The ratio of specific heats at constant pressure and at constant volume is \(\gamma\). To find out the…
The ratio of specific heats at constant pressure and at constant volume is \(\gamma\). To find out the degree of freedom, the expression is
\(f=\frac{2}{\gamma-1}\)
\(f=\frac{\gamma+1}{2}\)
\(f=\frac{2}{\gamma+1}\)
\(f=\frac{1}{\gamma+1}\)
Solution
We know that,
\(\gamma=\frac{C_p}{C_V}\) ...(i)
Energy of \(N\) molecules of gas,
\(U=\frac{f}{2} R T\)
where, \(f\) is degree of freedom.
\(\begin{array}{ll}
\therefore & C_V=\frac{d U}{d T}=\frac{d}{d T}\left(\frac{f}{2} R T\right)=\frac{f R}{2} \\
\therefore & C_p=C_V+R=\frac{f R}{2}+R \Rightarrow C_p=\frac{(f+2) R}{2}
\end{array}\)
\(\therefore\) From Eq. (i), we get
\(\begin{gathered}
\gamma=\frac{(f+2) R / 2}{f R / 2} \Rightarrow \gamma=\frac{f+2}{f} \\
\Rightarrow \quad f \gamma-f=2 \Rightarrow f(\gamma-1)=2 \Rightarrow f=\frac{2}{\gamma-1}
\end{gathered}\)