<P> Paul Ehrenfest classified phase transitions based on the behavior of the thermodynamic free energy as a function of other thermodynamic variables . Under this scheme, phase transitions were labeled by the lowest derivative of the free energy that is discontinuous at the transition . First - order phase transitions exhibit a discontinuity in the first derivative of the free energy with respect to some thermodynamic variable . The various solid / liquid / gas transitions are classified as first - order transitions because they involve a discontinuous change in density, which is the (inverse of the) first derivative of the free energy with respect to pressure . Second - order phase transitions are continuous in the first derivative (the order parameter, which is the first derivative of the free energy with respect to the external field, is continuous across the transition) but exhibit discontinuity in a second derivative of the free energy . These include the ferromagnetic phase transition in materials such as iron, where the magnetization, which is the first derivative of the free energy with respect to the applied magnetic field strength, increases continuously from zero as the temperature is lowered below the Curie temperature . The magnetic susceptibility, the second derivative of the free energy with the field, changes discontinuously . Under the Ehrenfest classification scheme, there could in principle be third, fourth, and higher - order phase transitions . </P> <P> Though useful, Ehrenfest's classification has been found to be an incomplete method of classifying phase transitions, for it does not take into account the case where a derivative of free energy diverges (which is only possible in the thermodynamic limit). For instance, in the ferromagnetic transition, the heat capacity diverges to infinity . The same phenomenon is also seen in superconducting phase transition . </P> <P> In the modern classification scheme, phase transitions are divided into two broad categories, named similarly to the Ehrenfest classes: </P> <P> First - order phase transitions are those that involve a latent heat . During such a transition, a system either absorbs or releases a fixed (and typically large) amount of energy per volume . During this process, the temperature of the system will stay constant as heat is added: the system is in a "mixed - phase regime" in which some parts of the system have completed the transition and others have not . Familiar examples are the melting of ice or the boiling of water (the water does not instantly turn into vapor, but forms a turbulent mixture of liquid water and vapor bubbles). Imry and Wortis showed that quenched disorder can broaden a first - order transition in that the transformation is completed over a finite range of temperatures, but phenomena like supercooling and superheating survive and hysteresis is observed on thermal cycling . </P>

Effect of temperature on change of state matter