4.5 Multicomponent Systems
187
E X A M P L E 4.19
Find the expression for the partial molar Helmholtz energy of a one-component ideal gas as
a function of temperature and pressure.
Solution
A A m µ − PV m µ ◦ + RT ln
P
P ◦
− P
RT
P
µ ◦ + RT ln
P
P ◦
− RT
According to Dalton’s law of partial pressures, each gas in a mixture of ideal gases
behaves as though it were alone in the container. Equation (4.5-23) applies to any
substance in an ideal gas mixture:
µ i µ ◦
i + RT ln
P i
P ◦
(substance i in an
ideal gas mixture)
(4.5-26)
where µ ◦
i is the chemical potential of substance i in the standard state at pressure P ◦
and P i is its partial pressure. All of the other equations for one-component ideal gases
apply as well.
E X A M P L E 4.20
Calculate µ i − µ ◦
i for argon gas in dry air at 298.15 K and 1.000 atm, assuming that the gases
are ideal. The mole fraction of argon is 0.00934.
Solution
µ i − µ ◦
i RT ln
P i
P ◦
(8.3145 J K −1 mol −1 )(298.15) ln
(0.00934 atm)(101325 Pa atm −1 )
100000 Pa
−11550 J mol −1
Exercise 4.17
a. Calculate µ i − µ ◦
i for argon gas at 298.15 K and a partial pressure of 1.000 atm.
b. Calculate µ i − µ ◦
i for argon gas at 298.15 K and a partial pressure of 10.00 atm.
In a mixture of gases that cannot be assumed to be ideal, we define f i , the fugacity
of component i, by the relation
µ i µ
◦
i + RT ln
f i
P ◦
(definition of f i )
(4.5-27)
where µ ◦
i is the same standard-state chemical potential as for the pure gas: the hypothetical ideal-gas state at pressure P ◦ and whatever temperature is being considered.
We will not discuss the evaluation of the fugacity in a mixture of nonideal gases.
187
E X A M P L E 4.19
Find the expression for the partial molar Helmholtz energy of a one-component ideal gas as
a function of temperature and pressure.
Solution
A A m µ − PV m µ ◦ + RT ln
P
P ◦
− P
RT
P
µ ◦ + RT ln
P
P ◦
− RT
According to Dalton’s law of partial pressures, each gas in a mixture of ideal gases
behaves as though it were alone in the container. Equation (4.5-23) applies to any
substance in an ideal gas mixture:
µ i µ ◦
i + RT ln
P i
P ◦
(substance i in an
ideal gas mixture)
(4.5-26)
where µ ◦
i is the chemical potential of substance i in the standard state at pressure P ◦
and P i is its partial pressure. All of the other equations for one-component ideal gases
apply as well.
E X A M P L E 4.20
Calculate µ i − µ ◦
i for argon gas in dry air at 298.15 K and 1.000 atm, assuming that the gases
are ideal. The mole fraction of argon is 0.00934.
Solution
µ i − µ ◦
i RT ln
P i
P ◦
(8.3145 J K −1 mol −1 )(298.15) ln
(0.00934 atm)(101325 Pa atm −1 )
100000 Pa
−11550 J mol −1
Exercise 4.17
a. Calculate µ i − µ ◦
i for argon gas at 298.15 K and a partial pressure of 1.000 atm.
b. Calculate µ i − µ ◦
i for argon gas at 298.15 K and a partial pressure of 10.00 atm.
In a mixture of gases that cannot be assumed to be ideal, we define f i , the fugacity
of component i, by the relation
µ i µ
◦
i + RT ln
f i
P ◦
(definition of f i )
(4.5-27)
where µ ◦
i is the same standard-state chemical potential as for the pure gas: the hypothetical ideal-gas state at pressure P ◦ and whatever temperature is being considered.
We will not discuss the evaluation of the fugacity in a mixture of nonideal gases.