What is the difference between \(K_{c} K_{p}\), and \(K_{x}\) ? For a gas-phase reaction, what is the relation between them?

There are various ways to express the equilibrium constant for a chemical reaction. The three constants mentioned in the exercise are defined as follows: 1. \(K_c\): The equilibrium constant expressed in terms of concentrations, where all species are expressed in molar concentrations (moles per liter). 2. \(K_p\): The equilibrium constant expressed in terms of partial pressures, where all gaseous species are expressed in partial pressure units (atmospheres, or any other pressure units). 3. \(K_x\): The equilibrium constant expressed in terms of mole fractions, where the concentration of each species is expressed in terms of its mole fraction within the mixture. The main difference between them is the way the concentrations of each species are expressed.

For a gas-phase reaction, the Ideal Gas Law, \(PV = nRT\), can be used to relate concentrations and partial pressures in terms of molar fraction. Let's derive the relation between \(K_c\), \(K_p\), and \(K_x\) for a generalized gas-phase reaction of the form: $$aA + bB \leftrightarrow cC + dD$$ The \(K_c\) expression for this reaction is given by: $$K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b}$$ Where \([X]\) represents the molar concentration of species \(X\) in moles per liter. For each gas, we can relate its molar concentration to its partial pressure using the ideal gas law as follows: $$[X] = \frac{P_X}{RT}$$ So, the \(K_c\) expression can be rewritten in terms of partial pressures: $$K_c = \frac{\left(\frac{P_C}{RT}\right)^c \left(\frac{P_D}{RT}\right)^d}{\left(\frac{P_A}{RT}\right)^a \left(\frac{P_B}{RT}\right)^b} = \frac{P_C^c P_D^d}{P_A^a P_B^b} \cdot \frac{1}{(RT)^{c+d-a-b}}$$ Now, the \(K_p\) expression for the reaction is given by: $$K_p = \frac{P_C^c P_D^d}{P_A^a P_B^b}$$ Comparing the new \(K_c\) expression and \(K_p\) expression, we can find a relation between them: $$K_p = K_c (RT)^{\Delta n}$$ Where \(\Delta n\) is the change in the number of moles of gas throughout the reaction, given by \(\Delta n = (c+d)-(a+b)\). To compute the relation between these constants and \(K_x\), we need to know the total pressure \(P_{total}\) of the system. Using mole fractions, we can express partial pressures as: $$P_X = x_X \cdot P_{total}$$ \(x_X\) is the mole fraction of species X. Now, from the above expressions: if we replace \(P_X\) with \(x_X \cdot P_{total}\) in the \(K_p\) expression, we get: $$K_p = \frac{(x_C \cdot P_{total})^c (x_D \cdot P_{total})^d}{(x_A \cdot P_{total})^a (x_B \cdot P_{total})^b}$$ Now, the \(K_x\) expression for the reaction is given by: $$K_x = \frac{x_C^c x_D^d}{x_A^a x_B^b}$$ Comparing these two expressions, we find: $$K_p = K_x (P_{total})^{\Delta n}$$ In conclusion, for a gas-phase reaction, the equilibrium constants are related according to the following expressions: 1. \(K_p = K_c (RT)^{\Delta n}\) 2. \(K_p = K_x (P_{total})^{\Delta n}\)