Express the change in internal energy in terms of molar specific heat capacity .

Explanation

If Q is the heat supplied to mole of a gas at constant volume and if the temperature changes by an amount `Delta T` we have
`Q = mu C_(v) Delta T`
By applying the first law of thermodynamics for this constant volume process (W=0 since dV =0) we have
Q = `Delta U -0`
By comparing the equation (1) and (2)
`Delta U = mu C_(v ) Delta T " or " C_(v ) = (Delta U )/(mu Delta T) " ( or ) " (1)/(mu) . (Delta U )/(Delta T)`
If the limit `Delta T` goes to zero we can write
`C_(v ) = (d U)/(mu dT) " ( or ) " (1)/(mu) . (d U)/( dT)`
Since the termperature and internal energy are state variables the above relation holds true for any process.