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What is the difference between molar specific heat and molar specific heat at constant pressure?

It seems that both of them have the same formula, i.e

$$\frac{1}{n}\left(\frac{\mathrm dQ}{\mathrm dT}\right)$$

Then what is the difference between these two and why do they have the same notation?

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    $\begingroup$ The difference is in the tiny index $\mathfrak p$ outside of the brackets. Why did you skip it? $\endgroup$ – Ivan Neretin Dec 25 '17 at 11:36
  • $\begingroup$ Yes but that $p$ does not change the value of the formula right ? So you mean that both of them are essentially the same ? How is that possible ? $\endgroup$ – Aditi Dec 25 '17 at 11:46
  • $\begingroup$ Yes it does change the result, and not by negligible amount. Ever heard about the specific heat at constant volume? $\endgroup$ – Ivan Neretin Dec 25 '17 at 11:49
  • $\begingroup$ Sorry , I think I understood what you meant to say. Is it that the subscript $p$ denotes that pressure is held constant during differentiation and that gives specific heat at constant pressure. But for specifically for molar specific heat we do not include a subscript which means nothing is held constant during differentiation I believe ? $\endgroup$ – Aditi Dec 25 '17 at 12:00
  • $\begingroup$ "Molar specific heat" is not a thing at all. Or rather, it is an umbrella term which means both $c_p$ and $c_V$ at once, much like the word "human" means you and me, and quite a few others. It has no value. Say, you know your weight, but what is the weight of a human? $\endgroup$ – Ivan Neretin Dec 25 '17 at 12:02
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First of all it is important to indicate which properties of your system stay constant during the partial differentiation. Therefore you get the molar specific heat at constant pressure as $$ \frac{1}{n}\left(\frac{\partial Q}{\partial T}\right)_p $$ and at constant volume $$ \frac{1}{n}\left(\frac{\partial Q}{\partial T}\right)_V $$

Now to the question why this even matters:
When putting energy (in form of heat) into the system its temperature (normally) rises. Considering a system with constant volume this is actually all that happens. Therefor all the energy put into the system gets transferred into a temperature rise of the system.

If you hold the pressure constant instead the system can expand while being warmed up. This requires it do do work against the outside pressure. Therefore some of the energy put into the system is used for the expansion and only the rest is transformed into a temperature rise.

Therefore the specific heat at constant pressure is always greater than the one at constant volume (more energy has to be put in it in order to achieve the same temperature rise).

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