Question

The equilibrium constant is \(0.0900\) at \(25^{\circ} \mathrm{C}\) for the reaction $$ \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{Cl}_{2} \mathrm{O}(g) \rightleftharpoons 2 \mathrm{HOCl}(g) $$ For which of the following sets of conditions is the system at equilibrium? For those that are not at equilibrium, in which direction will the system shift? a. \(P_{H, O}=1.00 \mathrm{~atm}, P_{\mathrm{CL}, \mathrm{O}}=1.00 \mathrm{~atm}, P_{\mathrm{HOC}}=1.00 \mathrm{~atm}\) b. \(P_{\mathrm{H}_{2} \mathrm{O}}=200\). torr, \(P_{\mathrm{Cl}_{2} \mathrm{O}}=49.8\) torr, \(P_{\mathrm{Ho} \mathrm{C}}=21.0\) torr c. \(P_{\mathrm{H}_{0} \mathrm{O}}=296\) torr, \(P_{\mathrm{C}_{6} \mathrm{O}}=15.0\) torr, \(P_{\mathrm{HO} \mathrm{C}}=20.0\) torr

Short answer

For the given reaction and equilibrium constant: a. \(Q_a = 1.00\). The system is not at equilibrium and will shift towards the reactants. b. \(Q_b = 0.342\). The system is not at equilibrium and will shift towards the reactants. c. \(Q_c = 0.0600\). The system is not at equilibrium and will shift towards the products.

Step-by-step solution

Write the expression for the reaction quotient Q and recall the meaning of Q, K, and equilibrium

For the given reaction, the reaction quotient Q is defined as follows: $$ Q = \frac{P_{\mathrm{HOCl}}^2}{P_{\mathrm{H}_{2}\mathrm{O}} \cdot P_{\mathrm{Cl}_{2}\mathrm{O}}} $$ We need to evaluate the reaction quotient for each case and compare Q to the equilibrium constant K. If Q > K, the reaction will shift towards the reactants. If Q < K, the reaction will shift towards the products. If Q = K, the system is at equilibrium.

Evaluate reaction quotient Q for each set of conditions

For a.: $$ Q_a = \frac{P_{\mathrm{HOCl},a}^2}{P_{\mathrm{H}_{2}\mathrm{O},a} \cdot P_{\mathrm{Cl}_{2}\mathrm{O},a}} = \frac{(1.00 \, \text{ atm})^2}{(1.00 \, \text{ atm}) \cdot (1.00 \, \text{ atm})} = 1.00 $$ For b.: (Note that we need to convert the pressures into atm by dividing by 760 torr/atm) $$ Q_b = \frac{P_{\mathrm{HOCl},b}^2}{P_{\mathrm{H}_{2}\mathrm{O},b} \cdot P_{\mathrm{Cl}_{2}\mathrm{O},b}} = \frac{(21.0 \, \text{torr})^2}{(200.0 \, \text{torr}) \cdot (49.8 \, \text{ torr})} \cdot \frac{(760 \, \text{torr/atm})^3}{(1 \, \text{atm})^3} = 0.342 $$ For c.: $$ Q_c = \frac{P_{\mathrm{HOCl},c}^2}{P_{\mathrm{H}_{2}\mathrm{O},c} \cdot P_{\mathrm{Cl}_{2}\mathrm{O},c}} = \frac{(20.0 \, \text{torr})^2}{(296.0 \, \text{torr}) \cdot (15.0 \, \text{ torr})} \cdot \frac{(760 \, \text{torr/atm})^3}{(1 \, \text{atm})^3} = 0.0600 $$

Compare Q to K and determine the equilibrium state and direction of shift

Recall that the equilibrium constant K is 0.0900. Comparing Q to K for each case: a. \(Q_a = 1.00 \Rightarrow Q_a > K\). The reaction will shift towards the reactants. b. \(Q_b = 0.342 \Rightarrow Q_b > K\). The reaction will shift towards the reactants. c. \(Q_c = 0.0600 \Rightarrow Q_c < K\). The reaction will shift towards the products. Hence, none of the given sets of conditions are at equilibrium. For cases (a) and (b), the system will shift towards the reactants, while for case (c), the system will shift towards the products.

