Explain the difference between \(K, K_{\mathrm{p}},\) and \(Q\).

\(K\) is the equilibrium constant. It is a dimensionless quantity that describes the relationship between the concentration of reactants and products in a chemical reaction at equilibrium. The expression for the equilibrium constant, \(K\), can be written as: \[K = \frac{[C]^c[D]^d}{[A]^a[B]^b}\] where: - \([A]\), \([B]\), \([C]\), and \([D]\) represent the equilibrium concentration of species A, B, C, and D, respectively. - \(a\), \(b\), \(c\), and \(d\) represent the stoichiometric coefficients of the balanced chemical equation.

\(K_\mathrm{p}\) is the equilibrium constant for a reaction involving gases, expressed in terms of partial pressures instead of concentrations. The expression for \(K_\mathrm{p}\) can be written as: \[K_\mathrm{p} = \frac{P_C^c P_D^d}{P_A^a P_B^b}\] where: - \(P_A\), \(P_B\), \(P_C\), and \(P_D\) represent the equilibrium partial pressure of the gaseous species A, B, C, and D, respectively. - \(a\), \(b\), \(c\), and \(d\) represent the stoichiometric coefficients of the balanced chemical equation. Note that for a reaction involving only gases, you can convert between \(K\) and \(K_\mathrm{p}\) using the following equation: \[K_\mathrm{p} = K(RT)^{\Delta n}\] where: - \(R\) is the ideal gas constant - \(T\) is the temperature in Kelvin - \(\Delta n\) is the change in the number of moles of gas in the reaction, given by \(\Delta n = (c + d) - (a + b)\).

Q is the reaction quotient, which is calculated similarly to the equilibrium constant but using the concentrations or partial pressures at any given point during the reaction (not necessarily at equilibrium). The equation for the reaction quotient can be written as: \[Q = \frac{[C]^c[D]^d}{[A]^a[B]^b}\] if the species are in the solution, or \[Q = \frac{P_C^c P_D^d}{P_A^a P_B^b}\] if the species are in the gas phase. Comparing the value of \(Q\) to the value of the appropriate equilibrium constant (\(K\) or \(K_\mathrm{p}\)) can help determine the direction in which a reaction will proceed. Specifically: - If \(Q < K\) (or \(Q < K_\mathrm{p}\), if the reaction involves gases), the reaction will proceed in the forward direction, converting more reactants into products. - If \(Q > K\) (or \(Q > K_\mathrm{p}\)), the reaction will proceed in the reverse direction, converting more products into reactants. - If \(Q = K\) (or \(Q = K_\mathrm{p}\)), the system is at equilibrium, and there will be no net change in the concentrations or partial pressures of the species in the reaction.