Question

Determine whether \(1.50 \mathrm{g} \mathrm{H}_{2} \mathrm{C}_{2} \mathrm{O}_{4}\) (oxalic acid: \(K_{\mathrm{a}_{1}}=\) \(\left.5.2 \times 10^{-2}, K_{\mathrm{a}_{2}}=5.4 \times 10^{-5}\right)\) can be dissolved in \(0.200 \mathrm{L}\) of \(0.150 \mathrm{M} \mathrm{CaCl}_{2}\) without the formation of \(\mathrm{CaC}_{2} \mathrm{O}_{4}(\mathrm{s})\left(K_{\mathrm{sp}}=1.3 \times 10^{-9}\right)\).

Short answer

The oxalic acid can dissolve in the solution without the formation of precipitate if the product of the equilibrium concentrations of \(Ca^{2+}\) ions and \(C_{2}O_{4}^{-2}\) ions is less than their given solubility product (\(K_{sp}\)). The detailed calculations need to perform for arriving this conclusion.

Step-by-step solution

Calculate initial concentrations

The first step is to convert the amount of oxalic acid from grams to moles by using the molar mass of oxalic acid and also calculate the initial concentration. The molar mass of oxalic acid is \(90.03 g/mol\). So, \(1.50g\) of oxalic acid corresponds to \(\frac{1.50g}{90.03 g/mol} = 0.0167 mol\). The initial concentration of \(H_{2}C_{2}O_{4}\) is then \(\frac{0.0167 mol}{0.200 L} = 0.0835 M\). The \(Ca^{2+}\) ions coming from the solution of \(CaCl_{2}\) have an initial concentration of \(0.150 M\).

Set up the reaction equations and equilibrium expressions

The dissolution of oxalic acid into \(H^{+}\) and \(HC_{2}O_{4}^{-}\) and the respective equilibrium constant expression for \(K_{a_{1}}\) can be written as: \(H_{2}C_{2}O_{4} \leftrightarrow H^{+} + HC_{2}O_{4}^{-} , K_{a_{1}} = \frac{[H^{+}][HC_{2}O_{4}^{-}]}{[H_{2}C_{2}O_{4}]}\) The second \(H^{+}\) ion dissociation is given by : \(HC_{2}O_{4}^{-} \leftrightarrow H^{+} + C_{2}O_{4}^{-2} , K_{a_{2}} = \frac{[H^{+}][C_{2}O_{4}^{-2}]}{[HC_{2}O_{4}^{-}]}\) The projected precipitation can be given as : \(Ca^{2+} + C_{2}O_{4}^{-2} \leftrightarrow CaC_{2}O_{4(s)} , K_{sp} = [Ca^{2+}][C_{2}O_{4}^{-2}]\) All these three equilibriums are connected and affect each other.

Calculate equilibrium concentrations

For simplification, approximations that \(K_{a_{1}} >> K_{a_{2}}\) and that the ionization from the second \(H^{+}\) contribution to the total \(H^{+}\) concentration is negligible can be made. Thus, the equilibrium concentrations can be found using \(K_{a_{1}}\) and the initial concentration of oxalic acid. Setting up and solving this for \(x=[H^{+}]= [HC_{2}O_{4}^{-}]\) we get: \(K_{a_{1}} = \frac{x^2}{[H_{2}C_{2}O_{4}]_0 - x}\) Solving the equation above for \(x\) gives a value for [H+], which can be used along with \(K_{a_{2}}\) to then find the concentration of \(C_{2}O_{4}^{-2}\).

Check for the precipitation

Once the concentration of \(C_{2}O_{4}^{-2}\) is calculated, it can be used along with the initial concentration of \(Ca^{2+}\) to find their product and then compare with \(K_{sp}\). If the product is greater than \(K_{sp}\), precipitation will occur. Otherwise, the salt can dissolve without formation of precipitate.

Key concepts

Equilibrium Constant Expression

Understanding the equilibrium constant expression is essential when analyzing chemical reactions that can reverse or reach a state of balance between the forward and reverse reactions. For oxalic acid, a diprotic acid (able to donate two protons), there are two dissociation steps, each with its own equilibrium constant, denoted by Ka1 and Ka2 for the first and second dissociations respectively.

The equilibrium constant expression for a reaction, such as the dissociation of oxalic acid \(H_2C_2O_4 \rightleftharpoons H^+ + HC_2O_4^-\), is written as \(K_{a1} = \frac{[H^+][HC_2O_4^-]}{[H_2C_2O_4]}\). It quantitatively describes the ratio of the concentrations of products to reactants at equilibrium. A high Ka value indicates a strong acid, meaning it largely dissociates into its ions.

Solubility Product Constant

The solubility product constant, Ksp, is critical when assessing whether a salt will precipitate out of solution. Ksp represents the maximum product of the ionic concentrations of a salt that can exist in solution without forming a solid precipitate.

For instance, the potential formation of calcium oxalate \(CaC_2O_4(s)\) from dissolved \(Ca^{2+}\) ions and oxalate \(C_2O_4^{-2}\) ions can be expressed by its solubility product constant: \(K_{sp} = [Ca^{2+}][C_2O_4^{-2}]\). When the ionic product exceeds Ksp, the solution is supersaturated, and precipitation occurs. Conversely, if the ionic product is less than Ksp, no precipitate forms and the solution remains unsaturated with respect to that particular salt.

Dissociation of Diprotic Acid

Diprotic acids, such as oxalic acid, can release two protons (\(H^+\)) in a stepwise manner, with each step having its respective equilibrium constant. For oxalic acid, this involves two dissociation steps: first, it releases a proton to form bicarbonate (HC2O4-), and then the bicarbonate further dissociates to release another proton and form oxalate (C2O4-2).

The dissociation expressions and their associated equilibrium constants for oxalic acid are:

• First dissociation: \(H_2C_2O_4 \rightleftharpoons H^+ + HC_2O_4^- \quad K_{a1}\)
• Second dissociation: \(HC_2O_4^- \rightleftharpoons H^+ + C_2O_4^{-2} \quad K_{a2}\)

In calculations, if Ka1 is significantly larger than Ka2, we often assume that the first dissociation is the primary source of protons and oxalate ions in the solution.

Precipitation Calculations

Precipitation calculations involve predicting whether a solid will form when solutions containing different ions are mixed. This can be determined by calculating the ionic product of the relevant ions and comparing it to the solubility product constant (Ksp).

In the exercise provided, the ionic product of calcium and oxalate ions, resulting from the dissolution of oxalic acid in the presence of calcium chloride, must be compared to the Ksp of calcium oxalate. The concentration of oxalate ions is calculated considering the equilibrium of the dissociation of oxalic acid, while the concentration of calcium ions originates from the dissolved calcium chloride. If the calculated ionic product exceeds Ksp, the reaction is supersaturated, and calcium oxalate will precipitate out of the solution.