Question

Write three different \(K_{P}\) relations for reacting ideal gas mixtures, and state when each relation should be used.

Short answer

Provide three different \(K_P\) relations for reacting ideal gas mixtures and determine when each relation should be used. 1. General \(K_P\) Formula: $$K_P = \frac{(P_{product1})^{a}(P_{product2})^{b}\cdots}{(P_{reactant1})^{p}(P_{reactant2})^{q}\cdots}$$ This formula should be used when the balanced chemical equation and partial pressures of reactants and products are available. 2. \(K_P\) and \(K_C\) Relationship: $$K_P = K_C(RT)^{Δn}$$ This relation should be used when you have \(K_C\) and need to find \(K_P\), or vice versa. 3. \(K_P\) for Reactions with Equal Moles of Reactants and Products: $$K_P = \frac{Π(P_{products})}{Π(P_{reactants})}$$ This relation should be used when the number of moles of gaseous reactants and products are equal in the balanced chemical equation.

Step-by-step solution

Relation 1: General \(K_P\) Formula

The general formula for \(K_P\) can be applied to any balanced chemical equation involving ideal gases. By considering the coefficients of individual reactants and products, the general formula can be expressed as: $$K_P = \frac{(P_{product1})^{a}(P_{product2})^{b}\cdots}{(P_{reactant1})^{p}(P_{reactant2})^{q}\cdots}$$ Where \(P_{product1}\), \(P_{product2}\), \(P_{reactant1}\), and \(P_{reactant2}\) are the partial pressures of respective products and reactants, and \(a\), \(b\), \(p\), and \(q\) are their coefficients in the balanced chemical equation. This formula should be used when the balanced chemical equation and partial pressures of reactants and products are available.

Relation 2: \(K_P\) and \(K_C\) Relationship

Sometimes, the equilibrium constant for a reaction is given in terms of concentrations (\(K_C\)) rather than partial pressures. In this case, you can convert between the two using the relationship: $$K_P = K_C(RT)^{Δn}$$ Where \(K_C\) is the equilibrium constant in terms of molar concentrations, R is the ideal gas constant, T is the temperature in Kelvin, and \(Δn\) is the change in the number of moles of gas between reactants and products as given by the balanced equation. This relation should be used when you have \(K_C\) and need to find \(K_P\), or vice versa.

Relation 3: \(K_P\) for Reactions with Equal Moles of Reactants and Products

For reactions in which the number of moles of gaseous reactants is equal to the number of moles of gaseous products (i.e., \(Δn = 0\)), the relationship between \(K_P\) and reaction quotient \(Q_P\) can be simplified as: $$K_P = \frac{Π(P_{products})}{Π(P_{reactants})}$$ where \(Π(P_{products})\) and \(Π(P_{reactants})\) represent the product of the partial pressures of the products and reactants, respectively. This relation should be used when the number of moles of gaseous reactants and products are equal in the balanced chemical equation.