Question

Graph shows a hypothetical speed distribution for a sample of $N$ gas particle (for $V>V_0;\frac{dN}{dV}=0,\frac{dN}{dV}$ is a rate of change of number of particles with change velocity).

• A. The value of $a{V}_0$ is $2N$
• B. The ratio $V_{avg}/V_0$ is equal to $2/3$
• C. The ratio $V_{rms}/V_0$ is equal to $1/\sqrt{2}$
• D. Three fourth of the total particle has a speed between $0.5V_0$ and $V_0$

Answer 1

The correct option is A The value of $a{V}_0$ is $2N$

The graph $\frac{dN}{dV}$ vs V is lincar, whose slope is given by :

$m=\frac{a}{v_0}$ [From graph]

So, equation of the graph can be through as :

$y=mx+c$ (Equation of straight line)

As graph passes through origin, c - 0 (y - intercept)]

So we have $y=\frac{dN}{dV}$ and $x=V$ in this graph.

$\Rightarrow \frac{dN}{dV}=\frac{a}{V_0}v$

$\Rightarrow dV=\frac{a}{V_0}vdv$

Integrating both sides in limit $0\to N$ as $V:0\to v_0$

$\Rightarrow {\int }_0^{N}dN={\int }_0^{v_0}\frac{a}{V_0}vdv$

$\Rightarrow [N{]}_0^N=\frac{a}{V_0}{\left[\frac{v^2}{2}\right]}_0^{v_0}\Rightarrow N=\frac{a}{V_0}.{\frac{V_0}{2}}^2$

That gives,

$\Rightarrow a{V}_0=2N$