Question

Graph shows a hypothetical speed distribution for a sample of N gas particles $(forv>v_0;\frac{dN}{dv}=0)$ :

• A. The value of $a{v}_0$ is 2N
• B. The ratio $\frac{v_{avg}}{v_0}$ is equal to $\frac{2}{3}$
• C. The ratio $\frac{v_{rms}}{v_0}$ is equal to $\frac{1}{\sqrt{2}}$
• D. Three fourth of the particle has a speed between $0.5v_0$ to $v_0$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A The value of $a{v}_0$ is 2N
B The ratio $\frac{v_{avg}}{v_0}$ is equal to $\frac{2}{3}$
C The ratio $\frac{v_{rms}}{v_0}$ is equal to $\frac{1}{\sqrt{2}}$
D Three fourth of the particle has a speed between $0.5v_0$ to $v_0$

Total number of particle $=\int \frac{dN}{dv}.dv$
$=N=\frac{1}{2}a.v_0$
$\Rightarrow a{v}_0=2N$ $⟹$The value of $a{v}_o$ is 2N
According to Graph
$\frac{dN}{dv}=\frac{a}{v_0}v$
$dN=\frac{a}{v_0}vdv$
Average velocity
$v_{avg}=\frac{\int vdN}{\int dN}=\frac{{\int }_0^{v_0}\frac{a}{v_0}{v}^{2}dv}{{\int }_0^{v_0}\frac{a}{v_0}vdv}$
$=\frac{\frac{a}{v_0}\frac{v_0^3}{3}}{\frac{a}{v_0}\frac{v_0^2}{2}}$
$v_{avg}=\frac{2v_0}{3}$ $⟹$The ratio $\frac{v_{avg}}{v_o}$ is equal to $\frac{2}{3}$
rms velocity
$v_{rms}=\sqrt{\frac{\int v^{2}dN}{\int dN}}=\sqrt{\frac{\frac{a}{v_0}.\frac{v_0^4}{4}}{\frac{a}{v_0}.\frac{v_0^2}{2}}}$
$r_{rms}=\frac{v_0}{\sqrt{2}}$ $⟹$The ratio $\frac{v_{rms}}{v_o}$ is equal to $\frac{1}{\sqrt{2}}$