One mole of an ideal gas is contained under a light...

One mole of an ideal gas is contained under a light frictionless piston of a vertical cylinder at a temperature $T$. The space above the piston opens into the atmosphere. Calculate the work done in increasing the volume of the gas to $n$ times the initial value by isothermally raising the piston in a very slow and controlled manner.

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Key Concepts

Integration of the Pressure-Volume Relation

This concept involves setting up and solving the integral of pressure with respect to volume. For an ideal gas undergoing an isothermal process, the pressure can be expressed as nRT/V, leading to an integral that yields a natural logarithm function when evaluated over the limits of the initial and final volumes.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, provides a relationship between pressure, volume, and temperature for an ideal gas. This principle is key in determining how the pressure changes as the volume changes when the temperature remains constant in an isothermal process.

Isothermal Process

An isothermal process is a thermodynamic process in which the temperature remains constant throughout. In such processes involving an ideal gas, the internal energy stays constant, and any work done by the gas is exactly balanced by heat exchange with the surroundings.

Work in Thermodynamic Processes

In thermodynamics, the work done by a system during expansion or compression is calculated by integrating the pressure with respect to the volume change. Specifically for an isothermal process, substituting the ideal gas law into the integral enables the derivation of work as nRT times the natural logarithm of the volume change ratio.

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