Energy and Entropy

Now that we have learned a little about what kinds of things exhibit phase transitions, we will explore topics related to why they happen, and what characterizes them. This second part is especially important as physicists must produce evidence that a new phase transition is observed, and by having a generally agreed upon understanding of what constitutes a phase transition there emerges a list of phenomena that should be observable during a phase transition

Of the thermodynamic variables previously considered, the three most important to understanding phase transitions are energy, temperature, and entropy. This is because the transition from order to disorder can be thought of as a competition between the minimization of energy, and the maximization of entropy. These are familiar principles. The minimization of energy is due to the fact that energy is conserved and it can be transferred from one variety to another. That entropy is maximized is a statistical statement that a system will always take the configuration that is most likely, or represented in the highest number of microstates.

The Ising model represents an idealized magnetic system composed of magnetic dipoles (bar magnets) arranged on the surface of a square lattice. The dipoles can only point in one of two directions (up or down) and interact with each other. When the so-called spins are pointing in the same direction their interaction energy is lower than when they are pointing in opposite directions, and thus there is a phase transition at sufficiently low temperature where the spins suddenly and overwhelmingly point in the same direction.

The energy stored in a material is understood in terms of the energy stored in the interactions between elements and the energy the elements possess independently. For a gas of water vapor the individual atoms possess energy in the form of kinetic and rotational energy, as they fly and spin and crash into one another. There is also an energy of interaction caused by electrostatic forces. The water molecule is bent in such a way that the electron density is not distributed evenly around the molecule. This results in the molecule possessing an electric field with which it can then interact with other water molecules. The interaction of two electric dipoles can be thought of as two arrows, each located on a water molecule, that want to point in the same direction. Thus, the effect of intermolecular interactions is to order the gas. This desire towards order is countered by the system's entropy.

Entropy is a measure of the number of ways the individual constituents of a system can be configured. If there are more atoms, or a larger volume for the atoms to occupy, the gas can be said to possess a higher entropy since there are more ways in which to arrange the atoms in the space. The temperature of the system relates the energy and the entropy, and the effect of these two competing interactions can be understood by considering the free energy of the system, F=E-TS. A collection of particles will always seek to minimize its free energy. That is how different phases arise. As the temperature of the system is lowered, the entropy will provide a decreasing contribution to the free energy, and the system will at some point fall into an ordered state.

A disordered state is categorized by a lack of correlation between the atoms. That is to say, knowing the state of one atom gives no information about nearby atoms. In the ordered state, there is very high correlation between atoms. In the transition from a liquid to a gas, for example, the atoms in the liquid phase are located almost at random, and any correlation decreases rapidly with the distance between any two atoms. In the solid phase, however, the atoms are arranged in a regular lattice and thus knowing the location of one gives immediate information about the location of another. This is a phase transition, and as we have seen, they do not occur gradually. A phase transition will occur suddenly at a particular set of conditions defined by the thermodynamic variables. The point at which the phase transition occurs is known as the critical point.
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