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For N particles in a two-energy-level system, with energy levels being 0 and $\Delta$, the internal energy tends to $N \Delta /2$ as temperature tends to infinity. At this point, the occupancy of each energy level is equal, and any further increase in internal energy would lead to a decrease in the entropy of the system.

Temperature is defined as $T = (\frac{\partial U}{\partial S})_V$. Using this definition, for a state with equal occupancies described above, the temperature should be negative. Where is the contradiction?

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    $\begingroup$ With equal occupancies, the temperature is infinite. It is the population inversion (higher occupancy for a higher-energy state) that makes temperature negative. There is no contradiction. $\endgroup$
    – Ivan Neretin
    Apr 8, 2017 at 7:35
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    $\begingroup$ and inversion is very common, for example in NMR experiments and in lasers $\endgroup$
    – porphyrin
    Apr 8, 2017 at 8:00
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    $\begingroup$ related chemistry.stackexchange.com/questions/36885 $\endgroup$
    – Mithoron
    Apr 8, 2017 at 17:43

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I think you would need to describe the derivative $\frac{\partial U}{\partial S}$ at that point, which should be infinity by your logic since the entropy is at a maximum with respect to internal energy ($\frac{\partial S}{\partial U}=0$). This is obtained using $$\frac{\partial U}{\partial S}=\frac{1}{\frac{\partial S}{\partial U}}$$

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