A water vapour filling the space under the piston of a cylinder is compressed (or expanded) so that it remaind saturated all the time, being just on the verge of condensation. Find the molar heat capacity `C` of the vapour in this process as a function of temperature `T`, assuming the vapour to be an ideal gas and neglecting the specific volume of water in comparion with that of vapour. Calculate `C` at a temperature `t = 100 ^@C`.

Explanation

When heat `dQ` is given to the vapour its temperature will change by `dT`, pressure by `dp` and volume by `dV `, it being assumed that the vapour remains saturated.
Then by `C-C` equation
`(dp)/(dT) = (q)/(TV) (V'_(vapour) gt gt V'_(Lip))`, or `dp = (q)/(TV') dT`
on the other hand, `pV' = (RT)/(M)`
So `pdV' + V' dp = (RdT)/(M)`,
Hence `pdV' = ((R)/(M) -(q)/(T))dT`
finally `dQ = CdT = dU + pdV`
=`C_V dT + ((T)/(M) - (q)/(T)) dT = C_p dT - (q)/(T) dT`
`(C_p, C_V` refer to unit mass here). Thus
`C = C_p - (q)/(T)`
For water `C_p = (R gamma)/(gamma - 1). (1)/(M)` with `gamma = 1.32` and `M = 18`
So `C_p = 1.90 J//gm K`
and `C = -4.13 J//gm^@K = - 74 J//"mole" K`.