Question

The equilibrium constant \(\left(K_{\mathrm{c}}\right)\) for the reaction $$ 2 \mathrm{HCl}(g) \rightleftharpoons \mathrm{H}_{2}(g)+\mathrm{Cl}_{2}(g) $$ is \(4.17 \times 10^{-34}\) at \(25^{\circ} \mathrm{C}\). What is the equilibrium constant for the reaction $$ \mathrm{H}_{2}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{HCl}(g) $$ at the same temperature?

Short answer

The equilibrium constant for the reaction \( \mathrm{H}_{2}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{HCl}(g) \) at the same temperature is \( \frac{1}{4.17 \times 10^{-34}} \).

Step-by-step solution

Understand the relationship between the equilibrium constants for forward and reverse reactions

The equilibrium constant for the reverse reaction is the reciprocal of the equilibrium constant for the forward reaction. This is because for a reaction in equilibrium, both the forward and reverse reactions are taking place, and if the reaction quotient for the forward reaction is \(K_c\), then the reaction quotient for the reverse reaction is always \( \frac{1}{K_c} \).

Apply the reciprocal rule to find the equilibrium constant for the reverse reaction

Given that the equilibrium constant \(K_c\) for the forward reaction is \(4.17 \times 10^{-34}\), the equilibrium constant for the reverse reaction will be calculated as \( \frac{1}{4.17 \times 10^{-34}} \).

Key concepts

Equilibrium Constant

The equilibrium constant, denoted as \( K_c \) or \( K_{eq} \), serves as a key indicator of the extent to which a chemical reaction proceeds before reaching the state of equilibrium. It is defined as the ratio of the concentration of products to reactants, each raised to the power of their stoichiometric coefficients in a balanced chemical equation, when the reaction has reached equilibrium at a constant temperature.

For a generic reaction where \( aA + bB \rightleftharpoons cC + dD \), the equilibrium constant would be expressed as:\[ K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b} \]
It’s essential to understand that \( K_c \) is constant only at a given temperature; any change in temperature could affect the value of the equilibrium constant, thus also affecting the position of the equilibrium.

Reciprocal Rule in Chemistry

The reciprocal rule in chemistry is an important concept when dealing with reversible reactions and their equilibrium constants. This rule states that if a reaction in one direction has an equilibrium constant \( K_c \), the equilibrium constant for the reaction in the reverse direction is \( 1/K_c \).

This is attributed to the fact that the reaction's dynamics are symmetrical; the forward process's propensity to create products at equilibrium is equally balanced by the reverse process's tendency to revert to reactants. Mathematically, if the forward reaction is represented as \( A \rightleftharpoons B \) with equilibrium constant \( K_{c1} \), the reverse \( B \rightleftharpoons A \) would have an equilibrium constant of \( K_{c2} = 1/K_{c1} \).

Chemical Equilibrium

Chemical equilibrium occurs in a reversible chemical reaction when the rates of the forward and reverse reactions equalize, resulting in no overall change in the concentrations of reactants and products over time. It is important to note that even though the macroscopic properties are not changing, the reaction remains dynamic; reactants and products continually form, albeit at the same rate.

At equilibrium, the reaction has reached a state of balance, but this does not mean the reactants and products are present in equal concentrations—rather, their ratios are stable. The concept of equilibrium plays a fundamental role in predicting product formation and reactant consumption in chemical reactions.

Reaction Quotient

The reaction quotient, denoted as \( Q \), is a numerical value that shows the relative amounts of products and reactants present during a reaction at any given point in time before equilibrium is reached. It is calculated using the same formula as for the equilibrium constant, but with the current concentrations rather than the equilibrium concentrations.

The value of \( Q \) in comparison to the equilibrium constant \( K_c \) allows prediction of the system’s direction to reach equilibrium. If \( Q < K_c \), the system tends to form more products, moving forward. If \( Q > K_c \), the system tends to form more reactants, shifting the reaction backward. When \( Q = K_c \), the system is already at equilibrium.