| Entropy and the second law of thermodynamics |
Q: Why the fast start? You took about 11 pages to get to entropy in that http://www.secondlaw.com . How come you’re putting it right up front here?
A: Some readers e-mailed me that their question, "What is entropy?", was the main thing they were interested in. Some objected pretty strongly that it was ‘buried way back there’. In writing that Web page, I thought some practical examples like burning paper or forest fires and rusting iron (or breaking surfboards – and bones) would be a great introduction to the second law before talking about that abstract and slippery idea called "entropy". Wasn’t that gradual approach fairly good for you?
Q: Yeh. I understood more than anyone in class when we came to thermodynamics. But entropy was the very first thing our prof and book talked about in the thermo chapter – no "gradual approach" there. Are you finally learning that’s the way to teach it?
A: No. The best teaching always is the kind you understand as the course goes along. Often that’s at a slower pace than a college class. Understanding isn’t always a fast process. This site you’re reading now is the fast track to a few basic ideas rather than the best track. But we’ll still do mainly explaining rather than equation-solving.
Q: OK, give me the bad news quick. What is entropy, really?
A: Straight question – and you make me embarrassed right away when you say that word "really"! The basic definition of entropy is an equation that doesn’t look too complicated, but what it means "REALLY" will take several pages. (Of course, you already know that thermodynamic entropy ONLY applies to chemicals and matter, atoms and molecules – not to economics or information or pollution or human problems, unless they are directly connected to atomic and molecular behavior, right?)
Q: Sure, I know that, and betcha I know the equation too. Our prof gave it first thing: DS = q/T where DS is the change in entropy and q is heat and T is absolute temp. And q is plus if heat comes into a system and minus if it goes out. Ha. I worked plenty of problems using it or the original differential equation for a reversible process. Isn’t that what you mean it MEANS?
A: Equations, shumations! Does dividing thermal energy transfer in a chemical reaction (the correct expression for "heat") OK, heat, by T give you a crystal clear picture of what DS or S IS? Does it tell you why it’s so important in understanding the way the world works? Of course not. It needs an awful lot of interpretation.
Here’s a start: Dividing a reaction's q by T gives you what is called entropy (joules per degree) just like dividing miles or kilometers by hours gives you what you call speed, mph or kmph. They’re both just ratios, just dividing something by something else.
What if you had been raised on an island totally outside of civilization? That ratio you now know so well for speed, mph or kmph, would be just as mysterious as entropy if you’d never in your life thought in terms of planes or cars going fast. Would you, the islander, recognize what the ratio for a car's speed depended on or was due to?! Of course not. What’s responsible for making something like a car go fast? Speed depends on the power of a car’s engine, of course. (Ignore gear ratios and rubber on the road!) You could talk professorish and say that speed measures the tendency of the engine to move the car rapidly – all assuming that you have gas and a heavy-footed driver.
Similarly, you deserve an answer to, "What makes that q/T ratio, entropy, come out with some number or other? What's the engine that it's tied into?" Easy answer: ENERGY. Entropy measures -- with a ratio -- the tendency of ENERGY to spread out, to diffuse, to become less concentrated in one physical location or one energetic state. Most often in chemistry, that spreading out is done by molecules (and atoms, but let's concentrate on molecules) because molecules above absolute zero always have energy inside of them. That's why they are incessantly speeding through space and hitting each other and rotating and vibrating in a gas or liquid. (For details, see Figure 1, a few pages ahead.) Even in solids where they can't move much, they are always vibrating.
Molecules tend to spread out their energy by moving and rotating/vibrating in as many different ways as they can. (The more ways, the less concentrated is their energy in just one "quantum state" or way. More about quantum states later.) Entropy is a measure of the second law of thermodynamics. Entropy doesn’t predict the speed of something happening, but it shows the direction and a quantity for the tendency of every chemical or physical event to happen on this earth (and maybe for every one in the whole universe).
That’s pretty sensational. That’s why entropy (OK, the change in entropy, DS) can be called "time’s arrow": The obvious spontaneous matter-energy events like hot pans cooling down and rocks rolling downhill and paper burning in air and shiny iron changing to red iron rust in air always go "thataway" -- in the energy-spreading-out direction that entropy points, never the opposite. (Cold pans in the cupboard never heat up by themselves, rocks never roll up mountains by themselves, ashes plus carbon dioxide and water never make paper by themselves, red rust never changes to solid iron plus oxygen gas by itself – big D S, "time’s arrow", points toward "cool, go downhill, burn, rust" not vice versa.)
Actually, the time’s arrow of DS applies to every change that involves matter and energy, spontaneous (i.e., by itself) or non-spontaneous (i.e., by being pushed by outside energy). I just grabbed examples of spontaneous events, those that happen by themselves. (The "outside" has to be included in talking about non-spontaneous changes, so we’ll save that for a few minutes.)
