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 "xi" 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 "pi" were either 1 or 0. We again obtain the entropy of mixing on multiplying by the Boltzmann constant formula_52.