Physical Chemistry Third Edition

6.3 Activity and Activity Coefficients 261

where we attach the superscript (I) to specify that we are using convention I.

Sinceμ

◦(I)

i is the chemical potential of the pure liquid,μ

∗

i, it is equal to the chem-

ical potential of the gaseous substance at a partial pressure equal to the equilibrium

vapor pressureP∗i:

μ∗iμ◦i(I)μ◦i(g)+RTln

(

Pi∗

P◦

)

(6.3-20)

We are ignoring a small correction inμ◦i(I)due to the fact that the standard state is

atPP◦1 bar, whereasμ∗i pertains to pressurePi∗. The relation of Eq. (6.3-19)

becomes

μ◦i(g)+RTln

(

Pi∗

P◦

)

+RTlna(I)i μ◦i(g)+RTln

(

Pi

P◦

)

(6.3-21)

Canceling equal terms and taking antilogarithms in Eq. (6.3-21), we obtain

PiPi∗a(I)i (6.3-22)

which resembles Raoult’s law except that the activitya

(I)

i occurs instead of the mole

reactionxi. The activity acts as an “effective” mole fraction in determining the partial

vapor pressure of the substance. Equation (6.3-22) is equivalent to

a(iI)

Pi

Pi∗

(6.3-23)

Theactivity coefficientin convention I is defined as the ratio of the activity to the

mole fraction:

γ(I)i 

a(iI)

xi

(definition of the activity coefficient) (6.3-24)

It is equal to the actual vapor pressure divided by the value of the vapor pressure that

is predicted by Raoult’s law:

γi(I)

Pi/P◦

Pi,ideal/P◦



Pi

Pi∗xi

(6.3-25)

The activity coefficient specifies how the substance deviates from Raoult’s law. If

γi>1, the partial vapor pressure of substanceiis higher than predicted by Raoult’s

law, and ifγi<1, it is lower than predicted by Raoult’s law.

EXAMPLE6.11

Find the value of the activity and the activity coefficient of 2,2,4-trimethyl pentane (com-

ponent 2) in ethanol at 25◦C at a mole fraction of 0.2748, according to convention I. The

partial vapor pressure is equal to 48.31 torr and the vapor pressure of the pure liquid is equal

to 59.03 torr.