Why is $c_p$ higher than $c_v$? why is heat capacity at constant pressure higher than heat capacity at constant volume?
It is supposed to be this way, since if you increase $C_p$ in volume work is being done, in other case not. This is not clear to me either...
 A: Recall two things...


*

*First that the 1st law is conservation of energy. 

*Second that temperature is a non-decreasing function of internal energy.


So if we take two identical samples of gas and add the same heat $Q$ to each (increasing their internal energies), but allow one to do work $W$ on the surrounding while the other does none, then the sample doing the work experiences a smaller increase in internal energy and consequently a smaller increase in temperature.
That's really all there is to it.
A: I still don't feel it on an intuitive level...
When a gas is heated at constant volume,all the heat supplied is used to increase the internal energy of the gas.(since the volume is constant the gas cannot expand so)
When a gas is heated at constant  pressure ,the gas expands.It does work against the external pressure .
The heat is used in two ways
1)partly to do work agaist external pressure.
2)partly to increase the internal energy.
so at constant pressure ,an additional amount of heat equivalent to work done is utilised.
Hence more amount of heat is required to increase the temperature of 1 mole of a gas at constant pressure than that at constant volume that is  $c_p$ higher than $c_v$
