A cylinder with a movable piston holds 2.75 mol of argon at...

A cylinder with a movable piston holds 2.75 mol of argon at a constant temperature of $295 \mathrm{K}$. As the gas is compressed isothermally, its pressure increases from 101 kPa to 121 kPa. Find $(a)$ the final volume of the gas, $(b)$ the work done by the gas, and $(c)$ the heat added to the gas.

Key Concepts

Isothermal Process

An isothermal process is one in which the temperature of the system remains constant throughout the process. For an ideal gas, this means that the internal energy remains constant since it depends only on temperature. As a result, any work done by or on the gas is exactly balanced by heat exchange with the surroundings, leading to no net change in internal energy.

Ideal Gas Law

The ideal gas law (PV = nRT) relates the pressure, volume, and temperature of an ideal gas. It is foundational for understanding gas behavior under different conditions. In situations like the one described, where the number of moles and temperature are constant, the product of pressure and volume also remains constant.

Work in Isothermal Processes

In an isothermal process for an ideal gas, the work done by the gas can be calculated by integrating the pressure-volume relationship, leading to the formula W = nRT ln(V_final/V_initial) or equivalently in terms of pressures. This expression captures the work done during compression or expansion under constant temperature, with the sign indicating whether the work is done by or on the gas.

Heat Transfer in Isothermal Processes

During an isothermal process, the change in internal energy of an ideal gas is zero, which implies that the heat added to the system (Q) is equal to the work done by or on the system. This concept is essential for energy conservation analysis in thermodynamic processes, ensuring that any energy entering or leaving the gas is accounted for by an equivalent amount of work.

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