Question

For an ideal gas mixture undergoing a reversible gaseous phase chemical reaction, the equilibrium constant:

• A. Decreases with pressure
• B. Increases with pressure
• C. Is independent of pressure
• D. Increases/decreases with pressure depending on the stoichiometric coefficients of the reaction

Hint:

Remember that the equilibrium constant \( K \) is a function of temperature only for an ideal gas reaction. Changes in pressure will shift the equilibrium position (i.e., the relative amounts of reactants and products at equilibrium) if the number of moles of gas changes during the reaction, but the value of \( K \) remains constant at a fixed temperature.

Answer 1

The Correct Option is C

Step 1: Understand the definition of the equilibrium constant.
The equilibrium constant \( K \) for a reversible reaction at a given temperature is a value that relates the amounts of reactants and products at chemical equilibrium. For a general reversible reaction: \[ aA + bB \rightleftharpoons cC + dD \] The equilibrium constant in terms of activities \( K_a \) is given by: \[ K_a = \frac{a_C^c a_D^d}{a_A^a a_B^b} \] where \( a_i \) represents the activity of species \( i \) at equilibrium. For ideal gases, the activity of a species is equal to its fugacity divided by the standard state fugacity. For an ideal gas, the fugacity is equal to its partial pressure. If the standard state is chosen as 1 bar, then the activity of an ideal gas is numerically equal to its partial pressure in bars. The equilibrium constant in terms of partial pressures \( K_p \) is given by: \[ K_p = \frac{(P_C)^c (P_D)^d}{(P_A)^a (P_B)^b} \] where \( P_i \) is the partial pressure of species \( i \) at equilibrium. The equilibrium constant in terms of concentrations \( K_c \) is given by: \[ K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b} \] where \( [i] \) is the molar concentration of species \( i \) at equilibrium.
Step 2: Analyze the dependence of the equilibrium constant on pressure.
The standard Gibbs free energy change \( \Delta G^\circ \) for a reaction is related to the equilibrium constant \( K_a \) by the equation: \[ \Delta G^\circ = -RT \ln K_a \] where \( R \) is the ideal gas constant and \( T \) is the absolute temperature. The standard Gibbs free energy change \( \Delta G^\circ \) depends only on temperature and the standard states of the reactants and products, not on the total pressure of the system. Since \( K_a \) is directly related to \( \Delta G^\circ \) and \( R \) and \( T \) are independent of pressure at a constant temperature, the equilibrium constant \( K_a \) is independent of pressure for an ideal gas mixture. The relationship between \( K_p \) and \( K_c \) is given by: \[ K_p = K_c (RT)^{\Delta n_g} \] where \( \Delta n_g = (c + d) - (a + b) \) is the change in the number of moles of gas in the reaction. While \( K_p \) and \( K_c \) are related, and their values might be affected by changes in partial pressures or concentrations due to a change in total pressure (according to Le Chatelier's principle, which shifts the equilibrium position), the values of \( K_p \) and \( K_c \) themselves remain constant at a constant temperature for an ideal gas reaction. They are defined based on the equilibrium conditions, and their constancy ensures that the relationship between reactants and products at equilibrium is maintained at a given temperature, regardless of the total pressure.
Step 3: Evaluate the given options.
Options 1 and 2 suggest that the equilibrium constant changes with pressure, which is incorrect for an ideal gas mixture at a constant temperature.
Option 4 suggests that the change depends on stoichiometric coefficients. While the equilibrium position shifts with pressure if \( \Delta n_g \neq 0 \), the equilibrium constant itself does not change.
Option 3 correctly states that the equilibrium constant is independent of pressure for an ideal gas mixture at a constant temperature.
Step 4: Select the correct answer.
For an ideal gas mixture undergoing a reversible gaseous phase chemical reaction, the equilibrium constant is independent of pressure at a constant temperature.