Problem 76 - Chemistry The Central Science In Si Units

Question

A \(1.0 \mathrm{M} \mathrm{Na}_{2} \mathrm{SO}_{4}\) solution is slowly added to \(10.0 \mathrm{~mL}\) of a solution that is \(0.20 \mathrm{M}\) in \(\mathrm{Ca}^{2+}\) and \(0.30 \mathrm{M}\) in \(\mathrm{Ag}^{+}\). (a) Which compound will precipitate first: \(\operatorname{CaSO}_{4}\left(K_{s p}=2.4 \times 10^{-5}\right)\) or \(\mathrm{Ag}_{2} \mathrm{SO}_{4}\left(K_{\mathrm{sp}}=1.5 \times 10^{-5}\right) ?(\mathbf{b})\) How \(\mathrm{much} \mathrm{Na}_{2} \mathrm{SO}_{4}\) solu- tion must be added to initiate the precipitation?

Step-by-Step Solution

Answer

CaSO₄ precipitates first; add about 1.2 mL of Na₂SO₄.

Step 1: Identify the ions and their reactions

We have two ions in the solution: \(\mathrm{Ca}^{2+}\) and \(\mathrm{Ag}^{+}\). The compounds that can form are \(\mathrm{CaSO}_{4}\) and \(\mathrm{Ag}_{2}\mathrm{SO}_{4}\). We will calculate when each will start to precipitate using their \(K_{sp}\) values.

Step 2: Calculate the ion product for CaSO4

For \(\mathrm{CaSO}_{4}\), the dissolution reaction is: \(\mathrm{CaSO}_{4}(s) \rightleftharpoons \mathrm{Ca}^{2+}(aq) + \mathrm{SO}_{4}^{2-}(aq)\). The ion product \(Q\) is given by \([\mathrm{Ca}^{2+}][\mathrm{SO}_{4}^{2-}]\). Initially, \([\mathrm{Ca}^{2+}] = 0.20 \, \mathrm{M}\). The precipitation begins when \(Q \geq K_{sp}\) of \(2.4 \times 10^{-5}\).

Step 3: Calculate [SO4^2-] needed for CaSO4

Set \(Q\) equal to \(K_{sp}\): \([0.20][\mathrm{SO}_{4}^{2-}] = 2.4 \times 10^{-5}\) to find \([\mathrm{SO}_{4}^{2-}]\).\[ [\mathrm{SO}_{4}^{2-}] \geq \frac{2.4 \times 10^{-5}}{0.20} = 1.2 \times 10^{-4} \, \mathrm{M} \]

Step 4: Calculate the ion product for Ag2SO4

For \(\mathrm{Ag}_{2}\mathrm{SO}_{4}\), the dissolution reaction is: \(\mathrm{Ag}_{2}\mathrm{SO}_{4}(s) \rightleftharpoons 2\mathrm{Ag}^{+} + \mathrm{SO}_{4}^{2-}\). The ion product \(Q\) is \([\mathrm{Ag}^{+}]^2[\mathrm{SO}_{4}^{2-}]\). The initial \([\mathrm{Ag}^{+}] = 0.30 \, \mathrm{M}\). The precipitation begins when \(Q \geq K_{sp}\) of \(1.5 \times 10^{-5}\).

Step 5: Calculate [SO4^2-] needed for Ag2SO4

Set \(Q\) equal to \(K_{sp}\): \([0.30]^2[\mathrm{SO}_{4}^{2-}] = 1.5 \times 10^{-5}\) to find \([\mathrm{SO}_{4}^{2-}]\).\[ [\mathrm{SO}_{4}^{2-}] \geq \frac{1.5 \times 10^{-5}}{(0.30)^2} = 1.67 \times 10^{-4} \, \mathrm{M} \]

Step 6: Compare the [SO4^2-] values

\(\mathrm{CaSO}_{4}\) requires \([\mathrm{SO}_{4}^{2-}] = 1.2 \times 10^{-4} \, \mathrm{M}\) to precipitate, and \(\mathrm{Ag}_{2}\mathrm{SO}_{4}\) requires \([\mathrm{SO}_{4}^{2-}] = 1.67 \times 10^{-4} \, \mathrm{M}\). Since \(1.2 \times 10^{-4} < 1.67 \times 10^{-4}\), \(\mathrm{CaSO}_{4}\) will precipitate first.

