A gas is formed of molecules each molecules possessing f degrees of freedom, then the value of `gamma=(C_(P))/(C_(V))` is equal to

Options

A. `2/f`
B. `1+(2)/(f)`
C. `1+(f)/(2)`
D. `f+(1)/(2)`

Answer

A

Explanation

According to law equipartition of energy, the internal energy associated per degree of freedom is 1/2 kT, where k is the boltzmann's constant.
Thus, internal energy associated per molecule `=f(1)/(2) kT`
If `N_(A)` is Avagadro's number, then internal energy of one mole of an ideal gas is
`U =N_(A)f (1)/(2) kT`
`=1/2 f(N_(A) k) T `
`=1/2 f RT`
where `R=N_(A)k` =gas constant
molar heat capacity at constant volume
`C_(v)=(dU)/(dT)=d/(dT)(1/2f RT) =1/2 fR`
molar heat capacity at constant pressure `C_(p)=C_(v)+R`
(Mayor's relation)
`=1/2 fR+R=(1/2 f+1) R`
Hence y=