Molecular Physics > Equation of state of an ideal gas > Problem 5.5.18

Statement

A heavy piston is in equilibrium in a cylindrical gas vessel. The mass of gas and its temperature above and below the piston are the same. The ratio of the internal volume of the upper part of the vessel to the internal volume of the lower part is . What will this ratio be if the gas temperature is doubled?

Solution

Suppose the pressure of the gas in the upper part is before the temperature change. Then, the pressure of the gas in the lower part is as its volume is one third of the volume of the gas in upper part, while the mass and the temperature of the gas in both parts are the same. The equilibrium of the piston means that

where is the mass of the piston and is the area of the cross section of the vessel. Let the volume of the gas in the upper part be when the temperature is doubled. Then, its pressure becomes . Similarly, the volume and the pressure of the gas in the lower part become and , respectivly. Considering the equilibrium of the piston, we have

which simplifies to the quadratic equation . As , the ratio of the volumes of the gas in the upper and lower parts is

Answer