Calculate the average energy of particles of an ideal gas at temperature T .

Question

Accepted Answer

The average kinetic energy of the particles is

[latex]\epsilon _{av}=\int_{0}^{\infty }\epsilon P(\epsilon) d\epsilon .[/latex]

Using Eqs. (8.26) and (8.27) to express the above integrand in terms of the variable u, we obtain

[latex]P(\epsilon )d\epsilon =\frac{2}{\sqrt{\pi } }\frac{\epsilon ^{1/2}d\epsilon }{(k_{B}T)^{3/2}} e^{-\epsilon /k_{B}T} .[/latex] (8.26)

[latex]u=\epsilon / k_{B}T .[/latex] (8.27)

[latex]\epsilon _{av}=\frac{2}{\sqrt{\pi } }k_{B}T \int_{0}^{\infty }u^{3/2}e^{-u}du.[/latex]

The above integral, which may be evaluated using a formula in Appendix G, is equal to [latex] 3\sqrt{\pi }/4[/latex]. We thus obtain

[latex]\epsilon _{av}=\frac{3}{2} k_{B}T .[/latex]

Question 8.4: Calculate the average energy of particles of an ideal gas at...