Equilibrium Pressure Dependence

The question asks which thermodynamic quantity for the reaction $\text{N}_2\text{(g)} + 3\text{H}_2\text{(g)} \rightleftharpoons 2\text{NH}_3\text{(g)}$ depends on the pressure at which equilibrium is reached. Let's examine each option.

Understanding Thermodynamic Quantities and Equilibrium

• $\Delta G^\circ$: This is the standard Gibbs free energy change. It is a standard state quantity, meaning it is measured under standard conditions (typically 1 bar pressure for gases and a specified temperature). $\Delta G^\circ$ depends only on the standard state properties of the reactants and products and the temperature, not the actual pressure at which equilibrium is achieved in a non-standard state system.
• $\Delta H^\circ$: This is the standard enthalpy change. Like $\Delta G^\circ$, it is a standard state quantity. It represents the heat change under standard conditions and depends primarily on temperature, not the equilibrium pressure of the system.
• $K_p$: This is the equilibrium constant expressed in terms of partial pressures. For a given reaction at a specific temperature, the value of $K_p$ is constant, regardless of the initial pressures or the total pressure at equilibrium. It is a true constant for the reaction at a fixed temperature.
• Ratio of mole fractions at equilibrium: This ratio, often denoted as $K_x$, is defined for the given reaction as $\frac{{{\rm{x}}_{{\rm{N}}{{\rm{H}}_3}}^{\rm{2}}}}{{{x_{{N_2}}}x_{{H_2}}^3}}$, where $x_i$ is the mole fraction of component $i$.

Relating $K_p$ and the Ratio of Mole Fractions ($K_x$)

The equilibrium constant $K_p$ and the ratio of mole fractions $K_x$ are related through the total pressure ($P_{total}$) and the change in the number of moles of gas ($\Delta n_g$) for the reaction.

The relationship is given by:

$\qquad K_p = K_x (P_{total})^{\Delta n_g}$

For the reaction $\text{N}_2\text{(g)} + 3\text{H}_2\text{(g)} \rightleftharpoons 2\text{NH}_3\text{(g)}$, the change in the number of moles of gas is:

$\qquad \Delta n_g = (\text{moles of gaseous products}) - (\text{moles of gaseous reactants})$

$\qquad \Delta n_g = 2 - (1 + 3) = 2 - 4 = -2$

Substituting $\Delta n_g = -2$ into the relationship between $K_p$ and $K_x$:

$\qquad K_p = K_x (P_{total})^{-2}$

This can be rearranged to express $K_x$ in terms of $K_p$ and $P_{total}$:

$\qquad K_x = K_p (P_{total})^{2}$

Since $K_p$ is a constant at a fixed temperature, this equation shows that $K_x$ is directly proportional to the square of the total pressure ($P_{total}$) at equilibrium. Therefore, the ratio of mole fractions at equilibrium depends on the total pressure at which equilibrium is established.

In summary:

• $\Delta G^\circ$ and $\Delta H^\circ$ are standard state quantities and do not depend on the equilibrium pressure.
• $K_p$ is the true equilibrium constant at a given temperature and does not depend on the equilibrium pressure.
• The ratio of mole fractions ($K_x$) is related to $K_p$ by the total pressure at equilibrium ($P_{total}$) and $\Delta n_g$. Since $\Delta n_g \neq 0$ for this reaction, $K_x$ depends on the equilibrium pressure.

Thus, the thermodynamic quantity among the given options that depends upon the pressure at which equilibrium is arrived at is the ratio of mole fractions at equilibrium.