220
5 Phase Equilibrium
the following way since the curve does not represent a single-valued function:
G m (e) − G m (a) 0
P b
P a
V m dP +
P c
P b
V m dP +
P d
P c
V m dP +
P e
P d
V m dP (5.4-7)
The area to the right of the vertical line segment between points a and c is called area 1
and is equal to
area 1
P b
P a
V m dP −
P b
P c
V m dP
P b
P a
V m dP +
P c
P b
V m dP
(5.4-8)
The area to the left of the line segment between points c and e is called area 2 and is
equal to
area 2
P c
P d
V m dP −
P e
P d
V m dP −
P d
P c
V m dP −
P e
P d
V m dP
(5.4-9)
Comparison of Eq. (5.4-7) with Eqs. (5.4-8) and (5.4-9) shows that when G m (e) −
G m (a) 0, area 1 and area 2 are equal to each other. The adjustment of the locations
of points a and e to make these areas equal gives a tie line between the coexisting liquid
and gas states and is known as the Maxwell equal-area construction.
This construction is named for the same
James Clerk Maxwell who devised the
Maxwell equations of electrodynamics
and the Maxwell relations of
thermodynamics and contributed to the
founding of gas kinetic theory.
The van der Waals equation of state provides a qualitatively correct description
of the liquid–vapor transition when the equal-area construction is applied to it. The
other common equations of state provide varying degrees of accuracy in describing the
liquid–vapor transition when the equal-area construction is applied to them. Gibbons
and Laughton obtained good agreement with experiment using their modification of
the Redlich–Kwong equation of state (see Table 1.1). 5
The Temperature Dependence of the Gibbs Energy Change
The temperature derivative of the Gibbs energy is given by Eq. (4.2-20):
∂G
∂T
P,n
−S
(5.4-10)
However, we cannot use this equation to calculate a value of ∆G for a temperature
change because the value of the entropy can always have an arbitrary constant added to
it without any physical effect. We can write an analogous equation for the temperature
dependence of ∆G for an isothermal process:
∂∆G
∂T
P,n
−∆S
(5.4-11)
The interpretation of this equation is that although ∆G pertains to an isothermal process,
it gives the comparison of isothermal processes at different temperatures.
A useful version of this equation can be written
∂(∆G/T )
∂T
P,n
1
T
∂∆G
∂T
P,n
−
∆G
T 2 −
∆S
T
−
∆G
T 2 −
∆H
T 2
(5.4-12)
5 R. M. Gibbons and A. P. Laughton, J. Chem. Soc., Faraday Trans. 2, 80, 1019 (1984).
5 Phase Equilibrium
the following way since the curve does not represent a single-valued function:
G m (e) − G m (a) 0
P b
P a
V m dP +
P c
P b
V m dP +
P d
P c
V m dP +
P e
P d
V m dP (5.4-7)
The area to the right of the vertical line segment between points a and c is called area 1
and is equal to
area 1
P b
P a
V m dP −
P b
P c
V m dP
P b
P a
V m dP +
P c
P b
V m dP
(5.4-8)
The area to the left of the line segment between points c and e is called area 2 and is
equal to
area 2
P c
P d
V m dP −
P e
P d
V m dP −
P d
P c
V m dP −
P e
P d
V m dP
(5.4-9)
Comparison of Eq. (5.4-7) with Eqs. (5.4-8) and (5.4-9) shows that when G m (e) −
G m (a) 0, area 1 and area 2 are equal to each other. The adjustment of the locations
of points a and e to make these areas equal gives a tie line between the coexisting liquid
and gas states and is known as the Maxwell equal-area construction.
This construction is named for the same
James Clerk Maxwell who devised the
Maxwell equations of electrodynamics
and the Maxwell relations of
thermodynamics and contributed to the
founding of gas kinetic theory.
The van der Waals equation of state provides a qualitatively correct description
of the liquid–vapor transition when the equal-area construction is applied to it. The
other common equations of state provide varying degrees of accuracy in describing the
liquid–vapor transition when the equal-area construction is applied to them. Gibbons
and Laughton obtained good agreement with experiment using their modification of
the Redlich–Kwong equation of state (see Table 1.1). 5
The Temperature Dependence of the Gibbs Energy Change
The temperature derivative of the Gibbs energy is given by Eq. (4.2-20):
∂G
∂T
P,n
−S
(5.4-10)
However, we cannot use this equation to calculate a value of ∆G for a temperature
change because the value of the entropy can always have an arbitrary constant added to
it without any physical effect. We can write an analogous equation for the temperature
dependence of ∆G for an isothermal process:
∂∆G
∂T
P,n
−∆S
(5.4-11)
The interpretation of this equation is that although ∆G pertains to an isothermal process,
it gives the comparison of isothermal processes at different temperatures.
A useful version of this equation can be written
∂(∆G/T )
∂T
P,n
1
T
∂∆G
∂T
P,n
−
∆G
T 2 −
∆S
T
−
∆G
T 2 −
∆H
T 2
(5.4-12)
5 R. M. Gibbons and A. P. Laughton, J. Chem. Soc., Faraday Trans. 2, 80, 1019 (1984).