Physical Chemistry Third Edition

6.3 Activity and Activity Coefficients 259

The activity is a dimensionless quantity that is equal to unity if the substance is in its

standard state. Each measure of the activity corresponds to a different standard state.

For example, the standard state for a component in an ideal solution or a solvent in

an arbitrary solution is the pure substance. The standard state for a solute that obeys

Henry’s law is the hypothetical pure substance that has a vapor pressure equal toki, the

Henry’s law constant. The standard state of a dilute solute in the molality description is

a hypothetical solute with a molality of 1 mol kg−^1 such that Henry’s law is obeyed at

this composition. The standard state of a dilute solute in the concentration description

is a hypothetical solute with a molar concentration of 1 mol m−^3 or 1 mol L−^1 such

that Henry’s law is obeyed at this composition.

The Activity of a Pure Solid or Liquid

The standard state of a pure solid or liquid is the pure substance at pressureP◦.Ifthe

substance is at a pressureP′not equal toP◦,

μi(P′)G∗m,i(P′)G

◦

m,i+

∫P′

P◦

Vm,∗idP≈μ◦i+Vm,∗i

(

P′−P◦

)

(6.3-12)

where we assume that the solid or liquid has a nearly constant volume. By comparison

with Eq. (6.3-6),

RTln(ai)Vm,∗i

(

P−P◦

)

(6.3-13)

where we drop the prime label (′)ontheP. The activity of a pure liquid or solid at

pressurePis given by

aiexp

(

Vm,∗i(P−P◦)

RT

)

(pure solid or liquid) (6.3-14)

The exponent in Eq. (6.3-14) is generally quite small unlessPdiffers greatly fromP◦.

For ordinary pressures we will assume that the activity of a pure solid or liquid is equal

to unity.

EXAMPLE6.10

Find the activity of pure liquid water at a pressure of 2.000 bar and a temperature of 298.15 K.

Solution

ln(a)

(

1. 805 × 10 −^6 m^3 mol−^1

)

( 1 .00 atm)

(

101325 N m−^2 atm−^1

)

(

8 .3145 J K−^1 mol−^1

)

( 298 .15 K)

 0. 0007378

ae^0.^0007378  1. 000728

Exercise 6.13

Find the pressure such that the activity of liquid water is equal to 1.0100 at 298.15 K.