Thermal Work (truly) Explained

(The following requires high school physics and a general idea of what thermal work is)

In an age when Maxwell’s equations were just a mathematical reckoning of laboratory experiments, thermodynamics powered trains that sped across continents.

But thermal power always comes at a cost--energy. Processes that are mechanically desirable are thermally inefficient, especially spontaneous ones. This is well-known and easy to understand—yet sufficient explanations are rarely offered.

In thermodynamics, achieving maxim work from thermodynamic state A to B is always path dependent: It doesn’t just depend on its initial and final states, but the path that it took from point A to B. A reversible transformation is an energy-exchanging process where the optimal amount of work is transferred to or from the system between two thermodynamic states--or values of temperature, pressure, and volume (all reversible processes are quasistatic, but a quasistatic process could still be irreversible due to intermolecular forces, friction against the walls of the piston, or non-quasistatic heat transfer). They’re called reversible because they can be returned to the same initial state without changing the heat content and entropy of the surroundings. Though reversible processes are impossible, thermal work couldn’t be calculated without using reversible processes as the baseline for calculations in the real-World. This article addresses the differences between reversible and irreversible processes, and includes two explanations for why non-quasistatic and spontaneous processes perform less work.

Irreversible processes are path-independent and occur much faster. During an expansion or compression, pressure usually differs throughout the gas, and even if the system’s average pressure could be calculated, it's value will always be less desirable than its reversible counterpart (between the same temperature and pressure), making real-life process mathematically unmanageable and energetically inefficient. Fortunately, the unambiguous calculations of reversible processes can still be used with simple modifications.

Since thermodynamic analyses are always performed from the viewpoint of the gas, it’s easy to forget that the relevant work is the average external force the gas pushes over a specified distance (e.g., a shaft connected to a piston); the integral of pdv of the gas is simply a means to that end. The higher the average external force over the same distance, the greater the work done. When an expansion occurs quickly, there’s necessarily a large average pressure differential between the gas and the piston, and, thus, force. However, in the quasistatic case, you can, and must, keep the differential arbitrarily close to zero. This allows you to maximize the average force moved by the gas. In the case of a fast compression, an engineer would want to minimize the work the piston (usually connected to a shaft) has to do to the gas. In order for the process to transpire quickly, the piston has to maintain a higher pressure differential between it and the gas, but the faster compression raises the pressure of the gas more, requiring greater and greater amounts of force over the same distance.   

The following thought experiments will further illustrate the importance of path.

Imagine a cylinder-piston system filled with gas. If the gas exchanges work with its surroundings, the collective bombardment of the gas’s molecules will overcome the piston’s inertia and push it forward or upwards. Since the identical piston and cylinder will be used for the upcoming reversible and irreversible processes and have the same area, you can think of pressure and force as one in the same. As before, the total change in volume—and, therefore, distance--will be the same, as well. If the gas performed work on the same vertically-oriented piston with varying weights of sand atop it in two separate processes--one reversible, one irreversible--the average sand mass would determine the difference in work transfer.*

   Note that because irreversible processes are path independent, any number of irreversible processes between A and B could reverse direction during the process while still performing the same net work as ours.

The expansion of a gas can be made quasistatic by the gradual removal of the sand grains, ensuring that the gas’s force is always just a hair greater than the weight above it. The opposite is the case in the irreversible. Since in the reversible process the pressure of the gas is always equal to the combined pressure of the piston + sand and the distance they move is equal to the change in length of the cylindrical space occupied by the gas, the thermodynamicist can measure the work transferred to the surroundings based on system variables alone, without requiring information about the surroundings.

From a molecular point of view, the force on the piston will depend on the collective strength of collisions. What the thermodynamicist is interested in is not momentum, but average effective momentum: mass (of the colliding molecules) multiplied by the average closing speed of the molecules and the piston (meaning how quickly they move toward each other). Closing speed, in turn, determines both the strength of individual collisions, and the number of collisions per unit time.

In a case where the gas expands and performs work on its surroundings, the slower the piston moves, the harder the collisions (due to greater average effective momentum) and the more of them per unit time. When the piston moves quickly, the molecules will be hitting something moving away from them, reducing closing speed—weaker collisions, less of them, and, therefore, less pressure. The processes can end at the same state because the non-quasistatic, and irreversible, process will exchange more or less heat with its surroundings. Lost work, or difference in work between the reversible process and actual process, will manifest itself in the form of lost heat; that is, additional heat in the surroundings, either due to more heat being dumped or less being extracted. 

 
It should be easy for the reader to deduce that a fast compression of a gas to a given thermodynamic state will require more work to be transferred to the gas--more collisions per unit time, greater momentum transfer per collision, greater resistance. This is undesirable from an engineering standpoint because gas power cycles require compression, and compression is negative work, meaning it reduces the total energy output of a power cycle. 

As mentioned, because quasistatic processes have virtually no power output, they aren’t mechanically desirable, but they are thermodynamically: All else being equal, the gas does more work and wastes less heat.

In the quasistatic case, the process could be reversed at any time: Because the gas pressure and the applied pressure are very nearly equal, the placement or removal of a tiny weight of sand can seamlessly reverse the direction of the piston’s (quasi-static) motion, and without leaving a measurable change in the surroundings’ supply of heat energy when the process is complete. This is important in engine cycles where gases are always returned to their initial states; any change in the surroundings’ quantity of heat equals the losses in work output due to irreversibilites. 

This thought experiment has a second profound consequence: It shows that it’s virtually impossible for a spontaneous thermodynamic process to perform anywhere near maximal work. The likelihood that the pressure differentials between gas and piston (or some equivalent) would remain close to zero is next to zero. Even in the case where the initial pressures and temperatures happened to be close, the process tends to alter the temperature of the gas (which isn’t contained in a perfect insulator), resulting in heat transfer (or additional heat transfer) to or from the surroundings and the gas itself; this, in turn, tends to alter pressure, which, in turn, affects temperature again, followed by yet another effect on pressure--and so on. Moreover, a real gas and cylinder system are never truly ideal: The latter will always have some intermolecular interactions, and the cylinder’s walls won’t be frictionless and will always permit heat exchange between the system and the surroundings.

 

While understanding quasistatic processes has little direct effect on calculations in rudimentary thermodynamics--given that calculations are based on the initial and final states of the system with necessary modifications--the above builds intuition for the dynamics of gases and pistons. Those who specialize in applied thermal-fluids physics, which spans multiple fields, don’t end their training and research with concepts as basic as those above. Ultimately, these thought experiments prime a student’s instincts for the more advanced subfields of gas dynamics, heat transfer/transport processes, statistical physics, chemical kinetics, nonequilibrium thermodynamics, and even plasma physics.