To determine the solubility of magnesium carbonate ([tex]\( \text{MgCO}_3 \)[/tex]) in water, we will follow these steps:
1. Write the Dissociation Equation:
[tex]\[
\text{MgCO}_3 \rightleftharpoons \text{Mg}^{2+} + \text{CO}_3^{2-}
\][/tex]
2. Establish the Solubility Product Expression:
The solubility product constant ([tex]\( K_{sp} \)[/tex]) is given by the expression:
[tex]\[
K_{sp} = [\text{Mg}^{2+}][\text{CO}_3^{2-}]
\][/tex]
3. Relate Ion Concentrations to Solubility:
If we let the solubility of [tex]\( \text{MgCO}_3 \)[/tex] be [tex]\( s \)[/tex] moles per liter, then at equilibrium, the concentrations of the ions will be:
[tex]\[
[\text{Mg}^{2+}] = s \quad \text{and} \quad [\text{CO}_3^{2-}] = s
\][/tex]
Thus, the solubility product expression becomes:
[tex]\[
K_{sp} = s \cdot s = s^2
\][/tex]
4. Substitute the Given [tex]\( K_{sp} \)[/tex] and Solve for [tex]\( s \)[/tex]:
We are given that [tex]\( K_{sp} = 3.5 \times 10^{-8} \)[/tex]. Therefore, we have:
[tex]\[
3.5 \times 10^{-8} = s^2
\][/tex]
To find [tex]\( s \)[/tex], take the square root of both sides:
[tex]\[
s = \sqrt{3.5 \times 10^{-8}}
\][/tex]
5. Calculate the Solubility:
Using a calculator to find the square root:
[tex]\[
s = 0.00018708286933869707
\][/tex]
6. Final Answer:
The solubility of [tex]\( \text{MgCO}_3 \)[/tex] in water is approximately [tex]\( 0.000187 \)[/tex] moles per liter (or [tex]\( 1.87 \times 10^{-4} \)[/tex] M).
