## APPENDIX

### Some aspects of entropy that are troublesome to students:

When q = 0, e.g., an ideal gas expanding into a vacuum, how can there be any entropy change?

This example can be very confusing because the first equation for entropy change that you see in most textbooks is S = q(rev)/T, and so it seems that the entropy change would be zero if q is zero. But you know from your experience with air whooshing out of tires that a gas would certainly expand into a vacuum chamber spontaneously! Therefore, since it's a spontaneous process with the higher pressure gas spreading out its energy over the new larger volume (the original volume plus that of the vac chamber), there must be an entropy increase.

Then, if you've read page six of this secondlaw (or a newer text that uses our definition of entropy change), you know the two ways in which entropy can increase: (1) how much energy has been dispersed to/from the system in a process, or (2) how spread out the original energy of the system has become after the change. Therefore, (2) fits this example of gas expanding its energy throughout all of a bigger volume.

But how can you measure the quantity of entropy change in gas expansion? The instructor and textbook in your course will tell you that "to find the entropy change, you have to restore the system to its original state reversibly" and then show you that if this process is done very very slowly (reversibly) this process depends on nRT ln (Vfinal/Vinitial), i.e., the work that is equivalent to a transfer of energy equals the (-) of the energy of expansion, q(rev). That's how a “(2)” kind of entropy change in which the initial energy becomes ‘spread out' but not increased or decreased can be measured in terms of q.

In other words, that S = q/T equation that looks so neat is only directly applied to finding entropy change when there is an actual “how much” thermal energy transfer ("heat") involved in a process that is "near-perfectly" reversible. Thus, if the melting/freezing temperature for ice is 273.00 K (actually it's 273.15 K, but let's use the even number) , you could slowly melt ice in a room that is 273.001 K, or you could change the room temp to 272.999 K and any water that had formed would reform into ice. (In other words, when you're measuring the q of the entropy of fusion of ice [its enthalpy of fusion], the T should be essentially constant; almost exactly 273 K. When you have situations in which the temperature changes by a degree or a hundred during a process, your text gives all the details necessary to find S exactly by a little calculus that essentially tells you the sum of reversible changes at a near-infinite number of tiny temperatures over that range.)

What is "positional" entropy? Is it different than 'plain vanilla' entropy?
The Simple Molecular Meaning of Entropy Change

Some textbooks explain the cause of the spontaneous expansion of a gas into a vacuum, or mixing of ideal gases or of ideal liquids by saying that these happen due to an increase in "positional" entropy or "configurational" entropy. The bigger the space available, the more "positions" there are in it and the bigger the "positional" entropy increase will be.

That sounds OK, but wait a minute! Entropy increase is this kind of expansion that is fundamentally due to the spreading out of molecular motional energy. What's happening to energy in this "positional" entropy stuff? Showing molecules as little dots and then spreading those dots out in different positions and saying that's entropy change, with no mention of energy, is like a magician diverting your attention from he's really doing -- a con job. Entropy change must involve energy spreading out, not just looking at positions in space!

The correct explanation of that ‘con job positional energy' (or any entropy change) first requires us to remember about the kinetic behavior of molecules. The molecules (or atoms, or ions) in all substances above 0 K move rapidly – in solids they vibrate fast even though they stay almost in one spot, in liquids and gases they move at an average of a thousand miles an hour (sometimes at 0 for an instant after colliding head on and, once in a while, going 2000 mph). Thus, every ‘system' above 0 K has motional energy in its molecules (and liquids or gases have the potential energy of phase change also). This innate energy in molecules is the enabling factor that makes entropy change possible.

Second, we should recall that all energy is quantized, i.e., it is on specific energy levels. The energy of slower moving molecules are on lower energy levels than the energy of faster moving molecules. If there are two or more atoms in the molecules of a substance, their energy of rotating and of vibrating involves different energy levels (see Figures 1 and 2 in http://2ndlaw.oxy.edu/entropy.html). Therefore, there are an enormous number of levels on which the different motional energy of each of the different molecules in a system may be at any one instant. And because they are constantly colliding and losing or gaining more energy in each collision, all of them could be on slightly different energy levels the next instant.) These possibilities for different distributions of molecular energies are the actualizing factor in entropy change. It is their existence that allows the total energy of a system to be in different energetic arrangements.

Now that we have some idea of the energetic movement of molecules and the levels on which their energies can be classified, we aren't fooled by anyone talking just about the ‘positions' of molecules in one bulb of a two-bulb apparatus that is shown in most chem textbooks. Sure, if you should open a stopcock between a bulb that has a gas in it and a bulb that has been evacuated, the molecules of the gas would move into that other bulb. They would have different positions. Sure. But there are two reasons why: First, as we said above, it is because those molecules are moving and colliding. And second, as mentioned above (but is not obvious) in a larger volume there are more energy levels available for the molecules' energies without any change in their total energy. It is this actualizing factor of more energy levels that is responsible for the increase in entropy of the energetic molecules in the larger volume, not just that there are “more positions for the molecules.”

