Kinetic Theory of Gases
11.0 Relation between Degree of Freedom and Specific Heat of Gas
11.0 Relation between Degree of Freedom and Specific Heat of Gas
Energy related with each degree of freedom$ = \frac{1}{2}{k_B}T $
Energy related with all degree of freedom $= \frac{f}{2}{k_B}T $
Internal energy of one mole of ideal gas (total KE) $$ U = \frac{f}{2}RT $$
For Isochoric Process (Volume constant) $dV=0$, $dW = 0$
By First law of Thermodynamics
$$dQ = dW + dU$$
or $${C_v}dT = dU$$
or $${C_v} = \frac{{dU}}{{dT}} $$
$$ {C_v} = \frac{{dU}}{{dT}} = \frac{f}{2}R $$
For Isobaric Process (constant pressure)
$$dQ = dW + dU$$
$$dU = \frac{{nf}}{2}RdT $$
$$dW = nRdT$$
$$Q = \frac{{nf}}{2}RdT + nRdT $$
As $$ Q = n{C_P}dT\;\; \Rightarrow n{C_P}dT = n\frac{f}{2}RdT + nRdT $$
$${C_P} = \frac{f}{2}R + R = \left( {\frac{{f + 2}}{2}} \right)R $$
Note: Relation between ${C_P}$ and ${C_V}$
As, $${C_V} = \frac{f}{2}R$$
and $${C_P} = \left( {\frac{{f + 2}}{2}} \right)R $$
$${C_P} = {C_V} + R$$
