A \(25-\mathrm{L}\) sample of ideal gas with \(\gamma=1.67\) is at \(250 \mathrm{K}\) and \(50 \mathrm{kPa} .\) The gas is compressed adiabatically until its pressure triples, then cooled at constant volume back to \(250 \mathrm{K},\) and finally allowed to expand isothermally to its original state. (a) How much work is done on the gas? (b) What is the gas's minimum volume? (c) Sketch this cyclic process in a \(p V\) diagram.

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In its simplest form, it considers how energy transforms from one form to another and how that affects matter. It's governed by four fundamental laws. The first law, also known as the law of energy conservation, states that energy can neither be created nor destroyed, only transformed from one form to another.

When applying this to an ideal gas, we often look at processes that change the state of the gas, such as pressure, volume, and temperature adjustments. The exercise you're studying involves a three-stage process: adiabatic compression (increasing pressure with no heat exchange), isochoric cooling (reducing temperature at constant volume), and isothermal expansion (increasing volume at constant temperature). The work done on the gas during these stages reflects changes in internal energy and energy flow in or out of the system, concepts at the heart of thermodynamics.

In contrast to adiabatic processes, isothermal processes occur at a constant temperature. For an ideal gas, this means that any work done on or by the gas is accompanied by an equal amount of heat transferred to or from the gas to keep the temperature steady. From the equation \(W = nRT ln\frac{V1}{V2}\) we see that work is dependent on the natural logarithm of the volume change, which is crucial when calculating the total work done during this isothermal phase.

Why isothermal and not adiabatic expansion?

Recall that our initial motivation is to return the gas to its original state. By allowing the gas to expand isothermally, we ensure it reaches the initial temperature and pressure, thereby completing a thermodynamic cycle. This phase is vital in applications such as heat engines, where cyclic processes enable the conversion of heat into work.

A pV diagram, or pressure-volume diagram, is a graphical representation of the changes in pressure and volume within a thermodynamic system. It serves as a map of the system's journey through various states. The area under the curve in a pV diagram often corresponds to work done by or on the system.

Visualizing the Process

In your exercise, the pV diagram helps visualize the three-stage process that the gas undergoes: the steep rise represents adiabatic compression (with pressure increasing), the vertical drop correlates to the isochoric cooling (volume remains constant), and the curve back to the starting point represents the isothermal expansion (constant temperature, pressure decreases). The total area inside this closed loop offers a glimpse at the net work done over the entire cycle—integral for understanding energy transfer during thermodynamic processes.