Derive Mayer's relation for an ideal gas.

Explanation

Meyer's relation
`(i)` Consider `mu` mole of an ideal gas in a container with volume `V` , pressure `P` and temperature `T`.
`(ii)` When the gas is heated at constant volume the temperature increases by `dT`.
`(iii)` As no work is done by the gas, the heat that flows into the system will increase only the internal energy. Let the change in internal energy be `dU`.
`(iv)` If `C_(v)` is the molar specific heat capacity at constant volume.
`dU=muC_(v)dT`........`(1)`
`(v)` Suppose the gas is heated at constant pressure so that the temperature increases by `dT`. If `'Q'` is the heat supplied in this process and `'dV'` the change in volume of the gas.
`Q=muC_(p)dT`..........`(2)`
`(vi)` If `W` is the work done by the gas in this process, then
`W=PdV`.......`(3)`
But from the first law of thermodynamics,
`Q=dU+W`....`(4)`
`(vii)` Substituting euations `(1)` , `(2)` and `(3)` in `(4)`, we get
`muC_(p)dT=muC_(v)dT+PdW`.......`(5)`
For mole of ideal gas , the equation of state is given by
`PV=muRTimpliesPdV+VdP=muRdT`..........`(6)`
Since the pressure is constant, `dP=0`
`:.C_(p)dT=C_(v)dT+RdT`
`:' C_(p)=C_(v)+R` (or) `C_(p)-C_(v)=R`..........`(7)`
This relation is called Meyer's relation.