Physical Chemistry Third Edition

6.4 The Activities of Nonvolatile Solutes 269

and themean ionic molality

m±

(

mν++mν−−

) 1 /ν

(6.4-7)

These are examples ofgeometric means. For a 1-3 electrolyte such as CrCl 3 ,m±is

equal to (27)^1 /^4 m 2  2. 2795 m 2 , wherem 2 is thestoichiometric molalityof the solute

(the molality that would occur if no dissociation occurred). The chemical potential of

the neutral electrolyte solute is given by

μ 2 ν+μ◦++ν−μ◦−+RTln(γ±ν

(

m±/m◦

)ν

)

ν+μ◦++ν−μ◦−+νRTln

(

γ±m±/m◦

)

(6.4-8)

The activity of the solvent can be expressed in terms of theosmotic coefficientφ,

defined by

φ−

ln(a 1 )

M 1 vm 2



μ◦ 1 −μ 1

RTM 1 vm 2

(definition) (6.4-9)

wherea 1 is the activity of the solvent, andM 1 is the molar mass of the solvent. If the

solute dissociates completely,νm 2 is equal to the sum of the molalities of the ions.

From Eq. (6.4-9)

μ 1 μ◦ 1 −RTM 1 vm 2 φ (6.4-10)

The chemical potential of the solute can be written

μ 2 μ◦ 2 +νRTln(γ±ν±m 2 /m◦) (6.4-11)

where

γ±

(

γ

v+

+γ

v−

−

) 1 /v

(6.4-12)

For constant pressure and temperature, the Gibbs–Duhem relation for a two-

component system is written in the form

n 1 dμ 1 +n 2 dμ 2  0 (6.4-13)

Since the molalitym 2 is equal to the amount of substance 2 divided by the mass of the

solvent (substance 1),

n 2 m 2 n 1 M 1 (6.4-14)

whereM 1 is the molar mass of the solvent. Use of Eqs. (6.4-10), (6.4-11), and (6.4-14)

in Eq. (6.4-13) gives

−n 1 νRTM 1 m 2 dφ+φdm 2 +m 2 n 1 M 1 νRTdln(γ 2 )+dln(m 2 /m◦) 0

Cancellation of the common factor and use of the identity

dln(m)(1/m)dm

gives

−m 2 dφ−φdm 2 +m 2 dln(γ 2 )+dm 2  0 (6.4-15)