Ignore the little "outside" heat of a match flame needed to start the paper burning spontaneously. That little kick is terribly important at first (see energy of activation in the second law site, http://www.secondlaw.com/three.html , if you’ve forgotten about Ea.). However, it’s usually trivial and ignorable in evaluating spontaneity for this reason: Once the reaction has started, energy is released, often very large amounts. This heat spreads to adjacent parts of the paper/system and quickly gives them far more energy than the little Ea needed for them to react. So, the process is overall spontaneous and continues by itself with no further energy from outside..
Q: Wait a minute, wait a minute! Back in www.secondlaw.com you said the second law of thermodynamics was "time’s arrow". Now you say entropy is it? How come?
A: They’re intertwined. Integration of the differential equation that defines entropy, dS = dq/T (that we take as DS = q/T) in a particular manner is an essential part of the energy-work cycle from which comes the technical statement of the second law. Thus, in informal talk – which is the only situation you use the phrase time’s arrow -- you can say either entropy or the second law is ‘the arrow’.
A more important point about that DS equation that I’m sure you remember: When entropy has a positive sign in a DS equation for ANY physical or chemical process, that process will go in the common-sense direction we talked about for pots cooling down, etc., in the simple spontaneous events mentioned above (or in the secondlaw.com Website.) Since a positive sign means an increase in entropy after the process, compared to before it happened, we often say "In the system of every spontaneous process there is an increase in entropy." (Actually, non-spontaneous processes in a system also are involved in an increase in the entropy of the universe, but it occurs in the surroundings outside the system that make the non-spontaneous process go. So, overall, every event in our earth-sun system increases the entropy of the universe.)
Q: You mean some people say entropy loosely for the second law and vice versa? Sounds possible. And the everyday examples you gave of entropy increasing – being more + every time that energy is more distributed in the world – what you call "spread out" or diffused -- are OK – but…
A: You’re generous? Giving me an OK……...
Q: You interrupted me! But how about strictly chemical stuff I never could understand in class? For example, how orderly crystals of all kinds of salts spontaneously dissolve in water to form disorderly solutions and yet the water cools down! How can something be spontaneous and not spread out heat if the second law and entropy predict all the time that things happen so energy spreads out? Well, "distributes itself" or "is distributed in molecular motion", whatever that really means?
A: Wow. You sure know how to load a question. To start, you have one very good point (with several complex implications), then one really confusing dumb point about disorder, and finally three intricate (though seemingly simple), tremendously important angles. All that in one question! Lean back in your chair, feet up on the desk. Let’s take them slowly.
The first good point about "entropy increasing" and "being more +". That’s a neat condensation of two ideas, namely: The qualitative power of the second law to predict in general what will happen, plus the added power we have to keep from making mistakes -- thanks to that quantitative equation defining entropy. Entropy becomes the measurable part of verbal predictions that come from our experience – a reliable signpost, time’s quantifiable arrow, the second law’s meter stick. Before, we might have had a correct "feeling" that energy spontaneously tended to become widely distributed or diffused rather than the reverse process. Now we can calculate exact numbers. (We’ll leave calculations to your class work.) The new idea is that every single time something spontaneous happens via energy spreading out to the surroundings – like the old example of the hot pan cooling down – entropy is +, positive. Every time something occurs that is non-spontaneous – LOOK OUT! Don’t draw the conclusion that entropy can become negative by itself!
Q: Whaddya mean by yelling like that – "LOOK OUT"? Why shouldn’t entropy become negative just as much as it can go to be positive or plus or "increased"?
A: Thanks for hearing me yell! Than means you caught on to another absolutely VITAL point (and that’s also a yell.): Whenever we think about entropy and measure it, we have to consider TWO small sections of our great big universe: (1) the "system" – in those examples, the hot pan or the stone itself , AND (2), the "surroundings" of the system.. In the case of the pan, the surroundings would be the cooler room; in the case of the stone up on a hill, it would be the downslope of the hill and the air. Surroundings are important!
That’s where people make their biggest mistakes in thinking about the second law or about entropy. They concentrate their attention only on the system that looks like it is doing something interesting, the pan or the rolling rock. When does anybody talk about the second law or do any entropy calculating? When there are changes from a given situation or state to a different state, i.e., State 1 (initial, before anything happens) to State 2 (final, after the event, whatever it is). To avoid coming to totally wrong conclusions, we have to include the system AND the surroundings in thinking about State 1 and 2. S2 (State 2, system + surroundings) minus S1 (State 1, system + surroundings) equals the DS for the change (system and surroundings).
