Ostwald´s Rule and the Principle of the Shortest Way

Abstract

AbstractWe consider a substance X with two monotropic modifications 1 and 2 of different thermodynamic stability ΔH1< ΔH2. Ostwald´s rule states that first of all the instable modification 1 crystallizes on cooling down liquid X, which subsequently turns into the stable modification 2. Numerous examples verify this rule, however what is its reason? Ostwald´s rule can be traced back to the principle of the shortest way. We start with Hamilton´s principle and the Euler-Lagrange equation of classical mechanics and adapt it to thermodynamics. Now the relevant variables are the entropy S, the entropy production P = dS/dt, and the time t. Application of the Lagrangian F(S, P, t) leads us to the geodesic line S(t). The system moves along the geodesic line on the shortest way I from its initial non-equilibrium state i of entropy Sito the final equilibrium state f of entropy Sf. The two modifications 1 and 2 take different ways I1and I2. According to the principle of the shortest way, I1< I2is realized in the first step of crystallization only. Now we consider a supercooled sample of liquid X at a temperature T just below the melting point of 1 and 2. Then the change of entropy ΔS1= Sf 1- Si 1on crystallizing 1 can be related to the corresponding chang of enthalpy by ΔS1= ΔH1/T. Now it can be shown that the shortest way of crystallization I1corresponds under special, well-defined conditions to the smallest change of entropy ΔS1< ΔS2and thus enthalpy ΔH1< ΔH2. In other words, the shortest way of crystallization I1really leads us to the instable modification 1. This is Ostwald´s rule.