DEVIATIONS FROM IDEALITY IN TERMS OF EXCESS THERMODYNAMIC FUNCTIONS
16.7 DEVIATIONS FROM IDEALITY IN TERMS OF EXCESS THERMODYNAMIC FUNCTIONS
Various functions have been used to express the deviation of observed behavior of solutions from that expected for ideal systems. Some functions, such as the activity coefficient, are most convenient for measuring deviations from ideality for a particu- lar component of a solution. However, the most convenient measure for the solution as a whole, especially for mixtures of nonelectrolytes, is the series of excess functions (1) (3), which are defined in the following way.
We have derived an expression for the free energy of mixing two pure substances to form one mole of an ideal solution [Equation (14.35)],
DG I mix,m ¼X 1 RT ln X 1 þX 2 RT ln X 2 (16 :54) In actual systems, the observed value for the free energy of mixing DG mix,m may
differ from DG I . We define this difference, the excess free energy G mix,m E m as DG E ¼ DG
l RT ln X 1 2 RT ln X 2 (16 :55) With a derivation analogous to that of Equation (14.35), we find that
m mix,m
mix,m ¼ DG mix,m
DG mix,m ¼X 1 RT ln a 1 þX 2 RT ln a 2 (16 :56) where the activities are based on Raoult’s-law standard states.
ACTIVITIES, EXCESS GIBBS FUNCTIONS, STANDARD STATES FOR NONELECTROLYTES
Thus,
G E a 1 m 2 ¼X 1 RT ln þX 2 RT ln
¼X 1 RT ln g 1 þX 2 RT ln g 2 (16 :57)
For the excess enthalpy H E m , because DH I mix is zero
H E m ¼ DH mix :m
I mix,m
¼ DH mix,m ¼X 1 (H m1
m1 † ) þX 2 (H m2 m,2 ) (16 :58)
For the excess volume V E m , because DV I mix is zero,
V m E ¼ DV mix,m ¼X 1 (V m1
m1 † ) þX 2 (V m2 m2 † ) (16 :59)
Similarly, we define the excess entropy S E m as
S E m I ¼ DS mix,m mix,m ¼ DS mix,m þX l R ln X 1 þX 2 R ln X 2 (16 :60)
It can be shown that the usual relationships between temperature coefficients of the Gibbs function and entropy or enthalpy, respectively, also apply if stated for excess functions. Thus,
Excess thermodynamic functions can be evaluated most readily when the vapor pressures of both solute and solvent in a solution can be measured.
Representation of G E m as a Function of Composition Redlich and Kister [4] suggested a convenient way to represent G E m as a function of
composition that permits convenient classification of various kinds of deviation from ideality. From Equation (16.57)
¼X 1 ln g
1 1 ) ln g 2 (16 :63)
RT
16.7 DEVIATIONS FROM IDEALITY IN TERMS OF EXCESS THERMODYNAMIC FUNCTIONS
375 If we differentiate Equation (16.63) with respect to X 1 , and simplify using the Gibbs –
Duhem relationship [Equation (9.34)], the result is
¼ ln 1 (16 :64)
dX 1 RT
As G E m must equal 0 when X 1 ¼ 0 ( pure solute) and when X 1 ¼ 1 ( pure solvent),
Thermodynamic consistency of data on solutions can be tested by plotting ln (g 1 /g 2 ) against X 1 and seeing whether the area between X 1 ¼ 0 and 0.5 is equal and opposite in sign to the area between 0.5 and 1.0. Such a plot is indicated in Figure 16.5 for solutions of methyl t-butyl ether and chloroform at 313.5 K (5).
Redlich and Kister suggested that solutions of different degrees of nonideality should be represented by a power series of the form
G E m ¼X 1 X 2 [B þ C(X 1 2 ) þ D(X 1 2 ) 2 RT
¼X X [B þ C(2X
Figure 16.5. A plot of ln(g 1 /g 2 ) for solutions of methyl t-butylether (1) and chloform (2) at 313.5 K (5).
ACTIVITIES, EXCESS GIBBS FUNCTIONS, STANDARD STATES FOR NONELECTROLYTES
where the factor X 1 X 2 is present so that the quantity will be equal to 0 at X 1 ¼ 0 and at
X 2 ¼ 0, and B, C, and D are coefficients that depend on the temperature and pressure but not on the composition. Such a series was used by Scatchard [6], after a sugges- tion of Guggenheim [7]. The factor (X 1 2X 2 ) is the variable of the power series because it is antisymmetrical with respect to interchange of components. If the solution is ideal, that is, it follows Raoult’s law, G E m ¼ 0, and all coefficients are equal to 0. If all coefficients other than B are equal to 0,
1 X 2 (16 :67) where B is related to the interaction energy between components 1 and 2. These sol-
¼ BX