Repeating: For the hot pan in a cooler room, the pan was the system and the room the surroundings. As it cooled, the pan lost heat – measuring at increments of tenths of a degree or so (to keep that T in the equation essentially constant) – and therefore q was minus in the equation and the change in entropy in the system in each increment, or in the whole process from "before cooling" to "after", was negative. A decrease in entropy occurred in the pan! BUT heat was gained by the cooler room, so its q/T was positive, an increase in entropy in the surroundings. Now another big "BUT" – the decrease in the pan was less than the increase that occurred in the room – the room gained more entropy than the pan lost because any spontaneous transfer of energy involves an increase in entropy DUE TO THE PROCESS ITSELF. Therefore, the overall change in hot pan (system, before) + cooler room (surroundings, before) to end up with cool pan (system, after) + slightly warmer room (surroundings, after) is a net increase in the total entropy of "system + surroundings".
For the stone rolling down hill, the stone was the system. The air and the hill were the surroundings. Energy changes? The potential energy in the stone was all spread out, distributed from the stone system to the hill/air surroundings when it was changed to kinetic energy – causing a little warming of the hill from friction and a slight movement of air. Entropy changes? None in the stone, the system. Zero. But, because its potential energy was diffused to the surroundings as kinetic energy and then to heat, the surroundings increased in entropy, a tiny tiny bit of warming. Final state? Total change in entropy of the whole "mini-universe"? An increase in entropy in the (system plus surroundings).
The reverse of the spontaneous rolling down the hill would be a non-spontaneous movement of a stone up a hill. Doesn’t ever happen? Sure it does, but not by itself because that’s exactly what non-spontaneous means. When you or I push that big stone (the system) from its initial state at the bottom of the hill to a final state somewhere up the hill, we’re acting as the surroundings! We have put a lot of potential energy in it. Where’d that energy come from? From us, the "surroundings", transferring energy to the system from the stored energy in our muscles (much of it from the chemical in them that’s abbreviated as ATP). There isn’t any change in the entropy of the stone. Zero. However, the person who does the pushing has had a really really big increase in his or her entropy – a lot of ATP in the muscles have been changed to ADP and heat and action of muscles (work) resulting in a higher body temperature, some increase in temp of the air, sweating, etc.,etc. The total DS for the process of S2 minus S1 is a great increase in the total entropy of the mini-universe-- zero in the system (the rock) but a big increase in the surroundings (mainly in the sweating person and the warmer air around him or her).
From those simple examples you can see how important it is always to consider both system and surroundings in thinking about entropy. That’s why I yelled "Look out" at you when you mentioned entropy being negative. There must have been a larger increase in entropy in the surroundings if ever there was a negative entropy change in a system. DS can’t be negative spontaneously, i.e., by itself. System plus surroundings. System plus surroundings. Never forget to look at both.
Q: I get it. I get it. (I can say things twice, too! Guess I’d better not forget or you’ll yell at me again.) What happened to what you called a confusing dumb point?
A: Sorry. I get excited about this stuff. I didn’t mean you were dumb. You were sold a bill of goods by some profs and authors who know better but think they are helping you by over-simplifying and using pretty little pictures as though you were children.. They go too far. Like encouraging you to believe in Santa Claus.
Q: Santa Claus? C’mon. White-beard in the red suit that parents tell children gives them gifts at Christmas when it’s really the parents and relatives that do the gift-giving?
A: Exactly.
The "confusing dumb point" is texts and profs telling you that "disorder" is the key to understanding entropy. "Disorder" can be a key sometimes, but it can be totally misleading (and always is a crutch, not a reason). Therefore, it should be used with great caution.
For a century "disorder" has been a codeword, a shorthand word, among professional thermodynamicists for the many ways that energy quanta (more about quanta in a minute) in molecules, atoms, etc., can be distributed among their motions. But the most distinguished themodynamicists have said clearly that other profs use that "disorder" codeword too loosely; it precisely applies only to three or four very specific molecular situations! So for gen chem texts to tell students disorder is a great picture to describe entropy in every situation is too much like Santa Claus being used as a codeword for all generosity because little kids can’t understand abstract ideas.
You aren’t a little kid. You can understand complex ideas or you and I wouldn’t be talking about entropy. No reason why you can’t be told that entropy is NOT defined by disorder, rather than talking down to you and confusing you by telling you about shuffled cards and messy desks and ions in solutions as examples of "disorder" and entropy. Nonsense. "Confusing dumb" because it can seriously mislead students beginning thermodynamics.
