192
4 The Thermodynamics of Real Systems
E X A M P L E 4.23
At constant temperature and pressure, the volume of a solution made from component 1 and
component 2 is represented by
V b 1 n 1 + b 2 n 2 + b 12 n 1 n 2 + b 11 n 2
1 + b 22 n 2
2
where n 1 and n 2 are the amounts of the two components in moles and the b’s are constants
at constant temperature and pressure. Find an expression for V 2 .
Solution
V 2
∂V
∂n 2
T ,P,n 1
b 2 + b 12 n 1 + 2b 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
V m x 1 V 1 + x 2 V 2
(4.6-14)
where the x’s are the mole fractions. Since x 2 1 − x 1 in a two-component system,
V m x 1 V 1 + (1 − x 1 )V 2 (V 1 − V 2 )x 1 + V 2
(4.6-15)
Figure 4.2 shows V m , the mean molar volume of a solution of ethanol (component
1) and water (component 2), as a function of x 1 . Let x
1 be a particular value of x 1
for which we desire the values of the partial molar quantities V 1 and V 2 . To apply
the method, we draw a tangent line to the curve at x 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 composition x 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):
∆V m,mix V m − (x 1 V
∗
m,1 + x 2 V
∗
m,2 ) (definition)
(4.6-16)
where V ∗
m,1 is the molar quantity of pure substance 1 and similarly for substance 2.
Since x 2 1 − x 1 , we can write
∆V m,mix V m − V
∗
m,2 + x 1 (V
∗
m,1 − V
∗
m,2 )
(4.6-17)
One plots experimental values of ∆V m,mix and constructs the tangent line at x 1 x
1 .
The intercepts of the tangent line are given by
left intercept V 1 (x
1 ) − V
∗
m,1
(4.6-18)
right intercept V 2 (x
1 ) − V
∗
m,2
(4.6-19)
4 The Thermodynamics of Real Systems
E X A M P L E 4.23
At constant temperature and pressure, the volume of a solution made from component 1 and
component 2 is represented by
V b 1 n 1 + b 2 n 2 + b 12 n 1 n 2 + b 11 n 2
1 + b 22 n 2
2
where n 1 and n 2 are the amounts of the two components in moles and the b’s are constants
at constant temperature and pressure. Find an expression for V 2 .
Solution
V 2
∂V
∂n 2
T ,P,n 1
b 2 + b 12 n 1 + 2b 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
V m x 1 V 1 + x 2 V 2
(4.6-14)
where the x’s are the mole fractions. Since x 2 1 − x 1 in a two-component system,
V m x 1 V 1 + (1 − x 1 )V 2 (V 1 − V 2 )x 1 + V 2
(4.6-15)
Figure 4.2 shows V m , the mean molar volume of a solution of ethanol (component
1) and water (component 2), as a function of x 1 . Let x
1 be a particular value of x 1
for which we desire the values of the partial molar quantities V 1 and V 2 . To apply
the method, we draw a tangent line to the curve at x 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 composition x 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):
∆V m,mix V m − (x 1 V
∗
m,1 + x 2 V
∗
m,2 ) (definition)
(4.6-16)
where V ∗
m,1 is the molar quantity of pure substance 1 and similarly for substance 2.
Since x 2 1 − x 1 , we can write
∆V m,mix V m − V
∗
m,2 + x 1 (V
∗
m,1 − V
∗
m,2 )
(4.6-17)
One plots experimental values of ∆V m,mix and constructs the tangent line at x 1 x
1 .
The intercepts of the tangent line are given by
left intercept V 1 (x
1 ) − V
∗
m,1
(4.6-18)
right intercept V 2 (x
1 ) − V
∗
m,2
(4.6-19)