How does one determine the order of phase transitions?
By the degree of continuity in the Gibbs function. If the Gibbs function is continuous, but its slope is discontinuous (say due to the latent heat of melting), the transition is first-order. If both the Gibbs function and it's derivative is continuous, but the second derivative is discontinuous, the phase change is a second-order. AFAIK, the conductor-superconductor transition in liquid He is the only known second-order transition, but I don't have any really up-to-date references.
There's also a 'lambda transition', in which some of the thermodynamics properties (specific heat, for example) diverges. The resolution of that led to renormalization and scaling concepts.
The concept of a phase transition has really expanded over the past 10 years or so- some of what I wrote may be out of date.
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