A cylinder that has a piston contains 2.00 mol of an ideal gas and undergoes a reversible isoth

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A cylinder that has a piston contains 2.00 mol of an ideal...

A cylinder that has a piston contains $2.00 \mathrm{~mol}$ of an ideal gas and undergoes a reversible isothermal expansion at $400 \mathrm{~K}$ from an initial pressure of $12.0$ atm down to $3.00$ atm. Determine the amount of work done by the gas.

Key Concepts

Isothermal Process

An isothermal process is one in which the temperature of the system remains constant. For an ideal gas undergoing such a process, the internal energy does not change, meaning that any work done by the gas is exactly balanced by heat transferred to or from the gas. This concept is essential in linking pressure and volume changes without altering the system's temperature.

Reversible Process

A reversible process is carried out infinitely slowly such that the system remains in thermodynamic equilibrium with its surroundings at all times. This ensures maximum work output during expansion or minimum work input during compression, as there is no dissipative loss. It underpins the method for calculating thermodynamic work as an integral along a path of equilibrium states.

Ideal Gas Law

The ideal gas law, represented as PV = nRT, relates the pressure, volume, temperature, and number of moles of a gas. This equation is foundational for describing the behavior of gases under various conditions, including isothermal processes, by providing the necessary relationships to solve for unknown quantities such as volume or pressure changes during expansion.

Work in Thermodynamic Processes

In thermodynamics, work is defined as the energy transferred by the system when it expands or contracts against an external pressure. For a quasi-static process, this work is calculated as the integral of pressure with respect to volume (w = -?PdV). In an isothermal process for an ideal gas, the dependence of pressure on volume leads to a logarithmic expression for work, highlighting the direct relationship between the work done and the ratio of initial and final volumes (or pressures).

Integration and Logarithmic Functions in Work Calculations

When calculating the work done in an isothermal, reversible expansion of an ideal gas, integration of the form -?(nRT/V)dV yields a logarithmic function. This mathematical treatment results in the expression w = -nRT ln(V2/V1), encapsulating how changes in volume (or equivalently pressure under constant temperature) dictate the work performed in the process.

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