Question

Suppose that the gas-phase reactions A \(\longrightarrow \mathrm{B}\) and \(\mathrm{B} \longrightarrow\) A are both elementary processes with rate constants of \(4.7 \times 10^{-3} \mathrm{s}^{-1}\) and \(5.8 \times 10^{-1} \mathrm{s}^{-1}\) , respectively. (a) What is the value of the equilibrium constant for the equilibrium \(A(g) \rightleftharpoons B(g) ?(\mathbf{b})\) Which is greater at equilibrium, the partial pressure of A or the partial pressure of \(B ?\)

Short answer

(a) The value of the equilibrium constant for the equilibrium \(A(g) \rightleftharpoons B(g)\) is approximately \(0.0081\). (b) At equilibrium, the partial pressure of A is greater than the partial pressure of B, because the reaction favors the formation of reactants.

Step-by-step solution

Write the relationship between equilibrium constant and rate constants

To find the value of the equilibrium constant, \(K_c\), we can use the relationship between the equilibrium constant and the rate constants of the forward and reverse reactions: \[K_c = \frac{k_{forward}}{k_{reverse}}\]

Substitute the values of rate constants

The given rate constants are \(k_{forward} = 4.7 \times 10^{-3} \mathrm{s}^{-1}\) and \(k_{reverse} = 5.8 \times 10^{-1} \mathrm{s}^{-1}\). Substitute these values into the formula to find \(K_c\): \[K_c = \frac{4.7 \times 10^{-3}}{5.8 \times 10^{-1}}\]

Calculate the value of the equilibrium constant

By dividing the given rate constants, we get the value of \(K_c\): \[K_c \approx 0.0081\]

Analyze the value of the equilibrium constant

The value of the equilibrium constant, \(K_c \approx 0.0081\), is less than 1. This means that the reaction favors the formation of reactants, in this case A. Therefore, at equilibrium, the partial pressure of A will be greater than the partial pressure of B.

Conclude the findings

(a) The value of the equilibrium constant for the equilibrium \(A(g) \rightleftharpoons B(g)\) is approximately \(0.0081\). (b) At equilibrium, the partial pressure of A is greater than the partial pressure of B, because the reaction favors the formation of reactants.

Key concepts

Chemical Equilibrium

Understanding chemical equilibrium is fundamental in the study of chemical reactions. It can be described as the state in a reversible reaction where the rate of the forward reaction equals the rate of the reverse reaction, leading to a consistent concentration of reactants and products over time. Importantly, this does not mean that the reactants and products are present in equal amounts but that their concentrations no longer change.

In the given problem, when reactant A converts to product B and vice versa at the same rate, an equilibrium is established. If the equilibrium constant, denoted as K_c, is less than one, it indicates that reactants are favored, and the system will have a higher concentration of the starting materials. Conversely, if K_c is greater than one, the products are favored in the equilibrium mixture.

Rate Constants

Rate constants are crucial parameters in the kinetics of a chemical reaction, representing the speed at which reactants turn into products. For a given reaction at a specific temperature, the rate constant can tell us how quickly a reaction proceeds. It's important to note that the rate constants for the forward and reverse reactions are independent of each other and are determined experimentally.

In the provided exercise, the rate constants for the forward and reverse reactions are given, allowing us to determine the equilibrium constant, K_c. The forward reaction has a rate constant, k_forward, of 4.7 x 10^-3 s^-1, and the reverse has a rate constant, k_reverse, of 5.8 x 10^-1 s^-1. The lower rate constant of the forward reaction hints at a slower rate of product formation compared to the rate at which the products revert to reactants.

Partial Pressure

The term partial pressure refers to the pressure exerted by a single gas in a mixture of gases. It is directly proportional to its concentration in the gas phase, making it a useful metric in reactions involving gases at equilibrium. According to Dalton's Law of Partial Pressures, the total pressure of a gas mixture is the sum of the partial pressures of each individual gas.

In our exercise, we compared the partial pressures of gas A and B at equilibrium to determine which one is greater. Because the equilibrium constant K_c is less than one, indicating a favoring of the reactants, gas A will have a greater partial pressure than gas B when the reaction reaches equilibrium.

Reactants and Products

Reactants are the starting materials in a chemical reaction, while products are the substances formed as a result of that reaction. The conversion of reactants to products and vice versa is what drives chemical reactions towards equilibrium.

By considering the provided information and calculating the equilibrium constant, we are able to understand the dynamics of the reactants and products involved. A small value of K_c tells us that the reaction system will have a higher concentration of reactants at equilibrium. Thus, we conclude that for the reaction A(g) ↔ B(g), reactant A will be present at a higher partial pressure than product B at equilibrium. This illustrates the direct connection between the value of the equilibrium constant and the relative concentrations or partial pressures of reactants and products.