Problem 32 - Chemistry and Chemical Reactivity

Question

The equilibrium constant, \(K_{c},\) for the following reaction is 1.05 at \(350 \mathrm{K}\) $$ 2 \mathrm{CH}_{2} \mathrm{Cl}_{2}(\mathrm{g}) \rightleftarrows \mathrm{CH}_{4}(\mathrm{g})+\mathrm{CCl}_{4}(\mathrm{g}) $$ If an equilibrium mixture of the three gases at \(350 \mathrm{K}\) contains \(0.0206 \mathrm{M} \mathrm{CH}_{2} \mathrm{Cl}_{2}(\mathrm{g})\) and \(0.0163 \mathrm{M} \mathrm{CH}_{4},\) what is the equilibrium concentration of \(\mathrm{CCl}_{4} ?\)

Step-by-Step Solution

Answer

The equilibrium concentration of \(\mathrm{CCl}_{4}\) is 0.0273 M.

Step 1: Write the Expression for Kc

The equilibrium constant expression, \( K_c \), for the given reaction is expressed as: \[ K_c = \frac{[\text{CH}_4][\text{CCl}_4]}{[\text{CH}_2\text{Cl}_2]^2} \] where the brackets denote the concentration of each species at equilibrium.

Step 2: Insert Known Values into Kc Expression

We know \( K_c = 1.05 \), \([\text{CH}_2 \text{Cl}_2] = 0.0206 \, M\), and \([\text{CH}_4] = 0.0163 \, M\). Let's substitute these into the \( K_c \) equation: \[ 1.05 = \frac{0.0163 \cdot [\text{CCl}_4]}{(0.0206)^2} \]

Step 3: Solve for Concentration of CCl4

To find \([\text{CCl}_4]\), first calculate \((0.0206)^2 = 0.00042436\). Substitute and solve: \[ 1.05 = \frac{0.0163 \cdot x}{0.00042436} \] Rearrange to find \(x\), the concentration of \(\mathrm{CCl}_{4}\): \[ x = \frac{1.05 \cdot 0.00042436}{0.0163} \] Calculating this gives \(x = 0.0273 \, M\).

Key Concepts

Equilibrium Constant (Kc)

The equilibrium constant, denoted as \( K_c \), is a vital concept when dealing with chemical equilibria. It represents the ratio of the concentrations of products to the concentrations of reactants, each raised to the power of their respective stoichiometric coefficients, at equilibrium. A balanced chemical reaction helps establish the expression for \( K_c \). For example, in our reaction: \[ 2 \text{CH}_2\text{Cl}_2(\text{g}) \rightleftarrows \text{CH}_4(\text{g}) + \text{CCl}_4(\text{g}) \] the equilibrium constant expression is derived directly from the reaction equation: \[ K_c = \frac{[\text{CH}_4][\text{CCl}_4]}{[\text{CH}_2\text{Cl}_2]^2} \] This equation tells us how the system behaves at equilibrium at a specific temperature. Notably, \( K_c \) is temperature-dependent; it will change if the temperature changes. Furthermore, the value of \( K_c \) does not have any units, as the concentrations cancel out accordingly. A large \( K_c \) value often indicates that products are favored at equilibrium.

Equilibrium Concentration

Equilibrium concentration refers to the concentrations of the reactants and products in a reaction mixture when chemical equilibrium is achieved. At this point, the rates of the forward and backward reactions are identical, meaning the concentrations of substances no longer change over time. Understanding the initial conditions and knowing the \( K_c \) value is crucial to computing these concentrations. To determine the equilibrium concentration of a specific substance, you might start with the given equilibrium concentrations of the other species, as outlined in the solution for the reaction: - \([\text{CH}_2 \text{Cl}_2] = 0.0206 \, M\) - \([\text{CH}_4] = 0.0163 \, M\) Then, apply the equilibrium constant expression, substitute known values, and solve for the unknown concentration. In this case: \[ 1.05 = \frac{0.0163 \cdot [\text{CCl}_4]}{(0.0206)^2} \] By re-arranging and solving this equation, you find \([\text{CCl}_4] = 0.0273 \, M\), which reflects the equilibrium concentration of carbon tetrachloride in the mixture.

Reaction Quotient

The reaction quotient, commonly denoted as \( Q_c \), is a measure that helps determine the direction in which a reaction will proceed to reach equilibrium. It is calculated similarly to the equilibrium constant \( K_c \), using the same equation: \[ Q_c = \frac{[\text{products}]}{[\text{reactants}]} \] The main difference is that \( Q_c \) represents the ratio of concentrations at any point in time, not necessarily at equilibrium.To deduce the status of the reaction with respect to equilibrium, compare \( Q_c \) and \( K_c \):

• If \( Q_c = K_c \), the system is at equilibrium.
• If \( Q_c < K_c \), the forward reaction is favored, and the system will shift to the right to reach equilibrium.
• If \( Q_c > K_c \), the reverse reaction is favored, and the system will shift to the left to achieve equilibrium.

These comparisons provide insights into whether the concentrations will remain the same or if adjustments are necessary to achieve equilibrium.