This remark proved prescient because in 1876-1878 Gibbs published his definitive work on thermodynamics in which he showed that thermodynamic equilibrium of a heterogeneous substance requires that, in addition to mechanical equilibrium (the same pressure for each component) and thermal equilibrium (the same temperature for each component), there must also be material equilibrium (the same chemical potential for each component). In the present instance of one substance and two phases in addition to p f = p g = p s and T f = T g = T , material equilibrium requires g f = g g (for the special case of one substance its chemical potential is the molar Gibbs function, μ μ = g ≡ ≡ G / N where g = u + p v − − T s ). This condition can be deduced by a simple physical argument as follows: the energy required to vaporize a mole is from the second law at constant temperature q v a p = T ( s g − − s f ) , and from the first law at constant pressure q v a p = h g − − h f , then equating these two and rearranging produces the result since h = u + p v . The conditions of material equilibrium lead to the famous Gibbs phase rule, D = n − − r + 2 , where n is the number of substances, r the number of phases, and D the number of independent intensive variables required to specify the state. In the case of one substance and two phases discussed here this gives, D = 1 , the experimentally observed number.