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Overview of critical phenomena

Consider the phase diagram of a typical substance:

Figure 1:

The boundary lines between phases are called the coexistence lines. Crossing a coexistence line leads to a first order phase transition, which is characterized by a discontinuous change in some thermodynamic quantity such as volume, enthalpy, magnetization.

Notice that, while the melting curve, in principle, can be extended to infinity, the gas-liquid/boiling curve terminates at a point, beyond which the two phases cannot be distinguished. This point is called a critical point and beyond the critical point, the system is in a supercritical fluid state. The temperature at which this point occurs is called the critical temperature tex2html_wrap_inline467 . At a critical point, thermodynamic quantities such as given above remain continuous, but derivatives of these functions may diverge. Examples of such divergent quantities are the heat capacity, isothermal compressibility, magnetic susceptibility, etc. These divergences occur in a universal way for large classes of systems (such systems are said to belong to the same universality class, a concept to be defined more precisely in the next lecture).

Recall that the equation of state of a system can be expressed in the form


i.e., pressure is a function of density an temperature. A set of isotherms of the equation of state for a typical substance might appear as follows:

Figure 2:

The dashed curve corresponds to the gas-liquid coexistence curve. Below the critical isotherm, the gas-liquid coexistence curve describes how large a discontinuous change in the density occurs during first-order gas-liquid phase transition. At the inflection point, which corresponds to the critical point, the discontinuity goes to 0.

As noted above, the divergences in thermodynamic derivative quantities occur in the same way for systems belonging to the same universality class. These divergences behave as power laws, and hence can be described by the exponents in the power laws. These exponents are known as the critical exponents. Thus, the critical exponents will be the same for all systems belonging to the same universality class. The critical exponents are defined as follows for the gas-liquid critical point:

The heat capacity at constant volume defined by


diverges with temperature as the critical temperature is approached according to


The isothermal compressibility defined by


diverges with temperature as the critical temperature is approached according to


On the critical isotherm, the shape of the curve near the inflection point at tex2html_wrap_inline469 is given by


The shape of the coexistence curve in the tex2html_wrap_inline471 -T plane near the critical point for tex2html_wrap_inline475 is given by


These are the primary critical exponents.

next up previous
Next: Review of the Van Up: No Title Previous: No Title

Mark Tuckerman
Fri Apr 30 17:00:54 EDT 1999