# Is this the Sackur-Tetrode equation or some other equation for entropy?

In my thermodynamic's course, this equation was presented as the Sackur-Tetrode equation:
$$\frac{S}{n}=R\log{\left(\frac{V/n}{V_{0}/n_{0}}\right)} +c_v^{*}\log{\left(\frac{T}{T_0}\right)} + \frac{S_0}{n_0}$$

An expression for entropy when $$N$$ is not constant.

However, researching on the Internet I discovered that, in fact, this is the Sackur-Tetrode equation: $$S = kN\left(\log{\left[\frac{V}{N}\left(\frac{U4 \pi m}{N3h^2}\right)^{\frac{3}{2}}\right]}+\frac{5}{2}\right)$$ Are they the same expression or is the first expression some other equation for entropy?

They are indeed the same relation, just written in different forms and with all the exact constants filled in in the second version. Note that, because of the logarithms, and the fact that the molar specific heat of a monoatomic ideal gas is $$c_{v}^{*}=\frac{3}{2}R$$, the first expression can be rewritten as $$\frac{S}{n}=R\log\left(\frac{VT^{3/2}}{N}\right)+X,$$ where $$X$$ is some combination of the constants $$V_{0}$$, $$n_{0}$$, etc.
It only takes a few more relations to give something equivalent to the second form. Note that the gas constant $$R$$ is just related to Boltzmann's constant $$k$$ by $$Nk=nR$$, where $$N$$ is the number of gas molecules and $$n$$ the number of moles. (The original question had capital $$N$$ instead of $$n$$ in the first formula, but I corrected that to standard notation.) Also, the energy in a monoatomic ideal gas is $$U=\frac{3}{2}NkT=c_{v}^{*}nRT$$. (And actually, $$U=c_{v}^{*}nRT$$ for any ideal gas, monoatomic or not). With these relations, we can further transform the first equation into $$\frac{S}{N}=k\log\left[\frac{V}{N}\left(\frac{U}{N}\right)^{3/2}\right]+Y.$$
This now has the same form as the second equation, with the constant $$Y$$ given by $$Y=\frac{3}{2}\log\left(\frac{4\pi m}{3h^{2}}\right)+\frac{5}{2}=\frac{3}{2}\log\left(\frac{4\pi e^{5/3} m}{3h^{2}}\right).$$ I would say that the second version, with no undetermined constants in the "real" Sackur-Tetrode relation, since it only depends on fundamental quantities, like $$m$$, the mass of the gass molecules, $$h$$, Planck's constant, etc. However, the first equation is the best you can do without using quantum statistical mechanics, and so it (or something very like it) is often introduced first.