Question

The pressure exerted by an ideal gas is numerically equal to_________ of the mean kinetic energy of translation per unit volume of the gas.

Answer 1

Pressure, ideal gas, and kinetic energy.

1. Pressure- The force applied perpendicular to an object's surface per unit area across which that force is spread is known as pressure.
2. An ideal gas-An ideal gas is a hypothetical gas made up of a large number of randomly moving point particles with no interparticle interactions.
3. Kinetic Energy- The energy possessed by an object due to its motion.

Mathematical expression.

According to kinetic theory of gases,

$P=\frac{1}{3}\rho {\nu }^2$ __________ (1)

Where,

$P$ is pressure.

$\rho$is density.

$\nu$ is r.m.s velocity.

Density of gas, $\rho =\frac{M}{V}$ ________ (2)

Where,

$M$ is Mass.

$V$ is volume.

By substituting value of density from equation (2) in equation (1)

$\begin{array}{l}P=\frac{M{\nu }^2}{3V}\end{array}$ _________ (3)

Kinetic Energy of the gas is expressed as,

$\begin{array}{l}K.E=\frac{1}{2}M{\nu }^2\end{array}$ _________ (4)

By dividing equation (3) by (4),

$\begin{array}{l}\frac{P}{K.E}=\frac{\frac{M{\nu }^2}{3V}}{\frac{1}{2}M{\nu }^2}\end{array}$

$\begin{array}{l}\frac{P}{K.E}=\frac{2}{3V}\end{array}$

$\begin{array}{l}P=\frac{2}{3}\times \frac{K.E}{V}\end{array}$

Hence, the pressure exerted by the ideal gas is two-thirds of the kinetic energy per unit volume.