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# Conversion of Solubility to Ksp

## Demonstrates how solubility constants can be derived from experimentally determined solubility.

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Conversion of Solubility to Ksp

Baking soda (sodium bicarbonate) is prepared by bubbling carbon dioxide gas through a solution of ammonia and sodium chloride. Ammonium carbonate is first formed which then reacts with the NaCl to form sodium bicarbonate and ammonium chloride. The sodium bicarbonate is less soluble than the other materials, so it will precipitate out of solution.

### Conversion of Solubility to $K_{sp}$

Solubility is normally expressed in g/L of saturated solution. However, solubility can also be expressed as the moles per liter. Molar solubility is the number of moles of solute in one liter of saturated solution. In other words, the molar solubility of a given compound represents the highest molarity solution that is possible for that compound. The molar mass of a compound is the conversion factor between solubility and molar solubility. Given that the solubility of Zn(OH) 2 is 4.2 × 10 -4  g/L, the molar solubility can be calculated as shown below:

$\frac{4.2 \times 10^{-4} \ \cancel{\text{g}}}{\text{L}} \times \frac{1 \ \text{mol}}{99.41 \ \cancel{\text{g}}}=4.2 \times 10^{-6} \ \text{mol/L} \ (\text{M})$

Solubility data can be used to calculate the  $K_{sp}$ for a given compound. The following steps need to be taken.

1. Convert form solubility to molar solubility.
2. Use the dissociation equation to determine the concentration of each of the ions in mol/L.
3. Apply the $K_{sp}$ equation.

#### Sample Problem: Calculating  $K_{sp}$ from Solubility

The solubility of lead(II) is found experimentally to be 0.533 g/L. Calculate the  $K_{sp}$ for lead(II) fluoride.

Step 1: List the known quantities and plan the problem .

Known

• solubility of PbF 2 = 0.533 g/L
• molar mass = 245.20 g/mol

Unknown

• $K_{sp}$ of PbF 2 = ?

The dissociation equation for PbF 2 and the corresponding  $K_{sp}$ expression

$\text{PbF}_2(s) \rightleftarrows \text{Pb}^{2+}(aq)+2\text{F}^-(aq) && K_{sp}=[\text{Pb}^{2+}][\text{F}^-]^2$

The steps above will be followed to calculate the  $K_{sp}$ for PbF 2 .

Step 2: Solve .

$\text{molar solubility} \qquad \frac{0.533 \ \cancel{\text{g}}}{\text{L}} \times \frac{1 \ \text{mol}}{245.20 \ \cancel{\text{g}}}=2.17 \times 10^{-3} \ \text{M}$

The dissociation equation shows that for every mole of PbF 2 that dissociates, 1 mol of Pb 2+ and 2 mol of F are produced. Therefore, at equilibrium the concentrations of the ions are:

$[\text{Pb}^{2+}]=2.17 \times 10^{-3} \ \text{M} \quad \text{and} \quad [\text{F}^-]=2 \times 2.17 \times 10^{-3}=4.35 \times 10^{-3} \ \text{M}$

Substitute into the expression and solve for the $K_{sp}$ .

$K_{sp}=(2.17 \times 10^{-3})(4.35 \times 10^{-3})^2=4.11 \times 10^{-8}$

The solubility product constant is significantly less than 1 for a nearly insoluble compound such as PbF 2 .

#### Summary

• Molar solubility calculations are described.
• Calculations of  $K_{sp}$ using molar solubility are described.

#### Practice

Read the material at the link below and do the problems at the end:

#### Review

1. What are the solution requirements for determining molar solubility?
2. Why do we need to convert mass to molarity to determine $K_{sp}$ ?
3. What  $K_{sp}$ values would you expect for very insoluble compounds?