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I am going through my lecture notes regarding thermodynamics and I saw this equation, called "Euler equation" that gives an expression for the internal energy of a thermodynamic system: $$U = TS - pV + \sum_{i = 2}^M x_iy_i$$ where $T$ ist the temperature, $S$ is the entropy, $p$ the pressure, $V$ the volume and $x_i$ are intensive and $y_i$ are extensive variables. I am now in confusion about what this equation is "trying to tell me".

The internal energy of the ideal gas can be written as $$U = \frac{3}{2}Nk_\mathrm{B}T$$ Does that imply, that by "equating coefficients" $S = \frac{3}{2}Nk_\mathrm{B} = \mathrm{const}$ and $pV = 0$ for the ideal gas (obviously, that cannot be correct)? Does this equation only apply under certain conditions? Additionally, it seems that the expression given by the Euler equation does not the "actual" equation for the ideal gas.

EDIT

There is a similar question here, but my question is slightly different.

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  • $\begingroup$ @A.V.S. No, there are no differentials in the equation. I have added a similar question for reference. $\endgroup$ – HerpDerpington Mar 4 '18 at 15:45
  • $\begingroup$ Could you explain why you think those relations would hold? $\endgroup$ – Javier Mar 4 '18 at 15:58
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    $\begingroup$ You are claiming that $S = \frac{3}{2}Nk_B$ because equating both expressions for $U$ on one side $S$ multiplies $T$ and on the other $\frac{3}{2}Nk_B$ multiplies $T$. You do the same for $p$. How would you justify this? It is simply incorrect to do it. $\endgroup$ – user1620696 Mar 4 '18 at 16:04
  • $\begingroup$ @user1620696 But shouldn't both equations give the same result? Or is the problem that $-pV$ can be rewritten such that it also gives an expression propotional to $T$? $\endgroup$ – HerpDerpington Mar 4 '18 at 16:08
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The way I see it, you have to be careful about the variables your functions depend on. If you write Euler equation explicitly, U, T and p appear as functions of S, V and N: $$ U(S,V,N) = T(S,V,N)\times S + p(S,V,N)\times V + \mu(S,V,N)\times N $$

So if you know $U(S,V,N) = \frac{3}{2}Nk_B T$, you won't go far because T is also a function of (S,V,N), so all you can say is: $$ \frac{3}{2}k_B T(S,V,N) = T(S,V,N)\times S + p(S,V,N)\times V + \mu(S,V,N)\times N $$ To further simply the expression, you need to express T, p and $\mu$ as a function of S, which is a bit of a pain...

It works better the other way around: you can check that the Sackur-Tetrode expression of the entropy $$ S(U,V,N) = Nk_b \log \left(\frac{U^{3/2}}{N^{3/2}}\frac{V}{N}\right)+nk_B\times c $$ does verify Euler equation $$ S(U,V,N) = \frac{1}{T(U,V,N)}\times U + \frac{p(U,V,N)}{T(U,V,N)}\times V - \frac{\mu(U,V,N)}{T(U,V,N)}\times N $$

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