Gibbs’ Paradox
Instatistical mechanics, a semi-classical derivation of theentropythat does not take into account the indistinguishability of particles, yields an expression for the entropy which is notextensive(is not proportional to the amount of substance in question). This leads to aparadoxknown as theGibbs paradox, afterJosiah Willard Gibbs. The paradox allows for the entropy of closed systems to decrease, violating thesecond law of thermodynamics. A related paradox is the "mixing paradox". If one takes the perspective that the definition of entropy must be changed so as to ignore particle permutation, the paradox is averted.
Illustration of the problem
Gibbs himself considered the following problem that arises if the ideal gas entropy is not extensive.[1]Two identical containers of an ideal gas sit side-by-side. There is a certain amount of entropySassociated with each container of gas, and this depends on the volume of each container. Now a door in the container walls is opened to allow the gas particles to mix between the containers. No macroscopic changes occur, as the system is in equilibrium. The entropy of the gas in the two-container system can be easily calculated, but if the equation is not extensive, the entropy would not be 2S. In fact, the non-extensive entropy quantity defined and studied by Gibbs would predict additional entropy. Closing the door then reduces the entropy again to 2S, in supposed violation of theSecond Law of Thermodynamics.
As understood by Gibbs,[2]and reemphasized more recently,[3][4]this is a misapplication of Gibb's non-extensive entropy quantity. If the gas particles are distinguishable, closing the doors will not return the system to its original state - many of the particles will have switched containers. There is a freedom in what is defined as ordered, and it would be a mistake to conclude the entropy had not increased. In particular, Gibbs' non-extensive entropy quantity for an ideal gas was not intended for varying numbers of particles. is averted by concluding theindistinguishability(at least effective indistinguishability) of the particles in the volume. This results in theextensiveSackur–Tetrode equationfor entropy, as derived next.
Calculating the entropy of ideal gas, and making it extensive
In classical mechanics, the state of anideal gasof energyU, volumeVand withNparticles, each particle having massm, is represented by specifying themomentumvectorpand the position vectorxfor each particle. This can be thought of as specifying a point in a 6N-dimensionalphase 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:
and be contained inside of the volume V (let's sayVis a box of sideXso thatV=X3):
for: i = 1…N and j = 1,2,3
The first constraint defines the surface of a 3N-dimensionalhypersphereof radius (2mU)1/2and the second is a 3N-dimensionalhypercubeof volumeVN. 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:
The entropy is proportional to the logarithm of the number of states that the gas could have while satisfying these constraints. In classical physics, the number of states is infinitely large, but according to quantum mechanics it is finite. Before the advent of quantum mechanics, this infinity was regularized by making phase space discrete. Phase space was divided up in blocks of volume h3N. The constant h thus appeared as a result of a mathematical trick and thought to have no physical significance. However, using quantum mechanics one recovers the same formalism in the semi-classical limit, but now with h being Planck's constant. One can qualitatively see this from Heisenberg's uncertainty principle; a volume in N phase space smaller thanh3N(his Planck's constant) cannot be specified.
To compute the number of states we must compute the volume in phase space in which the system can be found and divide that by h3N. This leads us to another problem: The volume seems to approach zero, as the region in phase space in which the system can be is an area of zero thickness. This problem is an artifact of having specified the energy U with infinite accuracy. In a generic system without symmetries, a full quantum treatment would yield a discrete non-degenerate set of energy eigenstates. An exact specification of the energy would then fix the precise state the system is in, so the number of states available to the system would be one, the entropy would thus be zero.
When we specify the internal energy to be U, what we really mean is that the total energy of the gas lies somewhere in an interval of length dUaround U. HeredU is taken to be very small, it turns out that the entropy doesn't depend strongly on the choice ofdU for large N. This means that the above "area"must be extended to a shell of a thickness equal to an uncertainty in momentum, so the entropy is given by:
where the constant of proportionality isk,Boltzmann's constant. UsingStirling's approximationfor theGamma functionwhich omits terms of less than orderN, the entropy for largeNbecomes:
This quantity is not extensive as can be seen by considering two identical volumes with the sameparticle numberand 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 increases when the barrier is removed by the amount
which is in contradiction to thermodynamics if you re-insert the barrier. This is the Gibbs paradox. The paradox is resolved 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 state. For example, if we have a 2-particle gas and we specifyABas a state of the gas where the first particle (A) has momentump1and the second particle (B) has momentump2, then this state as well as theBAstate where theBparticle has momentump1and theAparticle has momentump2should be counted as the same state.
