Physical Chemistry Third Edition

192 4 The Thermodynamics of Real Systems

EXAMPLE4.23

At constant temperature and pressure, the volume of a solution made from component 1 and

component 2 is represented by

Vb 1 n 1 +b 2 n 2 +b 12 n 1 n 2 +b 11 n^21 +b 22 n^22

wheren 1 andn 2 are the amounts of the two components in moles and theb’s are constants

at constant temperature and pressure. Find an expression forV 2.

Solution

V 2 

(

∂V

∂n 2

)

T,P,n 1

b 2 +b 12 n 1 + 2 b 22 n 2

The Method of Intercepts

This method is a graphical method for the determination of partial molar quantities in

a two-component solution. From Euler’s theorem, the mean molar volume is given by

Vmx 1 V 1 +x 2 V 2 (4.6-14)

where thex’s are the mole fractions. Sincex 2  1 −x 1 in a two-component system,

Vmx 1 V 1 +(1−x 1 )V 2 (V 1 −V 2 )x 1 +V 2 (4.6-15)

Figure 4.2 showsVm, the mean molar volume of a solution of ethanol (component

1) and water (component 2), as a function ofx 1. Letx′ 1 be a particular value ofx 1

for which we desire the values of the partial molar quantitiesV 1 andV 2. To apply

the method, we draw a tangent line to the curve atx 1 x′ 1 , as shown in the figure.

The intercepts of this line at the edges of the figure give the values of the two partial

molar quantities for the compositionx 1 x′ 1. A proof of the validity of this method is

contained in Appendix D.

A modified version of the method generally gives better accuracy. In this method

we make a graph of the change in the mean molar quantity on mixing (forming the

solution from the pure substances):

∆Vm,mixVm−(x 1 Vm,1∗ +x 2 Vm,2∗ ) (definition) (4.6-16)

whereVm,1∗ is the molar quantity of pure substance 1 and similarly for substance 2.

Sincex 2  1 −x 1 , we can write

∆Vm,mixVm−Vm,2∗ +x 1 (Vm,1∗ −Vm,2∗ ) (4.6-17)

One plots experimental values of∆Vm,mixand constructs the tangent line atx 1 x′ 1.

The intercepts of the tangent line are given by

left interceptV 1 (x′ 1 )−Vm,1∗ (4.6-18)

right interceptV 2 (x′ 1 )−Vm,2∗ (4.6-19)