Your prof ought to read "Shuffled Cards, Disorderly Desks, and Messy Dorm Rooms – Examples of Entropy Increase? Nonsense!" in the October 1999 Journal of Chemical Education, pp. 1385-87. (http://jchemed.chem.wisc.edu/Journal/Issues/1999/Oct/abs1385.html) But don’t tell him or her that!! Don’t EVER correct professors in public. The few noisy and arrogant ones have extremely tender egos that bruise easily. Others should be treated even more courteously. A wiser procedure is to photocopy the "Shuffled Cards…Entropy" article, write on it, "How about this??", with a red marker, and leave it on the seat of the prof’s chair. (Not on the desk. You know how easily stuff is shoved under other things on that desk.) Might be a helpful shock to him or her.
The closest thing to a clear picture of entropy – almost as good as a "photo" of DS – relates it to what atoms and molecules do with energy, specifically with thermal motion energy, heat. S measures the number of ways that the energy of a system can be distributed among the motions of its microparticles (its atoms, molecules, and/or ions). Then, if there are more ways for this energy distribution after some chemical reaction or physical event than there were before it, DS is positive.
Energetic motions of molecules are these: (1) whizzing around in space, i.e. translational motion; (2) rotating about axes (that you can visualize by drawing axes through their 3-D models), i.e., rotational; and (3) vibrating among adjacent microparticles in a solid, or vibrating between atoms in a molecule, whether they are molecules of a gas, liquid or solid. I.e., vibrational motion.
When you heat up any bit of matter above absolute zero, you are putting energy in the molecules in one or more of those energetic modes or motions. But why should all the energy be concentrated in one mode ("one place") more than another? Why not all over the neighborhood -- spread out ("distributed") as much as it can be among all the available energetic states of systems and their surroundings? That’s the key to understanding entropy change in most of chemistry. But before we get going in this enjoyable and profitable area, I have to ask you a question: What do you remember about quantization of energy, and about quantized energy levels in microparticles?
A &Q: Ho Ho!! Now I get to answer a question! Well, first I know that all energy, whether it’s the energy that atoms and molecules have (microparticles, like you say) or it’s radiation that zings through space, is actually always in bunches, little packages or quanta of energy. The size of the quanta depend on the kind of energy, big quanta for powerful energy (like x-rays), small-sized quanta for much weaker energy (like heat or as in molecules speeding around in a box.) The number of a given sized quanta indicates how much energy of that type is there. So, summarizing, the size of the quanta shows the intensity, the "quality" of the energy. The number of them shows the amount, the quantity.
Quantized energy levels? Do you mean that stuff about lower and higher energy levels that electrons occupy in atoms, with the lowest called the s orbitals and then the p orbitals and so forth?
A: Great summary on energy quanta and good start on levels. In chem reactions where bonds are broken and formed, the very large amounts of energy involved come from changes in the bonding, from electronic energy levels inside molecules. Then, energy from these changes can be distributed to cause all the molecules in the reaction to move faster or to vibrate more or to whirl around differently – and beyond that, in each of those vibrational and other modes there are many levels that the quanta of energy can occupy, depending on both the quantity and intensity of the energy packets.
The energy difference between levels for translational energy quanta is extremely small (around 10-14 kilojoules/mol), so small that "jumps" between translational energy levels are hard to detect experimentally. Thus, absorption/gain or emission/loss of energy appears virtually continuous rather than in the usual discrete steps or jumps between different energy levels. Rotational energy levels differ by about 10-2 kJ/mol, and vibrational energy levels in the range of 25 kJ/mol (e.g., from 6 to 40). (Electronic energy levels involve about 5 to 20 times greater energy for absorption or emission than do vibration levels.)
Q: So?
A: So now we have the clues as to WHAT entropy can be measuring. Energy diffuses or is distributed, if it can, by having the microparticles involved occupy as many of these different energy levels of translation or rotation or vibration as possible.
(What does "can occupy" mean? If the energy in the molecules isn’t intense enough, as would be indicated by a system's low temperature, its molecules certainly couldn’t be in the highest energy-demanding vibrational levels, for example. Those levels wouldn’t "be available" to the microparticles with inadequate energy "to occupy" them. Likewise, even though the microparticles may have "intense" energy quanta (indicated by a high temperature), if there is just a very small quantity present, only a few of the many available energy levels could be occupied or "filled".)
Such an increase in the number of ways that energy can be distributed by molecules or atoms in many quantized levels means an increase in entropy because entropy measures how much energy spreads out or can spread out.
If you let molecules of a gas into an empty container, or a gas or liquid into a container that has other molecules, you’ve given those molecules more volume to move in. Therefore, they have more closely-spaced translational/rotational energy levels in which to spread out their original energy. Their entropy increases – there are more combinations of ways for their many molecules to diffuse the energy in the larger "State 2" than the original "State 1".
