The state of chemical equilibrium describes a dynamic condition where the rates of the forward and reverse reactions are equal, meaning the concentrations of reactants and products remain constant over time. To quantify this equilibrium, chemists use an equilibrium constant, $K$, which is a ratio of product concentrations to reactant concentrations. For reactions involving gases, this constant is expressed in two primary ways: $K_c$, which uses molar concentrations, and $K_p$, which uses partial pressures. Converting between $K_c$ and $K_p$ is essential for analyzing gaseous reactions, and this conversion relies on a specific mathematical relationship derived from the behavior of ideal gases.
The Difference Between $K_p$ and $K_c$
The distinction between $K_c$ and $K_p$ lies in the units used to measure the amounts of the substances at equilibrium. $K_c$ uses molar concentration (moles per liter, M) and is applicable to all types of reactions, including those in solution and those involving gases. $K_p$ uses the partial pressures of the gaseous reactants and products and is exclusively used for reactions involving gases.
The partial pressure of a gas is directly related to its concentration, which is the fundamental reason a conversion formula exists between the two constants. This relationship is rooted in the Ideal Gas Law, which mathematically links pressure, volume, temperature, and the number of moles of a gas. Because pressure and concentration are linked, the numerical values of $K_c$ and $K_p$ are often different unless a specific condition is met.
The $K_p$ to $K_c$ Conversion Formula
The mathematical relationship for conversion is $K_p = K_c(RT)^{\Delta n}$. This formula is derived by substituting the concentration-pressure relationship from the Ideal Gas Law into the expression for $K_p$. Understanding the specific meaning and required units for each variable is necessary for correct usage.
The variable $R$ represents the Ideal Gas Constant; for this conversion, the value $R = 0.0821 \text{ L atm/mol K}$ must be used. This value aligns with the standard units of pressure (atmospheres) and volume (liters) used in the formula’s derivation. The variable $T$ is the absolute temperature of the reaction mixture, which must always be expressed in Kelvin (K).
The exponent $\Delta n$ is the change in the number of moles of gas during the reaction. This value is calculated by taking the total number of moles of gaseous products and subtracting the total number of moles of gaseous reactants. If the number of moles of gas does not change during the reaction, $\Delta n$ is zero, and since any value raised to the power of zero is one, $K_p$ and $K_c$ will be numerically equal.
Step-by-Step Calculation and Example
The first step in converting $K_p$ to $K_c$ is to write and balance the chemical equation. This balanced equation determines the stoichiometric coefficients needed to calculate the change in the number of moles of gas, $\Delta n$. Only species in the gaseous state are included in this calculation; solids, liquids, and aqueous species are ignored.
$\Delta n$ is calculated by summing the coefficients of the gaseous products and subtracting the sum of the coefficients of the gaseous reactants. For example, in the Haber process, $\text{N}_2(g) + 3\text{H}_2(g) \rightleftharpoons 2\text{NH}_3(g)$, the gaseous products have 2 moles, and the gaseous reactants have $1+3=4$ moles. Therefore, $\Delta n = 2 – 4 = -2$.
Ensure the temperature is in Kelvin; if given in Celsius, 273.15 must be added for conversion. Once $K_p$, $T$, and $\Delta n$ are known, substitute the values into the rearranged formula, $K_c = K_p(RT)^{-\Delta n}$, to solve for $K_c$. If the $K_p$ for the Haber process at $500 \text{ K}$ is $1.45 \times 10^{-5}$, the calculation is $K_c = (1.45 \times 10^{-5}) \times ((0.0821 \times 500)^{-(-2)})$. This yields $K_c = (1.45 \times 10^{-5}) \times (41.05)^2$, resulting in a $K_c$ value of approximately $0.0244$.