Page 331 - Physical Chemistry

lev38627_ch10.qxd 3/14/08 1:07 PM Page 312

312
Chapter 10 Debye and Hückel’s final result is
Nonideal Solutions
2
2
z AI 1>2 z AI 1>2

m
m

ln g , ln g (10.57)

1 BaI 1>2 1 BaI 1>2
m
m
where A, B, and I are defined as
m
e 2 3>2 2N r 1>2
A A
A 12pN r 2 1>2 a b , B e a b (10.58)
A A
4pe e kT e e kT
0 r,A
0 r,A
1
I a z m j (10.59)*
2
m
j
2 j
In these equations (which are in SI units), a is the mean ionic diameter, g and g are

the molality-scale activity coefficients for the ions M and X , respectively, N is the
z
z
A
Avogadro constant, k is Boltzmann’s constant [Eq. (3.57)], e is the proton charge, e 0
is the permittivity of vacuum (e occurs as a proportionality constant in Coulomb’s
0
law; see Sec. 13.1), r is the solvent density, e r,A is the solvent dielectric constant, and
A
T is the absolute temperature. I is called the (molality-scale) ionic strength; the sum
m
in (10.59) goes over all ions in solution, m being the molality of ion j with charge z .
j
j
Although the Debye–Hückel theory gives g of each ion, we cannot measure g or

g individually. Hence, we express the Debye–Hückel result in terms of the mean ionic

activity coefficient g . Taking the log of (g ) n n (g ) (g ) [Eq. (10.43)], we get
n
n

n ln g n ln g

ln g (10.60)

n n

Since the electrolyte M X is electrically neutral, we have
n n
n z n z 0 (10.61)

2
Multiplication of (10.61) by z yields n z n z z ; multiplication of (10.61) by

2
z yields n z n z z . Addition of these two equations gives

2
2
n z n z z z 1n n 2 z 0z 0 1n n 2 (10.62)

since z is negative. Substitution of the Debye–Hückel equations (10.57) into (10.60)

followed by use of (10.62) gives
AI 1>2
m
ln g z 0z 0 (10.63)

1 BaI 1>2
m
3
Using the SI values for N , k, e, and e , and e 78.38, r 997.05 kg/m for
r
0
A
H O at 25°C and 1 atm, we have for (10.58)
2
9
A 1.1744 1kg>mol2 1>2 , B 3.285
10 1kg>mol2 1>2 m 1
Substituting the numerical values for B and A into (10.63) and dividing A by 2.3026
to convert to base 10 logs, we get
1I >m°2 1>2
m
log g 0.510z 0z 0 dil. 25°C aq. soln. (10.64)

10
1 0.3281a>Å21I >m°2 1>2
m
where 1 Å 10 10 m and m° 1 mol/kg. I in (10.59) has units of mol/kg, and the
m
ionic diameter a has units of length, so log g is dimensionless, as it must be.

For very dilute solutions, I is very small, and the second term in the denomina-
m
tor in (10.64) can be neglected compared with 1. Therefore,
1>2
ln g z 0z 0AI very dil. soln. (10.65)

m
log g 0.510z 0z 0 1I >m°2 1>2 very dil. aq. soln., 25°C (10.66)

10
m