Residual entropy

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Residual entropy is the difference in entropy between a non-equilibrium state and crystal state of a substance close to absolute zero. This term is used in condensed matter physics to describe the entropy at zero kelvin of a glass or plastic crystal referred to the crystal state, whose entropy is zero according to the third law of thermodynamics. It occurs if a material can exist in many different states when cooled. The most common non-equilibrium state is vitreous state, glass.

A common example is the case of carbon monoxide, which has a very small dipole moment. As the carbon monoxide crystal is cooled to absolute zero, few of the carbon monoxide molecules have enough time to align themselves into a perfect crystal (with all of the carbon monoxide molecules oriented in the same direction). Because of this, the crystal is locked into a state with ${\displaystyle 2^{N}}$ different corresponding microstates, giving a residual entropy of ${\displaystyle S=Nk\ln(2)=nR\ln(2)}$, rather than zero.

Another example is any amorphous solid (glass). These have residual entropy, because the atom-by-atom microscopic structure can be arranged in a huge number of different ways across a macroscopic system.

The residual entropy has a somewhat special significance compared to other residual properties, in that it has a role in the framework of residual entropy scaling,^[1] which is used to compute transport coefficients (coefficients governing non-equilibrium phenomena) directly from the equilibrium property residual entropy, which can be computed directly from any equation of state.