Page 190 - Modern Analytical Chemistry

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Chapter 6 Equilibrium Chemistry 173

The true thermodynamic equilibrium constant, K sp , for the solubility of AgIO 3 ,
therefore, is
– + – –
K sp =(a Ag +)(a IO 3 ) = [Ag ][IO 3 ](g Ag +)(g IO 3 )
To accurately calculate the solubility of AgIO 3 , we must know the activity coeffi-
+
–
cients for Ag and IO 3 .
For gases, pure solids, pure liquids, and nonionic solutes, activity coefficients
are approximately unity under most reasonable experimental conditions. For reac-
tions involving only these species, differences between activity and concentration
are negligible. Activity coefficients for ionic solutes, however, depend on the ionic
composition of the solution. It is possible, using the extended Debye–Hückel the-
ory,* to calculate activity coefficients using equation 6.50
. 051 ´z 2 A ´ m
– log g A = 6.50
3
1 +3 . ´ a A ´ m

where Z A is the charge of the ion, a A is the effective diameter of the hydrated ion in
nanometers (Table 6.1), mis the solution’s ionic strength, and 0.51 and 3.3 are con-
stants appropriate for aqueous solutions at 25 °C.
Several features of equation 6.50 deserve mention. First, as the ionic strength
approaches zero, the activity coefficient approaches a value of one. Thus, in a solu-
tion where the ionic strength is zero, an ion’s activity and concentration are identi-
cal. We can take advantage of this fact to determine a reaction’s thermodynamic
equilibrium constant. The equilibrium constant based on concentrations is mea-
sured for several increasingly smaller ionic strengths and the results extrapolated

Table 6.1 Effective Diameters (a) for Selected Inorganic
Cations and Anions
Effective Diameter
Ion (nm)
H 3 O + 0.9
Li + 0.6
+ – – – –
Na , IO 3 , HSO 3 , HCO 3 , H 2 PO 4 0.45
– – – – – – –
OH , F , SCN , HS , ClO 3 , ClO 4 , MnO 4 0.35
+ – – – – – –
K , Cl , Br , I , CN , NO 2 , NO 3 0.3
+ + + +
Cs , Tl , Ag , NH 4 0.25
2+
Mg , Be 2+ 0.8
2+
2+
2+
2+
2+
2+
2+
Ca , Cu , Zn , Sn , Mn , Fe , Ni , Co 2+ 0.6
2+
2+
2+
2+
Sr , Ba , Cd , Hg , S 2– 0.5
2+ 2– 2–
Pb , CO 3 , SO 3 0.45
2+ 2– 2– 2– 2–
Hg 2 , SO 4 , S 2 O 3 , CrO 4 , HPO 4 0.40
3+
3+
Al , Fe , Cr 3+ 0.9
3– 3–
PO 4 , Fe(CN) 6 0.4
4+
4+
Zr , Ce , Sn 4+ 1.1
4–
Fe(CN) 6 0.5
Source: Values from Kielland, J. J. Am. Chem. Soc. 1937, 59, 1675.
*See any standard textbook on physical chemistry for more information on the Debye–Hückel theory and its
application to solution equilibrium