RELATIONSHIPS BETWEEN PARTIAL MOLAR QUANTITIES OF DIFFERENT COMPONENTS

9.4 RELATIONSHIPS BETWEEN PARTIAL MOLAR QUANTITIES OF DIFFERENT COMPONENTS

Extensive thermodynamic properties at constant temperature and pressure are homo- geneous functions of degree 1 of the mole numbers. From Euler’s theorem [Equation (2.33)] for a homogeneous function of degree n

For a general two-component system and any extensive thermodynamic property J, we can write

n 1 þn 2 ¼J (9 :26)

@n 1 T ,P,n 2 @n 2 T ,P,n 1

From the definition of partial molar quantities [Equation (9.12)], Equation (9.26) can

be written as

(9 :27) Like J, both J m1 and J m2 are functions of T and P and the system composition.

n 1 J m1 þn 2 J m2 ¼J

Although the function J is a homogeneous function of the mole numbers of degree

1, the partial molar quantities, J m1 and J m2 are homogeneous functions of degree 0; that is, the partial molar quantities are intensive variables. This statement can be proved by the following procedure. Let us differentiate both sides of Equation (2.32) with respect to x:

@f (lx, ly, lz)

n @f (x, y, z) ¼l

@x

@x

217 If we divide both sides by l, we obtain

9.4 RELATIONSHIPS BETWEEN PARTIAL MOLAR QUANTITIES

1 @f (lx, ly, lz)

We can rewrite Equation (9.29) in the common alternative notation for derivatives:

(9 :30) Equation (9.30) is an analoge of Equation (2.32) for the first derivative function, and

f 0 (lx, ly, lz)

¼l n f 0 (x, y, z)

it defines the degree of homogeneity of the partial derivative function f 0 and states that its degree of homogeneity is n 21, that is, one less than the degree of homogeneity n

of the original function f. Because they are homogeneous functions of the mole numbers of degree 0, the partial molar quantities, although still functions of n 1 and n 2 , are functions only of the ratio n 1 /n 2 , and thus, they are independent of the size of the system. Differentiation of Equation (9.27) leads to

dJ ¼n 1 dJ m1 þJ m1 dn 1 þn 2 dJ m2 þJ m2 dn 2 (9 :31) As at constant pressure and temperature J is a function of two variables, f (n 1 ,n 2 ), the

following equation is valid for the total differential:

¼J m1 dn 1 þJ m2 dn 2 (9 :32) If we equate Equations (9.31) and (9.32), we obtain

n 1 dJ m1 þJ m1 dn 1 þn 2 dJ m2 þJ m2 dn 2 ¼J m1 dn 1 þJ m2 dn 2

or

(9 :33) Equation (9.33) is one of the most useful relationships between partial molar quan-

n 1 dJ m1 þn 2 dJ m2 ¼0

tities. When applied to the chemical potential, it becomes

(9 :34) which is called the Gibbs-Duhem equation at constant temperature and pressure. This

n 1 dm 1 þn 2 dm 2 ¼0

equation shows that in a two-component system, only one of the chemical potentials can vary independently at constant T and P.

THERMODYNAMICS OF SYSTEMS OF VARIABLE COMPOSITION

It follows from Equation (9.33) that

This equation is very useful in deriving certain relationships between the partial molar quantity for a solute and that for the solvent. An analogous equation can be written for

the derivatives with respect to dn 2 .

Partial Molar Quantities for Pure Phase If a system is a single, pure phase, a graph of any extensive thermodynamic property

plotted against mole number at constant temperature and pressure gives a straight line passing through the origin (again neglecting surface effects). The data for the volume of water are given in Figure 9.1. The slope of the line gives the partial molar volume

Figure 9.1. Volume of a pure phase at specified temperature and pressure. Data for water at 273.16 K and 100 kPa from the NIST WebBook on saturation properties for water. (http: // www.webbook.nist.gov /chemistry)

219 in which V † m is the molar volume of the pure phase. Similarly, for any extensive

9.5 ESCAPING TENDENCY

thermodynamic property J of a pure phase

J J m ¼ ¼J † m

9.5 ESCAPING TENDENCY Chemical Potential and Escaping Tendency

G. N. Lewis proposed the term “escaping tendency” to give a strong kinetic-molecular flavor to the concept of the chemical potential. Let us consider two solutions of iodine,

in water and carbon tetrachloride, which have reached equilibrium with each other at a fixed pressure and temperature (Fig. 9.2). In this system at equilibrium, let us carry out

a transfer of an infinitesimal quantity of iodine from the water phase to the carbon tet- rachloride phase. On the basis of Equation (9.17), we can say that

m I 2 (H 2 O) dn I 2 (H 2 O) þm I 2 (CCl 4 ) dn I 2 (CCl 4 ) ¼0 (9 :38) In this closed system, any loss of iodine from the water phase is accompanied by an

equivalent gain in the carbon tetrachloride thus, Hence

I 2 (H 2 O) ¼ dn I 2 (CCl 4 )

m I 2 (H 2 O) dn I 2 (H 2 O) þm I 2 (CCl 4 ) [

I 2 (H 2 O) ] ¼0 (9 :40)

Figure 9.2. Schematic diagram of equilibrium distribution of iodine between water and carbon tetrachloride at fixed temperature and pressure.

THERMODYNAMICS OF SYSTEMS OF VARIABLE COMPOSITION

It follows that

(9 :41) for this system in equilibrium at constant pressure and temperature. Thus, at

m I 2 (H 2 O) ¼m I 2 (CCl 4 )

equilibrium, the chemical potential of the iodine is the same in all phases in which it is present, or the escaping tendency of the iodine in the water is the same as that of the iodine in the carbon tetrachloride. We can return to the analogy with gravita- tional potential; stating that the iodine in the two phases have the same chemical potential is analogous to saying that two bodies at the same altitude have the same gravitational potential.

It also may be helpful to consider the situation in which the iodine will diffuse spontaneously (at constant pressure and temperature) from the water into the carbon tetrachloride, a case in which the concentration in the water phase is greater than that which would exist in equilibrium with the carbon tetrachloride phase. From Equation (9.17), we can write

m I 2 (H 2 O) dn I 2 (H 2 O) þm I 2 (CCl 4 ) dn I 2 (CCl 4 ) ,0 (9 :42) For the spontaneous diffusion of iodine, Equation (9.39) is valid in this closed

system. Hence,

I 2 (H 2 O) ],0 (9 :43) or

m I 2 (H 2 O) dn I 2 (H 2 O) þm I 2 (CCl 4 ) [

(9 :44) As the water loses iodine,

[m I 2 (H 2 O)

I 2 (CCl 4 ) ]dn I 2 (H 2 O) ,0

(9 :45) That is, dn is a negative number. In such a case, Equation (9.44) is valid only if the

dn I 2 (H 2 O) ,0

difference in chemical potentials is a positive number. Therefore

(9 :46) Thus, we may say that the escaping tendency of the iodine is greater in the water than

m I 2 (H 2 O) .m I 2 (CCl 4 )

in the carbon tetrachloride phase, and the chemical potentials in the two phases describe the spontaneous direction of transport from one phase to the other.

In general, when the chemical potential of a given species is greater in one phase than in a second, we also shall say that the escaping tendency is greater in the former case than in the latter. The escaping tendency thus is a qualitative phrase, which cor- responds to the property given precisely by the chemical potential. Therefore, the escaping tendency or the chemical potential can be used to determine the spontaneous direction of transport.

9.6 CHEMICAL EQUILIBRIUM IN SYSTEMS OF VARIABLE COMPOSITION