Relation Between CP and CV

When the volume is kept constant, no work is done, so the amount of heat absorbed is completely used to raise the temperature of system. But when pressure is kept constant so work of expansion will be done. Thus, the amount of heat absorbed by system is not wholly used to raise the temperature but some part of heat is used to do some work of expansion. Thus when pressure is constant, more heat is to be given to system to raise its temperature by 1^oC then when volume is kept constant.

Thus,   C_P > C_V  

The relation between C_P and C_V can be calculated by finding difference between two so, we have

                C_P  –  C_V   = dH/dT   -   dE/dT    ––––––––   (i).

H    =     E   +    PV

                and     PV  =    nRT

                \       H     =   E  +  nRT

                For 1 mole of gas (molar heat is for 1 mole.)

                           H     =   E  +  RT  ––––––––––  (ii)    

                Differentiating (ii) w. r. to ‘T’, we get     

                                         dH/dT   =  dE/dT   +    R

                                Or    dH/dT    -   dE/dT   =    R      

                Substituting in (i), we get

                                                C_P  –  C_V   =  R