176
4 The Thermodynamics of Real Systems
Solution
Let 1 atm be denoted by P ◦atm with a similar symbol for the 1-atm standard-state molar
Gibbs energies. We write
G m (T , P) G ◦
m (T ) + RT ln (P/P ◦ )
G ◦
m (T ) + RT ln (P ◦atm /P ◦ ) + RT ln (P/P ◦atm )
This equation has the correct form if
G ◦atm
m
G ◦
m (T ) + RT ln (P ◦atm /P ◦ )
G ◦
m (T ) + RT ln (1.01325)
G ◦
m (T ) + (0.013 16)RT
At 298.15 K,
G ◦atm
m
− G ◦
m (T ) 32.16 J mol −1 0.03216 kJ mol −1
The Gibbs Energy of a Real Gas. Fugacity
When a gas requires corrections for nonideality we write a new equation in the same
form as Eq. (4.4-5), replacing the pressure by the fugacity, f , which has the dimensions
of pressure:
G m (T , P) G ◦
m (T ) + RT ln
f
P ◦
(definition of the fugacity f )
(4.4-6)
The fugacity plays the same role in determining the molar Gibbs energy of a real gas as
does the pressure in determining the molar Gibbs energy of an ideal gas. The quantity
G ◦
m (T ) is the molar Gibbs energy of the gas in its standard state. The standard state of
a real gas is defined to be the corresponding ideal gas at pressure P ◦ . We can obtain an
expression for the molar Gibbs energy of a real gas as follows:
G m,real (T , P
) − G
◦
m (T ) G m,real (T , P
)
− lim
P →0
G m,real (T , P
) − G m,ideal (T , P
)
− G
◦
m (T )
(4.4-7)
where we have added two terms that cancel because the real gas and the corresponding
ideal gas become identical in the limit of zero pressure. From Eq. (4.4-1), the first two
terms on the right-hand side of Eq. (4.4-7) represent ∆G for changing the pressure of
the real gas from 0 to P at constant temperature:
(first two terms)
P
0
V m,real dP
(4.4-8)
From Eq. (4.4-3), the last two terms in the right-hand side of Eq. (4.4-7) are equal to
(last two terms)
0
P ◦
V m,ideal dP
0
P ◦
RT
P
dP
(4.4-9)
4 The Thermodynamics of Real Systems
Solution
Let 1 atm be denoted by P ◦atm with a similar symbol for the 1-atm standard-state molar
Gibbs energies. We write
G m (T , P) G ◦
m (T ) + RT ln (P/P ◦ )
G ◦
m (T ) + RT ln (P ◦atm /P ◦ ) + RT ln (P/P ◦atm )
This equation has the correct form if
G ◦atm
m
G ◦
m (T ) + RT ln (P ◦atm /P ◦ )
G ◦
m (T ) + RT ln (1.01325)
G ◦
m (T ) + (0.013 16)RT
At 298.15 K,
G ◦atm
m
− G ◦
m (T ) 32.16 J mol −1 0.03216 kJ mol −1
The Gibbs Energy of a Real Gas. Fugacity
When a gas requires corrections for nonideality we write a new equation in the same
form as Eq. (4.4-5), replacing the pressure by the fugacity, f , which has the dimensions
of pressure:
G m (T , P) G ◦
m (T ) + RT ln
f
P ◦
(definition of the fugacity f )
(4.4-6)
The fugacity plays the same role in determining the molar Gibbs energy of a real gas as
does the pressure in determining the molar Gibbs energy of an ideal gas. The quantity
G ◦
m (T ) is the molar Gibbs energy of the gas in its standard state. The standard state of
a real gas is defined to be the corresponding ideal gas at pressure P ◦ . We can obtain an
expression for the molar Gibbs energy of a real gas as follows:
G m,real (T , P
) − G
◦
m (T ) G m,real (T , P
)
− lim
P →0
G m,real (T , P
) − G m,ideal (T , P
)
− G
◦
m (T )
(4.4-7)
where we have added two terms that cancel because the real gas and the corresponding
ideal gas become identical in the limit of zero pressure. From Eq. (4.4-1), the first two
terms on the right-hand side of Eq. (4.4-7) represent ∆G for changing the pressure of
the real gas from 0 to P at constant temperature:
(first two terms)
P
0
V m,real dP
(4.4-8)
From Eq. (4.4-3), the last two terms in the right-hand side of Eq. (4.4-7) are equal to
(last two terms)
0
P ◦
V m,ideal dP
0
P ◦
RT
P
dP
(4.4-9)