# Specific heat capacity at constant volume VS constant pressure

For an ideal gas, why is the specific heat capacity at constant volume lower than the specific heat capacity at constant pressure?

• What about the Wikipedia article on relations between heat capacities does not satisfy you? – ACuriousMind Aug 21 '15 at 19:34
• I have not yet learned about isothermal compressibility, or that much about partial derivatives in detail. I was hoping for a simpler explanation, possibly including the ideal gas equation $PV=nRT$ and/or the heat equation involving specific heat capacity $Q=mc \Delta t$. @ACuriousMind – mnmakrets Aug 21 '15 at 19:44

The simple answer, which is what I think you're hoping for is the following: At constant volume, the system (by definition) is not able to do work on the surroundings because work involves a change in volume. All the heat you put in is spent raising the temperature (internal energy).

At constant pressure, some of the energy you put in goes into raising the temperature (internal energy) and some of it goes into doing work by expanding the ideal gas.

Thus, the temperature increase is smaller in the constant pressure case than in the constant volume case. This is equivalent to saying that the specific heat capacity at constant pressure is larger than the specific heat capacity at constant volume.

First law of thermodynamics $$\delta Q=\delta E+p\delta V$$ $$\therefore\frac {\delta Q}{\delta T} = \frac{\delta E}{\delta T}+nR$$ As $p\delta V$ can be written as $nR\delta T$

When volume is constant $$\delta Q= \delta E$$ $$\therefore\frac {\delta Q}{\delta T} = \frac{\delta E}{\delta T}$$ $\frac {\delta Q}{n\delta T}$ is Molar heat capacity. $$\therefore C_\textrm{constant pressure}=C_\textrm{constant volume} + R$$ Heat capacity at constant volume lower than heat capacity at constant pressure.

i think ,heat energy is required to change the temperature of the gas as well as for the the expansion of the gas against external pressure hence,change in temperature at constant pressure ,therefore Cp>Cv

• Don't add, i think; it reflects your ignorance of the matter and nothing else; if you're not sure, then don't post your view as an answer. – user36790 Jan 25 '16 at 5:59

If you cannot understand, you are not alone; even teachers find difficulty to grab. The difficulty of understanding is due to the rubbish ways of explanation in academic/ textbook. Let's see the actual concept. Cv is heat capacity at constant volume. Constant volume has no direct meaning in heat retaining capacity. Constant volume indirectly means work done is prevented. In this situation, all the heat, if added, is contained within the system volume.

On the other case, Cp is heat capacity at constant pressure. Constant Pressure is of no meaning in heat containing capacity. But the fact is: when heat is added, system pressure raises leading to expansion. During expansion, some of energy is expelled i.e. transported to boundary/ periphery and resides over the boundary as potential energy between the system and neighbour/ surrounding. Here, heat is kept in two locations - one) within volume, two) Over-periphery.

While Cv is referring capacity within volume, Cp is referring capacity within volume as well as on periphery. Hence Cp is more than Cv. Cv may be redefined as heat capacity within volume and Cp may be redefined as heat capacity within upto periphery.

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