Physical Chemistry Third Edition

6.4 The Activities of Nonvolatile Solutes 273

EXAMPLE6.14

Calculate the value ofγ±for a 0.0100 mol kg−^1 solution of NaCl in water at 298.15 K.

Assume that the distance of closest approach is 3.05× 10 −^10 m, which gives a value ofβa

equal to 1.00 kg^1 /^2 mol−^1 /^2.

Solution

ln

(

γ±

)

−

(

1 .171 kg^1 /^2 mol−^1 /^2

)(

0 .01 mol kg−^1

) 1 / 2

1 +

(

1 .00 kg^1 /^2 mol−^1 /^2

)(

0 .01 mol kg−^1

) 1 / 2 −^0.^1065

γ± 0. 899

The Debye–Hückel result has been shown experimentally to be an accuratelimiting

law. That is, it becomes more nearly exact as the concentration approaches zero. In

practice, it is sufficiently accurate for ionic strengths up to 0.01 mol kg−^1 , but is not

necessarily accurate for larger ionic strengths. Figure 6.10 shows experimental values

of the mean ionic activity coefficient of several electrolyte solutes in water at 298.15 K

as a function of

√

m. The correct limiting behavior is shown by the fact that the curves

of each set of ions with the same value ofz+|z−|approach unity with the slope that is

given by the Debye–Hückel theory.

1.50

1.25

HCl

NaCl

KCl

CaSO 4

LaCl 3 Na 2 SO 4

BaCl 2

SrCl 2

CaCl 2

(a)

LiCl

1.00

0.75

0.50

0.25

0 0.4 0.8 1.2 1.6 2.0

^6

œwm/(mol kg^21 )1/2

(b)

œwm/(mol kg^21 )1/2

1.50

1.25

1.00

0.75

0.50

0.25

0 0.4 0.8 1.2 1.6 2.0

^6

Figure 6.10 The Mean Ionic Activity

Coefficients of Several Electrolyte

Solutes as a Function of the Square

Root of the Ionic Strength.

Ifa≈ 3. 0 × 10 −^10 m, thenβa≈ 1 .00 kg^1 /^2 mol−^1 /^2 and we can write an approx-

imate equation

ln(γ±)−z+|z−|

αI^1 /^2

1 +(I/m◦)^1 /^2

(6.4-28)

wherem◦1 mol kg−^1 (exactly). For small values of the ionic strength theβaI^1 /^2 term

in the denominator of Eq. (6.4-27) is relatively small compared with unity and can be

neglected. The resulting equation is similar to the first term in an empirical equation

of Brønsted that predated the theory of Debye and Hückel.^3 For a 1-1 electrolyte like

NaCl, this equation is

ln(γ±)−am^1 /^2 +bm (1-1 electrolyte) (6.4-29)

wheremis the molality of the solute (note thatmIfor a 1-1 electrolyte) and where
aandbare constants.
Johannes Nicolas Brønsted,
1879–1947, was a Danish physical
chemist who made various
contributions, including the
Brønsted–Lowry definition of acids and
bases. He was elected to the Danish
parliament in 1947, but died before
taking office.

Much of the disagreement of the Debye–Hückel result with experiment for moderate

concentrations has been attributed to the formation ofion pairs, which consist of two

ions of opposite charge held together by electrostatic attraction.^4 These ion pairs are

not chemically bonded like complex ions such as AgCl− 2 , and the two ions can have

solvent molecules between them. Much work has been done to extend the Debye–

Hückel theory, beginning in 1926 with a theory of Bjerrum^5 that explicitly included

ion pairing. Some later research is based on theoretical work of Mayer,^6 in which

(^3) J. N. Brønsted,J. Am. Chem. Soc., 44 , 938 (1922).
(^4) See R. W. Clark and J. M. Bonicamp,J. Chem. Educ., 75 , 1182 (1998) for a discussion of the inclusion
of this and other factors in solubility equilibria.
(^5) N. Bjerrum,Kgl. Danske Vidensk. Selskab., 7 , 9 (1926).
(^6) J. E. Mayer,J. Chem. Phys., 18 , 1426 (1950); K. S. Pitzer,Acc. Chem. Res., 1 0 , 317, (1977).