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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$\begingroup$ Possibly related: How do I ask homework questions on Physics Stack Exchange? $\endgroup$– Peter MortensenCommented May 3, 2022 at 18:16
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$\begingroup$ Is there any information about presumed properties for the substance? For example, is the state of matter the same at the two temperatures? $\endgroup$– Peter MortensenCommented May 3, 2022 at 18:17
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2 Answers
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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'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'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 $\omega$, you have the following formula (using the canonical ensemble) for $C$, the heat capacity: $$ C = k_B \left(\frac{x}{\sinh x}\right)^2 $$ with $x=\hbar\omega/2k_BT$, which gives $C=k_B$ in the high temperature limit (ie $T\gg \hbar\omega/k_B$) 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.
Hope this helps and tell me if you need more detail.
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3$\begingroup$ or the constant heat capacity of ideal gases // Note that ideal gases do not have generally constant heat capacity. Ideal gases with the constant heat capacity are called perfect gases, as a subset of ideal gases. $\endgroup$– PoutnikCommented May 3, 2022 at 8:27
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$\begingroup$ @lpz with what relation specific heat varies with temperature at low temperatures? $\endgroup$ Commented May 3, 2022 at 11:48
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$\begingroup$ It depends a lot on your model, there is no general formula. There are some generic examples, if you have a gapped ground state, you have an exponential decrease just as in the above example with the harmonic oscillator. If you have many modes, you typically have a power law like for the Debye model or blackbody radiation. $\endgroup$– LPZCommented May 3, 2022 at 11:58
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$\begingroup$ @Ipz You mentioned that the heat capacity flattens out in the high temperature limit due to the equipartition theorem, what does this mean? $\endgroup$ Commented Dec 19, 2022 at 8:04
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Yes if the specific heat capacity of a substance is same.
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3$\begingroup$ This doesn't really answer the question. Can you edit this answer to include more detail? $\endgroup$ Commented May 3, 2022 at 13:46