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derive a relationship between Cp&Cv

Heat Capacity C.
The quantity of heat required to raise the temperature of a substance by one degree is called its heat capacity.
If one gram of the substance is considered it is named as specific heat capacity.
If one mole of the substance, it is termed as molar heat capacity.
                
According to I law of Thermodynamics,
             Sq = dE + pdv
            
Specific heat capacity of constant volume C_v:

At constant volume
                         dv = 0
         
Heat Capacity at constant pressure C_p:
   By definition,
At constant P, dH = dE + pdv.
     
Relation between C_p and C_v: Based on work of expansion:

C_p.dT - C_vdT = pdv.      ? ( 3)
For one mole of an ideal gas,
                        pv =  RT

Differentiating the above equation.
         pdv  + vdp   = R.dT.
At constant P, vdp = 0
                    pdv  = RdT   ? (4)

Substituting  (4) in (3),
               ( C_p - C_v ) dT = R. dT