lnKp

lnKp is the natural logarithm of the equilibrium constant Kp for a gas-phase reaction, where Kp is defined in terms of the partial pressures of the reacting species. In a general reaction aA + bB ⇌ cC + dD, Kp = (pC^c pD^d) / (pA^a pB^b), with partial pressures p expressed in a standard state of 1 bar. Under standard conventions Kp is treated as dimensionless, which makes lnKp well defined.

lnKp is related to the standard Gibbs free energy change of the reaction by the equation ΔG° = -RT ln Kp, where T is temperature and R is the gas constant. Consequently, lnKp = -ΔG°/(RT). This ties the position of equilibrium to thermodynamic data and explains why Kp changes with temperature.

Kp is related to Kc, the equilibrium constant in terms of concentrations, by Kp = Kc (RT)^{Δn}, where Δn is the change in moles of gas during the reaction (n_gas(products) − n_gas(reactants)). Therefore lnKp = lnKc + Δn ln(RT). This relation is often used to convert between pressure-based and concentration-based data.

Temperature dependence is commonly described by the van’t Hoff equation. Assuming ΔH° is approximately constant over the temperature range, ln(Kp2/Kp1) = -ΔH°/R (1/T2 - 1/T1). This allows prediction of how Kp, and thus lnKp, varies with temperature.

Practical notes: lnKp assumes ideal gas behavior and standard states at 1 bar. At high pressures or in nonideal conditions, fugacities or activities may be more appropriate. lnKp can be negative if Kp < 1, indicating the reaction favors reactants at the given temperature.