Derive Mayer's equation from the I law of thermodynamics.

Explanation

We know that, dQ=dU+dW (First law of thermodynamics)
At a constant volume, heat supplied to 1 mole of ideal gas enclosed in a conducting cylinder fitted with a conducting piston will be utilised to increase the internal energy of the system and dW=0
`:.(dQ)_("volume")=dU=C_(V)dT`
where `C_(v)` is molar specific heat of a gas at constant valume.
At a constant pressure heat supplied will be used to increase the internal energy as well to do external work by the gas
`:.(dQ)_("pressure,")=dU+dW`
For the same rise in the temperature in both the cases at a constant volume and pressure, `dU=C_(V)dT`
For 1 mole of gas,
`C_(P)dT=C_(V)dT+dW`
where, `dW=Fdx=PAdx=PdV`
where, .Adx. is a small change in the volume of the gas.
Also from PV=RT, at constant pressure,
Pressure, `P(V+dV)=R(T+dT)`
i.e., PdV=RdT
Hence, `C_(P)dT=C_(V)dT+RdT`
i.e., `C_(P)=C_(V)+R`
This equation is known as Mayer.s equation.