# Thermal: Entropy of Ideal Gas (Sackur-Tetrode equation)

1. May 19, 2014

### PsychonautQQ

okay so I suck at La-Tex so i'm not going to put the actual equation, but it's not important for my question.

In the equation the entropy is dependent on the natural log with mass in the numerator of the argument. Why is mass involved when talking about entropy at all?

I mean I think of entropy as being related to multiplicity of states or whatever, which is independent of mass? is rest mass energy being involved here?

2. May 19, 2014

### WannabeNewton

The mass (which yes means the rest mass) is one of the parameters in the thermal wavelength of the gas that appears in the Sackur-Tetrode equation $\lambda = (\frac{\beta \hbar^2}{2\pi m})^{1/2}$ and it defines the characteristic length scale of the thermal system. In other words it sets the scale for which the gas can actually be considered classical i.e. for which the Sackur-Tetrode equation is actually valid. So in this sense it is almost self-evident that the mass should show up in the expression for the entropy.

Now as far as multiplicity goes, there is no reason to expect mass not to be a factor in the multiplicity. Recall the multiplicity is the number of accessible microstates of a system in between two infinitesimally separated average energy levels of the system and as such the multplicity will involve some characteristic spacing between average energy levels in the determination of this volume; this characteristic spacing will of course depend on various scales of the system. For a two-level system of spins with magnetic moment $\mu$ in an external uniform magnetic field $B$ we have $\Omega \propto \frac{\delta E}{2\mu B}$ where the proportionality is the number of microstates for a given average energy. This characteristic spacing clearly depends on properties of the system such as the applied magnetic field and the intrinsic magnetic moment of the spins. For an ideal gas it is the thermal wavelength that we are interested in and in particular for a classical ideal gas we must have the intermolecular spacings much larger than the thermal wavelength so this will definitely show up in the entropy.