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 resolved 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. Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Ice melting - a classic example of entropy increasing In thermodynamics, thermodynamic entropy (or simply entropy) is an important state function of a thermodynamic system: that is, a property depending only on the current state of the system, independent of how that state came to be achieved. ... The Maxwell-Boltzmann distribution is a probability distribution with applications in physics and chemistry. ... In physics and chemistry, an extensive quantity (also referred to as an extensive variable) is a physical quantity whose value is proportional to the size of the system it describes. ... A physical paradox is an apparent contradiction relating to physical descriptions of the universe. ... Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. ... An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of negligible volume, with no intermolecular forces. ... The Sackur-Tetrode equation is an expression for the entropy of a classical ideal gas which uses quantum considerations to arrive at an exact formula. ...

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 Gibb’s Paradox. The paradox is resolved 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.

Calculating the Gibbs paradox

If we have an ideal gas of energy U, 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 specified by the constraint that the gas will have a particular energy: An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of negligible volume, with no intermolecular forces. ... In classical mechanics momentum (pl. ... Phase space of a dynamical system with focal stability. ...

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

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 (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: In mathematics, a hypersphere is a sphere which has dimension 3 or higher. ... In geometry, the tesseract, or hypercube, is a regular, convex polychoron with eight cubical cells. ...

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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 thickness h, so we therefore write the entropy as: Image File history File links Stop_hand. ... In quantum physics, the Heisenberg uncertainty principle, sometimes called the Heisenberg indeterminacy principle, expresses a limitation on accuracy of (nearly) simultaneous measurement of observables such as the position and the momentum of a particle. ...

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: The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... The relative difference between (ln x!) and (x ln x - x) approaches zero as x increases. ... The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ...

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

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: Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American mathematical physicist who contributed much of the theoretical foundation that led to the development of chemical thermodynamics and was one of the founders of vector analysis. ...

which can be easily shown to be extensive. This is the Sackur-Tetrode equation. The Sackur-Tetrode equation is an expression for the entropy of a classical ideal gas which uses quantum considerations to arrive at an exact formula. ...