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In chemical equilibrium, the equilibrium constant (often denoted as \(K\) or \(K_p\) for gas-phase reactions) represents the ratio of the concentrations or partial pressures of the products to those of the reactants. This is under equilibrium conditions for a given reaction at a specific temperature.
For the reaction \( \mathrm{C}(s) + 2 \mathrm{H}_{2}(g) \leftrightharpoons \mathrm{CH}_{4}(g) \), the equilibrium constant in terms of partial pressures \(K_p\) is given by:

• \(K_p = \frac{P_{\mathrm{CH_4}}}{(P_{\mathrm{H_2}})^2}\)

Here, \(P_{\mathrm{CH_4}}\) represents the partial pressure of methane, while \(P_{\mathrm{H_2}}\) is the partial pressure of hydrogen.

The constant \(K_p\) at 500°C is given as \(2.69 \times 10^{3} \mathrm{bar}^{-1}\). This value tells us how much product we have compared to the reactants at equilibrium. A high \(K_p\) indicates that the reaction favors the formation of products under these conditions. In this particular scenario, with a high \(K_p\), methane is significantly favored at equilibrium.

Partial pressure refers to the pressure a particular gas in a mixture would exert if it were alone in the container. It is an essential concept when dealing with gaseous reactions and equilibrium.

In the given reaction, we have the partial pressures of hydrogen and methane.

• Partial pressure of hydrogen, \(P_{\mathrm{H_{2}}} = 0.20\) bar
• Partial pressure of methane, \(P_{\mathrm{CH_{4}}} = 3.0\) bar

Each gas contributes to the total pressure proportionally to its amount in the mixture. Knowing partial pressures helps use the equilibrium expression to determine the extent to which a reaction has progressed or predict how it will shift to reach equilibrium.
In this case, these initial pressures are crucial for calculating the reaction quotient and deciding if the reaction is at equilibrium or will proceed in a particular direction.

The reaction quotient, denoted as \(Q\), is calculated in the same way as the equilibrium constant but uses the initial concentrations or partial pressures of the reactants and products.
This measure helps predict the direction a reaction will shift to achieve equilibrium. For our reaction, the reaction quotient \(Q\) is:

• \(Q = \frac{P_{\mathrm{CH_4}}}{(P_{\mathrm{H_2}})^2}\)

Substituting the given initial values:\[Q = \frac{3.0}{(0.20)^2} = 75.0\]
A comparison between \(Q\) and \(K_p\) dictates the shift in the reaction:

• If \(Q < K_p\), the reaction proceeds forward to produce more products.
• If \(Q > K_p\), the reaction will go backward, converting products back to reactants.
• If \(Q = K_p\), the system is at equilibrium, and no shift is necessary.

In this scenario, since \(Q = 75.0\) is greater than \(K_p = 2.69 \times 10^{3}\), the reaction will proceed in the reverse direction, meaning reactants will form from the products to reach equilibrium.