# Entropy -- implications of the 2nd law of thermodynamics

#### Prerequisites

How does the very general statement about probabilities and states that we developed in our discussion of the second law explain why cold objects don't spontaneously transfer heat to hot objects, or why resting chairs don't spontaneously pick up heat from the floor and start moving? Let's go through our examples on our motivating page (The 2nd Law of Thermodynamics) and see how adding the concept of micro and macrostates and the idea of entropy helps to explain why things run spontaneously in the direction observed.

### Revisiting thought experiment 1: A hot object and a cold object

Let's consider the case of a system composed of a hot object placed in contact with a cold object made of the same material and of the same mass. We do this so we can talk about total amount of energy and not worry about the important but complicating factors of density of energy and specific heat. Remember, it's temperature — the density of thermal energy, not the total — that tells us which way energy flows. And remember, each kind of material translates heat into temperature in its own way. With the subtleties of density vs total energy and energy translated into temperature suppressed by our choice of example, we can just look at how much energy is transferred between our two identical objects.

Another complexity is that energy is a continuous variable. To "count" the number of ways energy can be distributed is very tricky. It requires breaking the energy up into tiny pieces, creating sums, taking limits as the pieces go to zero, and changing the sums to integrals. But the sense of the process can be seen by modeling the energy as coming in finite-sized blocks that can be shared between the two objects in different ways. This is much easier to think about, and, in fact, the key reasoning is identical to the more formally correct reasoning using limits.

So our model is a simple one: Two isolated identical objects that can share a set of energy blocks between them, but that have no exchange of energy or matter with anything else in the universe. Let's say that the cold object starts with 2 blocks of energy and the hot object starts with 8 blocks of energy, and let's assume that one block of heat is transferred from the cold object toward the hot one. (Remember: "Heat" refers to blocks of energy transferred by thermal contact. It's not "just" the energy itself.) The result is that the hot object now has 9 "units" of energy and the cold object is left with just 1 block.

What does such a process do to the entropy of the system?  Clearly, the entropy of the system has gone down! There are fewer arrangements consistent with the 9/1 energy distribution than there are arrangements consistent with 8/2 energy distribution, just like 9 H/1T is less likely in flipping a coin 10 times than is 8H/2T. Now consider, instead, that the hot object transfers one "unit" of heat to the cold object. In this case, the system's entropy has gone UP, since the 7/3 distribution is consistent with more arrangements than is the 8/2 distribution.

We see, then, that in this simple case of a hot object placed next to a cold object, the probabilistic statement of the Second Law of Thermodynamics correctly tells us the direction in which heat is likely to transfer. As the temperature difference between the objects gets greater, the Second Law says that the direction of heat flow becomes almost solely toward the colder object.

We've just described this with a simple one-step model to see how it works. This calculation is worked out in more explicit detail in the follow-on examples

There is another, more mechanistic way of understanding what happens when a hot object is placed in contact with a cold object.  Instead of considering probabilities, think for a moment about what happens when a hot molecule comes in contact with a cold molecule via a collision.  In such a collision, the hot molecule is much more likely to transfer some of its kinetic energy to the cold molecule, thereby producing two molecules whose temperatures (energies) are more similar to each other than they were prior to the collision. (For the slow molecule to speed up the fast one it would have to be in just the right place to catch the fast one from behind — possible, but very unlikely.)  The result, therefore, is the same as what one would obtain using probabilistic considerations of entropy alone: the two objects move toward an equilibrium temperature lying somewhere between the two extreme temperatures.

### Revisiting thought experiment 2: The sliding chair

Consider now the case of the chair sliding across a floor with friction and coming to a stop, an example of a process that exhibits a uni-directionality not explained using just the 1st Law of Thermodynamics. How can we understand this irreversibility in light of the 2nd Law's statement about entropy?

What would have to happen for the chair to collect heat from the floor and start sliding across the room in a straight line? All of the molecules that were vibrating randomly about their equilibrium positions in the floor would have to, for at least a moment, all start vibrating in the same direction, so that the legs of the chair would develop motion in a particular direction. The likelihood that all of the floor molecules will suddenly align themselves in a particular direction is incredibly small.

Put another way, the entropy of the floor's molecules is MUCH larger when they are vibrating randomly than when they are all moving in a single direction.  So while it is exceedingly likely that the floor will cause the sliding chair to stop moving, it is exceedingly UNlikely that the floor will act in reverse, causing the chair to start moving again.

### Revisiting thought experiment 3: The smoke-filled room

Can you now do this analysis yourself for the case of smoke diffusing out through a room? You are now in a position to understand, at both a mechanistic and probabilistic level, why it is that smoke emanating from the corner of a room spreads out to fill the room while the reverse process does not spontaneously occur. The randomly oscillating air molecules in the room are incredibly unlikely to direct their vibrating efforts in only one spatial direction; in doing so they would be lowering their own entropy tremendously. Do you see why this makes it incredibly unlikely that the smoke will re-coalesce in the corner? This shows that the 2nd law is the fundamental physical principle that explains why diffusion happens.

Even our qualitative probabilistic version of the 2nd Law of Thermodynamics has gotten us pretty far down the road toward understanding why some thermodynamic processes occur and others do not. We've unpacked a number of mysteries and seen why, at least in the macroscopic world we all inhabit, certain processes do not occur because they would violate the Second Law of Thermodynamics. These conceptual ideas can be quantified and lead to powerful tools for analyzing what is going to happen in chemical and biological systems.

In the follow-on pages, we discuss a variety of more quantitative approaches to working with entropy.

Ben Geller and Joe Redish 12/8/11

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