Calculation of the Common Ion Effect on Solubility

Understanding the Calculation of the Common Ion Effect on Solubility

The common ion effect calculation determines how solubility decreases in the presence of a shared ion. This article explores detailed methods and formulas for precise solubility predictions.

Readers will find comprehensive tables, formula derivations, and real-world examples illustrating the common ion effect’s impact on solubility. Advanced techniques and applications are thoroughly explained.

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  • Calculate the solubility of AgCl in a 0.01 M NaCl solution considering the common ion effect.
  • Determine the solubility of CaF2 in a solution containing 0.02 M NaF.
  • Find the solubility of PbI2 in a 0.005 M KI solution using the common ion effect.
  • Evaluate the change in solubility of BaSO4 when 0.01 M Na2SO4 is added.

Comprehensive Tables of Common Ion Effect Parameters and Solubility Values

SaltCommon IonKsp (at 25°C)Typical Initial Ion Concentration (M)Solubility Without Common Ion (mol/L)Solubility With Common Ion (mol/L)
AgClCl–1.8 × 10-100.01 (NaCl)1.3 × 10-51.8 × 10-7
CaF2F–3.9 × 10-110.02 (NaF)2.0 × 10-41.0 × 10-5
PbI2I–7.1 × 10-90.005 (KI)8.4 × 10-41.4 × 10-4
BaSO4SO42-1.1 × 10-100.01 (Na2SO4)1.0 × 10-51.0 × 10-6
Mg(OH)2OH–1.8 × 10-110.001 (NaOH)1.3 × 10-44.2 × 10-6
CuBrBr–6.3 × 10-90.01 (NaBr)7.9 × 10-57.9 × 10-7
SrSO4SO42-3.2 × 10-70.005 (Na2SO4)5.7 × 10-41.3 × 10-4
ZnSS2-2.0 × 10-250.001 (Na2S)1.4 × 10-131.4 × 10-15

Fundamental Formulas for Calculating the Common Ion Effect on Solubility

Calculating the common ion effect on solubility requires understanding the equilibrium between the solid salt and its dissociated ions in solution. The solubility product constant (Ksp) governs this equilibrium.

1. Basic Solubility Product Expression

For a generic salt MX that dissociates as:

MX (s) ⇌ M+ (aq) + X– (aq)

The solubility product expression is:

Ksp = [M+] × [X–]

Where:

  • Ksp = solubility product constant (unitless or mol2/L2 depending on salt)
  • [M+] = molar concentration of the cation
  • [X–] = molar concentration of the anion

2. Solubility Without Common Ion

When no common ion is present, the solubility s is equal for both ions:

Ksp = s × s = s2

Therefore, solubility is:

s = √Ksp

3. Solubility With Common Ion Present

If the solution already contains a common ion at concentration c, the solubility s of the salt decreases. For the salt MX:

Ksp = [M+] × [X–] = s × (c + s)

Since s is usually much smaller than c, the equation simplifies to:

Ksp ≈ s × c

Solving for s:

s = Ksp / c

4. Solubility for Salts with Multiple Ions

For salts with formula MmXn, dissociating as:

MmXn (s) ⇌ m Mz+ + n Xy-

The solubility product is:

Ksp = [Mz+]m × [Xy-]n

Assuming solubility s mol/L, the ion concentrations are:

  • [Mz+] = m × s
  • [Xy-] = n × s + c (if common ion present)

Thus, the equation becomes:

Ksp = (m s)m × (n s + c)n

This often requires solving polynomial equations for s numerically or via approximation.

5. Effect of Ionic Strength and Activity Coefficients

In real solutions, ion activities (a) replace concentrations due to non-ideal behavior:

Ksp = aMm × aXn = (γM[M])m × (γX[X])n

Where:

  • γ = activity coefficient (dimensionless)
  • [M], [X] = molar concentrations

Activity coefficients depend on ionic strength (I), calculated as:

I = 0.5 × Σ ci zi2

Where:

  • ci = molar concentration of ion i
  • zi = charge of ion i

Debye-Hückel or extended models estimate γ values, refining solubility calculations especially in concentrated solutions.

Detailed Explanation of Variables and Typical Values

  • Ksp: The equilibrium constant for the dissolution of a salt. Values vary widely, from extremely low (e.g., ZnS ~10-25) to relatively high (e.g., NaCl ~10-1), indicating solubility.
  • s: Solubility of the salt in mol/L, representing the concentration of dissolved ions from the salt.
  • c: Concentration of the common ion already present in solution, typically from added salts sharing the ion.
  • m, n: Stoichiometric coefficients of ions in the salt formula.
  • γ: Activity coefficients, usually close to 1 in dilute solutions but decrease with ionic strength.
  • I: Ionic strength, a measure of total ion concentration weighted by charge squared.

Real-World Applications and Case Studies

Case Study 1: Solubility of Silver Chloride in Seawater

Seawater contains approximately 0.55 M chloride ions, primarily from NaCl. The solubility of AgCl in pure water is limited by its Ksp of 1.8 × 10-10. However, the high chloride concentration drastically reduces AgCl solubility due to the common ion effect.

Step 1: Write the dissolution equilibrium:

AgCl (s) ⇌ Ag+ + Cl–

Step 2: Express Ksp:

Ksp = [Ag+] × [Cl–] = s × (0.55 + s) ≈ s × 0.55

Step 3: Calculate solubility s:

s = Ksp / 0.55 = (1.8 × 10-10) / 0.55 ≈ 3.27 × 10-10 mol/L

This solubility is significantly lower than in pure water (~1.3 × 10-5 mol/L), demonstrating the strong common ion effect in natural waters.

Case Study 2: Calcium Fluoride Solubility in Fluoride-Containing Dental Products

Calcium fluoride (CaF2) is used in dental treatments to provide fluoride ions. The presence of fluoride ions in solution affects CaF2 solubility via the common ion effect.

Step 1: Dissolution equilibrium:

CaF2 (s) ⇌ Ca2+ + 2 F–

Step 2: Ksp expression:

Ksp = [Ca2+] × [F–]2 = s × (c + 2s)2

Where c is the fluoride ion concentration from the dental product, e.g., 0.02 M.

Step 3: Approximate since 2s ≪ c:

Ksp ≈ s × c2

Step 4: Calculate solubility s:

s = Ksp / c2 = (3.9 × 10-11) / (0.02)2 = (3.9 × 10-11) / 4 × 10-4 = 9.75 × 10-8 mol/L

This solubility is drastically reduced compared to pure water (~2.0 × 10-4 mol/L), indicating that fluoride presence suppresses CaF2 dissolution, relevant for controlled fluoride release in dental care.

Additional Considerations and Advanced Topics

  • Temperature Dependence: Ksp values vary with temperature, affecting solubility and common ion effect magnitude. Thermodynamic data should be consulted for precise calculations.
  • Complex Ion Formation: Some ions form complexes, altering free ion concentrations and thus solubility. For example, Ag+ forms complexes with NH3, affecting AgCl solubility.
  • pH Influence: For salts involving hydroxide or other pH-sensitive ions, solution pH impacts ion speciation and solubility.
  • Activity Coefficients in Concentrated Solutions: At higher ionic strengths, deviations from ideality require use of extended Debye-Hückel or Pitzer models for accurate activity coefficients.

References and Further Reading