Physical Chemistry Third Edition

7.4 Equilibria in Solutions of Strong Electrolytes 329

where we now omit the label aq on aqueous HCl. At chemical equilibrium,

μ◦(HCl)+RTln(a(HCl))μ◦(H+)+RTln

[

γ(H+)meq(H+)/m◦

]

+μ◦(Cl−)+RTln

[

γ(Cl−)meq(Cl−)/m◦

]

(7.4-5)

To eliminatem′from our equations we specify that

μ◦(HCl)μ◦(H+)+μ◦(Cl−)μ◦++μ◦− (7.4-6)

where we abbreviateμ◦(H+)byμ◦+andμ◦(Cl−)byμ◦−. This equation determines

m′, even though we do not know its value. Combining Eq. (7.4-6) with Eq. (7.4-5)

and taking antilogarithms, we obtain an expression for the activity of the unionized

aqueous HCl:

a(HCl)(γ(H+)meq(H+)/m◦)(γ(Cl−)meq(Cl−)/m◦)

γ+γ−m+m−/m◦^2 γ±^2 m+m−/m◦^2 (7.4-7)

whereγ±is the mean ionic activity coefficient defined in Eq. (6.4-6) and where we

have abbreviatedmeq(H+)bym+andmeq(Cl−)bym−. Equation (7.4-7) expresses

a(HCl) in terms of measurable quantities.

Equation (7.4-6) is the same as requiring that

∆G◦ 0 (7.4-8)

for the ionization reaction of Eq. (7.4-2), which means that the equilibrium constant

for the reaction is given by

K

(γ+m+/m◦)(γ−m−/m◦)

γ(HCl)meq(HCl)/m′

 1 (7.4-9)

Equation (7.4-9) doesnotmean that the molality of the unionized HCl is roughly equal

to the product of the molalities of the ions, becausem′is not equal tom◦.

We can now relate the partial vapor pressure of gaseous HCl to the molality of the

solution with which it is at equilibrium. If the vapor is assumed to be an ideal gas

mixture,

μ(HCl,g)μ◦(HCl,g)+RTln

P(HCl)

P◦

(7.4-10)

From Eqs. (7.4-1) and (7.4-5),

μ◦(HCl,g)+RTln(P(HCl)/P◦)μ◦++μ◦−+RTln(γ+m+γ−m−/m◦^2 )

(7.4-11)

Equation (7.4-11) is equivalent to

P(HCl)k±(m)γ+γ−m^2 k(±m)γ±^2 m^2 (7.4-12)

wheremis thestoichiometric molalityof HCl (the molality of HCl that would occur

if no ionization occurred) and where

k(±m)(P◦/m◦^2 ) exp

[

μ◦++μ◦−−μ◦

(

HCl,g

)

RT

]

(7.4-13)