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15  Adiabatic Processes

15.1  Multiple Definitions of “Adiabatic”

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The word adiabatic is another term that suffers from multiple inconsistent meanings. The situation is summarized in figure 15.1.

adiabatic
Figure 15.1: Multiple Definitions of Adiabatic
  1. Some thoughtful expertns use “adiabatic” to denote a process where no entropy is transferred across the boundary of the region of interest. This was probably the original meaning, according to several lines of evidence, including the Greek etymology: α + δια + βατoς = not passing across. As a corollary, we conclude the entropy of the region does not decrease.
  2. Other thoughtful experts refer to the adiabatic approximation (in contrast to the sudden approximation) to describe a perturbation carried out sufficiently gently that each initial state can be identified with a corresponding final state, and the occupation number of each state is preserved during the process. As a corollary, we conclude that the entropy of the region does not change.
  3. Dictionaries and textbooks commonly define “adiabatic” to mean no flow of entropy across the boundary and no creation of entropy.

In the dream-world where only reversible processes need be considered, definitions (1) and (2) are equivalent, but that’s not much help to us in the real world.

Also note that when discussing energy, the corresponding ambiguity cannot arise. Energy can never be created or destroyed, so if there is no transfer across the boundary, there is no change.

As an example where the first definition (no flow) applies, but the second definition (occupation numbers preserved) does not, see reference 44. It speaks of an irreversible adiabatic process, which makes sense in context, but is clearly inconsistent with the second meaning. This is represented by point (1) in the figure.

As an example where the second definition applies but the first definition does not, consider the refrigeration technique known as adiabatic demagnetization. The demagnetization is carried out gently, so that the notion of corresponding states applies to it. If the system were isolated, this would cause the temperature of the spin system to decrease. The interesting thing is that people still call it adiabatic demagnetization even when the spin system is not isolated. Specifically, consider the subcase where there is a steady flow of heat inward across the boundary of the system, balanced by a steady demagnetization, so as to maintain constant temperature. Lots of entropy is flowing across the boundary, violating the first definition, but it is still called adiabatic demagnetization in accordance with the second definition. This subcase is represented by point (2) in the diagram.

As an example where the second definition applies, and we choose not to violate the first definition, consider the NMR technique known as “adiabatic fast passage”. The word “adiabatic” tells us the process is slow enough that there will be corresponding states and occupation numbers will be preserved. Evidently in this context the notion of no entropy flow across the boundary is not implied by the word “adiabatic”, so the word “fast” is adjoined, telling us that the process is sufficiently fast that not much entropy does cross the boundary. To repeat: adiabatic fast passage involves both ideas: it must be both “fast enough” and “slow enough”. This is represented by point (3) in the diagram.

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

15.2  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 both senses of the word. The process is diagrammed in figure 15.2.

isentropic-expansion-prob
Figure 15.2: 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 15.3. 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.

isothermal-expansion-prob
Figure 15.3: 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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