Question

At \(100^{\circ} \mathrm{C}\) the equilibrium constant for the reaction \(\mathrm{COCl}_{2}(g) \rightleftharpoons \mathrm{CO}(g)+\mathrm{Cl}_{2}(g)\) has the value \(K_{c}=\) \(2.19 \times 10^{-10}\). Are the following mixtures of \(\mathrm{COCl}_{2}, \mathrm{CO}\), and \(\mathrm{Cl}_{2}\) at \(100^{\circ} \mathrm{C}\) at equilibrium? If not, indicate the direction that the reaction must proceed to achieve equilibrium. (a) \(\left[\mathrm{COCl}_{2}\right]=2.00 \times 10^{-3} \mathrm{M}, \quad[\mathrm{CO}]=3.3 \times 10^{-6} \mathrm{M}\) \(\left[\mathrm{Cl}_{2}\right]=6.62 \times 10^{-6} M ; \quad\) (b) \(\left[\mathrm{COCl}_{2}\right]=4.50 \times 10^{-2} M\) \([\mathrm{CO}]=1.1 \times 10^{-7} \mathrm{M},\left[\mathrm{Cl}_{2}\right]=2.25 \times 10^{-6} \mathrm{M} ;\) (c) \(\left[\mathrm{COCl}_{2}\right]=\) \(0.0100 M,[\mathrm{CO}]=\left[\mathrm{Cl}_{2}\right]=1.48 \times 10^{-6} \mathrm{M}\)

Short answer

(a) The reaction will shift to the left, forming more reactants (COCl₂). (b) The reaction will shift to the right, forming more products (CO and Cl₂). (c) The reaction is already at equilibrium.

Step-by-step solution

Write down the balanced chemical equation and the equilibrium expression

We are given the balanced chemical equation: COCl₂(g) ⇌ CO(g) + Cl₂(g) The equilibrium expression for the reaction is given by: \(K_c = \frac{[CO][Cl_2]}{[COCl_2]}\) For each mixture, we will calculate the reaction quotient (Qc) and compare it to the equilibrium constant (Kc) to determine if the mixture is at equilibrium or not.

Calculate Qc for mixture (a)

Given concentrations for mixture (a): [COCl₂] = 2.00 × 10⁻³ M [CO] = 3.3 × 10⁻⁶ M [Cl₂] = 6.62 × 10⁻⁶ M Calculate Qc for mixture (a): \(Q_c = \frac{[CO][Cl_2]}{[COCl_2]} = \frac{(3.3 \times 10^{-6})(6.62 \times 10^{-6})}{(2.00 \times 10^{-3})} = 1.09 \times 10^{-8}\)

Compare Qc to Kc for mixture (a)

Given Kc = 2.19 × 10⁻¹⁰, Since Qc (1.09 × 10⁻⁸) > Kc (2.19 × 10⁻¹⁰), the reaction will shift to the left, forming more reactants (COCl₂).

Calculate Qc for mixture (b)

Given concentrations for mixture (b): [COCl₂] = 4.50 × 10⁻² M [CO] = 1.1 × 10⁻⁷ M [Cl₂] = 2.25 × 10⁻⁶ M Calculate Qc for mixture (b): \(Q_c = \frac{[CO][Cl_2]}{[COCl_2]} = \frac{(1.1 \times 10^{-7})(2.25 \times 10^{-6})}{(4.50 \times 10^{-2})} = 5.46 \times 10^{-12}\)

Compare Qc to Kc for mixture (b)

Given Kc = 2.19 × 10⁻¹⁰, Since Qc (5.46 × 10⁻¹²) < Kc (2.19 × 10⁻¹⁰), the reaction will shift to the right, forming more products (CO and Cl₂).

Calculate Qc for mixture (c)

Given concentrations for mixture (c): [COCl₂] = 0.0100 M [CO] = [Cl₂] = 1.48 × 10⁻⁶ M Calculate Qc for mixture (c): \(Q_c = \frac{[CO][Cl_2]}{[COCl_2]} = \frac{(1.48 \times 10^{-6})(1.48 \times 10^{-6})}{(0.0100)} = 2.19 \times 10^{-10}\)

Compare Qc to Kc for mixture (c)

Given Kc = 2.19 × 10⁻¹⁰, Since Qc (2.19 × 10⁻¹⁰) = Kc (2.19 × 10⁻¹⁰), the reaction is already at equilibrium for mixture (c). In conclusion, for mixture (a) the reaction will shift to the left, for mixture (b) it will shift to the right, and for mixture (c), it is already at equilibrium.

Key concepts

Equilibrium Constant (Kc)

The equilibrium constant, denoted as \(K_c\), is a key concept in understanding chemical equilibrium. It defines the ratio of the concentrations of products to reactants, raised to the power of their stoichiometric coefficients, at equilibrium. For a given reaction such as - \(\text{COCl}_2(g) \rightleftharpoons \text{CO}(g) + \text{Cl}_2(g)\), the equilibrium constant expression is written as:- \[K_c = \frac{[\text{CO}][\text{Cl}_2]}{[\text{COCl}_2]}\]This equation helps determine the position of equilibrium; whether it favors the reactants or the products.

• A \(K_c\) value much greater than one indicates that, at equilibrium, products are favored.
• A \(K_c\) value much less than one suggests that reactants are favored.

Understanding \(K_c\) is crucial as it provides insight into how a reaction will behave under certain conditions. It is important to note that \(K_c\) remains constant for a given reaction at a specific temperature.

Reaction Quotient (Qc)

The reaction quotient, \(Q_c\), is a useful tool for predicting the direction in which a reaction will proceed at any given point in time. It is calculated using the same formula as the equilibrium constant, \(K_c\). However, \(Q_c\) is computed with initial concentrations:- \[Q_c = \frac{[\text{CO}][\text{Cl}_2]}{[\text{COCl}_2]}\]By comparing \(Q_c\) to \(K_c\), one can determine how close a system is to equilibrium:

• If \(Q_c = K_c\), the system is at equilibrium.
• If \(Q_c < K_c\), the system will shift towards the products to reach equilibrium.
• If \(Q_c > K_c\), the system will shift towards the reactants to reach equilibrium.

Thus, \(Q_c\) serves as a snapshot to help predict how a reaction will adjust its concentrations in order to find equilibrium, providing practical insights into reaction dynamics.

Le Chatelier's Principle

Le Chatelier's Principle is a key concept in chemical equilibrium that describes how a system at equilibrium reacts to disturbances. According to this principle, if an external change is applied to a system at equilibrium, the system will adjust itself in such a way as to counteract that change and achieve a new equilibrium state. There are several common disturbances that can affect a chemical reaction:

• **Concentration Change:** Increasing the concentration of a reactant or product will cause the system to shift in the direction that consumes the added substance. Conversely, decreasing one will cause the system to shift to produce more of that substance.
• **Temperature Change:** An increase in temperature for an endothermic reaction will shift the equilibrium towards the products, while for an exothermic reaction, it will shift towards the reactants.
• **Pressure Change (for gaseous reactions):** Increasing pressure by decreasing volume will cause the equilibrium to shift towards the side with fewer moles of gas.

Understanding Le Chatelier's Principle helps predict how equilibrium systems will respond to changes, providing a predictive insight into the behavior of chemical reactions under various conditions.