Question

In a cubical box of volume $V$, there are $N$ molecules of a gas moving randomly. If $m$ is mass of each molecule and $v^2$ is the mean square of $x$ component of the velocity of molecules, then the pressure of the gas is

• A. $P=\frac{1}{3}\frac{mN{v}^2}{V}$
• B. $P=\frac{mN{v}^2}{V}$
• C. $P=\frac{1}{3}mN{v}^2$
• D. $P=mN{v}^2$

Answer 1

The correct option is B $P=\frac{mN{v}^2}{V}$

Let the mass of each molecule be m.
$\therefore$ Pressure $P=\frac{1}{3}\rho v_{rms}^2$
$=\frac{1}{3}\frac{mN}{V}.v_{rms}^2.....\left(1\right)$
Root mean square velocity is
$v_{rms}^2=v_x^2+v_y^2+v_2^2$
$v_x=v_y=v_z=v$
$v_{rms}^2=3v^2$
$\therefore$ From equation (1).
$P=\frac{1}{3}\frac{mN}{V}.3v^2$
$P=\frac{mN{v}^2}{V}$