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Prove that for an ideal gas, CP is greater than CV .

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1. It is clear that two heat capacities are not equal and CP is greater than CV by a factor which is related to the work done.

2. At a constant pressure, a part of heat absorbed by the system is used up in increasing the internal energy of the system and the other for doing work by the system.

3. At constant volume, the whole of heat absorbed is utilized in increasing the temperature of the system as there is no work done by the system. Thus CP is greater than CV .

C= dH / dT ; C= dU / dT

4. By definition, H = U + PV for 1 mole of an ideal gas. 

H = U + RT 

By differentiating this equation with respect to temperature T. we get,

dH / dT = dU / dT + R

CP = CV + R 

CP – CV = R

Thus for an ideal gas, CP is greater than CV by the gas constant R.

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