Physical Chemistry Third Edition

176 4 The Thermodynamics of Real Systems

Solution

Let 1 atm be denoted byP◦atmwith a similar symbol for the 1-atm standard-state molar

Gibbs energies. We write

Gm(T,P)G◦m(T)+RTln (P/P◦)

G◦m(T)+RTln (P◦atm/P◦)+RTln (P/P◦atm)

This equation has the correct form if

G◦matmG◦m(T)+RTln (P◦atm/P◦)

G◦m(T)+RTln (1.01325)

G◦m(T)+(0.013 16)RT

At 298.15 K,

G◦matm−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 thefugacity,f, which has the dimensions

of pressure:

Gm(T,P)G◦m(T)+RTln

(

f

P◦

)

(definition of the fugacityf) (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 pressureP◦. We can obtain an

expression for the molar Gibbs energy of a real gas as follows:

Gm,real(T,P′)−G◦m(T)Gm,real(T,P′)

− lim

P′′→ 0

[

Gm,real(T,P′′)−Gm,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∆Gfor changing the pressure of

the real gas from 0 toP′at constant temperature:

(first two terms)

∫P′

0

Vm,realdP (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◦

Vm,idealdP

∫ 0

P◦

RT

P

dP (4.4-9)