Derive Mayer's relation for an ideal gas.

Explanation

Mayer.s relation: Consider `mu` mole of an ideal gas in a container with volume V, pressure P and temperature T.
When the gas is heated at constant volume the temperature increases by dT. 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 intemal energy be dU.
If `C_(v)` is the molar specific heat capacity at constant volume, from equation.
`C_(V)=(1)/(mu)(dU)/(dT)" ...(1)"`
`dU=muC_(V)dT" ...(2)"`
Suppose the gas is heated at constant pressure so that the temperature increases by dT. If .Q. is hte heat supplied in the process and .dV. the change in volume of the gas.
`Q=muC_(P)dT" ...(3)"`
If W is t he workdone by the gas in this process, then
`W=PdV" ...(4)"`
But from the first law of thermodynamics,
`Q=dU+W" ...(5)"`
Substituting equations (2), (3) and (4) in (5), we get,
`muC_(P)dT=muC_(V)dT+PdV`
For mole of ideal gas, the equation of state is given by
`PV=muRT rArr PdV+VdP=muRdT`
Since the pressure is constantm, `dP=0`
`therefore" "C_(P)dT=C_(V)dT+RdT`
`therefore" "C_(P)=C_(V)+R(or) C_(P)-C_(V)=R" ...(6)"`
This relation is called Mayer.s relation It implies that the molar specific heat capacity of an ideal gas at constant pressure is greater than molar specific heat capacity at constant volume. The relation shows that specific heat at constant pressure `(S_(P))` is always greater than specific heat at constant volume `(S_(V))`.