Physical Chemistry Third Edition

4.5 Multicomponent Systems 187

EXAMPLE4.19

Find the expression for the partial molar Helmholtz energy of a one-component ideal gas as

a function of temperature and pressure.

Solution

AAmμ−PVmμ◦+RTln

(

P

P◦

)

−P

RT

P

μ◦+RTln

(

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:

μiμ◦i+RTln

(

Pi

P◦

)

(substanceiin an

ideal gas mixture)

(4.5-26)

whereμ◦iis the chemical potential of substanceiin the standard state at pressureP◦

andPiis its partial pressure. All of the other equations for one-component ideal gases

apply as well.

EXAMPLE4.20

Calculateμi−μ◦ifor 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

μi−μ◦iRTln

(

Pi

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.Calculateμi−μ◦ifor argon gas at 298.15 K and a partial pressure of 1.000 atm.

b.Calculateμi−μ◦ifor 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 definefi, the fugacity

of componenti, by the relation

μiμ◦i+RTln

(

fi

P◦

)

(definition offi) (4.5-27)

whereμ◦iis the same standard-state chemical potential as for the pure gas: the hypo-

thetical ideal-gas state at pressureP◦and whatever temperature is being considered.

We will not discuss the evaluation of the fugacity in a mixture of nonideal gases.