Relation between the ratio of specific heats (γ) of gas and degree of freedom ' f ' will be

Question

Relation between the ratio of specific heats (&gamma;) of gas and degree of freedom ' f ' will be (A) &gamma;= f +2 (B) (1/&gamma;)=(1/ f )+(1/2) (C) f =(2/&gamma;-1) (D) f =2(&gamma;-1)

Tardigrade Admin · Accepted Answer

&gamma;=( C P / C V )=( C V + R / C V ) =((1/2) fR + R /(1)2 fR )=( f +2/ f ) ⇒ &gamma; f - f =2 ⇒ f =(2/&gamma;-1)

Q. Relation between the ratio of specific heats $(γ)$ of gas and degree of freedom ' $f$ ' will be

$γ = f + 2$

$\frac{1}{γ} = \frac{1}{f} + \frac{1}{2}$

$f = \frac{2}{γ - 1}$

$f = 2 (γ - 1)$

$γ = \frac{C _{P}}{C _{V}} = \frac{C _{V} + R}{C _{V}}$
$= \frac{\frac{1}{2} f R + R}{\frac{1}{2} f R} = \frac{f + 2}{f}$
$\Rightarrow γ f - f = 2$
$\Rightarrow f = \frac{2}{γ - 1}$

Q. Relation between the ratio of specific heats (γ) of gas and degree of freedom ' f ' will be

γ=f+2

γ1​=f1​+21​

f=γ−12​

f=2(γ−1)