The Second Law of Thermodynamics

The Second Law clicked today. It took two hours of work at a chalk board along with conversations with a professor (who happens to be very generous with his time), but it clicked in my head, and the interpretation that helped it click was the statistical formulation of the Second Law. So, for me, the most confusing part of the second law is NOT how esoteric it is -- it's far from esoteric. It makes perfect sense and matches up with what we observe. To me, describing the Second Law
as "In spontaneous processes the entropy of the universe tends to increase, where entropy is the measure of disorder" is the confusing part. This statement makes sense, but only if you're familiar with the jargon. And even then, I was still left with wondering "So... why is this, again...?" While you can always ask why (and one ought to), the statistical interpretation satiates the confusing "Why?" for the "Hm, I wonder Why?" kind of why -- bridging the gap from frustrating unfamiliarity to curiosity.


But stating the statistical formulation takes a lot more room. I'll still take a go at it, however.

Suppose you have a chunk of energy. You split that energy into 10 equal parts to observe how it behaves, and you have two metal blocks that can absorb that energy. Placing all 10 equal parts into one of the metal blocks (We'll say so that the energy heats up the block, since I am referring to thermodynamics here) and sitting it next to the other metal block, you sit and wait to see what happens. The heat from the first block should heat up the second block until they're about the same temperature. For our purposes, this is no different than when you let your soup cool off to room temperature, or your ice melts in a glass of water, or when you cuddle up with someone when you feel cold. Eventually heat will be transferred until you reach the same temperature. At this point, heat transfer seems to stop. Ice does not later boil, the soup does not freeze, and you and your partner remain at about the same temperature (though there are some extra complications involved with cuddling, since human bodies produce their own heat, but for rough analogy and everyday experience, it works). Something stops the transfer of heat from continuing in the same direction that is initially observed. Something also stops the transfer of heat from going back to where it used to be (Hot soup, ice cubes, you stay cold). This "Something" is the Second Law of Thermodynamics. From the 10 pieces of energy analogy above:

You have two blocks of metal. However, those blocks of metal have places to store this energy -- atoms. Everything has atoms that it can store energy in. The question really becomes which atoms hold what amount of energy. This is a question that can be addressed mathematically with a concept termed "Multiplicity". Multiplicity is the number of ways you can store those 10 units of energy in however many atoms are present in the metal block. You can place all 10 in the first atom you touch, or spread them out in 10 different atoms, or put 5 in one atom and 5 in another. These are all different ways to arrange this amount of energy. Even so, if all 10 of the energy units are still in the first block, this would mean that the block is at the same temperature (if you'll recall that our energy units tell us how hot our blocks are) no matter how they are arranged within the individual atoms that make up the block. This is something called a "Macrostate" -- a mathematical description of what we observe, namely, the temperature of the block. However, the "Microstate", or the mathematical description of how the energy units are distributed amongst the individual atoms in the block, still plays a crucial role. See, if we take into consideration the second block of metal we just touched to the first block (Let's suppose that both of the blocks are the same size), we essentially double the number of atoms our 10 units of energy can spread between. We also increase the number of macrostates from the single one before (Where our block stayed at the temperature of 10 units of energy that we placed there) to 11 different possible macrostates -- 9 units of energy in the first block, 1 unit of energy in the second block, or 8 units of energy in the first block and 2 units of energy in the second block, so on and so forth.

So the question becomes: Which macrostate is the most likely one to observe? From common experience, we know that things tend to have the same temperature as one another if given enough time, such as the soup cooling off in a room example above. So we should expect that what we observe will be 5 units of energy in the first and 5 units of energy in the 2nd block, given enough time. But why? That is where the term for "Microstates" comes in. It turns out that when you have 5 in the first and 5 in the second, you have more possible ways of distributing the energy throughout the different atoms than you do with any other macrostate. So, it just becomes a statistical issue: There are more possible ways for the Macrostate 5/5 to be observed, therefore it is the one most often observed. There may be some oscillations about this point, but we still observe this more often than anything else.

