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Chapter 16: Problem 70

What is the solubility of \(\mathrm{CaF}_{2}\) in a buffer solution containing \(0.30 \mathrm{M}\) \(\mathrm{HCHO}_{2}\) and \(0.20 \mathrm{M} \mathrm{NaCHO}_{2}\) ? Hint: Consider the equation $$ \mathrm{CaF}_{2}(s)+2 \mathrm{H}^{+}(a q) \longrightarrow \mathrm{Ca}^{2+}(a q)+2 \mathrm{HF}(a q) $$ and solve the equilibrium problem.

Short Answer

Expert verified
Given the step-by-step solution to find the solubility of CaF2 in a buffer solution containing HCHO2 and NaCHO2, identify the concentration of H⁺ ions and the solubility product constant expression for this reaction.

Step by step solution

01

Identifying the Weak Acid and Its Ionization Constant

The given buffer solution is composed of \(\mathrm{HCHO}_{2}\) and \(\mathrm{NaCHO}_{2}\). The weak acid is \(\mathrm{HCHO}_{2}\) and it is in equilibrium with the conjugated base, \(\mathrm{CHO}_{2}^{-}\), coming from the salt \(\mathrm{NaCHO}_{2}\). The equation for the dissociation of \(\mathrm{HCHO}_{2}\) is: $$ \mathrm{HCHO}_{2}(a q) \longleftrightarrow \mathrm{H}^{+}(a q) + \mathrm{CHO}_{2}^{-}(a q) $$ We can use the Henderson-Hasselbalch equation to find the concentration of \(\mathrm{H}^{+}\) in the buffer solution, but first, we need to find the ionization constant for the weak acid, \(\mathrm{HCHO}_{2}\). Let's denote it as \(K_{a}\).
02

Applying the Henderson-Hasselbalch Equation

Now that we have \(K_{a}\) for the weak acid, we can determine the concentration of \(\mathrm{H}^{+}\) ions in the buffer solution. The Henderson-Hasselbalch equation is given by: $$ pH = pK_{a} + \log{\frac{[\mathrm{A}^{-}]}{[\mathrm{HA}]}} $$ In our problem, the concentration of \([\mathrm{HA}] = [\mathrm{HCHO}_{2}] = 0.30\,\text{M}\), and the concentration of \([\mathrm{A}^{-}] = [\mathrm{CHO}_{2}^{-}] = 0.20\,\text{M}\). We can plug in these values along with \(pK_{a}\) which can be calculated as \(-\log{K_{a}}\), to obtain the concentration of \(\mathrm{H}^{+}\) ions using the relation: $$ [\mathrm{H}^{+}] = 10^{-pH} $$
03

Setting up the Solubility Product Constant Expression

We have the main equation for the dissolution of \(\mathrm{CaF}_{2}\), which is: $$ \mathrm{CaF}_{2}(s)+2 \mathrm{H}^{+}(a q) \longrightarrow \mathrm{Ca}^{2+}(a q)+2 \mathrm{HF}(a q) $$ The solubility product constant expression for this reaction is given by: $$ K_{sp} = [\mathrm{Ca}^{2+}][\mathrm{HF}]^{2} $$ We need to find the solubility product constant, \(K_{sp}\), for \(\mathrm{CaF}_{2}\).
04

Solving for the Solubility of \(\mathrm{CaF}_{2}\)

We have the concentration of \(\mathrm{H}^{+}\) ions from the previous steps, and we know the solubility product constant \(K_{sp}\) for \(\mathrm{CaF}_{2}\). We can plug in these values into the solubility product constant expression: $$ K_{sp} = x(2x + [\mathrm{H}^{+}])^{2} $$ Here, \(x\) represents the solubility of \(\mathrm{CaF}_{2}\) in \(\text{mol}\,L^{-1}\). We can now solve this equation for \(x\) to obtain the solubility of \(\mathrm{CaF}_{2}\) in the given buffer solution.

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Most popular questions from this chapter

Write the equilibrium equations on which the following \(K_{s p}\) expressions are based. (a) \(\left[\mathrm{Ca}^{2+}\right]\left[\mathrm{CO}_{3}{ }^{2-}\right]\) (b) \(\left[\mathrm{Co}^{3+}\right]\left[\mathrm{OH}^{-}\right]^{3}\) (c) \(\left[\mathrm{Ag}^{+}\right]^{2}\left[\mathrm{~S}^{2-}\right]\) (d) \(\left[\mathrm{Pb}^{2+}\right]\left[\mathrm{Cl}^{-}\right]^{2}\)

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