Question

A container has an ideal gas at pressure $P$ in thermal equilibrium. Assuming the mass of a molecule is $m$ and all the molecules are moving wih same speed $v$ in random directions, the expression for number of colloisions per second which the molecules make with unit area of container's wall is $\frac{\sqrt{xP}}{ymv}$. Find $(x+y)$

Answer 1

Find change in momentum per second.

Formula used: $p=mv,v^2=v_x^2+v_y^2+v_z^2$
Below figure represent a particle's collision with one of the surface of container


Since collision is elastic
momentum change/collision is $2m{v}_x...\left(i\right)$
Also, $v^2=v_x^2+v_y^2+v_z^2$
Since we know $v_x^2=v_y^2=v_z^2$
$v_x^2=\frac{v^2}{3}$
$\therefore v_x=\frac{v}{\sqrt{3}}$
$\therefore$ Momentum change/collision $=2m\frac{v}{\sqrt{3}}$

Find number f collisions per second per unit area.
Formula used : $P=\frac{F}{A}=\frac{\Delta p}{\Delta tA}$

Let there be $n$ collisions with unit area of wall per unit time.
$\therefore P=2m\frac{v}{\sqrt{3}}\times n$
$\therefore n=\frac{P\sqrt{3}}{2mv}$
Comparing wih $\frac{\sqrt{xP}}{ymv}$

$x=3,y=2$
$\therefore x+y=5$