(a) With the aid of a labelled diagram, describe an experiment to illustrate the relationship between the volume and the temperature of a given mass of air ...

(a) Experiment

A uniform capillary tube is sealed at one end. A short pellet of mercury traps a fixed mass of dry air between the sealed end and the mercury pellet. The tube is held horizontally in a stirred water bath beside a half-metre rule, with a thermometer in the bath. The mercury pellet is free to move, and the pressure on its outer face remains constant.

At a steady temperature \(\theta\), read the temperature from the thermometer and the length \(L\) of the trapped air column from the rule. Heat the water bath slowly, stirring continuously, and repeat the readings at several steady temperatures. Since the capillary tube has a uniform bore, \(V\propto L\).

A plot of \(L\) against \(\theta\) is a straight line. When produced backwards, it cuts the temperature axis at about \(-274\ ^\circ\mathrm{C}\). Thus, for a fixed mass of air at constant pressure, its volume increases uniformly with temperature.

Precautions: Use dry air; stir the water to maintain a uniform temperature; take readings only when the temperature is steady; and avoid parallax when reading the thermometer and rule.

(b) Since the tube is uniform, \(V\propto L\). Therefore, the cubical expansivity is

\[\gamma=\frac{V_{100}-V_0}{V_0(100-0)}=\frac{L_{100}-L_0}{L_0\times100}\]

\[\gamma=\frac{18.7-13.7}{13.7\times100}=\frac{5.0}{1370}=3.65\times10^{-3}\ \mathrm{K^{-1}}.\]

(c) Increasing the temperature increases the average kinetic energy and speed of the gas molecules. The molecules would then strike the walls more forcefully and tend to increase the pressure. At constant pressure, the gas expands so that the molecules travel greater distances between collisions with the walls. Hence the volume of the fixed mass of gas increases.