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Kinetic theory of gases. Boltzmanns Law and Maxwell distribution

Problem 2.86

In the case of gaseous nitrogen find: (a) the temperature at which the velocities of the molecules \(v_{1}\) \(=300 \mathrm{~m} / \mathrm{s}\) and \(v_{2}=600 \mathrm{~m} / \mathrm{s}\) are associated with equal values of the Maxwell distribution function \(F(v)\) (b) the velocity of the molecules \(v\) at which the value of the Maxwell distribution function \(F(v)\) for the temperature \(T_{0}\) will be the same as that for the temperature \(\eta\) times higher.

Reveal Answer
 (a) T=m(v22−v12)4kln⁡(v2/v1)=330 K (b) v=3kT0mηln⁡ηη−1\text { (a) } T=\frac{m\left(v_{2}^{2}-v_{1}^{2}\right)}{4 k \ln \left(v_{2} / v_{1}\right)}=330 \mathrm{~K} \text { (b) } v=\sqrt{\frac{3 k T_{0}}{m} \frac{\eta \ln \eta}{\eta-1}}
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