Chapter U3.5, Problem 12E

Interpretation Introduction

Interpretation: The reason for gases to expand on heating and shrink on cooling needs to be explained.

Concept Introduction: The kinetic theory of gases is a simple thermodynamic model that aims to describe the macroscopic properties of gases like the pressure, temperature, etc in terms of microscopic parameters like the speed of the gas particles. The kinetic theory is based on a number of postulates concerning the size and the random motion of the gas particles in a sample and the derivations from the kinetic theory are approximate at best.

Answer to Problem 12E

Gases expand on heating and shrink on cooling due to an increase or decrease in the average kinetic energy of the gas particles upon heating or cooling.

Explanation of Solution

The ideal gas law is governed by the kinetic theory of gases. The kinetic theory of gases is based on a number of assumptions. The theory assumes that the gas particles are hard spheres and in constant random motion in straight line paths with respect to each other. The gas particles are massless and do not exert any force on each other or on the walls of the container except during collisions. The collisions between the gas particles or with the walls of the container are completely elastic, i.e, energy is conserved in these collisions. Moreover, the absolute temperature of the gas affects the average kinetic energy of the gas particles.

The mean free path of a gas is defined as the distance that a gas particle travels between successive collisions. When a sample of a gas is heated, the average kinetic energy of the gas increases. Thus, the average speed of the gas particles increases. As the average speed increases, the gas particles move more randomly and thereby the frequency of collisions decreases. Since fewer gas particles collide upon heating, the mean free path of the gas increases which leads to an increase in the volume of the gas.

The reverse is the situation on cooling of a gas. As the temperature of a sample of gas is lowered, the average kinetic energy decreases, thereby reducing the average speed. Thus, the particles move slowly and the frequency of collisions increases. Therefore, the mean free path and hence, the volume of the gas decreases.

Conclusion

Heating of a gas causes a volume increase due to the increase in average kinetic energy of the gas while cooling the gas leads to a decrease in the volume.