Imagine that the helium balloon from the previous problem...

Imagine that the helium balloon from the previous problem was placed in an evacuated container of volume $0.020 \mathrm{m}^{3}$ and that the balloon popped when it touched a sharp edge on the inside of the container. (a) How much work is done on the helium gas? (b) What happens to the temperature of the helium, its density, the gas pressure, the average kinetic energy of each particle, and the thermal energy of the helium gas? Provide quantitative answers.

Key Concepts

Free Expansion

Free expansion is a thermodynamic process in which a gas expands into a vacuum without performing work against an external pressure. In this process, since the external pressure is zero, no work is done by or on the gas. For an ideal gas, free expansion is characterized by no change in internal energy, meaning that the temperature and average kinetic energy of the gas particles remain constant.

Work in Thermodynamics

Work in thermodynamics is the energy transfer associated with a force acting over a displacement. In the context of a gas expanding into a vacuum, there is no opposition from any external pressure, which results in zero work done by the gas. This is an important concept that underpins the understanding of energy changes in various gas expansion processes.

Ideal Gas Behavior

The behavior of an ideal gas is explained by the ideal gas law, which relates pressure, volume, and temperature. For an ideal gas, the internal energy depends solely on temperature. During a free expansion, even as the gas's volume increases suddenly, the temperature remains unchanged, ensuring that the average kinetic energy of the gas particles remains constant. This property supports the conclusion that the thermal energy is also unchanged.

Density and Pressure Changes

When a gas expands into a larger volume, its density decreases because there are fewer gas molecules per unit volume. According to the ideal gas law, if the number of particles and the temperature remain constant, an increase in volume will result in a decrease in pressure. This demonstrates how the macroscopic properties of a gas, such as density and pressure, are affected by changes in volume during processes like free expansion.

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