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 internal 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 heateq at constant pressure so that the temperature increases by elf. If .Q. is the heat supplied in this process and .dV. the change in volume of the gas
`Q = mu C_P dT` ...(3)
If W is the 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,
`mu C_P dT = mu C_v dT + PdV`
For mole of ideal gas, the equation of state is given by
`PV = mu RT rArr PdV + VdP - mu RdT`
since the pressure is constant dP = 0
` therefore C_v dT = C_v dT + RdT`
` therefore C_P - C_V + R " or " C_P - C_V = R` ....(6)
This relation is called mayers relation . The relation show that specific heat at constant pressure `(s_P)` is alaways greater than specific heat at constant volume `(s_P)` .