That’s why molecules of a liquid spontaneously mix with others of similar kind when you pour a liter of one kind with a liter of another related kind in a beaker. It’s not that there is any big deal kind of attraction between them, nor only that there is more volume -- two liters of liquid instead of one in the beaker.
The volume increase is important only because it allows many more ways for the energetic molecules to be distributed (in "many more energy levels"). Therefore entropy increases in "State 2" (after) compared to "State 1" (before). Is that "disorderly" compared to "orderly"? Dumb question. Who cares? (To put it very nicely.) What's most fundamental? The driving force of energy changes is what entropy is measuring not cc or liters. Entropy is looking at the distribution of energy not at pretty patterns from Martha Stewart. The important FUNDAMENTAL fact is that there are more energy locations, more "places" on energy levels for molecules, etc, etc. – not places in space -- in S2 than in S1 and therefore there is an increase in entropy as occurs in all spontaneous processes.
Q: Hmm. Got that. Now, maybe you forgot – seems like an hour ago -- I asked about soluble salts dissolving in water spontaneously but the temperature of the beaker and the solution dropping. How does that fit in to all the stuff you’ve been talking about?
A: Didn’t forget it. Had to take a look at all the background ideas before we could talk about it.
Crystals just dissolving in water -- seems like it should be a simple process, right? But there really are opposing tendencies fighting each other, even though it happens spontaneously, by itself, without any energy being supplied from outside the system of salt and water. The temperature of the solution usually decreases, as you said, when most common salts dissolve. (In some, the temp rises rather than decreases. This is a key to seeing that there's probably a conflict between two or more tendencies. Sometimes one beats the other; sometimes the reverse. Sorry that it's not simple! Back to that in a minute.) Ammonium nitrate plus water gets so cold so fast that it is used as an instant athletic "ice-pack" for sprains and bruises.
YES, about "order" at the beginning of dissolving: Perfect crystals are very orderly and symmetrical and even ordinary imperfect NaCl and sodium sulfate and ammonium nitrate are neatly arranged in their crystalline form. YES, about "disorder" after dissolving (but only a half-S yes, because the crystal ions/molecules get disordered while solvent molecules get more ordered!): The ions from any of those rigid orderly salts when they dissolve move every whichways in the aqueous solution, disorderly and flickering around like a bunch of drunken mosquitoes. But wait a minute. Before the crystal ions came along to get in the swim, the solvent molecules were free as birds (or mosquitoes), happily disorderly. In the dissolving process, the solvent molecules associate around each of the ions and they are NOT as disorderly as they were before in pure solvent. They lose in the game of entropy being "disorder". So what? Wrong game. Entropy is determined by ENERGY distribution/diffusion, not by some arbitrary lack of neatnik patterns (above absolute zero). All right, let’s see what happens energetically when we think about crystals dissolving. It will still be complicated by a conflict between the solid getting free and the solvent getting more tied up being around the ions, but at least we'll be thinking straight about what happens and we'll have a clue about entropy changes.
In this case, the crystals are doing something, the water something different. Then, we can't forget the surroundings, the beaker and the air around it. So, to minimize confusion, let’s call the crystals System 1 (S1), the water and resulting solution System 2 (S2), and the beaker and air near it Surroundings (Surr).
Now, in S1 at the start, the crystals of NaCl consist of sodium and chloride ions constantly vibrating back and forth in their individual tiny regions. That’s the only way they can behave with their particular energy quanta and their close neighbors. If the ions could move into the volume where the water molecules are, they would have other energy states that they could occupy -- primarily because of the more closely spaced translationa/rotational states. Because that would be an increase in the ways of distributing the original amount of energy, it would be an increase in entropy. It should occur spontaneously. So, according to the second law, there’s no reason why dissolving shouldn’t occur easily.
But look at what happens to entropy changes in S2, the water into which the ions go. Before any salt was in contact with the water, as we said, the water molecules were zipping around quite freely. (Oh yes, they had hydrogen bonds between numbers of them, that formed and broke constantly, so they weren't as free as gas molecules. But still, a pretty free life.) When the ions of the salt came into the water to make a solution, the polar water molecules gathered relatively tightly around each ion because of the ion's concentrated electrical charge. Entropy-wise, that restricted the movement of the water molecules and decreased the number of readily available energetic states. So in S2 (after) compared to S2 (before), there is a decrease in entropy. It tends to oppose the positive increase in entropy of ions (in DS1) dissolving in water. Overall, the entropy increase in DS1 is usually slightly greater than the decrease in DS2 so there is a modest entropy increase when salts dissolve.
Oops. There are two much more powerful energy factors we have to look at when we are thinking about salts dissolving. A tremendous amount of energy has to be put into S1, a crystal like NaCl, in order to break the strong bonds among the ions and to let them float into the water and be surrounded by water molecules. However, as polar water molecules surround all those ions, they would form huge numbers of weak bonds with them -- and making so many of these bonds would give out large amounts of energy from S2. Which "system" wins? Bond-breaking energy demands from the crystals in S1, or bond-forming energy release in S2?
