Question

At \(25^{\circ} \mathrm{C},\) the reaction $$\mathrm{CaCrO}_{4}(s) \rightleftharpoons \mathrm{Ca}^{2+}(a q)+\mathrm{CrO}_{4}^{2-}(a q)$$ has an equilibrium constant \(K_{c}=7.1 \times 10^{-4} .\) What are the equilibrium concentrations of \(\mathrm{Ca}^{2+}\) and \(\mathrm{CrO}_{4}^{2-}\) in a saturated solution of \(\mathrm{CaCrO}_{4} ?\)

Short answer

The equilibrium concentrations of \(\mathrm{Ca}^{2+}\) and \(\mathrm{CrO}_{4}^{2-}\) in a saturated solution of \(\mathrm{CaCrO}_{4}\) are approximately 0.0267 M for each ion.

Step-by-step solution

Write the expression for the equilibrium constant

The equilibrium constant is defined as the ratio of the concentrations of the products to the concentrations of the reactants, each raised to the power of their coefficients in the balanced chemical equation. For the given reaction, the expression is: \[K_c = \frac{[\mathrm{Ca}^{2+}][\mathrm{CrO}_{4}^{2-}]}{[\mathrm{CaCrO}_{4}]}\] However, as \(\mathrm{CaCrO}_{4}\) is a solid, its concentration remains constant and is not included in the equilibrium constant expression. The revised expression for the equilibrium constant is: \[K_c = [\mathrm{Ca}^{2+}][\mathrm{CrO}_{4}^{2-}]\]

Define variables and relationships

Let us denote the equilibrium concentrations of \(\mathrm{Ca}^{2+}\) and \(\mathrm{CrO}_{4}^{2-}\) as "x." In a saturated solution, an equal amount of \(\mathrm{Ca}^{2+}\) and \(\mathrm{CrO}_{4}^{2-}\) is formed. Therefore, the equilibrium concentrations are the same: \[[\mathrm{Ca}^{2+}] = [\mathrm{CrO}_{4}^{2-}] = x\]

Substitute the expressions into the equilibrium constant equation

Now, substitute the equilibrium concentrations into the expression for the equilibrium constant, \(K_c\): \[K_c = x^2\]

Solve for the equilibrium concentrations

We are given the value of \(K_c\) as \(7.1 \times 10^{-4}\). Substitute this value into the equation and solve for x: \[7.1 \times 10^{-4} = x^2\] Take the square root of both sides: \[x = \sqrt{7.1 \times 10^{-4}}\] \[x \approx 0.0267\, \mathrm{M}\]

Interpret the results

The equilibrium concentrations of \(\mathrm{Ca}^{2+}\) and \(\mathrm{CrO}_{4}^{2-}\) in a saturated solution of \(\mathrm{CaCrO}_{4}\) are approximately 0.0267 M for each ion.

Key concepts

Equilibrium Constant Expression

Understanding the equilibrium constant expression (\textbf{K}) is essential for studying chemical equilibrium. This term quantifies the ratio of the concentrations of products over reactants at the state of equilibrium, which is a balance where the forward and reverse reactions occur at the same rate. For the given reaction \(\mathrm{CaCrO}_{4}(s) \rightleftharpoons \mathrm{Ca}^{2+}(aq)+\mathrm{CrO}_{4}^{2-}(aq)\), the equilibrium constant expression excludes the solid \(\mathrm{CaCrO}_{4}\), as its concentration doesn't change during the reaction.

It is vital to recognize that an equilibrium constant is specific to a particular reaction at a specific temperature. Altering the temperature can change the value of \(K\), which indicates a shift in the position of equilibrium. In chemistry, this gives us a broad understanding of how systems react to environmental changes and how the proportions of reactants and products are affected.

Saturation Concentration

Saturation concentration represents the point at which a solution contains the maximum amount of a solute that can be dissolved at a given temperature and pressure. Beyond this point, any added solute won't dissolve and will remain as a precipitate. This is closely tied to the concept of solubility, which quantifies that maximum amount.

When discussing the saturation concentration in relation to equilibrium, it refers to a dynamic state where the rate of the solute dissolving is equal to the rate of the solute precipitating out of the solution. For example, in the case of \(\mathrm{CaCrO}_{4}\), the saturation concentration is where the solution transitions into a saturated one and any additional calcium chromate wouldn't be able to dissolve. This is crucial because the saturation concentration is the starting point for determining the concentrations of ions present at equilibrium in a saturated solution.

Equilibrium Concentration

The equilibrium concentration is the concentration of a reactant or product in a solution at equilibrium. For the ions \(\mathrm{Ca}^{2+}\) and \(\mathrm{CrO}_{4}^{2-}\) in our reaction, the equilibrium concentrations tell us the amounts present when the system is undisturbed and in a state of equilibrium.

To find these concentrations, we follow a series of steps that begins with writing the equilibrium constant expression. From there, we can solve for 'x', which in step-by-step solutions is used to denote the exact concentrations of these ions. Recognizing that these ions are produced in a 1:1 ratio in this reaction, we can deduce that their equilibrium concentrations are equal. This is a result of the stoichiometry from the balanced equation, which in the given example is simple, as each reactant and product has a coefficient of one.