Chapter 13, Problem 54A

Solution

Interpretation: The increase in the temperature of a gas confined to a rigid container resulting an increase in the pressure needs to be explained in terms of kinetic molecular theory.

Concept Introduction: - The kinetic molecular theory of gases explains the gases are composed of molecules and these molecules are always in motion.

On the basic of the assumptions made an equation has been given known as kinetic gas equation.

  $PV=\frac{1}{3}mn{v}^2$

Where,

P = pressure exerted by gas.

V = volume of gas.

m = mass of each molecule

n = total number of molecules

v = root mean square velocity.

On increasing temp, kinetic energy of molecules increases due to which collision of molecules increases which increase the pressure.

The molecules are always in a state of rapid random motion in all directions.

These molecules are perfectly elastic i.e., there is no loss of energy on collisions. The average kinetic energy of the molecules is directly proportional to the absolute temperature (T) of the gaseous system. On increasing temperature Kinetic energy increases. This means that the particles collide the sides of container more often and with greater force. Both these factors cause the pressure of the gas to increase. If the speed of the particles is high, they will combine more frequently with the walls of the container and exert more force.

This cause the force on the walls of the container to increase and so pressure increases.

Kinetic energy $=\frac{1}{2}mn{v}^2\propto \mathrm{T}$

  $\begin{array}{l}\frac{1}{2}mn{v}^2=kT\{k=constant\}\\ =\frac{3}{2}\times \frac{1}{3}mn{v}^2=kT\\ \frac{1}{3}mn{v}^2=\frac{2}{3}kT\end{array}$

But,

  $\begin{array}{l}\frac{1}{3}mn{v}^2=PV\left\{kineticgastheory\right\}\\ PV=\frac{2}{3}kT\\ \frac{PV}{T}=\frac{2}{3}k\end{array}$

For 1 mole of a gas constant represented by R

$\begin{array}{l}\frac{PV}{T}=R\\ PV=RT\end{array}$

This is general gas equation.

On increasing temperature, kinetic energy increases due to which collision of gas molecules on the walls of container increases. This causes increase in pressure.