Gibbs paradox

In a simple derivation based on the ideal gas law, the entropy S is not an extensive variable as it must be, leading to an apparent paradox known as the Gibbs paradox. The difficulty is resolved by letting the particles be indistinguishable. The resulting equation for the entropy of an ideal Maxwell-Boltzmann gas is known as the Sackur-Tetrode equation.

Calculating the Gibbs paradox

If we have an ideal gas of energy E, volume V and with N particles, then we can represent the state of the gas by specifying the 3D momentum vector and the 3D position vector for each of the N particles. This can be thought of as specifying the coordinates of a point in a 6N-dimensional phase space, where each of the axes corresponds to one of the momentum or position coordinates of one of the particles. The set of all possible points that the gas could ever occupy in phase space is given by the constraints that it have a particular energy:

<math>E=\frac{1}{2m}\sum_{i=1}^{N} \sum_{j=1}^3 p_{ij}^2<math>

and be contained inside of the volume V (let's say V is a box of side X so that X³=V):

<math>0 \le x_{ij} \le X<math>

where [p_i1, p_i2, p_i3] and [x_i1, x_i2, x_i3] are the vector momentum and position of particle i. The first constraint defines the surface of a 3N-dimensional hypersphere of radius (2mE)^1/2 and the second is a 3N-dimensional hypercube of volume V^N. These combine to form a 6N dimensional "hypercylinder" and just as the area of the wall of a cylinder is the circumference of the base times the height, so the area φ of the wall of this hypercylinder is:

<math>

\phi(E,V,N) = V^N \left(\frac{2\pi^{\frac{3N}{2}}(2mE)^{\frac{3N-1}{2}}}{\Gamma(3N/2)}\right)~~~~~~~~~~~(1) <math>

The entropy is proportional to the logarithm of the number of states that the gas could have while satisfying these constraints. Another way of stating Heisenberg's uncertainty principle is to say that we cannot specify a volume in phase space smaller than h^3N where h is Planck's constant. We therefore write the entropy as:

<math>\left.\right.

S=k\,\textrm{log}(\phi/h^{3N}) <math>

where k is the constant of proportionality, Boltzmann's constant. Using Stirling's approximation for the Gamma function, and keeping only terms of order N the entropy becomes:

<math>

S (=?)~~k N \log \left[ V \left(\frac EN \right)^{\frac 32}\right]+ {\frac 32}kN\left( {\frac 53}+ \log\frac{4\pi m}{3h^2}\right) <math>

This quantity is not extensive as can be seen by considering two identical volumes, with the same particle number and the same energy. Suppose the two volumes are separated by a barrier in the beginning. Removing or reinserting the wall is reversible, but the entropy difference after removing the barrier is

<math>

\delta S = k \left[ 2N \log(2V) - N\log V - N \log V \right] = 2 k N \log 2 > 0 <math>

which is in contradiction to thermodynamics. This is the Gibbs paradox. It was resolved by J.W. Gibbs himself, by postulating that the gas particles are in fact indistinguishable. This means that all states that differ only by a permutation of particles should be considered as the same point. For example, if we have a 2-particle gas and we specify AB as a state of the gas where the first particle (A) has momentum p₁ and the second particle (B) has momentum p₂, then this point as well as the BA point where the B particle has momentum p₁ and the A particle has momentum p₂ should be counted as the same point. It can be seen that for an N-particle gas, there are N! points which are identical in this sense, and so to calculate the volume of phase space occupied by the gas we must divide Equation 1 by N!. This will give for the entropy:

<math>

S = k N \log \left[ \left(\frac VN\right) \left(\frac EN \right)^{\frac 32}\right]+ {\frac 32}kN\left( {\frac 53}+ \log\frac{4\pi m}{3h^2}\right) <math>

which can be easily shown to be extensive. This is the Sackur-Tetrode equation, and the value of S/kN is the Sackur-Tetrode constant.