Summary

The simple physical observations and succinct mathematical models set forth in this chapter provide powerful tools for current chemical applications and excellent examples of model development that we would all do well to emulate. This chapter has illustrated applications of physical reasoning, dimensional analysis, asymptotic analysis, and parameter estimation that have set the standard for many modern engineering developments.

Furthermore, the final connection has been drawn between the molecular level and the macroscopic scale. In retrospect, the microscopic definition of entropy is relatively simple. It follows naturally from the elementary statistics of the binomial distribution. The qualitative description of molecular interaction energy is also simple; it was discussed in the introductory chapter. Last, but not least, the macroscopic description of energy is easy to understand; it gives the macroscopic energy balance. What is not so simple is the connection of the qualitative description of molecular energies with the macroscopic energy balance. This is the significant development of this chapter. Having complete descriptions of the molecular and macroscopic energy and entropy, all the “pieces to the puzzle” are now in our hands. What remains is to put the pieces together. This final step requires a fair amount of mathematics, but it is largely a straightforward application of tools that are readily available from elementary courses in calculus and the background of Chapter 6.

Important Equations

Several equations stand out in this chapter because we apply them repeatedly going forward. These are Eqns. 7.2, 7.12, and 7.15–7.19. Eqn. 7.2 is the definition of acentric factor (ω), which provides a convenient standard vapor pressure (cf. back flap) and a crude characterization of the molecular shape. Eqn. 7.12 is the van der Waals equation of state, one of the greatest model equations of all time. Eqns. 7.15–7.19 describe the Peng-Robinson equation of state, a remarkably small evolution from the van der Waals model considering the 100 years of intervening research. We refer to these often as the basis for further derivations and applications to energy and entropy balances for chemicals other than steam and refrigerants.

Finally, Eqns. 7.51 and 7.52 convey the foundation for understanding the connection between the molecular scale and the macroscopic scale. There is no simpler way to see the connection between u and U than Eqn. 7.51. Understanding the molecular interactions becomes essential when we consider why some mixtures behave ideally while others do not. This becomes apparent when we extend Eqn. 7.51 to mixtures.