Question

Calculate the molar solubility of \(\mathrm{Cd}(\mathrm{OH})_{2}, K_{\mathrm{sp}}=5.9 \times 10^{-11}\).

Short answer

The molar solubility of \(\mathrm{Cd}(\mathrm{OH})_{2}\) is approximately \(1.63 \times 10^{-4} \,M\).

Step-by-step solution

Write the balanced chemical equation for the dissolution of Cd(OH)2

The balanced chemical equation for the dissolution of \(\mathrm{Cd}(\mathrm{OH})_{2}\) can be written as: \[ \mathrm{Cd}(\mathrm{OH})_{2} \rightleftharpoons \mathrm{Cd}^{2+} + 2\mathrm{OH}^-\] As the solid dissolves in water, it creates Cd2+ and OH- ions.

Define the solubility product expression and molar solubility

For the dissolution of \(\mathrm{Cd}(\mathrm{OH})_{2}\), let the molar solubility be represented by the variable s. Then, the concentration of the ions in the saturated solution can be expressed as follows: \[\mathrm{[Cd}^{2+}] = s\] \[\mathrm{[OH}^\mathrm{-}] = 2s\] The solubility product expression is given by the product of the concentrations of the dissolved ions: \[K_{\mathrm{sp}} = \mathrm{[Cd}^{2+}] \times \mathrm{[OH}^\mathrm{-}]^{2}\]

Substitute the solubility product constant and molar solubility into the equation

Substitute the given values of \(K_{\mathrm{sp}}\) and the expressions for the ion concentrations in terms of molar solubility into the equation from Step 2: \[5.9 \times 10^{-11} = (s) \times (2s)^2\]

Solve the equation for molar solubility (s)

First, simplify the equation obtained in Step 3: \[5.9 \times 10^{-11} = 4s^3\] Now, divide by 4: \[s^3 = \frac{5.9 \times 10^{-11}}{4}\] Next, take the cube root of both sides to get: \[s = \sqrt[3]{\frac{5.9 \times 10^{-11}}{4}}\] Calculate the value of s: \[s \approx 1.63 \times 10^{-4}\] So the molar solubility of \(\mathrm{Cd}(\mathrm{OH})_{2}\) is approximately \(1.63 \times 10^{-4} \,M\).