The thermodynamic entropy is only due to positional uncertainty, so we may take the "alphabet" to be any of the r different species in the gas, and, at equilibrium, the probability that a given particle is of type i is simply the mole fraction x i for that particle. Since we are dealing with ideal gases, the identity of nearby particles is irrelevant. Multiplying by the number of particles N yields the change in entropy of the entire system from the unmixed case in which all of the p i were either 1 or 0. We again obtain the entropy of mixing on multiplying by the Boltzmann constant k B .