Now the real kicker is that when dealing with the real world, one deals with more than 10 energy units. We deal with billions upon billions of energy units. And, as atoms are awfully small, we also deal with billions upon billions of atoms. So, with such large numbers the oscillations about the midpoint become immeasurable. So, while oscillations are dictated by probability to occur, as every possible way to arrange the energy in the atoms is just as likely as any other way, we don't notice them due to the sheer improbability of that happening. Like, much more than 10^23. I'm not sure how to express how improbable it is to feel an object heat up without anything heating it up(as it is REALLY FRIGGEN IMPROBABLE), but as you've never experienced it in your life, and I am confident in saying that, you too can feel confident that the 2nd Law is pretty sound stuff! Cool factoid: another common experience unrelated to heat, table salt dissolving in water is an entropy driven process, which is to say that without the 2nd Law of Thermodynamics, table salt wouldn't dissolve.

The Stories of Problems, and visuals

While visualizability is far from a necessary component in a physical system, I still find fictional visualizations beneficial to working problems. I imagine energy as a sinusoidal beam, heat as a cloud of these beams, and electron probabilities as a static mist. I think it helps me to create a narrative of the events, which can make arranging appropriate questions to ask myself easier in the mental array of problem solving techniques. I have recently started developing a visualization for circuits by using water pipes. Except, not. I imagine they're large, already filled pipes that require motors to both pull and push the water, because the fluid is just that dense; or, I try to think of it as a steam like substance under pressure, but so high in mass that it's very stubborn to move, so it just needs two motors. I try to avoid thinking about liquid water, because water is blue, and I imagine that electrons are blue, so I'm trying to keep the visual for the flow of positive charge separate from the visual of electrons that I use, say, when comparing electronegativities, because their stories are different. Maybe something more "Yellow"-like

I highly recommend it. Even if the visualizations are somewhat false, I've found them to be helpful in the problem-solving area.

Teaching Experience, 3

This is an experience I've noticed over my tutoring that happens with most students, in general.

If you ever ask someone, "Does that make sense?" they will always, always, always answer "Uh-huh" (or "Yes", or another general colloquial affirmation). I could say "the delta G favors dissociation" to someone memorizing the solubility rules, and they'll only start to nod their heads, look a little confused, but they will answer "Yes" with at least a .99 probability -- I haven't tested that, but I hypothesize that it would happen.

I wouldn't say that I suddenly get the pass on this one, either. If I'm struggling with a concept, I'll often just blurt the first thing that comes to mind to see if it sticks and see if I'm anywhere near the right track. If someone asks if I understand, I'll say "Yes", wait a minute, and then ask a question directly related to what I was just told. Sometimes the answer will be the exact same thing that they just said.

So this got me to thinking about a general possible maxim for teaching: Never ask your students if they understand. Always assume that they do not understand. When they look bored, then that is the point at which they understand.

This isn't always necessary, as sometimes an individual's body language will let you know whether or not they understand the concept. But some people, including myself, are tricky at hiding it... in the hopes that they don't embarrass themselves (at least, that's my personal motivation), and in the hopes that something later will make it all click together.

I am going to start testing this tomorrow.

EDIT: The phrase is a habit. I totally fail.

"Uncertainty" by David Cassidy

Last night I finished Cassidy's biography on Heisenberg, and so wanted to write a brief review.

The author is a scientist-turned-historian writing a biography on a great scientist. As such, the book is really writing three stories that all occur simultaneously. The obvious one is the life that Heisenberg led. You also get a brief synopsis of his scientific achievements as they were developed and published. To put both of these stories in context, however, the third story being told is a pseudo-personal history of Germany. To give the reader a better understanding of this history, Cassidy will give brief anecdotes about the figures that appear in Heisenberg's life that Heisenberg would not have known, such as the activities of Oppenheimer during the second world war, or the actions of influential Nazi party individuals that, entirely unknown to Heisenberg, essentially saved his life.

There is a historical controversy about Heisenberg dealing with his actions during World War II. The author takes great pains to tread around this with tact, and succeeds at doing so while giving information around the controversial events. He lays out why certain pieces of evidence are suspect, historically speaking, but because these pieces of evidence seemed wrapped up in the controversy, he gives the evidence and its subsequent argument.

While I do not mean to denigrate the efforts of historians, as a scientist-in-training I personally think that the interest in those controversial events lies not in the exact truth of them, but rather in the ethical implications attached either way. If this book can be said to have a theme outside of the main subject matter, the "ethics of science" is the most prominent. This is far from surprising, as World War II really encompasses that question as a whole. I honestly don't think the question was considered before the fall of Nazi Fascism and the bombing of Hiroshima and Nagasaki. However one falls on the question of ethics, the life of Heisenberg is an excellent first stepping stone for addressing the intersection between ethics and science, and as such, this is a book any scientist (or ethical philosopher) ought to be interested in reading.

