A substance at 200-degree Celsius is given some amount of heat to raise its temperature by one degree Celsius and the same substance when at -200 degrees Celsius is given some amount of heat to increase its temperature to -199 degrees Celsius. Is the amount of heat required for both the processes same?

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Preston Espinoza · Accepted Answer

In general, heat capacity depends on temperature, so the answer is no, the amount is different. However, due to the equipartition theorem, at sufficiently high temperature (compared to the typical temperature scale given by quantum mechanics) the temperature dependence flattens out. Famously, this gives rise to Dulong-Petit&#039;s law for solids or the constant heat capacity of ideal gases that you must have seen. This is why to observe temperature dependence you need to go to low temperature (take for example Debye&#039;s heat capacity for solids), or include many degrees of freedom (take blackbody radiation). As a consequence of the 3rd law of thermodynamics or more generally a finite value of residual entropy, you also need to have a vanishing of the heat capacity at low temperature.The archetypical example is the harmonic oscillator. At pulsation   ω, you have the following formula (using the canonical ensemble) for C, the heat capacity:  C  =      k    B              (              x                  sinh          ⁡          x                    )        2  with   x  =  ℏ  ω      /    2      k    B    T, which gives   C  =      k    B   in the high temperature limit (ie   T  ≫  ℏ  ω      /        k    B  ) as predicted by the equipartition theorem. However, in the low energy limit, quantum effects dominate, and you have the signature exponential decrease of C due to the energy gap of the groundstate.

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