In the case of NaCl, the energy required to break the ionic bonds of the crystalline salt, S1, is very close to the energy evolved from the new bonds between water and the ions in S2. They're so nearly equal that the S1 entropy factor dominates. S1 Entropy factor? Yes, the ions having more ways to distribute themselves energetically in the solution if they can be helped to escape from the crystalline salt, S1. But in most common salts, this process requires so much energy that the bond forming in S2 isn't enough to do the whole job and that means taking some energy from the surroundings, the beaker and the nearby atmosphere that we called "Surr".
The example of ammonium salts fits what we've said. We would expect the entropy factor to be about the same. But the fact that the ammonium ion is larger than sodium means that its + charge would be spread over a larger surface area, not be as intense, not as strongly attract water molecules. Therefore, the energy given out by bond formation in S2 would be much less than what would be needed to break the bonds between the ammonium and its negative ions, such as chloride or nitrate. But the process does occur; ammonium salts easily dissolve in water -- with a great deal of cooling of "Surr", the beaker and the surrounding air. This means that the tendency of the ammonium and its negative ions to move into the water in more energetic levels (greater entropy) is so dominant that energy is taken from the surroundings.
On the other hand, some salts like calcium chloride form such strong bonds with water molecules that this bond-formation in S2 is far greater than the energy needed to break the salt's bonds. In these cases the solutions gets hot as the salt dissolves.
Q: Man, it sure is easier to dissolve salt than to explain it! Oh. Wait a minute. What hit me a second ago? Oh, this – what about your talking about calcium chloride or salts like it that form more bonds or stronger ones with water and actually get hot. Isn’t that a chemical reaction? And shouldn’t you always use the Gibbs equation, DG = DH - TDS, when you work with chemical reactions and entropy and heat evolved and stuff?
A. Now you’re getting to the goodies! YES – all chemistry that involves energy (and what doesn’t?) is dominated by energy flow – the second law – and so entropy is the key to measuring it. But this introductory section about entropy is getting too long so let’s cut it after just two more short items. Then in a later section, "Entropy and Gibbs free energy…", let’s talk more about how terrifically important entropy-thinking is in all of chemistry.
This may rattle your brain about the Gibbs equation a bit: Divide that equation by T and what do you get? Three terms, each having the dimensions of entropy. The Gibbs is ALL entropy really!! How about that?
First short item: A neat example of entropy as a measure of the number of ways energy can be distributed in a system that embarrasses a lot of profs who use that misleading "order-disorder" stuff about entropy: supersaturated solutions.
Q: Supersaturated solutions?. Yeh. I remember a beautiful picture in my book about crystals forming fast and almost filling a beaker when you just drop a "seed" of a crystal into that kind of a clear solution. What has that to do with entropy or being embarrassed?
A: Come on now! Where have you been the last ten minutes?? Let’s say you have just been promoted to professor and are teaching gen chem. A week or so ago you told your class about spontaneous processes occurring with an increase in entropy. (Assume you hadn’t been properly educated (!) and you’re an "entropy is a measure of disorder" believer.) Quote from Professor You (glasses down on your nose, to look wise and authoritative), "Order never spontaneously arises from disorder."
So can you guess how foolish you look if you are demonstrating to a class and show a supersaturated clear aqueous solution – that they all know is disorderly "just like all solutions are" -- drop a single crystal in it, and zoom! the whole beaker fills up with beautiful sparkling orderly crystals and it gets warm?? What if a student, as feisty as you are now, raises his hand then and says "But sir, that’s going from disorder to visible order… Then that’s a spontaneous negative entropy change, right? And it heats up, TOO?" Blush, blush, sweat, sweat Professor You. "Order-disorder" has made you look like an idiot.
(Always say "sir" or "maam" when you ask an embarrassing question. NO! Better, don’t ever do it in class. It really isn’t courteous – and you want always to be treated courteously, right? So, just let it go by, scribble the question on a piece of paper and ask a senior or grad student in private later.)
From an energy standpoint, there’s no sweat or blushing needed in answering the question. The crystallization is spontaneous entropy-wise for these energetic reasons: (1) the process gives out heat (because of so many strong bonds in the crystals being formed), and thus there is a positive entropy change in the surroundings from the thermal energy transfer from the system; (2) the water is initially in a thermodynamically unstable state when supersaturated because more ions are surrounding/surrounded by water molecules and constricting the water’s movement than in an equilibrium state. Thus, the water becomes more free to move when the excess of ions crystallize out and there are more available translational, rotational, and vibrational modes for it than before in the supersaturated state – also a positive entropy change. Voila!! We can see that OF COURSE there is an increase in entropy in this system + surroundings because entropy changes depend on occupancy of energetic states!. It has nothing to do with how neat or disorderly the beaker’s contents look to us before and after the event.
