Gibbs paradox

File:Thermodynamicist Willard Gibbs.jpg

Josiah Willard Gibbs
(1839 – 1903)

In thermodynamics the Gibbs paradox (Gibbs' paradox or Gibbs's paradox) involves the discontinuous nature of the entropy of mixing. It was first considered by Josiah Willard Gibbs in his paper On the Equilibrium of Heterogeneous Substances. Suppose we have a box divided in half by a movable partition. On one side of the box is an ideal gas A, and on the other side is an ideal gas B at the same temperature and pressure. When the partition is removed, the two gases mix, and the entropy of the system increases because there is a larger degree of uncertainty in the position of the particles. The paradox is the discontinuous nature of the entropy of mixing. It can be shown that the entropy of mixing multiplied by the temperature is equal to the amount of work one must do in order to restore the original conditions: gas A on one side, gas B on the other. If the gases are the same, no work is needed, but given a tiniest difference between the two, the work needed jumps to a large value, and furthermore it is the same value as when the difference between the two gases is great. Entropy of mixing of liquids, solids and solutions can be calculated in a similar fashion and Gibbs paradox can be applied to liquids, solids and solutions in condensed phases as well as the gaseous phase.

Similarity and entropy of mixing

When Gibbs paradox is discussed, the correlation of the entropy of mixing with similarity is always very controversial and there are three very different opinions regarding the entropy value as related to the similarity (Figures a, b and c). Similarity may change continuously: similarity Z=0 if the components are distinguishable; similarity Z=1 if the parts are indistinguishable. Entropy of mixing does not change continuously in the Gibbs paradox.

There are many claimed resolutions and all of them fall into one of these three kinds of entropy of mixing-similarity relationship (Figures a, b and c).

Generally speaking, a so-called resolution leading to the entropy of mixing and similarity relation as shown in Figure (a) cannot be accepted as such because the paradox is still there. This kind of resolution is the explanation of the Gibbs paradox.

John von Neumann provided a real resolution of the Gibbs paradox by removing the discontinuity of the entropy of mixing: it decreases continuously with the increase in the property similarity of the individual components (See Figure b). More recently Shu-Kun Lin provided another relationship (See Figure c). They will be explained in detail in a following section.

Entropy discontinuity

Classical explanation in thermodynamics

Gibbs himself posed a solution to the problem which many scientists take as Gibbs's own resolution of the Gibbs paradox. The crux of his resolution is the fact that if one develops a classical theory based on the idea that the two different types of gas are indistinguishable, and one never carries out any measurement which reveals the difference, then the theory will have no internal inconsistencies. In other words, if we have two gases A and B and we have not yet discovered that they are different, then assuming they are the same will cause us no theoretical problems. If ever we perform an experiment with these gases that yields incorrect results, we will certainly have discovered a method of detecting their difference.

This insight suggests that the concept of thermodynamic state and entropy are somewhat subjective. The increase in entropy as a result of mixing multiplied by the temperature is equal to the minimum amount of work we must do to restore the gases to their original separated state. Suppose that the two different gases are separated by a partition, but that we cannot detect the difference between them. We remove the partition. How much work does it take to restore the original thermodynamic state? None—simply reinsert the partition. The fact that the different gases have mixed does not yield a detectable change in the state of the gas, if by state we mean a unique set of values for all parameters that we have available to us to distinguish states. The minute we become able to distinguish the difference, at that moment the amount of work necessary to recover the original macroscopic configuration becomes non-zero, and the amount of work does not depend on the magnitude of the difference.

Unfortunately the paradox is still there. This should be taken as Gibbs's explanation of the Gibbs paradox.

Explanation in statisctical mechanics and quantum mechanics, N! and entropy extensivity

A large number of scientists believe that this paradox is resolved in statistical mechanics (attributed also to Gibbs) or in quantum mechanics by realizing that if the two gases are composed of indistinguishable particles, they obey different statistics than if they are distinguishable. Since the distinction between the particles is discontinuous, so is the entropy of mixing. The resulting equation for the entropy of a classical ideal gas is extensive, and is known as the Sackur-Tetrode equation.

The state an ideal gas of energy U, volume V and with N particles, each particle having mass m, is represented by specifying the momentum vector p and the position vector x for each particle. This can be thought of as specifying 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 points in phase space that the gas could occupy is specified by the constraint that the gas will have a particular energy:

$U={\frac {1}{2m}}\sum _{i=1}^{N}\sum _{j=1}^{3}p_{ij}^{2}$

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

$0\leq x_{ij}\leq X$

for i=1..N and j=1..3.

The first constraint defines the surface of a 3N-dimensional hypersphere of radius (2mU)^1/2 and the second is a 3N-dimensional hypercube of volume V^N. These combine to form a 6N-dimensional hypercylinder. 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:

$\phi (U,V,N)=V^{N}\left({\frac {2\pi ^{\frac {3N}{2}}(2mU)^{\frac {3N-1}{2}}}{\Gamma (3N/2)}}\right)$

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. The above "area" must really be a shell of a thickness equal to the uncertainty in momentum $\Delta p$ so we therefore write the entropy as:

$\left.\right.S=k\,\ln(\phi \Delta p/h^{3N})$

where the constant of proportionality is k, Boltzmann's constant.

We may take the box length X as the uncertainty in position, and from Heisenbergs uncertainty principle, $X\Delta p=\hbar /2$. Solving for $\Delta p$, using Stirling's approximation for the Gamma function, and keeping only terms of order N the entropy becomes:

$S=kN\log \left[V\left({\frac {U}{N}}\right)^{\frac {3}{2}}\right]+{\frac {3}{2}}kN\left(1+\log {\frac {4\pi m}{3h^{2}}}\right)$

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

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

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:

$S=kN\log \left[\left({\frac {V}{N}}\right)\left({\frac {U}{N}}\right)^{\frac {3}{2}}\right]+{\frac {3}{2}}kN\left({\frac {5}{3}}+\log {\frac {4\pi m}{3h^{2}}}\right)$

which can be easily shown to be extensive. This is the Sackur-Tetrode equation. If this equation is used, the entropy value will have no difference after mixing two parts of the identical gases.

Once again, strictly speaking, this cannot fairly be taken as a resolution because the paradoxical discontinuity of entropy still exists.

External links

• Gibbs paradox and its resolutions — a list of publications.