THERMODYNAMICS OF TRANSFER OF A COMPONENT FROM ONE IDEAL SOLUTION TO ANOTHER

14.3 THERMODYNAMICS OF TRANSFER OF A COMPONENT FROM ONE IDEAL SOLUTION TO ANOTHER

We can represent the transfer process by the equation component i(A) ¼ component i(A 0 )

(14 :21) where A represents one solution and A 0 represents the other. The thermodynamic

properties of the two solutions are shown in Table 14.1. If we transfer dn i moles of component i from solution A to solution A 0 , we can write from Equation (11.15) that

dG m ¼m 0 i dn i

i ¼ (m i 0 dn i

i )dn i (14 :22)

THE IDEAL SOLUTION

TABLE 14.1. Thermodynamic Properties of Two Ideal Solutions (A and A 0 ) of Different Mole Fractions, X i and X i 0 , Prepared from the Same Components

Property of Component i A A 0 Mole fraction

X i 0 Fugacity

f i 0 ¼f i † X i 0 Chemical potential

f i ¼f † i X i

m i 0 ¼ m8 i þ RT ln ( f i 0 /f 8) Enthalpy

m i ¼ m8 i þ RT ln ( f i /f 8)

H 0 mi ¼H † mi Entropy

@n i T ,P,X i

Then DG m for the transfer process can be obtained by integrating Equation (14.23) from n ¼ 0 to n ¼ 1. That is,

0 T ,P,X i

If we assume that the solutions are of large enough volume that the transfer does not change the compositions and chemical potentials, that is the infinite copy model, then the integral in Equation (11.24) can be evaluated and

Furthermore, the enthalpy change is

DH ¼ H 0 mi

mi

(14 :26) Similarly, the volume change is

¼H †

mi † mi ¼0

DV ¼ V 0 mi

mi

¼V †

mi † mi ¼0

325 Finally, the entropy change is

14.4 THERMODYNAMICS OF MIXING

and from Equation (14.7) and Equation (14.28)

14.4 THERMODYNAMICS OF MIXING In a similar way we can consider an integral mixing process for the formation of an

ideal solution from the components, as illustrated in Figure 14.2. The mixing process can be represented by the equation

n 1 moles component 1(pure) þn 2 moles component 2(pure) ¼ solution containing [n 1 moles component 1(X 1 )

(14 :30) Thus, DJ ¼ J final

þn 2 moles component 2(X 2 )]

Then the change in the Gibbs function is

DG ¼ n 1 m 1 þn 2 m 2 1 m † 1 2 m † 2

¼n 1 [m8 1 þ RT ln X 1 † ] þn 2 [m8 2 þ RT ln X 2 1 † 2 ] ¼n 1 RT ln X 1 þn 2 RT ln X 2 (14 :32)

because m 1 † ¼ m8 1 and m † 2 ¼ m8 2 .

For the enthalpy change

DH ¼ n 1 H m1 þn 2 H m2

H m1 † 2 m2

(14 :33) because H mi ¼H mi † .

THE IDEAL SOLUTION

Figure 14.2. Thermodynamics of formation of ideal solution from pure components.

As, for an isothermal change, from Equation (7.26)

1 R ln X 1 2 R ln X 2 (14 :34) The values for the formation of an ideal solution are identical with those we derived

for mixing ideal gases in Chapter 10. Thus, a mixture of ideal gases is a special case of an ideal solution. The equations that we have derived are equally applicable to solid, liquid, and gaseous solutions as long as no phase change occurs in the

mixing process. For the special case when n 1 þn 2 ¼ 1, the thermodynamic changes are

n 1 RT ln X 1 þn 2 RT ln X 2 DG m ¼ n 1 þn 2

¼X 1 RT ln X 1 þX 2 RT ln X 2 (14 :35)

DH m ¼0

327 and

14.5 EQUILIBRIUM BETWEEN A PURE SOLID AND AN IDEAL LIQUID SOLUTION

DS m

1 R ln X 1 2 R ln X 2 (14 :37)