The ratio `C_(P)//C_(V)` represented by `gamma` correspends ot the atomicity of an ideal gas. Which of following ratios corresponds to a diatomic molecule such a oxygen?

Options

A. `C_(P) // C_(V) = 1.66`
B. `C_(P) // C_(V) = 1.30`
C. `C_(P) // C_(V) = 1.40`
D. `C_(P) // C_(V) = 1.22`

Answer

C

Explanation

For a linear polyatomic molecule (such as `CO_(2))`, there are 2 degrees of rotational motion, leaving `3 n - 5` degrees of freedom for vibrational motion. For an ideal diatomic gas, `3 (2) - 5 = 1`. Since the vibrational degrees of freedom have negligible contribution at normal temperature (they contribute only at high temperature), we have (for on mole of an ideal diatiomc gas),
`U = 3 ((1)/(2) Rt) + 2 ((1)/(2) RT)`
`(5)/(2) RT`
`C_(V) = (Delta U)/(Delta T) = ((5)/(2) R Delta T)/(Delta T) = (5)/(2) R`
Since `C_(P) - C_(V) = R` (for 1 mol),
`C_(P) = R + C_(V) = R + (5)/(2) R = (7)/(2) R`
`:. (C_(P))/(C_(V)) = ((7//2) R)/((5//2) R) = 1.40`
Note that for a nonlinear polyatomic molecule, there are 3 degrees of rotational motion, leaving `3n - 6` degrees of freedom for vibrational motion.