A saturated solution of mangement(II) carbonate (M=114.95) contains `5.44xx10^(-5)`g of `MnCO_(3)` per 100mL at `25^(@)`C. What is its `K_(sp)` at this temperature?

Options

A. `4.7xx10^(-6)`
B. `3.0xx10^(-9)`
C. `2.2xx10^(-11)`
D. `2.2xx10^(-13)`

Answer

To find the solubility product constant (\(K_{sp}\)) of manganese(II) carbonate (\(MnCO_3\)), we can follow these steps:

### Step 1: Calculate the concentration of \(MnCO_3\)
We start with the mass of \(MnCO_3\) given in the problem:
- Mass of \(MnCO_3\) = \(5.44 \times 10^{-5}\) g
- Molecular mass of \(MnCO_3\) = 114.95 g/mol
- Volume of the solution = 100 mL = 0.1 L

First, we need to calculate the number of moles of \(MnCO_3\):
\[ \text{Number of moles} = \frac{\text{mass}}{\text{molecular mass}} = \frac{5.44 \times 10^{-5} \text{ g}}{114.95 \text{ g/mol}} \approx 4.73 \times 10^{-7} \text{ mol} \]

Next, we calculate the concentration (molarity) of \(MnCO_3\):
\[ \text{Concentration} = \frac{\text{Number of moles}}{\text{Volume in L}} = \frac{4.73 \times 10^{-7} \text{ mol}}{0.1 \text{ L}} = 4.73 \times 10^{-6} \text{ M} \]

### Step 2: Write the dissociation equation
The dissociation of \(MnCO_3\) in water can be represented as:
\[ MnCO_3 (s) \rightleftharpoons Mn^{2+} (aq) + CO_3^{2-} (aq) \]

### Step 3: Relate solubility to ion concentrations
Let the solubility of \(MnCO_3\) be \(S\). From the dissociation equation:
- The concentration of \(Mn^{2+}\) ions = \(S\)
- The concentration of \(CO_3^{2-}\) ions = \(S\)

Thus, at equilibrium:
\[ [Mn^{2+}] = S \quad \text{and} \quad [CO_3^{2-}] = S \]

### Step 4: Write the expression for \(K_{sp}\)
The solubility product constant (\(K_{sp}\)) is given by:
\[ K_{sp} = [Mn^{2+}][CO_3^{2-}] = S \times S = S^2 \]

### Step 5: Substitute the value of \(S\) into the \(K_{sp}\) expression
From our earlier calculation, we found that:
\[ S = 4.73 \times 10^{-6} \text{ M} \]
Now substituting this value into the \(K_{sp}\) expression:
\[ K_{sp} = (4.73 \times 10^{-6})^2 = 2.23 \times 10^{-11} \]

### Final Result
Thus, the solubility product constant \(K_{sp}\) of manganese(II) carbonate at 25°C is:
\[ K_{sp} \approx 2.23 \times 10^{-11} \]