Undergraduate Research

While it may be a pain in the ass for the professor involved, I have to say that I'm happy that this class is a required part of my undergraduate degree. Especially when put in contrast to the upper-level science courses I am currently taking, which half the time cease to have a lab component complementing the theory -- not that theoretical classes are bad unto themselves, as there's a lot of material out there from which one has to play catch-up with. But I've been forced to learn about a subject I've never had a class in by way of teaching myself from current literature. I haven't done a single experiment, I've only given myself a beginning background in an area. And the ability to utilize things like scifinder or pubmed or the ACS website, and teach yourself (with a little help from my advisor, I must admit) about a topic... I can't help but think these are invaluable skills for work that I hope to be doing in the future. And they aren't skills I ever used in a class room setting, because their you're more concerned with problem solving, memorization, and finding answers in your text-book index.

Additionally, there's an emotional satisfaction to it all -- becoming familiar with an area in order to do original research. But I wouldn't argue that is prime reason for including things in curriculum.

Hydrogen Bonding

Last I mentioned wanting to go over the reason why drinking alcohol, despite being heavier, has a lower boiling point than water. The explanation lies not just in chemical bonding, but in a specific type of chemical bond: The hydrogen bond. In order to understand hydrogen bonding, however, I think one needs to understand chemical bonding in general.

A chemical bond is what holds molecules together. When you have something like H2O, a chemical bond holds the two hydrogen atoms to the oxygen atom. By this definition a hydrogen bond isn't strictly a chemical bond, as it does not hold molecules together, but rather is a way to describe the interaction between a large group of molecules. However, they are related, as a sort of "Bonding" occurs between multiple molecules. Behold, the molecular shape of water!


You'll notice that, with respect to the atoms involved, it has what is called a "Bent" shape that resembles the shape of the letter "V". The four dots around the oxygen atom represent electrons that the oxygen carries around with it. The important thing to know about those electrons in this case is that they are negatively charged, like magnets, which have both a positive and a negative side.


Or a North and South side, as in this picture. Same idea. In fact, if you've played with magnets, water behaves in much the same way: The negative side of water is attracted to the positive side of water. The negative side of water is the side with the oxygen, because oxygen "likes" to carry around electrons (relative to hydrogen). The positive side of water is around the two hydrogen atoms for two reasons: Hydrogen atoms are single protons, which have a positive charge, and as stated before, the oxygen atom "likes" to carry around negative charge much more than hydrogen does. So the oxygen atom not only has the four electrons that it normally carries around, but it will also carry both hydrogens' electrons around. This causes the entire water molecule to become "polar" in the same way that the bar magnets above are polar: With a North and a South side.

Hydrogen bonding is this sort of interaction: Where one side of a molecule will have hydrogen atoms attached to atoms, like Oxygen, which will carry much more negative charge than hydrogen will. This causes a polarity on the molecule, and then large groups of that molecule will interact with itself, where the negative side will be attracted to the positive side. This won't cause true chemical bonds, as they aren't new molecules, but the interaction is enough to have an effect on macroscopic observations, such as boiling point.

To relate this back to the post on distillation: you'll notice from the diagrams in the previous post that drinking alcohol also happens to have an oxygen atom with a hydrogen atom attached to it, which makes it suspiciously similar to alcohol. In fact, this is the case, and some hydrogen bonding occurs in drinking alcohol. However, you'll also notice that there are two hydrogen atoms attached to the oxygen in water, and only one in the case of drinking alcohol. This allows for multiple hydrogen bonds to form, which makes water more attracted to itself than alcohol is to itself. Because water is attracted to itself than alcohol is to itself, it takes more energy (and hence a greater temperature) to cause it to boil. So in the distillation process, alcohol will evaporate before water because of the effects of hydrogen bonding are greater on boiling point than the molecular weights.

This sort of explanation is the essence of chemistry. There are a number of physical things one can measure. There are a number of attributes to a given compound. But the desired end goal is to find a molecular explanation for a macroscopic observation -- something that the hydrogen bond easily does in this case. Also, for further reading, check out the effects of the hydrogen bond on DNA configuration and the density of ice. It has a great deal of explanatory power across several differing areas of study, as well as theoretical justification in physics. These are all the makings of great scientific facts.