Thermodynamics and Propulsion

7.6 Summary and Conclusions

1. Entropy as defined from a microscopic point of view is a measure of randomness in a system.
2. The entropy is related to the probabilities $p_i$ of the individual quantum states of the system by
   $$S =-k\sum_i p_i\ln p_i,$$
   where $k$ , the Boltzmann constant, is given by $\mathbf{R}/ N_\textrm{Avogadro}$ .
3. For a system in which there are $\Omega$ quantum states, all of which are equally probable (for which the probability is $p_i =1/\Omega$ ), the entropy is given by
   $$S=k\ln\Omega.$$
   The more quantum states, the more the randomness and uncertainty that a system is in a particular quantum state.
4. From the statistical point of view there is a finite, but exceedingly small possibility that a system that is well mixed could suddenly ``unmix'' and that all the air molecules in the room could suddenly come to the front half of the room. The unlikelihood of this is well described by Denbigh [Principles of Chemical Equilibrium, 1981] in a discussion of the behavior of an isolated system:

   ``In the case of systems containing an appreciable number of atoms, it becomes increasingly improbable that we shall ever observe the system in a non-uniform condition. For example, it is calculated that the probability of a relative change of density, $\Delta \rho/\rho$ , of only $0.001\%$ in $1\textrm{ cm}^3$ of air is smaller than $10^{-10^{8}}$ and would not be observed in trillions of years. Thus, according to the statistical interpretation the discovery of an appreciable and spontaneous decrease in the entropy of an isolated system, if it is separated into two parts, is not impossible, but exceedingly improbable. We repeat, however, that it is an absolute impossibility to know when it will take place.''

5. The definition of entropy in the form $S =-k\sum_i p_i\ln p_i$ arises in other aerospace fields, notably that of information theory. In this context, the constant $k$ is taken as unity and the entropy becomes a dimensionless measure of the uncertainty represented by a particular message. There is no underlying physical connection with thermodynamic entropy, but the underlying uncertainty concepts are the same.
6. The presentation of entropy in this subject is focused on the connection to macroscopic variables and behavior. These involve the definition of entropy given in Chapter 5 of the notes and the physical link with lost work, neither of which makes any mention of molecular (microscopic) behavior. The approach in other sections of the notes is only connected to these macroscopic processes and does not rely at all upon the microscopic viewpoint. Exposure to the statistical definition of entropy, however, is helpful as another way not only to answer the question of ``What is entropy?'' but also to see the depth of this fundamental concept and the connection with other areas of technology.