Finally, an amusing illustration of the inadequacy of "order-disorder" in "explaining" phenomena is the melting of ice and freezing of water (or any other phase change). If "order to disorder" is fundamentally involved in entropy explanations, how is it that such a radical thing as an absolute stop in temperature increase occurs at 0o C and you can’t raise the temp a hundredth of a degree until all the ice melts? If you were commanded to change an orderly pattern like a checkerboard (or a three-dimensional crystal) very very gradually toward a disorderly jumble, you would just move one square or tiny bit and then another and then another, ditto, ditto, ditto, right? No stop in the process, of course, because your job is to move a little, move a little, move a little until whoever made you do it says it's all a disorderly heap.
OK, if the fundamental process in nature is just "pattern changing" to disorder, shouldn't it occur incrementally, gradually, continually just like step by step checkerboard mixing-up without any halt or break? Orderly ice should slowly become more and more disorderly water as thermal energy is continually transferred to the system and the temperature gradually rises from – 0.1o C to 0.0o C and then moves right along to increase to + 0.1o C. Wait a minute. You can’t warm ice and ice-water and see the temp steadily rise those two-tenths of a degree from –0.1 to +0.1 as you could from, say, –0.3 to –0.1o! Unlike steady incremental pattern changing, steadily pouring energy into the ice-water system doesn't continue the process of warming when the temperature reaches 0o . Solely on an "orderly-disorderly" basis, there is no reason why an incremental changing of patterns in ice should magically, arbitrarily stop at some temperature or other and wait wait wait? (So the prof has to sneak in an extraneous reason, i.e., outside of order-disorder concepts.)
There isn’t any answer to these simple questions in the "order-disorder" visualization because it is a gimmick and not fundamental. In contrast, the reason for a stop at 0.0o becomes clear if we think about the microenergetics of what happens as we transfer more and more thermal energy and see the temperature rise. Below 0.0o.C, the energy quanta are only large enough to change the least energy-requiring vibrational-rotational modes between the water molecules that are attracted to one another in crystalline ice. Only when the temperature (an indication of the "quality", the intensity of the energy quanta present in the system) reaches 0.0o C are there larger quanta that molecules can acquire so the molecules are able to occupy greater rotational - translational modes that are characteristic of liquid water. Thus, only then can molecules can break free of the crystalline ice to move into the water phase. No temperature rise can occur -- i.e. a huge number of rotational/translational/vibrational modes in the molecules in ice can’t be filled – until enough incoming energy at 0o has been supplied, that is quantity (or number of quanta) of the required quality (or intensity of quanta). Then, all the molecules distribute the incoming energy by rotating and translating enough and break the weak bond holding them in a solid form and become liquid water.
Here's an analogy (even though it's only mediocre): The need for energy to be intense enough (quality) and sufficient (quantity) to occupy rotational and translational modes before melting or other phase transitions can occur is a little bit like a small creek across which we have just thrown a dam. The slowly increasing depth of the water behind the dam is comparable to the slow filling of rotational/etc modes as thermal energy flows into ice at 00 C. The creek can’t go over the dam – i.e., in analogy, there isn't sufficient energy in the system to make all the crystals change to liquid -- until the dam is full. THEN, when the "energetic dam" is filled, all the molecules in the ice have absorbed enough energy to change to liquid water and further input of thermal energy makes them move faster. That means the temperature can rise to +0.01o C and above – just as the creek coming from way upstream can now flow freely over the dam. (OK, OK , no analogy is perfectly parallel to what really happens!)
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SUMMARY
Important stuff to remember
BUT
(As an excellent method of increasing your 'mental muscles', when your prof makes an error, think to yourself, "Knowing what I know about entropy, what should he/she have said? If I were teaching, what would I say correctly?" and scribble it in your notes, But keep it to yourself!)
| A generalization for classical thermodynamics (macro thermo, Clausius): |
| Entropy measures the energy dispersion in a system divided by temperature (Dq/t) |
| A generalization for statistical thermodynamics (micro thermo, Boltzmann): | |
| Entropy measures the energy distribution by molecules on quantized levels in a system: the number of ways in which energetic molecules in a system are distributed among the energy levels available to them. (DS = R/N ln (Wfinal)/(Winitial) | |
Entropy is NOT a driving force; it measures the driving force.
| The driving force that entropy measures in systems of atoms and molecules | |
| (microparticles) is the tendency of the energy within the microparticles to spread out. It does this by making them occupy as many microenergetic quantum levels as are available to them -- translational, vibrational, rotational (and electronic) -- considering the intensity and amount of energy that the microparticles have. | |
| Thus, entropy is a measure of the number of ways (the microenergetic levels) in |
| which the energy of a system can be distributed. |
Entropy is NOT "disorder".
