where T is the temperature, P is pressure, V is volume, and μ is the chemical potential. Boltzmann's equation S = k ln ⁡ ⁡ W is the realization that the entropy is proportional to ln ⁡ ⁡ W with the constant of proportionality being the Boltzmann constant. Using the ideal gas equation of state (PV = NkT), It follows immediately that β β = 1 / k T and α α = − − μ μ / k T so that the populations may now be written: