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In statistical mechanics, a simple derivation of the entropy of an ideal gas, based on the Boltzmann distribution yields an expression for the entropy which is not an extensive variable as it must be, leading to an apparent paradox known as the Gibbs paradox. The difficulty is averted by requiring the particles be indistinguishable which results in "correct Boltzmann counting". The resulting equation for the entropy of a classical ideal gas is extensive, and is known as the Sackur-Tetrode equation.

If you have a fixed volume of an ideal gas, the measurement of the total entropy within the volume should not change if you were to divide the volume into two (or more) equal partitions, then remove the partitions. However, if you calculate the total entropy by measuring (and adding up) the position and momentum of each individual particle inside the volume (and keep track of each particle separately), then the entropy you calculate will change depending on whether you add and remove partitions inside the fixed volume. This discrepancy is called the Gibbs Paradox. The paradox is averted by concluding that every particle is indistinguishable from every other particle in the volume, thus you cannot measure entropy by measuring particles as if each were individually identifiable.

[edit] Calculating the Gibbs paradox

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):

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

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:

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

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 $Δ p$ 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 Heisenberg's uncertainty principle, $X\Delta p=\hbar/2$ . Solving for $Δ p$ , using Stirling's approximation for the Gamma function, and keeping only terms of order N the entropy becomes:

$S = k N \log \left[ V \left(\frac UN \right)^{\frac 32}\right]+ {\frac 32}kN\left( 1+ \log\frac{4\pi m}{3h^2}\right)$

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

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

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:

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

which can be easily shown to be extensive. This is the Sackur-Tetrode equation.

[edit] The mixing paradox

A closely related paradox is the mixing paradox. If we have a box with a partition in it, with gas A on one side, gas B on the other side, and both gases are at the same temperature and pressure then we have two possibilities: First that the two gases are identical, in which case removing the partition results in no entropy change, or that the two gases are different, in which case there is an entropy increase as the gases mix, called the entropy of mixing.

The paradox is the discontinuous nature of the entropy of mixing. The slightest detectable difference between the two gases yields the same entropy of mixing as does a large difference.

Gibbs himself posed a solution to the problem. 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 detects this fact, 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 and recalculating the entropy increase when the partition is removed.

This insight suggests that the idea 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 that difference.

[edit] References

Gibbs, J. Willard (1875-1878). On the Equilibrium of Heterogeneous Substances. Connecticut Acad. Sci.. Reprinted in
- Gibbs, J. Willard (October 1993). The Scientific Papers of J. Willard Gibbs (Vol. 1). Ox Bow Press. ISBN 0-918024-77-3.
- Gibbs, J. Willard (February 1994). The Scientific Papers of J. Willard Gibbs (Vol. 2). Ox Bow Press. ISBN 1-881987-06-X.
Jaynes, E.T. (1996). "The Gibbs Paradox" (PDF). http://bayes.wustl.edu/etj/articles/gibbs.paradox.pdf. Retrieved November 8, 2005.

[edit] External links

Gibbs paradox and its resolutions

Contents

[edit] Calculating the Gibbs paradox

[edit] The mixing paradox

[edit] References

[edit] External links