We have quickly introduced here the ideas of quantization of molecular energies on energy levels and the relation of that to a cause for entropy increase when a gas spontaneously increases its volume. A next step in understanding molecular thermodynamics is to think about something that is really impossible to visualize, but from talking about it we can get a sense of the molecular energetic situations that we deal with in chemistry:

Imagine that you had a mole of gas in a glass bulb and you could take some kind of an instantaneous freeze-photo of all the arrangements of the molecules' energy on all the energy levels that they could be (Then in the next instant, of course, there would be a different energetic arrangement even if only two molecules collided and came out with different speeds.). The name for this knowledge ( the “instantaneous photo”) of the total energy distribution of all the particles in a system is a microstate. Although there are a really gigantic number of microstates for a mole of any substance – whether gas, liquid, or even solid – that number can be calculated and the number of microstates of a system at a given temperature and pressure and volume determines the entropy of the system. The correlation of entropy with the number of microstates is shown by the Boltzmann equation of S = kB ln W, where W is the number of microstates. (kB is Boltzmann's constant that is equal to R/N and R is the gas constant with N being Avogadro's number)

An entropy change would be S = kB ln WFinal /WInitial and the generality is “The greater the number of microstates in only one of which at any one instant, the total energy of the system might be, the greater is the entropy.” That very specific phrase, “in only one of which”, is important because it is easy to infer that “the more the states, the more the energy is smeared out over all of them”, but this is impossible. All of the energy of a system can only be in one microstate at any time. That a system always changes, if it can, to a state with a greater number of accessible microstates (and thus increases in entropy) means simply that then there are more choices in any one of which the system can be at the next instant.

Now, recapping “Case (2)” of how spread out is the original energy of a system in terms of our new ideas about microstates:

When the volume of a system is increased with no change in energy, its energy levels become closer together, i.e., more energy levels become accessible to energetic molecules that are within that original energy range. Thus, the original molecular motional energy isn't any larger in the larger volume but, having more energy levels means that the system has more accessible microstates. Thus, because there are more choices in one of which the total energy can be the entropy increases. It is no special new variety of "positional" entropy; it is just the same old entropy increase that is due to enabling motional energy becoming actualized because of additional numbers of accessible microstates.Then, in “Case 1” situations of how much energy has been dispersed to/from the system, i.e., when the total energy actually increases or decreases, in terms of microstates:

When a system is heated, its motional energy is increased and, because of this, many new higher energy levels become accessible to the more energetic molecules in that system. Therefore, there are a huge number of additional microstates, a greatly increased number of choices in any one of which the total energy can be at any instant. Thus, since WFinal in the Boltzmann equation has increased, the entropy increases. This isn't some special "thermal" entropy increase; just as we found out above about "positional" entropy increase, it is just plain entropy change! If the molecular motional energy of a system has more accessible microstates available to it for any reason, its entropy increases. (Of course, a decrease in the number of accessible microstates results in a decrease in entropy.)

There is only one kind of entropy change, a change in the number of accessible microstates, in any one of which the energy of a system might be at any instant. This is the meaning of energy dispersal in terms of microstates .

Entropy change is due to energy dispersal to, from, or within a system (as a function of temperature, i.e., temperature must always be considered in evaluating entropy.)

When a system is heated, energy becomes more dispersed because additional higher energy microstates become accessible, in any one of that greater number, the total energy of the system might be at any instant.

Even though no temperature changes in a system, and the energy of the system is unchanged, the motional energy of its molecules becomes more dispersed (because there are additional accessible microstates in any one of which that original might be at any instant). This pertains to (1) ideal gases expanding into a vacuum, (2) ideal gases or ideal liquids mixing, (3) phase change, or (3) solvents dissolving ideal solutes.

The processes differ widely but energy dispersal (in any one of the number of microstates) and its quantitation in S = kB ln [microstates Final /microstates Initial] are common to all.

Why does cream mix in coffee?

Without any additional energy input, cream mixes in coffee, a drop of ink in water drifts and eventually colors a whole glassful, perfume (or skunk odor) moves slowly throughout the completely still air in a closed room. Gases like the nitrogen and oxygen in air mix spontaneously when brought together, without any energy change, but never unmix spontaneously, just as similar liquids like water and rubbing alcohol, when poured together, mix but will not unmix.

All of these phenomena are simply understood on the basis of the preceding section: they all involve no change in the original energy of the system -- but a large spreading out of the molecules occurs because then more microstates become available to the energy of those molecules! Fundamentally, it is a spontaneous process because the enabling factor of molecular motional energy is actualized (allowed to operate) by the presence of many additional accessible microstates in the larger volume. (You might say, "Why drag in that idea of microstates?? It's obvious -- if molecules are moving fast and bumping into each other, of course they'll go wherever they can -- into a vacuum or mixing with other stuff." That's crudely “in the right direction” but you'd be wrong scientifically. There is no reason that bumping and moving fast won't cause the molecules to stay right where they are, jumping from one microstate at one instant to another the next -- unless there is the chance to have more choices, to have more microstates in any one of which they might be. It isn't the larger volume that let's the molecules move; it's where their energy can have more choices in which to be, and that means additional accessible microstates. )

### Acknowledgement

I thank Norman C. Craig, Emeritus Professor of Chemistry, Oberlin College, for suggesting the word “enable” to describe the influence of kinetic molecular energy on the process of entropy change and for aiding me to see that the kinetic energy of colliding molecules per se cannot result in volume expansion of a gas into a vacuum nor in an increase in entropy.