Question

Consider the reaction: \begin{equation}\begin{aligned} \mathrm{SO}_{2} \mathrm{Cl}_{2}(g) \Longrightarrow & \mathrm{SO}_{2}(g)+\mathrm{Cl}_{2}(g) \\ & K_{\mathrm{p}}=2.91 \times 10^{3} \mathrm{at} 298 \mathrm{K} \end{aligned}\end{equation} In a reaction at equilibrium, the partial pressure of \(\mathrm{SO}_{2}\) is 137 torr and that of \(\mathrm{Cl}_{2}\) is 285 torr. What is the partial pressure of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) in thismixture?

Short answer

The partial pressure of \( \mathrm{SO}_2\mathrm{Cl}_2 \) in the mixture is approximately 13.42 torr.

Step-by-step solution

Write the expression for the equilibrium constant Kp

The equilibrium constant for the given reaction in terms of partial pressures (Kp) is given by: \[ K_{\mathrm{p}} = \frac{P_{\mathrm{SO}_{2}} \cdot P_{\mathrm{Cl}_{2}}}{P_{\mathrm{SO}_2\mathrm{Cl}_2}} \] where \( P_{\mathrm{SO}_2} \), \( P_{\mathrm{Cl}_2} \), and \( P_{\mathrm{SO}_2\mathrm{Cl}_2} \) are the partial pressures of the substances at equilibrium.

Plug the known values into the equation

Substitute the given values of Kp, P_{SO_2}, and P_{Cl_2} into the equilibrium expression: \[ 2.91 \times 10^{3} = \frac{137 \times 285}{P_{\mathrm{SO}_2\mathrm{Cl}_2}} \]

Solve for the partial pressure of SO2Cl2

Rearrange the equation and solve for \( P_{\mathrm{SO}_2\mathrm{Cl}_2} \): \[ P_{\mathrm{SO}_2\mathrm{Cl}_2} = \frac{137 \cdot 285}{2.91 \times 10^{3}} \] Perform the calculations with the given values to determine the partial pressure of \( \mathrm{SO}_2\mathrm{Cl}_2 \).

Convert torr to atm if necessary

If the calculation for Kp is required in atm units then convert the pressure from torr to atm using the conversion factor: 1 atm = 760 torr. In this problem, all pressures are already given in torr, which is consistent with the value of Kp, therefore no conversion is necessary.

Calculate the result

After calculations, the partial pressure of \( \mathrm{SO}_2\mathrm{Cl}_2 \) is found to be: \[ P_{\mathrm{SO}_2\mathrm{Cl}_2} = \frac{137 \cdot 285}{2.91 \times 10^{3}} = \frac{39045}{2910} \approx 13.42 \text{ torr (rounded to two decimal places)} \]

Key concepts

Equilibrium Constant

When a chemical reaction reaches equilibrium, the rates of the forward and reverse reactions are equal, resulting in a constant ratio of product to reactant concentrations. This ratio is represented by the equilibrium constant, denoted as K or Kp when expressed in terms of partial pressures. The magnitude of the equilibrium constant gives us insight into the position of the equilibrium and, consequently, the relative concentrations of reactants and products.

The formula for the equilibrium constant in terms of partial pressures (Kp) generally takes the form \[ K_{p} = \frac{\prod (P_{products})^{\text{stoichiometric coefficient}}}{\prod (P_{reactants})^{\text{stoichiometric coefficient}}} \] where \(P\) denotes the partial pressure of each gas, and the exponents are the coefficients from the balanced chemical equation. A large Kp value indicates that products are favored at equilibrium, while a small Kp value suggests reactants are favored.

Understanding Kp is crucial in predicting how a system will react to changes in conditions, such as pressure or concentration variations, as well as temperature changes, which can all affect the equilibrium position.

Partial Pressure

The partial pressure of a gas in a mixture is the pressure the gas would exert if it occupied the entire volume on its own. It's analogous to the concentration of a gas in a mixture. Dalton's Law of Partial Pressures dictates that the total pressure of a gas mixture is equal to the sum of the partial pressures of each individual gas.

For example, the partial pressure of a substance in a reaction can be used in the equilibrium constant expression (Kp). In the provided exercise, we're given the individual partial pressures for \(\mathrm{SO}_2\) and \(\mathrm{Cl}_2\), and asked to solve for \(P_{\mathrm{SO}_2\mathrm{Cl}_2}\), the pressure of \(\mathrm{SO}_2\mathrm{Cl}_2\).

To solve problems involving partial pressures, one should always crosscheck the units to ensure they are consistent across all terms in the equation. As with concentrations, the partial pressures provide a direct measurement of a reactant or product's molar quantity within the volume of the system, which is essential in determining the progress and extent of the reaction at equilibrium.

Le Chatelier's Principle

Le Chatelier's Principle specifies that if a dynamic equilibrium is disturbed by changing the conditions, such as concentration, temperature, or pressure, the system adjusts to counteract the change and restore a new equilibrium. This concept helps us predict the direction a reaction will shift to regain balance.

For example, increasing the concentration of reactants will shift the equilibrium toward the products to consume the added reactants. Conversely, removing products will also drive the reaction toward the product side to replenish them. A change in pressure impacts the equilibrium of gas phase reactions, particularly those with different numbers of moles of gases on each side of the balanced chemical equation.

When applying Le Chatelier's Principle, remember that temperature effects are contingent on the exothermic or endothermic nature of the reaction. Increasing the temperature in an exothermic reaction shifts equilibrium toward the reactants, whereas, for an endothermic reaction, the equilibrium shifts toward the products.

Le Chatelier's Principle is an invaluable tool in optimizing industrial chemical processes, such as ammonia synthesis in the Haber process, by maximizing product yields through careful control of reaction conditions.