Physical Chemistry Third Edition

270 6 The Thermodynamics of Solutions

which is the same as

dln(γ 2 )dφ+

φ− 1

m 2

dm 2 (6.4-16)

If data for the osmotic coefficient are available over this molality range, Eq. (6.4-16)

can be integrated fromm 2 0tom 2 m′ 2 , a particular value ofm 2 :

∫m 2 m′

2

m 2  0

dln(γ 2 )

∫m 2 m′

2

m 2  0

dφ+

∫m′

2

0

φ− 1

m 2

dm 2 (6.4-17)

The integral on the left-hand side of this equation yields zero at its lower limit, since

the activity coefficient approaches unity asm 2 approaches 0. It can be shown thatφ

approaches unity asm 2 approaches zero, so that the integral converges and we can

write

ln(γ 2 (m′ 2 ))φ(m′ 2 )− 1 +

∫m′

2

0

φ− 1

m 2

dm 2 (6.4-18)

The integral is approximated numerically from the data, giving the value ofγ 2 atm′ 2.

Exercise 6.16

Using the relationx 1  1 −x 2  1 −n 2 /n 1 at high dilution, show thatφapproaches 1 asm 2

approaches 0.

The Debye–Hückel Theory

This is a theory for the activity coefficients of dilute electrolyte solutes. Ions exert large

forces on each other, and these forces are calledlong-range forcesbecause they act

over large distances compared with other intermolecular forces. The electrostatic force

Coulomb’s law is named for Charles on a chargeQ 1 due to a chargeQ 2 is given byCoulomb’s law,
Augustin de Coulomb, 1736–1806, the
French physicist who discovered the
law. F 12 er

Q 1 Q 2

4 πεr 122

(6.4-19)

wherer 12 is the distance between the charges and where the unit vectorerpoints from

object 1 toward object 2. If the two charges have the same sign the force is a repulsion,

and if they have opposite signs the force is an attraction. The quantityεis thepermittivity

of the medium between the charges. The permittivity of a vacuum is denoted byε 0

and is equal to 8. 854519 × 10 −^12 C^2 N−^1 m−^2. The ratio of the permittivity of a given

material to that of a vacuum is called thedielectric constantand is denoted byεrel:

εrel

ε(material)

ε 0

(6.4-20)

The dielectric constant of water is larger than that of most substances and at 25◦Cis

equal to 78.54. Even though the electrostatic forces between ions in an aqueous solution

are weaker than those in a vacuum by the reciprocal of this factor, the electrostatic forces

between ions in a solution are strong compared with other intermolecular forces, and

they act over a large distance compared to other intermolecular forces.