Problem 43 - Chemistry The Central Science In Si Units

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) \(\left[\mathrm{COCl}_{2}\right]=[\mathrm{CO}]=1.00 \times 10^{-4} \mathrm{M},\left[\mathrm{Cl}_{2}\right]=7.2 \times 10^{-6} \mathrm{M}\) (b) \(\left[\mathrm{COCl}_{2}\right]=2.20 \times 10^{-2} \mathrm{M},[\mathrm{CO}]=2.2 \times 10^{-7} \mathrm{M}\) \(\left[\mathrm{Cl}_{2}\right]=3.0 \times 10^{-6} \mathrm{M}\) (c) \(\left[\mathrm{COCl}_{2}\right]=0.0100 \mathrm{M},[\mathrm{CO}]=\left[\mathrm{Cl}_{2}\right]=7.2 \times 10^{-6} \mathrm{M}\)

Step-by-Step Solution

Answer

(a) Not at equilibrium, shifts left; (b) Not at equilibrium, shifts right; (c) Not at equilibrium, shifts left.

Step 1: Determine Reaction Quotient Formula

The reaction quotient \( Q_c \) for the reaction \( \mathrm{COCl}_2(g) \rightleftharpoons \mathrm{CO}(g) + \mathrm{Cl}_2(g) \) is given by the formula:\[Q_c = \frac{[\mathrm{CO}] \times [\mathrm{Cl}_2]}{[\mathrm{COCl}_2]}\] This formula helps to compare the current state of the system with the equilibrium state.

Step 2: Calculate Qc for Mixture (a)

For mixture (a), substitute the given concentrations:\[Q_c = \frac{(1.00 \times 10^{-4})(7.2 \times 10^{-6})}{1.00 \times 10^{-4}}\]Simplifying gives \( Q_c = 7.2 \times 10^{-6} \).

Step 3: Compare Qc with Kc for Mixture (a)

Compare \( Q_c = 7.2 \times 10^{-6} \) with \( K_c = 2.19 \times 10^{-10} \). Since \( Q_c > K_c \), the system is not at equilibrium and will shift to the left (towards the reactant) to reach equilibrium.

Step 4: Calculate Qc for Mixture (b)

For mixture (b), substitute the given concentrations:\[Q_c = \frac{(2.2 \times 10^{-7})(3.0 \times 10^{-6})}{2.20 \times 10^{-2}}\]Simplifying gives \( Q_c \approx 3.0 \times 10^{-11} \).

Step 5: Compare Qc with Kc for Mixture (b)

Compare \( Q_c = 3.0 \times 10^{-11} \) with \( K_c = 2.19 \times 10^{-10} \). Since \( Q_c < K_c \), the system is not at equilibrium and will shift to the right (towards the products) to reach equilibrium.

Step 6: Calculate Qc for Mixture (c)

For mixture (c), substitute the given concentrations:\[Q_c = \frac{(7.2 \times 10^{-6})(7.2 \times 10^{-6})}{0.0100}\]Simplifying gives \( Q_c \approx 5.18 \times 10^{-9} \).

Step 7: Compare Qc with Kc for Mixture (c)

Compare \( Q_c = 5.18 \times 10^{-9} \) with \( K_c = 2.19 \times 10^{-10} \). Since \( Q_c > K_c \), the system is not at equilibrium and will shift to the left (towards the reactant) to reach equilibrium.

Key Concepts

Reaction Quotient

In chemical equilibrium, the reaction quotient, represented as \( Q_c \), is a helpful tool to determine whether a system is at equilibrium. The reaction quotient is calculated in a similar manner to the equilibrium constant \( K_c \), but uses the current concentrations of reactants and products in the formula. For our example reaction \( \text{COCl}_2(g) \rightleftharpoons \text{CO}(g) + \text{Cl}_2(g) \), it is calculated as follows:
\[Q_c = \frac{[\text{CO}] \times [\text{Cl}_2]}{[\text{COCl}_2]}\]
It's important to note that \( Q_c \) can change as a reaction progresses, unlike \( K_c \) which is constant at a given temperature.
After calculating \( Q_c \), it is compared to \( K_c \), the equilibrium constant, to assess the system's state:

• If \( Q_c = K_c \), the system is at equilibrium.
• If \( Q_c > K_c \), there are more products compared to reactants than at equilibrium, so the reaction will shift towards the reactants to reach equilibrium.
• If \( Q_c < K_c \), the products are less than the equilibrium concentration, necessitating a shift towards product formation to attain equilibrium as in Mixture (b) from the example.

This comparison provides insights into the direction in which a reaction needs to adjust to achieve equilibrium.

Equilibrium Constant

The equilibrium constant \( K_c \) is a crucial concept in understanding chemical equilibrium. It represents the ratio of the concentrations of products to reactants, each raised to the power of their respective coefficients, at equilibrium. It's a fixed value at a given temperature.
For the reaction \( \text{COCl}_2(g) \rightleftharpoons \text{CO}(g) + \text{Cl}_2(g) \), the equilibrium constant is expressed as:
\[K_c = \frac{[\text{CO}] \times [\text{Cl}_2]}{[\text{COCl}_2]}\]
The value of \( K_c \) is given for the reaction at \( 100^{\circ} \text{C} \) as \( 2.19 \times 10^{-10} \). This small value indicates that at equilibrium, the concentrations of reactants are favored over products, as there are fewer molecules of \( \text{CO} \) and \( \text{Cl}_2 \) relative to \( \text{COCl}_2 \).
On comparing the calculated \( Q_c \) values for the mixtures in the exercise with \( K_c \), one can determine in which direction the reaction needs to shift. This helps in predicting how the system reacts to changes in concentration, temperature, and pressure, as it strives for equilibrium.

Le Chatelier's Principle

Le Chatelier's Principle is a guiding concept for predicting how a chemical system at equilibrium will respond to changes. It states that if a change or stress is applied to a system at equilibrium, the system will adjust itself to counteract that change and restore a new equilibrium.
In the context of the reaction \( \text{COCl}_2(g) \rightleftharpoons \text{CO}(g) + \text{Cl}_2(g) \), if the concentration of one of the components is changed, the system will strive to adjust the concentrations to offset that change, thereby moving towards a new equilibrium state.

• Addition of more \( \text{COCl}_2 \) will shift the reaction to the right (forming more products) to alleviate the stress.
• Conversely, adding more \( \text{CO} \) or \( \text{Cl}_2 \) will push the reaction to the left (forming more reactant) to decrease the product concentration.

This principle also applies to changes in temperature and pressure, affecting the direction in which a reaction proceeds. It is a valuable tool for understanding and predicting the behavior of equilibria in chemical systems.