# Heat Capacity at Constant Pressure or Volume

Q:
why CP always greater than CV for gases? can u explain with better example?
- P.Mouryan (age 17)
INDIA
A:

Cp > Cv for all materials, unless the coefficient of thermal expansion is zero, in which case heating at constant pressure and constant volume are the same thing. What's interesting is that this inequality holds both for materials that expnd on heating at constant pressures (like gases and most liquids and solids) and those that contract on heating at constant pressure (like some solids and liquids, e.g water at 0°C to 4°C). That means that the usual simple explanation for ideal gases- that they expand on heating and therefore heat input is needed to balance they work they do on expanding- must be missing something, since by itself it would give the wrong answer for materials that contract. The full ressult depends also on how the energy of the material changes as it expands or contracts, and that can be quite a large effect for liquids and solids.

Why does the net effect always end up with Cp > Cv, if there's any expansion or contraction? I haven't thought of a simple way to explain why this must be so, but the formal derivation of the exact amount
Cp-Cv in terms of the thermal expansion coefficient and the compressibility is very straightforward: https://en.wikipedia.org/wiki/Relations_between_heat_capacities (https://en.wikipedia.org/wiki/Relations_between_heat_capacities). The key point is that it depends on the square f the thermal expansion coefficient.

Mike W.

(published on 05/12/2018)

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