub-lattices exist and the atoms of one sublattice are completely independent of the atoms of the other. The model corresponds to the mixing of solid ionic salts. The strong coulombic forces existing in such a solid solution lead to an alternation of charges such that cations are surrounded by anions and vice versa, just as exist in the pure component salts. Although solid solutions of ionic salts do exist1 they are not too common. However, the same model should be a good approximation for molten salt mixtures, which are well known. Temkin2 was the first to derive the thermodynamic properties of ionic melts assuming a random distribution of cations on the cationic sublattice and of anions on the independent anionic sublattice. The derivation is given below. Consider the case of mixing A& with B,D,, where A and B mix on one sub-lattice and C and D on the other. This is a special "quasi-binary" case of mixing A&,, B,D,, and A,D, which permits independent variation of N,,, NB, Nc, and N D . The thermodynamic probability is now given by the product of the thermodynamic probabilities of the individual sub-lattices. W
=
~ N A
+ Nn)!
NA!NB!
+
(No No)! Nc!ND!
By using Stirling's approximation and the Boltemann relation
04
06
as
Xw3
1.0
Mn7C3
Figure 3. Activity of Mn& in Mn&-Fc& solid mlutions illvrtroting volidity of Temkin'r law. Data is from reference given in footnote 3.
where
is the cationic fraction of A and X+B the cationic fraction of B. X-C and X-D are the anionic fractions of C and D. Now in the quasi-binary A,C, - B,D, NA = zN~,c,
Similar relations hold for N B , NO,and N o and hence --=
k
xNA,~, In X,"
+ xNn,o, in X + B + y N ~ =Inq X.C
+ Y N B ~111DX~P
=
N A , ~In , (Xi'P(X.C)v
+ N R ~In"(X+BP(X.D)~
For the partial configurational entropy of one of the component salts
If an ideal ionic solution is defined as one whose heat of mixing is zero and whose entropy of mixing is given by the above equations then the activity of one of the components is given by a*&, = (X+A)'(X-C)r
This solution law of Temkin only becomes identical with Raoult's law if either X+* or X-C is unity and if x and y are unity. NaC1-IICl, for example, should obey Raoult's law if Ap = 0 and ASM is that corresponding to random mixing. However, Na20I'i20and NaC1-KBr would not be expected to obey Raoult's law even if the heats and entropies of mixing were ideal. Several examples might be cited which illustrate Temkin's ideal solution law for mixtures of this type. In Figure 3 results are shown for the mixing of F e G with n.In7C3.3 I t is anticipated that mixtures of these carbides will be approximately ideal since Fe-Mn solid solutions show only small
' HOVI,V., Acta Melallu~gica,6, 254 (1958). ' TEMKIN, IN., Acta Physieoehim URSS, 20, 411 (1945). Figure 2.
Model of o solution used in deriving Temkin'r law. 0, A atoms; atom$; //,D otomr.
X, B otoms; 0 , C
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BUTLER, J. F., MCCABE,C. L.,A N D PAXTON, 1%.W., T. Met.
Sac. A.I.M.E., 221, 479 (1961).
Figure 4. Model of a solution used in deriving Langrnuir's law. ~titiolrite; @, occupied intentitiol site.
0, inter-
deviations from ~ a o u M ' s law. Note that these carbides are by no means ionic. The important property from the mixing law point of view is the independent metal and carbon sub-lattices. The Temkin equation for the activity of M n G is and it may be seen in Figure 3 that the solutions approximately obey this' law. The deviations from Raoult's law are quite marked. Systems involving two different species on both sublattices are generally more complicated. Cationanion interactions are such that definite neighbor preferences will normally occur, and hence large deviations from random mixing are expected. Complications can also arise when dealing with salts of different charge type, e.g., in KC1-CdC12, for every Cd2+ added from ClC12 to replace two I
