A cylinder fitted with a piston contains 5.00 L of a gas at a pressure of 4.00 atm . The entir

step 1 To calculate the work that is done by the gas as it expands at a pressure of one atmosphere, sta

A cylinder fitted with a piston contains 5.00 L of a gas at...

A cylinder fitted with a piston contains $5.00 \mathrm{~L}$ of a gas at a pressure of $4.00 \mathrm{~atm} .$ The entire apparatus is contained in a water bath to maintain a constant temperature of $25^{\circ} \mathrm{C}$. The piston is released and the gas expands until the pressure inside the cylinder equals the atmospheric pressure outside, which is 1 atm. Assume ideal gas behavior and calculate the amount of work done by the gas as it expands at constant temperature.

Key Concepts

Isothermal Process

An isothermal process is a thermodynamic process in which the temperature remains constant throughout. For an ideal gas undergoing an isothermal process, the internal energy remains unchanged, so the work done by or on the gas is directly related to the heat exchanged with the surroundings. This concept is fundamental when examining expansions or compressions at a constant temperature, as it simplifies the analysis by directly linking pressure and volume changes.

Work in Thermodynamic Processes

In thermodynamics, work is defined as the energy transfer resulting from a force acting through a distance. For gases, work is often calculated by integrating pressure over changes in volume (?P dV). In processes such as isothermal expansion or compression, this integration takes into account the continuously varying pressure, typically yielding a logarithmic relationship between the initial and final volumes.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure, volume, temperature, and amount of an ideal gas. It serves as a cornerstone in understanding gas behavior under various conditions. In problems involving gas expansion or compression, this law is used to relate changes in volume and pressure while keeping the temperature constant or changing in a controlled way.

Logarithmic Integration for Isothermal Work

For an ideal gas undergoing an isothermal process, substituting the ideal gas law into the work integral leads to an expression that involves the natural logarithm of the ratio of the final and initial volumes. This logarithmic relationship encapsulates how the work done by the gas depends on the extent of the expansion or compression, reflecting the varying pressure during the process.

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