JEE Main & Advanced Physics Kinetic Theory of Gases Specific Heat in Terms of Degree of Freedom

(1) \[{{C}_{V}}:\] For a gas at temperature T, the internal energy \[U=\frac{f}{2}\mu RT\] \[\Rightarrow \] Change in energy \[\Delta U=\frac{f}{2}\mu R\Delta T\]... (i)

Also, as we know for any gas heat supplied at constant volume \[{{(\Delta Q)}_{V}}=\mu {{C}_{V}}\Delta T=\Delta U\]                                ... (ii)

From equation (i) and (ii) \[{{C}_{V}}=\frac{1}{2}fR\]

(2) \[{{C}_{P}}:\] From the Mayer?s formula \[{{C}_{p}}-{{C}_{v}}=R\]

\[\Rightarrow \] \[{{C}_{P}}={{C}_{V}}+R=\frac{f}{2}R+R\]\[=\left( \frac{f}{2}+1 \right)\,R\]

(3) Ratio of \[{{C}_{p}}\] and \[{{C}_{v}}\,(\gamma ):\] \[\gamma =\frac{{{C}_{p}}}{{{C}_{v}}}=\frac{\left( \frac{f}{2}+1 \right)R}{\frac{f}{2}R}=1+\frac{2}{f}\]

(i) Value of \[\gamma \] is different for monoatomic, diatomic and triatomic gases.\[{{\gamma }_{mono}}=\frac{5}{3}=1.6,\,{{\gamma }_{di}}=\frac{7}{5}=1.4,\,{{\gamma }_{tri}}=\frac{4}{3}=1.33\]

(ii) Value of \[\gamma \]  is always more than 1. So we can say that always \[{{C}_{P}}>{{C}_{V}}\].