Find the number of degrees of freedom of molecules in a gas. Whose molar heat capacity
(a) at constant pressure `C_p = 29 J mol^-1 K^-1`
(b) `C = 29 J mol^-1 K^-1` in the precess (pT) = constant.

Answer

(a)5 (b)3

Explanation

(a) We have `C_(P)-C_(u) = R or 1-(1)/(gamma) = (R)/(C_(P))` or `gamma = (C_(P))/(C_(P) -R)`
But `g = 1 + (2)/(r)` where r = degree of Freedom
`therefore 1+(2)/(r) =C_(P)/(C_(P) -R)` or `r = 2(C_(P)/(R)-1)`
`therefore` Here `r= 2((29)/(8.3)-1) =5`
(b) In a polytropic process `(pV^n = constant)C = C_v -(R)/(n-1)`
Here `pT = constant or p^2 V =, constatn (thereforepV = RT always)`
or `pV^(1//2) = constant so n = 1/2`
`therefore C= (R)/(gamma-1)-(R)/((1)/(2)-1) = (R)/(gamma-1)+2R or gamma =1+(R)/(C-2R)`
But `gamma = 1 + (2)/(r)(always)`
Hence `1 + (R)/(C-2R) = 1+(2)/(r)` " " or `r=(2(C-2R))/(R)`
`=(2(29-2xx8.3))/(8.3)=3`