Compute the entropy of a mole of helium at room temperature and atmospheric pressure, pretending that all the atoms are distinguishable. Compare to the actual entropy, for indistinguishable atoms, computed in the text.
We have to find the entropy of helium atom at room temperature and atmospheric pressure.
The entropy of an ideal gas is given by
where V is the volume
U is the energy
N is the number of molecules
m is the mass of a single molecule
h is Planck’s constant.
The molecules are indistinguishable, so for any configuration of the molecules in position and momentum space, interchanging any of the molecules makes no difference. This assumption introduces the factor of N! in the denominator of the multiplicity function :
If we assume that the molecules are all distinguishable, we can follow through the derivation, but without the N!, we start with
We find that the term in 1 changes to and the factor in the logarithm loses its N, so we get