Physical Chemistry
Lecture 37
Solving Ionic-Equilibrium Problems
Ionic equilibria in solution
Ionic materials dissociate (at least partially) in
solution
An equilibrium problem that can be solved by
estimation of the activity coefficient
Examples

Solubility of sparingly soluble salts
Mν + X ν − ( solid ) → v + M z+ (ai ) + v − X z− (ai )

Partial dissociation of weak acids and bases
H n A (ao) → n H + (ai ) + An − (ai )
B(OH ) m (ao) ⇔
B m + (ai ) + m OH − (ai )
Equilibrium constants and free
energies
As with other reactions, a relation between standard
free energy of reaction and the equilibrium constant
 ∆Gθ
K a (T ) = exp  −
 RT



Free-energy difference can be found from standard
free energies of formation
∆Gθ
=
θ
ν
∆
G
∑ k k , formation
k
Free energies of formation of ions at the Henry-law
standard state are found in tables
Free energies of formation
Standard state usually the
Henry’s Law extrapolation
at 1 molal
Standard state depends
on temperature
Scale determined by the
free energy of formation
of H+ in solution
Free Energies of Formation at 25°C
(Henry’s Law at 1.0 molal)
Species
∆fGθ (kJ/mol)
H+
0.00
Cu+
49.98
Cu2+
65.49
Na+
-261.91
OH-
-157.24
Cl-
-131.23
HCO2-
-351.00
CH3CO2-
-369.31
HCO2H
-372.30
CH3CO2H
-396.46
CO2
-385.98
H2S
-27.83
Example calculation
Dissolution of AgCl
AgCl ( solid ) →
Ag + (ai ) + Cl − (ai )
Find free energies of formation from tables
∆GθAgCl
= −109.789 kJ / mol
∆GθAg +
= 77.107 kJ / mol
θ
∆GCl
−
= −131.228 kJ / mol
Equilibrium constant
Ka


3


55.668 ×10 joule
 = 1.77 ×10 −10
= exp −
 8.3144349 joule 298.15 K 


K


Equilibrium solubilities of
sparingly soluble salts
Example: AgCl
Use


AgCl ( solid ) →
Ag + (ai ) + Cl − (ai )
Equilibrium constant
Debye-Hueckel equation
for γ±
Solve iteratively

Use self consistency to
determine convergence
Result



m = 1.34x10-5 mol/kg
γ± = 0.996
Two iterations
Ka
= γ ±2 m 2
γ±
 1.177 m 


= exp −

 1+ m 
Presence of other ionic species
Solutions may have several ions present
Presence of other ions changes solubility

Changes ionic strength, which changes the activity
coefficient
 Example: solubility of AgCl in a solution containing
NaNO3

Common-ion effect if one of the ions is added
from a separate source
 Example: solubility of AgCl in a solution containing NaCl
Must take these factors into account in
determining the solution of the equilibrium
equation
Ionic-strength effects
Example: Solubility of
AgCl in 0.01 molal
NaNO3
Include NaNO3 in the
ionic strength
Repeat iterative
procedure to
convergence
Converges to


m = 1.477x10-5 mol/kg
γ± = 0.898
Ka
= γ ±2 m 2
γ±
 1.177 m + 0.01 

= exp −

 1 + m + 0.01 
Common-ion effect
Solubility of AgCl in 0.01
molal NaCl
Presence of NaCl affects


Ionic strength
Equilibrium expression
Ka
= γ ±2 m(m + 0.01)
γ±
 1.177 m + 0.01 

= exp −

m
1
0
.
01
+
+


Set up and solve iteratively

Use self-consistency test
Results


m = 2.18 x 10-8 mol/kg
γ± = 0.899
Weak-acid dissociation
Example: acetic acid
H 3CCO2 H (ao) ⇔
H 3CCO2− (ai ) + H + (ai )
Equilibrium constant: Ka = 1.75x10-5
Must consider nonideality


Assume all nonideality lies in the ions
γHAc = 1
Solve by methods similar to solubility
problems
Determination of dissociation
of acetic acid
Fractional dissociation, α, determines the acid
strength
H 3CCO2 H (ao) ⇔
H 3CCO2− (ai ) + H + (ai )
(1 − α )m
αm
αm
Solve equilibrium equation iteratively
Ka
ln γ ±
2
α
m
2
= γ±
1−α
= −
1.177 αm
1 + αm
Dissociation of acetic acid
pH of Acetic Acid Solutions at 25 C
Solve iteratively at each
concentration to find α
Dissociation increases as the
concentration goes down
pH of the solution changes
with concentration
pH
3.5
3
2.5
2
0
1
2
3
4
5
6
Molality
= − log10 (aH + )
Dissociation of Acetic Acid at 25 C
0.07
0.06
Dissociation Fraction
pH
4
0.05
0.04
0.03
0.02
0.01
0
0
0.02
0.04
0.06
Molality
0.08
0.1
0.12
Summary
Equilibrium in ionic solutions requires consideration of
activity coefficients of the ions


Solubility of sparingly soluble salts
Weak acid (base) equilibria
Must solve equilibrium equation subject to selfconsistency of the definition of the activity coefficients
At sufficiently low concentrations, the activity
coefficients are determined by the DHG limiting law
Additional complication when other ions are added


Change of ionic strength
Addition of common ion
Calculate concentrations of ions, knowing the activity
coefficients