Library / Solution chemistry, equilibria and kinetics / SCH-28
Solution chemistry, equilibria and kinetics
Common-ion solubility of a 1-to-2 salt by approximation
Solubility of MX₂ in a solution that already has free X at concentration C, assuming that the salt adds little (2s ≪ C).
- Ref
- SCH-28
- Rev
- v1
- Field
- Solution chemistry, equilibria and kinetics
- Result
- s [mol/L]
In a solution that already has free X at concentration C, the solubility of MX₂ satisfies Ksp = s(C + 2s)². When the added X from the salt is small (2s ≪ C), the equation simplifies to s ≈ Ksp/C². Always check 2s/C afterwards: if it is not small, solve the cubic. The common ion can lower the solubility by orders of magnitude compared with pure water.
Solubility of MX₂ in a solution that already has free X at concentration C, assuming that the salt adds little (2s ≪ C). Check the condition after computing s. If it fails, solve s(C + 2s)² = Ksp.
Calculator
- Check ratio, 2s/C (should be small)1×10−6
- Solubility in pure water, (Ksp/4)^(1/3)1×10−4 mol/L
- Reduction factor against pure water10,000
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With 0.02 mol/L of common ion the solubility drops to 1×10−8 mol/L, 10,000 times less than the 1×10−4 mol/L in pure water.
- The check ratio 2s/C = 1×10−6 is small, so the approximation holds.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mol/L) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Ksp Solubility product | 1e-20 to 1e-4 | 2.5e-17 | 0.25 | 0.25 | +10% | +10 % |
| C Free concentration of the common ion X | 0.001 to 2 mol/L | 1e-12 | 4e-6 | 4e-6 | +10% | −17.4 % |
Sweep of Free concentration of the common ion X
| Free concentration of the common ion X (mol/L) | Molar solubility (mol/L) |
|---|---|
| 0.001 | 4e-6 |
| 0.002586 | 5.981e-7 |
| 0.006687 | 8.944e-8 |
| 0.01729 | 1.337e-8 |
| 0.04472 | 2e-9 |
| 0.1156 | 2.991e-10 |
| 0.2991 | 4.472e-11 |
| 0.7734 | 6.687e-12 |
| 2 | 1e-12 |
Sweep one variable
Variables and how each one contributes
Molar solubility
What the equation gives you. Everything else feeds it.
Solubility product
The solubility product of MX₂. Raises the solubility in direct proportion. Raising it by 10 % (to 4.4×10−12) moves the result by +10 %, from 1×10−8 to 1.1×10−8 mol/L.
Free concentration of the common ion X
The free concentration of the common ion X, in mol/L. A higher common-ion concentration lowers the solubility with the square. Raising it by 10 % (to 0.022) moves the result by −17.4 %, from 1×10−8 to 8.264×10−9 mol/L.
Worked example with the values above
- Square the common-ion concentration: C² = 0.02² = 4×10−4.
- Divide: Ksp/C² = 4×10−12 / 4×10−4.
- s ≈ 1×10−8 mol/L.
Assumptions and limits
- Pure-water dissolution, or the stated common ion, with no complexation, hydrolysis or ion pairing unless stated.
- Concentration-based Ksp, with the solid present at equilibrium.
Common mistakes
- Using the formula-unit coefficients in the wrong places in the Ksp expression.
- Applying the approximation when the added solubility is not small compared with the common-ion concentration.
- Ignoring complexation, which can raise the solubility by orders of magnitude.
Statistics and uncertainty
Solubility products vary widely between sources, often by a factor of two or more, and activity effects at ionic strengths of 0.1 M change solubility by 20 % to 50 %.
Where it is used
- Predicting precipitation and the separation of ions.
- Water treatment, hydroxide precipitation and gravimetric analysis.
Maths you need
- Solubility product: Ksp, the product of ion concentrations at saturation
- Common-ion effect: adding a shared ion lowers solubility
- Ion product: Q, compared with Ksp to predict precipitation