A 1.00 -mol sample of H2 gas is contained in the left side...

A 1.00 -mol sample of $\mathrm{H}_{2}$ gas is contained in the left side of the container shown in Figure $P 18.41,$ which has equal volumes on the left and right. The right side is evacuated. When the valve is opened, the gas streams into the right side. (a) What is the entropy change of the gas? (b) Does the temperature of the gas change? Assume the container is so large that the hydrogen behaves as an ideal gas.

A 1.00 -mol sample of $\mathrm{H}_{2}$ gas is contained in the left side
of the container shown in Figure $P 18.41,$ which has equal volumes on the left and right. The right side is evacuated. When the valve is opened, the gas streams into the right side. (a) What is the entropy change of the gas? (b) Does the temperature of the gas change? Assume the container is so large that the hydrogen behaves as an ideal gas.

Key Concepts

Ideal Gas Behavior

Ideal gas behavior is characterized by the ideal gas law, PV = nRT, which relates a gas's pressure, volume, and temperature. This law is particularly applicable when intermolecular forces and the volume occupied by the gas molecules themselves can be neglected, making the behavior predictable during processes such as expansion into a vacuum.

Free Expansion

Free expansion, also known as Joule expansion, is a process where a gas expands into an evacuated space. In this process, there is no external work done by the gas and no heat exchange with the surroundings. This results in the process being inherently irreversible, although for an ideal gas the internal energy remains unchanged.

Entropy Change in Isothermal Expansion

For an ideal gas undergoing an isothermal expansion, the change in entropy is given by ?S = nR ln(V_final/V_initial). This formula quantifies the increase in disorder as the gas molecules spread out to occupy a larger volume. Even though the free expansion is irreversible, if the temperature remains constant, the entropy change can be evaluated using this relationship.

Isothermal Process

An isothermal process involves a change in the volume and pressure of a system while the temperature remains constant. For an ideal gas undergoing free expansion in a sufficiently large container, even though the expansion is irreversible, the temperature does not change because the internal energy of an ideal gas is solely a function of temperature.

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