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Why do we use $\Delta E_{int}=nC_{V}\Delta T$ when finding internal energy for a diatomic gas that undergoes an adiabatic process? Isn't this supposed to be used only when the volume is constant? And why is it do we choose to use this instead of $\Delta E_{int}=nC_{P}\Delta T$, even though neither is the volume nor the pressure is constant? I just don't fully understand why, and the textbook I am using doesn't really explain why.

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  • $\begingroup$ The internal energy of an ideal gas is a function only of temperature, so it doesn’t matter whether the volume varies. $\endgroup$ May 4 '18 at 16:36
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    $\begingroup$ This equation confuses so many thoughtful thermo students that I've dubbed it "the cruelest equation in introductory thermodynamics" and discuss it at length in that note, along with its treatment in various textbooks. $\endgroup$ May 4 '18 at 18:59
  • $\begingroup$ @Chemomechanics Thank you very much, that was actually very very helpful! $\endgroup$
    – Friedrich
    May 7 '18 at 8:56
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The internal energy is proportional to the temperature and independent of the volume (for an ideal gas). If volume is constant, then heat added is equal to change in internal energy (because there is no work--nowhere else for that energy to go). In an adiabatic process, there is some work, and more heat must be added to cause an equivalent increase in internal energy than in the constant volume case, but the change in internal energy for a given increase in temperature is still the same.

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