Step 7: Calculate the volume of Na2SO4 needed

Let \(V\) be the volume of \(1.0 \, \mathrm{M} \, \mathrm{Na}_{2}\mathrm{SO}_{4}\) added. Using the equation \([\mathrm{SO}_{4}^{2-}] = \frac{1.0V}{10+V}\) and setting it equal to \(1.2 \times 10^{-4} \):\[ 1.0V = 1.2 \times 10^{-4}(10 + V) \]Solve for \(V\):\[ V = \frac{1.2 \times 10^{-4} \times 10}{1 - 1.2 \times 10^{-4}} \approx 1.2 \times 10^{-3} \, \mathrm{L} \text{ or } 1.2 \, \mathrm{mL} \]

Step 8: Conclusion

\(\mathrm{CaSO}_{4}\) precipitates first when approximately \(1.2 \, \mathrm{mL}\) of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) is added.

Key Concepts

Solubility Product Constant (Ksp)

The solubility product constant, or \(K_{sp}\), is a critical concept in understanding when a salt will dissolve in a solution or begin to precipitate out. It's a value specific to each salt and reflects how much of that substance can exist in equilibrium in a saturated solution before a solid begins to form.

When a salt like \(\text{CaSO}_4\) dissolves in water, it dissociates into its constituent ions: \(\text{Ca}^{2+}\) and \(\text{SO}_4^{2-}\). The \(K_{sp}\) is mathematically expressed as the product of the concentrations of these ions raised to the power of their stoichiometric coefficients in the dissolution equation.

For example, the dissolution of \(\text{CaSO}_4\) is described by the equation:
\[\text{CaSO}_4(s) \rightleftharpoons \text{Ca}^{2+}(aq) + \text{SO}_4^{2-}(aq)\]
The expression for the \(K_{sp}\) of \(\text{CaSO}_4\) is:
\[K_{sp} = [\text{Ca}^{2+}][\text{SO}_4^{2-}]\]
This value, \(2.4 \times 10^{-5}\) for \(\text{CaSO}_4\), tells us how much product of the ions' concentrations can remain in solution before precipitation occurs. Understanding \(K_{sp}\) helps predict whether ions in a solution will stay dissolved or will form a solid precipitate.

Ion Product (Q)

The ion product, or \(Q\), helps determine the likelihood of precipitation for a solution containing ions. While \(K_{sp}\) represents the equilibrium condition of a saturated solution, \(Q\) indicates the current state of the solution.

You can think of \(Q\) as a snapshot of the product of ion concentrations at any given moment. It is calculated using the same method as \(K_{sp}\), based on their concentrations when mixed, rather than those at equilibrium.

• If \(Q < K_{sp}\), the solution is unsaturated, and more solute can dissolve.
• If \(Q = K_{sp}\), the solution is saturated, meaning it is in equilibrium.
• If \(Q > K_{sp}\), the solution is supersaturated, and precipitation is likely to occur.

Let's examine the case of \(\text{Ag}_2\text{SO}_4\). For precipitation to begin, the following condition needs to be met:
\[Q = [\text{Ag}^{+}]^2[\text{SO}_4^{2-}] > K_{sp}\]
By comparing \(Q\) and \(K_{sp}\), you can predict whether a precipitate will form, and in which order compounds will begin to precipitate if multiple ions are present.

Chemical Equilibria

Chemical equilibria play a pivotal role in understanding precipitation reactions. Equilibrium describes the state where the rate of a forward reaction equals the rate of its reverse, leading to a constant concentration of reactants and products.

In the context of solubility, equilibrium is achieved in a saturated solution when the dissolved ions remain at a steady state in the solution without forming more precipitate.

• The dissolution of \(\text{CaSO}_4\) reaches equilibrium when the rate of \(\text{CaSO}_4\) dissolving equals the rate of ions coming together to form more solid \(\text{CaSO}_4\).
• If more \(\text{Na}_2\text{SO}_4\) is added to a solution containing \(\text{Ca}^{2+}\) ions, it increases the \([\text{SO}_4^{2-}]\) in the solution."

When equilibrium is disturbed by adding more of a solute, or by removing some solvent, Le Chatelier's Principle suggests that the system will shift to counteract this change. Understanding how solutions reach and maintain equilibrium can help predict reactions such as the likelihood of precipitate formation when additional salts are added.