Spontaneity, Reversibility, Equilibrium

Last week, right after my P-Chem II class, a bright student came up to ask me to help clear up and issue comparing reversible processes that actually move a process forward (you might expect a change in free energy or overall entropy) and equilibrium (you expect delta-G to be zero). Since we only have ten minutes in between when one class gets out and the next one comes in to the room, I did what many professors do – I gave a handwaving explanation. I said something about an overall system being at equilibrium macroscopically (equal rates of forward and reverse reactions), and separated that from a “reversible” process where you’re moving something along via infinitesimal steps. One’s the overall “system”, the other is looking at a series of steps for a specific process, I said. Clearly, or maybe obtusely, I was hedging.

 

Part of the issue here is that in real life you’d never run a process in infinitesimal steps because it would take an infinite amount of time. Essentially nothing is changing if each step takes eternity. I did tell students that, in practice, no process is truly reversible in this infinitesimal-step sense if in fact it actually takes place. Students understand this issue of practically. But since P-Chem II is calculus-infused, we can do the math by taking limits. In particular we go to the limit of the infinitesimal step and use d’s instead of deltas. This leads to beautifully simple calculations to calculate the mechanical work or heat transfer in idealized “reversible” cases, while comparing them to “irreversible” cases that are “less efficient”. In the reversible case, work in equals work out if you did this ideally. In the irreversible case, you get less work out compared to what you put in. Most students are satisfied by all of this, but for this student, there was a bee in her bonnet, and rightly so.

 

Actually, I don’t know if she found my hedging answer satisfactory. I should ask her. I was unimpressed by my own answer even though it was practical in the interest of time (she had to run to another class). In fact, I told her that I needed to be more careful in how I defined the terms “equilibrium” and “reversible process” and that sometimes I’m not careful enough. I provide the students with the formal definitions in our lecture notes, but I slide my way from one term to another in the “heat of the moment” when I’m trying to be dynamic and lively in class. (“Heat” is another of those tricky terms.) In class I only look sporadically at my own notes. Sometimes I remember to emphasize something to watch out for, and other times I simply forget.

 

In my quest to figure out how to not confuse future students, I started scouring the primary literature for inspiration. The article I’ve found most useful thus far is by John Norton titled “The impossible process: Thermodynamic reversibility” (Studies in History and Philosophy of Modern Physics 2016, 55, 43-61). It provides a historical slant and includes many examples where much more famous physicists have also elided their way through. Now I don’t feel so bad about my quick handwave. I think I can tighten up my definitions a little better. And I need to spend a little time talking about the issue of using calculus to take the limit for infinitesimal steps. Many chemistry students are a little calculus-phobic so I try not to emphasize the mechanics of calculus and instead concentrate on the chemistry. But I’m reminded that I need to be more careful in this regard. Norton also points out that on the molecular scale, thermal fluctuations make this ideal-calculus-limit taking a problem. In a big-picture Mack view, all this might be okay, but tiny Mike would protest that there’s a problem! (Mack and Mike represent macroscopic and microscopic views.)

 

I also need to be very clear when I use each of these terms: reversible, equilibrium, and spontaneous. I make a very big deal (multiple times) in both my G-Chem and P-Chem classes that thermodynamic spontaneity has nothing to do with how fast a reaction might take place. All it tells you is which way the reaction is likely to proceed absent any external intrusions on the system. Thankfully, we’re on the verge of changing our G-Chem textbook to, in my opinion, a superior one that excises the confusing term “spontaneity” and instead uses “thermodynamic favorability”. I’m all in favor of that change. I’ll just have to remind students to be careful when they encounter “spontaneity” on the internet because, sadly, that’s where many of them go to look up things rather than their textbook.

 

I think I should stop using the term “reversible” in G-Chem as a thermodynamic definition. I should limit myself to discussing forward and reverse reactions (in the kinetic sense), that both occur, and that if one waits long enough eventually the rates of the forward and reverse reactions are equal. That’s when dynamic equilibrium is reached. If the change in system free energy or the change in the entropy of the thermodynamic universe is zero, then the system is overall at equilibrium. That’s it. No need to belabor the point.

 

In P-Chem I’m considering using the term quasi-reversible to emphasize that taking the infinitesimal limit is actually an impossible situation at the molecular level. Perhaps I should always say quasi-reversible process. This may help emphasize the distinction between the macroscopic system as a whole (which may or may not be at equilibrium) and considering a specific process in getting from one state to another state. I’m not sure I want to go into the language of “a series of connected equilibrium states” since this muddies the waters. Since I don’t use a textbook in P-Chem, I can just change all the notes that I provide students to tighten up these definitions. I will restrict using the word equilibrium to the usage I mentioned above in G-Chem. When I get to the stat mech version of discussing equilibrium, I will focus it on the equilibrium constant as a ratio of the number of product molecules versus reactant molecules, while reminding the students that the state of being at equilibrium is a macroscopic description. There will be a tricky part when I get to transition-state theory in thinking about the transition state as a quasi-equilibrium state; not sure how to handle that terminology-wise. We’ll see how this all works out the next time I teach P-Chem II.

