Graph shows a hypothetical speed distribution for a sample of N gas particles (for v>v₀; $\frac{\mathrm{dN}}{\mathrm{dv}}$=0)
 

The graph is a linear relation between $\frac{\mathrm{dN}}{\mathrm{dV}}$ and V
$\text{a) }\mathrm{N}=\int \mathrm{dN}=\frac{{\mathrm{av}}_0}{2},{\mathrm{V}}_0=\frac{2\mathrm{N}}{9}$
b) < v >= average velocity
$<\mathrm{v}>=\frac{\int \mathrm{v}\left(\mathrm{dN}\right)}{\int \mathrm{dN}}$
$\therefore$for the graph
$$\begin{gathered} \begin{array}{l}\frac{\mathrm{dN}}{\mathrm{dv}}=\left[\frac{\mathrm{a}}{{\mathrm{v}}_0}\right]\mathrm{v}\\ \mathrm{dN}=\left(\frac{\mathrm{a}}{{\mathrm{v}}_0}\right)\mathrm{vdv}----2\\ <\mathrm{v}>=\frac{\mathrm{a}}{{\mathrm{Nv}}_0}{\int }_0^{{\mathrm{v}}_0} {\mathrm{v}}^2\left(\mathrm{dv}\right)\\ =\frac{\left({\mathrm{v}}_0^3\right)}{2{\mathrm{v}}_0}\\ =\frac{{\mathrm{v}}_{0}9}{2}\end{array} \\ \text{c) }{\mathrm{v}}_{\mathrm{rm}}={\left[\frac{\int {\mathrm{v}}^2\left(\mathrm{dN}\right)}{\int \mathrm{dN}}\right]}^{1/2}--------3 \end{gathered}$$
2 and 3
${\mathrm{v}}_{\mathrm{rm}}=\frac{{\mathrm{v}}_0}{\sqrt{2}}$
d)
$\begin{array}{l}\int \mathrm{dN}=\mathrm{\Delta N}={\int }_{{\mathrm{v}}_0} \frac{\mathrm{a}}{{\mathrm{v}}_0}\mathrm{vdv}\\ =\frac{\mathrm{a}}{{\mathrm{v}}_0}{\left[\frac{{\mathrm{v}}^2}{2}\right]}_{{\mathrm{v}}_{0/2}}^{{\mathrm{v}}_0}\\ =\frac{\mathrm{a}}{2{\mathrm{v}}_0}\left[\frac{3{\mathrm{v}}_0^2}{4}\right]\\ =\frac{3{\mathrm{av}}_0}{8}=\frac{3\mathrm{N}}{4}\\ {\mathrm{\Delta N}}_0=\frac{3\mathrm{N}}{4}\end{array}$