For anN-particle gas, there areN!states which are identical in this sense, if one assumes that each particle is in a different single particle state. One can safely make this assumption provided the gas isn't at an extremely high density. Under normal conditions, one can thus calculate the volume of phase space occupied by the gas, by dividing Equation 1 byN!. Using theStirling approximation again for largeN, ln(N!) ≈Nln(N) -N, the entropy for largeNis:
which can be easily shown to be extensive. This is theSackur–Tetrode equation.
Sackur–Tetrode Equation
TheSackur–Tetrode equationis an expression for theentropyof amonatomicclassicalideal gas which incorporates quantum considerations which give a more detailed description of its regime of validity.The Sackur–Tetrode equation is named forHugo Martin Detrude[1](1895–1931) andOtto Sackur[2](1880–1914), who developed it independently as a solution of Boltzmann's gas statistics and entropy equations, at about the same time in 1912.
The Sackur–Tetrode equation is written:
whereVis the volume of the gas,Nis the number of particles in the gas,Uis the internal energy of the gas,kisBoltzmann's constant,mis the mass of a gas particle,hisPlanck's constantand ln() is thenatural logarithm. SeeGibbs paradoxfor a derivation of the Sackur–Tetrode equation. See also theideal gasarticle for the constraints placed upon the entropy of an ideal gas by thermodynamics alone. The Sackur–Tetrode equation can also be conveniently expressed in terms of thethermal wavelength L.
Note that the assumption was made that the gas is in the classical regime, and is described byMaxwell–Boltzmann statistics(with "correct Boltzmann counting"). From the definition of thethermal wavelength, this means the Sackur–Tetrode equation is only valid for
and in fact, the entropy predicted by the Sackur–Tetrode equation approaches negative infinity as the temperature approaches zero.
The mixing paradox
A closely related paradox is the mixing paradox. Again take 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. If gas A and B are different gases, there is an entropy that arises due to the mixing. If the gases are the same, no additional entropy is calculated. The additional entropy from mixing does not depend on the character of the gases. The paradox is that the two gases can be arbitrarily similar, but the entropy from mixing does not disappear unless they are the same gas.
The resolution is provided by a careful understanding of entropy.
In particular, as explained concisely by Jaynes,[2]there is an arbitrariness in the definition of entropy.
A central example in Jaynes' paper relies on the fact that, if one develops a theory based on the idea that the two different types of gas are indistinguishable, and one never carries out any measurement which contradicts this fact, then the theory will have no internal inconsistencies. In other words, if there are two gases A and B and we have not yet discovered that they are different, then assuming they are the same will cause no theoretical problems. If ever an experiment is performed 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 differential increase in entropy (dS), as a result of mixing dissimilar element sets (the gases), multiplied by the temperature (T) 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. As soon as we become able to distinguish the difference, 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.
This line of reasoning is particularly informative when considering the concepts ofindistinguishable particlesandcorrect Boltzmann counting. Boltzmann's original expression for the number of states available to a gas assumed that a state could be expressed in terms of a number of energy "sublevels" each of which contain a particular number of particles. While the particles in a given sublevel were considered indistinguishable from each other, particles in different sublevels were considered distinguishable from particles in any other sublevel. This amounts to saying that the exchange of two particles in two different sublevels will result in a detectably different "exchange macrostate" of the gas. For example, if we consider a simple gas withNparticles, at sufficiently low density that it is practically certain that each sublevel contains either one particle or none (i.e. a Maxwell–Boltzmann gas), this means that a simple container of gas will be in one ofN!detectably different "exchange macrostates", one for each possible particle exchange. Just as the mixing paradox begins with two detectably different containers, and the extra entropy that results upon mixing is proportional to the average amount of work needed to restore that initial state after mixing, so the extra entropy in Boltzmann's original derivation is proportional to the average amount of work required to restore the simple gas from some "exchange macrostate" to its original "exchange macrostate". If we assume that there is in fact no experimentally detectable difference in these "exchange macrostates" available, then using the entropy which results from assuming the particles are indistinguishable will yield a consistent theory. This is "correct Boltzmann counting". It is often said that the resolution to the Gibbs paradox derives from the fact that, according to the quantum theory, like particles are indistinguishable in principle. By Jaynes' reasoning, if the particles are experimentally indistinguishable for whatever reason, Gibbs paradox is resolved, and quantum mechanics only provides an assurance that in the quantum realm, this indistinguishability will be true as a matter of principle, rather than being due to an insufficiently refined experimental capability.