Entropy is NOT a "measure of disorder" (except in three atomic/molecular situations)
| Entropy is NEVER a measure of disorder in the arrangement of macro objects -- |
| from playing cards and messy desks to bricks and boulders. |
Disorder in macro objects is caused by energetic agents (wind, heat, earthquakes, driving rain, people, or, in a quite different category, gravity) acting on them to push them around to what we see as "disorderly" arrangements, their most probable locations after the agent is not actively moving them. The agents (other than gravity!) undergo an increase in entropy in the process. The objects are unchanged in entropy if they are simply rearranged.
If an object is broken, there is no measurable change in entropy until the number of bonds broken is about a thoudsandths of those unchanged in the object. This means that one fracture or even hundreds make no significant difference in an object's entropy. (It is only when something is ground to a fine powder that a measurable increase or decrease in entropy occurs -- the sign of change depending on the kinds of new bonds formed after the break compared to those in the original object.)
Even though breaking a mole-sized crystal of NaCl in half involves slight changes in hundreds to thousands of the NaCl units adjacent to the fracture line, in addition to those actually on such a line, there are still at least 106 bonds totally unaffected. Thus we can see why a single fracture of a ski (unhappy as it is to the skier), or a house torn apart in to ten thousand pieces by a hurricane (disastrous as it is to the homeowner), represent truly insignificant entropy changes. The only notable entropy change occurs in the agent causing the breaks. Human concepts of order are misplaced in evaluating entropy.
| "Disorder" or "positional/configurational entropy" -- as judged from an | |
| increase in the number of molecules, an increase in volume or in temperature or molecular complexity, etc. -- is often a useful gimmick for guessing at an entropy increase in a chemical process, but it is not a fundamental thermodynamic attribute. When it is applied blindly by beginning students to all situations, it can mislead them. (Glaring examples of errors in a whole area of physical events are those involving the "disorderly patterns" when macro objects are moved, but other examples involving molecular behavior have been mentioned in the previous pages.) | |
| "Disorder" ("positional entropy") is indisputably useful in quick estimates of entropy changes (really necessary in some complex situations, e.g., in protein folding) by skilled students and pros. It is fun -- but not fundamental. | |
| Energy is the capacity of a system to do work. |
| Every transfer of energy within a system or with its surroundings involves an increase in |
| entropy, a decrease in ability to do work, a "spreading out" or "diffusion" of energy, an |
| increase in "waste heat". |
| All physical and chemical processes involve an increase in entropy in the combination of |
| (system + surroundings). |
| System plus surroundings. System plus surroundings. System plus surroundings! |
References
Professor John P. Lowe’s superb explanation of the importance of the occupancy of energy levels as a genuine basis for entropy (rather than "randomness" or "disorder") via informal QandA is in the Journal of Chemical Education, 1988, 65 (5), 403 – 406. Pages 405 and 406 are especially pertinent and very readable.
An excellent introduction to Professor Norman C. Craig’s procedure of attacking entropy problems is in "Entropy Analyses of Four Familiar Processes", Journal of Chemical Education, 1988, 65 (9), 760 – 764. Professor Craig’s 200 page paperback Entropy Analysis (John Wiley, New York, 1992), is the best short technical introduction to the laws of thermodynamics and the correct utilization of entropy in print. It is accessible to a diligent first-year college student, and especially valuable to a student beginning physical chemistry thermo (as well as to mature chemists who really never understood entropy in their thermodynamics classes).
Most current college general chemistry texts have excellent sections on calculations involving entropy. Use them for extending your understanding to this essential quantitative area.
Links
A valid, but brief, lecture is http://chem.brown.edu/chem31/1998/lectures-98/lecture15.html (Reading the Oxtoby/Gillis/Nachtrieb text on which it is based would be far more helpful to any student.) As of 11/99, other links to second law URLs are too sketchy, too non-helpful or have too many errors to be useful.
http://www.secondlaw.com , mentioned in this section, is a practical introduction to the second law from which illustrations in this present site were taken, but the treatment of entropy there is much less detailed except in the appendix where endothermic reactions are mentioned.
http://jchemed.chem.wisc.edu/Journal/Issues/1999/Oct/abs1385.html is the abstract of my article showing why shuffled cards, messy desks and disorderly dorm rooms are NOT examples of entropy increase: Journal of Chemical Education, 1999, 76 (10), 1385 – 1387.