The Optimality of Forgetting

In the education business, we’re often emphasize the business of remembering. Remembering what you learned is good. Forgetting what you learned is bad. Students may wish they had better memories to remember all the stuff I’m telling them. Heck, I often wish for better memory as I age and forgetfulness increases in frequency. So why do we forget when improved remembering seems like what we want? If remembering was so adaptively so much better than forgetting, evolution should have selected for the best memorizers!

 

What has our memory evolved for? And why might forgetting be just important as remembering? One possibility is that in a noisy and ever-changing environment, having specific detailed memories that persist make it difficult to learn new things and adapt appropriately to analogous yet different situations. I didn’t come up with this myself. I just spent the last hour reading a perspective article: “The Persistence and Transience of Memory” by Richards and Frankland (Neuron 2017, 94, 1071-1084). Parts of the article were slow-going because I lack the background related to the experimental work being reviewed, but I think I got the gist of it. And that’s the point! Getting the gist may be what matters adaptively.

 

The authors argue that the interplay between persistence (remembering) and transience (forgetting or erasing memories) is key. In particular, transience “enhances flexibility, by reducing the influence of outdated information on memory-guided decision making, and prevents overfitting to specific past events, thereby promoting generalization.” There are supporting experiments in rats and fruit flies for this hypothesis. Neural network models also suggest a congruence with the experiments: Injecting “noise” into the network, reducing weighting factors, encoding sparsely rather than densely, seem to improve the network’s ability to handle generalized situations.

 

When teaching physical chemistry (and to a lesser extent in general chemistry), I try to emphasize the models underlying the equations we used. The simpler the model, the simpler the equation and the more generalizable it is: the ideal gas law equation (PV = nRT) is an example of a very powerful equation that works for any gas, as long as it behaves close to ideally. The model of an ideal gas imagines a large number of particles moving randomly in a box with plenty of empty space with all collisions being elastic. That’s a good approximation for N₂, O₂, CO₂ and Ar which constitute over 99% in dry air. We can elaborate the model further for “real” gases through the two-parameter van der Waals equation or a multi-parameter virial equation. A mathematical model is powerful because its quantitative aspect allows it to make predictions of future situations to be encountered.

 

But putting in too many parameters can result in over-fitting, which can then result in incorrect predictions. So if we go through life encoding every moment in dense detail, it might actually hamper our ability to see the forest from the trees and adapt to new situations. Everything is a detail and the big picture is lost. The article’s introduction mentions the oft-quoted story of a patient with seemingly photographic memory of his entire life, but had plenty of problems navigating life because of this. I’m also reminded of how we learn when encountering something new. If you’re a novice, you try to absorb as much as you can but you have no idea which “details” are important and which are not. But if you already have some background, you’re able to ignore the artifacts and focus on abstracting the most crucial features. How exactly that happens, I don’t know. But I see it every day in my teaching. I constantly have to remind myself that I have the curse of knowledge in that I can’t quite remember or fathom how hard it was for me to build my chemistry scaffold oh so many years ago.

 

We humans haven’t had enough time to evolve towards learning academic subjects. Or even the seemingly simple acts of reading, writing and arithmetic. I don’t remember how I learned to read. I improved my writing through sheer practice and repetition. I have a vague “memory” that algebra was completely obtuse when I first encountered it; but I had an aha(!) moment at some point in life and somehow grasped it in a gestalt experience. Now algebra is obvious to me, at sometimes I’m at a loss helping students work a chemistry problem and realize they don’t get algebra. (This is a very small number, but I’ve noticed a few more post-pandemic.) Learning is still mysterious to me.

 

What can I do to help students learn chemistry? In class and through homework and practice, I try to emphasize the things students need to remember. I repeat the salient points a lot such that I sound like a broken record, but I think it’s crucial to keep the students attending to the main thing. The first time I say something the strongest students may grasp the salience but the majority of the class hasn’t yet. So I need to keep repeating and emphasizing the most general principles. But I have to do this in the context of multiple examples that look different from each other. Same principle, different example. This is the key to “transfer”, the ability to effectively apply something you’ve learned in a different situation; and this includes knowing the limits of applicability!

 

I also add a lot of tidbits (history, broader applications, interdisciplinary connections) to my lectures. I hope that the students find them interesting, possibly strengthening a neural connection; but even if students forget these, that’s okay. For the things I need them to remember and use, there’s no substitute for repetition to strengthen the memory (both conceptual and procedural). If the students don’t practice retrieving these memories and using them, they will forget. It’s not a bad thing. Transience and persistence go together and I wouldn’t want my students to be maladaptive to new situations. So I’m not looking for them to have better memories (even though they might wish for it), but I’m trying to strengthen the neural connections they do have and maybe even replace some incorrect misconceptions they might have. Forgetting has its place in learning!