Library / Solution chemistry, equilibria and kinetics / SCH-30
Solution chemistry, equilibria and kinetics
Metal precipitation recovery at a fixed free ligand concentration
Percentage of a metal precipitated as MXᵦ when the free precipitating ion concentration X is held constant, from the Ksp and the initial total metal concentration C₀.
- Ref
- SCH-30
- Rev
- v1
- Field
- Solution chemistry, equilibria and kinetics
- Result
- recovery [%]
If free precipitating ion X is held at a fixed concentration, the dissolved metal at saturation is Ksp/Xᵇ. When the initial metal concentration C₀ exceeds that, the fraction recovered is 1 − Ksp/(Xᵇ C₀), which here is given as a percentage. Below saturation nothing precipitates, so the recovery is zero. It assumes that no other reactions, such as complexation, hold metal in solution.
Percentage of a metal precipitated as MXᵦ when the free precipitating ion concentration X is held constant, from the Ksp and the initial total metal concentration C₀. The result is zero below saturation.
Calculator
- Dissolved metal at equilibrium, min(C₀, Ksp/Xᵇ)1×10−7 mol/L
- Metal precipitated per litre9.999×10−4 mol/L
- Precipitant needed for 99.9 % recovery0.001 mol/L
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 0.01 mol/L of free precipitant, 99.99 % of the metal is recovered, leaving 1×10−7 mol/L in solution.
- Reaching 99.9 % would need 0.001 mol/L of precipitant.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (%) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Ksp Solubility product | 1e-20 to 1e-4 | 0 | 100 | 100 | +10% | −0.001 % |
| b Precipitant ions per formula unit | 1 to 4 | 0 | 99.99 | 99.99 | +1 | −0.99 % |
| C0 Initial metal concentration | 1e-5 to 1 mol/L | 99 | 100 | 1 | +10% | +9.09e-4 % |
| X Free precipitant concentration | 1e-4 to 1 mol/L | 99 | 100 | 0.9999 | +10% | +9.09e-4 % |
Sweep of Free precipitant concentration
| Free precipitant concentration (mol/L) | Metal recovered (%) |
|---|---|
| 1e-4 | 99 |
| 3.162e-4 | 99.68 |
| 0.001 | 99.9 |
| 0.003162 | 99.97 |
| 0.01 | 99.99 |
| 0.03162 | 100 |
| 0.1 | 100 |
| 0.3162 | 100 |
| 1 | 100 |
Sweep one variable
Variables and how each one contributes
Metal recovered
What the equation gives you. Everything else feeds it.
Solubility product
The solubility product of MXᵦ. A larger Ksp leaves more metal dissolved, so the recovery falls. Raising it by 10 % (to 1.1×10−9) moves the result by −0.001 %, from 99.99 to 99.99 %.
Free precipitant concentration
The free precipitant concentration, held constant, in mol/L. A higher precipitant level pushes the metal out more completely, so the recovery rises. Raising it by 10 % (to 0.011) moves the result by +9.09×10−4 %, from 99.99 to 99.99 %.
Precipitant ions per formula unit
The number of precipitant ions per formula unit. A larger b makes the precipitant’s concentration matter more, so the recovery rises. Raising it by 10 % (to 1.1) moves the result by −0.00585 %, from 99.99 to 99.98 %.
Initial metal concentration
The initial metal concentration, in mol/L. More metal at the start raises the recovered fraction, since the dissolved remainder is fixed. Raising it by 10 % (to 0.0011) moves the result by +9.09×10−4 %, from 99.99 to 99.99 %.
Worked example with the values above
- Dissolved metal at saturation: Ksp/Xᵇ = 1×10−9 / 0.01 = 1×10−7 mol/L.
- Fraction left: 1×10−7 / 0.001 = 1×10−4.
- Recovery = 100 × (1 − that) = 99.99 %.
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