For the reaction:
aA(g) + bB(g) + ...cC(g) + dD(g) + ...
The equilibrium constant based on partial pressures is
From the ideal gas law:
PAeV = nART
nA is the number of moles of A
R is the ideal gas constant = 0.0821 Latm/molK
T the absolute temperature in K
P is the pressure in atm
V the system volume in L
Similar expressions can be written for each gas phase component.
but nA/V is just the molar concentration = [A]e
Substituting into the expression for Kp (for each gas phase component) gives
Collecting terms gives
The left part of the fraction is Kc, so
Kp = Kc × (RT)(c+d+...)(a+b+...)
The exponent in RT is the sum of the stoichiometric coefficients for the reactants subtracted from the sum of the stoichiometric coefficients for the products, defined as n.
Kp = Kc × (RT)n
Because the derivation goes through the ideal gas law, the proper units for R in this case are Latm/molK (i.e., R = 0.0821 Latm/molK).