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We consider a substance X with two monotropic modifications 1 and 2 of different thermodynamic stability ΔH 1  &lt; ΔH 2 . 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 S i  to the final equilibrium state f of entropy S f . The two modifications 1 and 2 take different ways I 1  and I 2 . According to the principle of the shortest way, I 1  &lt; I 2  is 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 ΔS 1  = S f 1  - S i 1  on crystallizing 1 can be related to the corresponding chang of enthalpy by ΔS 1  = ΔH 1 /T. Now it can be shown that the shortest way of crystallization I 1  corresponds under special, well-defined conditions to the smallest change of entropy ΔS 1  &lt; ΔS 2  and thus enthalpy ΔH 1  &lt; ΔH 2 . In other words, the shortest way of crystallization I 1  really leads us to the instable modification 1. This is Ostwald´s rule.

Ostwald´s Rule and the Principle of the Shortest Way