Molar specific heat

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?




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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