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.

Explanation

(a) `C_p = 29 (J)/(.^@K "mole") = (2.9)/(8.3) R`
`C_v = (20.7)/(8.3) R, gamma = (29)/(20.7) = 1.4 = (7)/(5)`
`i = 5`
(b) In the process `pT = `const.
`(T^2)/(V) =` const, So `2 (dT)/(T) - (dV)/(V) = 0`
Thus `CdT = C_V dT + pdV = C_V dT + (RT)/(V) dV = C_V dT + (2 RT)/(T) dT`
or `C= C_V + 2 R = ((29)/(8.3)) R` so `C_V = (12.4)/(8.3) R = (3)/(2) R`
Hence `i = 3` (monoatomic).