If a gas has 'f' degrees of freedom, find the ratio of `C_(p) and C_(v)`.

Explanation

Suppose a polyatomic gas molecule has .T. degree of freedom.
`U = f xx 1/2 k_(B) T xx N_(A) = f/2 RT`
`As C_(v) = (dU)/(dT)`
`:. C_(v) = d/(dT)(U) = (d)/(dT) (f/2 RT) = f/2 R`
As `C_(p) = C_(v) + R :. C_(p) = f/2 R+R = (f/2 + 1) R`
Ratio of `C_(p) and C_(v)` is
`gamma = (C_p)/(C_v) = ((f/2+1)R)/(f/2 R) = 2/f ((f+1)/(2))`
`implies gamma = (C_p)/(C_v) = (1+2/f)`
This is the relation between number of degrees of freedom (f) and ratio of `C_(p)` and `C_(v) & (gamma)`.