Key concepts

Equilibrium Constant

The equilibrium constant, represented as K, is a vital concept in the study of chemical reactions. It is a number that provides a quantitative measure of the position of equilibrium for a reversible reaction at a given temperature. When a system reaches equilibrium, the rates of the forward and reverse reactions are equal, and the concentrations of the reactants and products remain constant over time.

The equilibrium constant is determined by the specific balance between the concentrations (for solutions) or partial pressures (for gases) of the reactants and products. The expression for the equilibrium constant for a reaction such as
$$A + B \rightleftharpoons C + D$$
is given by
$$K = \frac{[C]^{c}[D]^{d}}{[A]^{a}[B]^{b}}$$
where [A], [B], [C], and [D] are the molar concentrations or partial pressures of the reactants and products, and a, b, c, d are their respective stoichiometric coefficients in the balanced chemical equation.

The value of K can indicate the extent to which a reaction favors the formation of products. A high K value means that at equilibrium, the concentration (or pressure) of the products is much greater than that of the reactants, indicating a reaction that lies to the right or is 'product-favored'. Conversely, a low K value suggests a reactant-favored equilibrium, lying to the left.

Reaction Quotient

The reaction quotient, denoted as Q, plays an essential role in predicting the direction of a chemical reaction. Unlike the equilibrium constant K that is only true at equilibrium, Q can be calculated at any given moment during a reaction. The expression for Q is identical to that of K, but it uses the current concentrations or partial pressures instead of those at equilibrium.

$$Q = \frac{[C]^{c}[D]^{d}}{[A]^{a}[B]^{b}}$$
The comparison of Q with the known value of K serves as a predictive tool:

• If Q < K, the forward reaction is favored, and the system will shift towards the products to reach equilibrium.
• If Q > K, the reverse reaction is favored, and the system will shift towards the reactants.
• If Q = K, the system is already at equilibrium, and no shift will occur.

Using this approach, chemists can determine how changes in concentration or pressure will influence the progress of the reaction, and predict the conditions needed to achieve equilibrium.

Le Chatelier's Principle

Le Chatelier's principle is a key concept that explains how a system at equilibrium responds to external changes in conditions, such as changes in concentration, temperature, or pressure. The principle states that if a stress is applied to a system at equilibrium, the system will adjust, or shift, to partially counteract the imposed change and re-establish equilibrium.

For instance, if there is an increase in the concentration of reactants, the system will shift to produce more products. Similarly, a decrease in the concentration of products will result in a shift towards the formation of more products from reactants. Le Chatelier's principle can also be applied to changes in pressure for reactions that involve gases. If the pressure increases by decreasing the volume, the reaction will shift towards the side with fewer moles of gas in order to reduce pressure.

Temperature changes can also affect equilibrium. For exothermic reactions, an increase in temperature shifts the equilibrium towards the reactants, treating heat as a product. Conversely, an endothermic reaction will shift towards products when heated, treating heat as a reactant. Le Chatelier's principle is an essential tool for chemists in controlling reaction conditions to obtain the desired product yield.

Partial Pressure

In chemical thermodynamics, the term 'partial pressure' refers to the pressure exerted by a single gas component in a mixture of gases. It is the pressure that the gas would have if it alone occupied the entire volume of the mixture at the same temperature.

The total pressure exerted by a gas mixture is the sum of the partial pressures of each individual gas component, as described by Dalton's Law of Partial Pressures:$$P_{total} = P_{1} + P_{2} + P_{3} + ... = \sum P_{i}$$
where P_{1}, P_{2}, P_{3}, ... are the partial pressures of the gases. In the context of chemical equilibrium involving gaseous systems, the equilibrium expression is often written in terms of partial pressures.

For example, for the reaction given in the exercise, the equilibrium constant in terms of partial pressure is expressed as:$$K = \frac{{P_{HOCl}}^2}{P_{H_2O} \times P_{Cl_2O}}$$
Partial pressures are used when predicting the direction of a reaction's shift based on changes in pressure, following Le Chatelier's principle. It is important for students to be comfortable with converting between pressure units, such as from atmospheres (atm) to torr, as pressures can be measured or reported in various units.