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16  Adiabatic Processes

16.1  Various Definitions of “Adiabatic”

The word adiabatic is another term that suffers from multiple inconsistent meanings. The situation is summarized in figure 16.1. The underlined terms capture some of the possible meanings of adiabatic.

Figure 16.1: Some Concepts Related to Adiabatic
a) Some thoughtful experts use the term adiabatic to denote a process where no entropy is transferred across the boundary of the region of interest. In other words, the system is thermally insulated. This was probably the original meaning, according to several lines of evidence, including the Greek etymology: α + δια + βατoς = not passing across. This implies that the process must be fast enough so that the inevitable small heat leaks are insignificant. As a corollary, we conclude the entropy of the region does not decrease. This is represented by regions 2, 3, 6, and 7 in the diagram.
b) Other thoughtful experts use the term adiabatic approximation (in contrast to the sudden approximation) to describe a transformation carried out sufficiently gently that each initial state can be identified with a corresponding final state, and the probabilities (aka occupation numbers) are not changed – at least not changed by the transformation process itself.
c) Dictionaries and textbooks commonly define “adiabatic” to mean no flow of entropy across the boundary and no creation of entropy. This is represented by regions 6 and 7 in the diagram. As a corollary, this implies that the system is isentropic. That is to say, its entropy is not changing.

Here are some illustrations of the relationships between the various definitions:

My recommendation is to avoid using the term adiabatic whenever possible. Some constructive suggestions include:

16.2  Adiabatic versus Isothermal Expansion

Suppose we have some gas in a cylinder with a piston, and we gradually move the piston so that the gas expands to twice its original volume.

Further suppose that we do this fast enough that there is no thermal energy transport through the walls of the cylinder ... yet slow enough that there is a 1-to-1 correspondence between the states before and after. So this is adiabatic in every sense of the word. The process is diagrammed in figure 16.3.

Figure 16.3: Isentropic Expansion

We consider 20 states, as shown by the dots in the diagram. These are, for practical purposes, the only accessible states. That is to say, all the higher-energy states are unoccupied, to a good approximation. In the diagram, there are three panels. The left panel shows the situation before expansion, and the right panel shows the situation after. The middle panel is a “energy level diagram” that shows how the energy of each mode changes during the expansion.

You can see that within each pair of corresponding dots, the probability is the same before and after. Therefore the entropy of the gas is exactly the same. The energy of the gas has gone down, and the temperature has gone down in equal measure. The slope of the red line is an indication of temperature, in accordance with the Boltzmann factor.

We can contrast this with the isothermal expansion shown in figure 16.4. The gas in this case is in contact with a heat bath, so that during the expansion the temperature does not decrease. The energy of each mode goes down as before, but the occupation numbers do not stay the same. The lion’s share of the probability is now distributed over twice as many states. Therefore the entropy of the gas goes up. Within each pair of corresponding states, each state is very nearly half as likely after, as compared to before.

Figure 16.4: Isothermal Expansion

The red line is shifted to the right by one unit, reflecting the 2× lower probability of each state, but it keeps the same slope, representing the constant temperature.

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