Work, Heat Capacity and Reversibility

In Lecture 6 we fleshed out the ideas of work and heat a little more, showing how to take an infinitesimal quantity like dw and turn it into a macroscopic, measurable value. In the process, we distinguished between external and internal pressure, developed the heat capacity parameter and reminded ourselves that both heat and work are path functions.

Then we calculated general expressions for the work performed in two processes: free expansion into a vacuum and compression/expansion against constant external pressure, both irreversible processes. After defining reversibility, we addressed a third process: reversible, isothermal compression/expansion of a gas.

Quiz 1 on Monday. Any questions?

Mathematics, Reality and Energy

The fact that mathematics works so well to depict the phenomena of thermodynamics, and other physical behavior, is -- to me at least -- nothing short of remarkable. This correlation between math and reality is so familiar, so apparently self-evident, that we take it for granted, not stopping to think whether or not it is even valid. Of course the pragmatist might argue that it obviously works well enough, otherwise we wouldn't have been able to do things like land people on the moon, program computers or construct MRIs. But a fundamental part of pchem is examining the inherent assumptions of a given model and analyzing their validity. We will not likely spend any more time on this concept, unless it slips out of my philosophically-knotted brain and into lecture. I do think anyone majoring in one of the physical sciences should be exposed to this question at least once -- hopefully I've yanked the carpet out from under your brain just a little. I will end this thread with some quotes:

The study of physics has driven us to the positivist conception of physics. We can never understand what events are, but must limit ourselves to describing the pattern of events in mathematical terms: no other aim is possible .... the final harvest will always be a sheaf of mathematical formulae. (Sir James Jeans)

How can it be that mathematics, a product of human thought independent of experience, is so admirably adapted to the objects of reality? (Albert Einstein)

Mathematics has the completely false reputation of yielding infallible conclusions. Its infallibility is nothing but identity. Two times two is not four, but it is just two times two, and that is what we call four for short. But four is nothing new at all. And thus it goes on and on in its conclusions, except that in the higher formulas the identity fades out of sight. (Johann Wolfgang Von Goethe)

Lecture Five found us [re]examining the concept of energy, momentarily reflecting on the fact that it is, at heart, a defined rather than measured quantity. So the principle of the Conservation of Energy is, simply, the statement that humanity has stumbled onto some number that happens to never change for the universe. This is a direct consequence that the laws of physics do not appear to change with time.

This conservation law leads directly to the development of the First Law: dU = dq + dw. We reminded ourselves of the differences between state and path functions -- a distinction which will be key to later developments. Further, the idea of work and generalized force was developed, enumerating six or seven examples to be used in subsequent lectures.

Compression Factors and Corresponding States

In Lecture 4 we introduced Z, the compression factor, a useful parameter that relates the molar volume of a real gas to that expected for ideal behavior. Regions where Z are greater or less than one represent the relative strengths of attractive (intermolecular) forces or repulsive (volume-based) forces. After discussing three general trends of Z vs P plots, we defined the Boyle temperature as that temperature at which the initial slope is equal to zero.

In order to connect this concept to stuff we already know, we saw how to relate a particular equation of state to this Boyle temperature. After manipulating the equation of state so that it fits into the Z=PV/nRT construct, we take the derivative with respect to P, take the limit as P approaches 0 and then see what we get. This procedure was straightforward with the virial equation (expanded in terms of P) but the van der Waals equation (and others) is a little bit more work. Still, a procedure was developed that connects these real gas empirical parameters to measurable and important quantities like the Boyle temperature.

Lastly we introduced van der Waals' Principle of Corresponding States, the hypothesis that a single equation could describe fluid behavior in which material-dependent parameters like a, b and B were not used, instead being expressed in terms of reduced thermodynamic parameters. Of considerable historical importance, this analysis was used to help guide early scientists into cryogenic work and represents a universal equation for substances that do not possess high directionality (e.g. polarity).

Question 09 in hw.1 has a typo: Instead of alpha and beta (the symbols from our previous textbook) that should read beta and kappa.

We will spend perhaps 15 minutes finishing up Chapter 7 on Wednesday [delaying the discussion of fugacity until later] before hopping back to Chapter 2 and Thermodynamics proper.

Only 9 weeks (+10 +10) of pchem left!

So far in pchem: We finally met each other and went through the syllabus (including thursday exams and face the query). We discussed systems [open, closed, isolated], variables [intensive and extensive], equilibrium, SI units and equations of state.

The second lecture focused on the four assumptions of the ideal gas law. This led us into the van der Waals equation, which attempts to correct for nonzero volume and intermolecular attractive forces, briefly introduced in Chapter 1.5 and further developed in Chapter 7.

Isotherms and partial derivatives were the main topics in Lecture 3, particularly in correlating real gas equations of state (which are largely empirical) to observable parameters like the critical point. Figure 7.2 is a clearer example of real gas isotherms than what I chickenscratched on the board. It was pointed out to me after class that I was missing a 2 in that derivation so be sure to work through it yourself (no peeking at Example Problem 7.1).

To get a better feel for critical points, visit Table 7.2 in Appendix A for values, Wikipedia for a general background, while Table 7.4 across the page has a bunch of real gas parameters. If you want more information on partial derivatives you can either visit Appendix B.6 or, again, Wikipedia.

If you are feeling rusty in your maths, I highly recommend this book by Barrante. It basically distills down an 800-page calculus book to a paperback full of parts useful in pchem.

For this coming week we will examine the compression factor and the law of corresponding states then dive right into thermodynamics proper. To placate you, I have posted solutions to hw.zero and have updated the schedule with quiz dates.