## Monday, February 13, 2012

### Nothing is Static. Everything Changes

Nothing is static. Everything changes. The 3rd Law of Thermodynamics demands it. The 3rd was always my favorite Law, although most prefer Newton's 2nd Law of Motion and The 2nd Law of Thermodynamics best.

The third law of thermodynamics is sometimes stated as follows:
The entropy of a perfect crystal at absolute zero is exactly equal to zero.
At zero temperature the system must be in a state with the minimum possible energy, and this statement of the third law holds true if the perfect crystal has only one state with zero energy. Entropy is related to the number of possible microstates, and with only one microstate, the entropy is exactly zero.
A more general form of the third law that applies to systems such as glasses that may have more than one minimum energy state:
The entropy of a system approaches a constant value as the temperature approaches zero.
The constant value (not necessarily zero) is called the residual entropy of the system.[1]

## History

The third law was developed by the chemist Walther Nernst during the years 1906-1912, and is therefore often referred to as Nernst's theorem or Nernst's postulate. The third law of thermodynamics states that the entropy of a system at absolute zero is a well-defined constant. This is because a system at zero temperature exists in its ground state, so that its entropy is determined only by the degeneracy of the ground state. It means that "it is impossible by any procedure, no matter how idealised, to reduce any system to the absolute zero of temperature in a finite number of operations".
An alternative version of the third law of thermodynamics as stated by Gilbert N. Lewis and Merle Randall in 1923:
If the entropy of each element in some (perfect) crystalline state be taken as zero at the absolute zero of temperature, every substance has a finite positive entropy; but at the absolute zero of temperature the entropy may become zero, and does so become in the case of perfect crystalline substances.
This version states not only ΔS will reach zero at 0 K, but S itself will also reach zero as long as the crystal has a ground state with only one configuration. Some crystals form defects which causes a residual entropy. This residual entropy disappears when the kinetic barriers to transitioning to one ground state are overcome. [2]
With the development of statistical mechanics, the third law of thermodynamics (like the other laws) changed from a fundamental law (justified by experiments) to a derived law (derived from even more basic laws). The basic law from which it is primarily derived is the statistical-mechanics definition of entropy for a large system:
$S = k_B \log \, \Omega \$
where S is entropy, kB is the Boltzmann constant, and Ω is the number of microstates consistent with the macroscopic configuration.

## Overview

In simple terms, the third law states that the entropy of a perfect crystal approaches zero as the absolute temperature approaches zero. This law provides an absolute reference point for the determination of entropy. The entropy determined relative to this point is the absolute entropy.
The entropy of a perfect crystal lattice as defined by Nernst's theorem is zero (provided that its ground state is unique, whereby ln(1)k = 0).
An example of a system which does not have a unique ground state is one containing half-integer spins, for which time-reversal symmetry gives two degenerate ground states (an entropy of ln(2) kB, which is negligible on a macroscopic scale). Some crystalline systems exhibit geometrical frustration, where the structure of the crystal lattice prevents the emergence of a unique ground state. Ground-state helium (unless under pressure) remains liquid.
In addition, glasses and solid solutions retain large entropy at 0K, because they are large collections of nearly degenerate states, in which they become trapped out of equilibrium. Another example of a solid with many nearly-degenerate ground states, trapped out of equilibrium, is ice Ih, which has "proton disorder".
For the third law to apply strictly, the magnetic moments of a perfectly ordered crystal must themselves be perfectly ordered; indeed, from an entropic perspective, this can be considered to be part of the definition of "perfect crystal". Only ferromagnetic, antiferromagnetic, and diamagnetic materials can satisfy this condition. Materials that remain paramagnetic at 0K, by contrast, may have many nearly-degenerate ground states (for example, in a spin glass), or may retain dynamic disorder (a spin liquid [3]).