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Is there a mistake in this logic :

  1. Lets say we 2 substances: A and B. Substance A has a greater s.h.c. (specific heat capacity) than substance B.
  2. Both substances start at 20 Kelvin, and are heated to 40 Kelvin. Substance A needs more energy to achieve this.
  3. Assuming no phase change, all energy was converted to K.E. (kinetic energy).
  4. Therefore substance A has more K.E. than substance B, even though they are the same temperature.

Conclusion : substances with different s.h.c. have different K.E. at the same temperature.

I tired searching this online but couldn't find anything to support it.

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    $\begingroup$ The (total) internal KE depends on the amount of substance. You did not specify if the two have the same mass or the same number of particles or whatever. $\endgroup$
    – nasu
    Commented Sep 8, 2023 at 16:25
  • $\begingroup$ Notice that heat capacity per mole is the same for all monatomic gases. $\endgroup$ Commented Sep 8, 2023 at 16:27
  • $\begingroup$ You cannot find it online because your reasoning contains at least two flaws. The most important one is that, even if one considers perfect gases (i.e., there are only kinetic energy terms in the Hamiltonian), one may have different specific heats (@MariusLadegårdMeyer) if one ideal gas is monoatomic and the other is not. In this case, the connection between (kinetic) energy and temperature contains a constant depending on the number of degrees of freedom. This explains how two systems may get the same temperature with different added energies. $\endgroup$ Commented Sep 8, 2023 at 16:39
  • $\begingroup$ By substances, do you include liquids and solids? Which specific heat capacity, constant volume or constant pressure? $\endgroup$ Commented Sep 8, 2023 at 22:56

1 Answer 1

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Assuming no phase change, all energy was converted to K.E. (kinetic energy).

This is actually incorrect.

The average thermal energy per internal degree of freedom is $\frac{1}{2}k_B T$ where $k_B$ is the Boltzmann constant and $T$ is the thermodynamic temperature. This implies that if an internal degree of freedom is kinetic energy then it will have that amount of energy (on average).

However, it does not imply that all internal degrees of freedom are purely kinetic energy. The translational and rotational degrees of freedom qualify as purely KE. But there are also torsional and vibrational degrees that are a mix of KE and PE, and there are also electronic and nuclear excitations that are PE.

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