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When we're talking thermo, we're talking about systems. A system could be the gas in a room, a chunk of metal, a beaker full of water---anything with a bunch of atoms. A system has volume, pressure, temperature, and internal energy. But, since we've started running into a problem with reversibility, you might start to wonder whether there's something important missing from this list.

Let's look at the Carnot cycle and the Otto cycle side-by-side:

Can you spot the difference?

The difference is when we're adding or subtracting heat in the Otto cycle, we let the temperature change. We never do that in the Carnot engine. And since heat only flows from hot to cold, these two steps (2 and 4) are irreversible. If you ran the film backwards, the air in the piston would dump a bunch of its heat into the hot bath to reduce its temperature, which never happens according to the 2nd law (heat flows from hot to cold). At the same time, while we were heating we lost the opportunity to do work. So, what we're looking for is a quantity that only shows up when we missed that opportunity. Let's see if we can quantify "lost opportunity to do work", or irreversibility. The quantity we're looking for will be zero for the Carnot engine, and nonzero for the Otto engine.

We're going to look at heat divided by temperature. That is, for some stage of a process, we'll take the heat added and divide that by the temperature it was added at. This is a simple thing to do. For the Carnot engine, the compression in step 1 has no heat added or subtracted. Step 3 also has no heat. In Step 2 the temperature stays the same, and some heat is added. In step 4 the temperature stays the same and heat is thrown out.

Whatever heat divided by temperature is, if you add all of them up for a Carnot cycle, you get zero. Now, that's for the engine. But suppose we want to ask what the heat divided by temperature is for the entire universe. That is also zero! So, the Carnot engine does not change heat divided by temperature for the entire universe. It's zero. This reflects the fact that no opportunity to do work was lost, and that whatever we did can be undone. (In this case, we just put the work done back in, and all the heat will be put back where it was. The Carnot engine is reversible, so it can always be run as a refrigerator if we run it backwards by putting work into it.)

Now let's look at the same thing for the Otto cycle. Again, Steps 1 and 3 have no heat, so is zero. In stages 2 and 4, the heat can be calculated for the engine.

Now, for the Otto cycle, we should also check what heat over temperature is for the universe.

Let's go ahead and give "change in heat over temperature" a name: it's the change in entropy. The entropy of a system, always denoted by , is something that depends on its temperature, pressure, volume, etc. For an engine, just like all the other variables I mentioned, must be reset at the end of the cycle. And, if there is ever a change in for the universe, that represents that somebody did something irreversible, and that they didn't do the maximum possible work. From then on, nobody will ever be able to undo that act: the total entropy of the universe is a big number, and when you increase it, it never decreases again.

There are lots of processes in thermo that we can think of that are irreversible. That means that if you ran a film of the process backwards, what you see would never happen. When you put an ice cube into lukewarm tea, the ice cube melts. If you run the film backwards, you see a cold drink suddenly warm up while simultaneously forming an ice cube. This doesn't violate energy conservation, but it never happens.

Or consider mixing two gases of different types. If you have nitrogen in one bottle, and oxygen in another, and you hook them together, the gases both fill the total available space and mix. But if you run the film backwards, you see a mixture of gas spontaneously segregate: all the nitrogen rushes to one and all the oxygen to the other. This never happens. You could wait billions of years and never see it.

Earlier we discussed The Carnot engine, which had 4 stages (called strokes, if you're interested):

1. expansion at doing work
2. expansion where temp changed from to .
3. compression at
4. compression from to

A normal car engine has the same number of stages

1. Air and gasoline are let into the piston and compressed.
2. The mixture is ignited, rapidly increasing the temperature.
3. The piston extends, doing work.
4. The combusted gases are expelled

When the gasoline/air is ignited in step 2, the hydrocarbons undergo combustion, making carbon dioxide and water vapor, plus a bunch of heat.

Now, here's a funny question: Could either of these two engines can be run backwards?

For the Carnot engine the answer is: absolutely. It would work the following way. The most important thing to observe (which may not be totally obvious) is that the net effect is to take in work from somebody and move heat from the cold bath to the hot bath.

1. Start with the gas at and allow the piston to push out until the temp falls to . No heat is absorbed.
2. Continue allowing it to expand in contact with the cold bath. It will absorb heat to keep its temp from dropping.
3. Now push it in while not letting heat in or out, which will increase its temperature to .
4. Push while it's in contact with the hot bath, so that it gives heat to the bath to remain at .

The net effect is to take in work and move heat from the cold bath to the hot one, which is what a refrigerator does. For this reason, this is called a Carnot refrigerator. It's the same as the diagram for the Carnot engine, but with all the directions reversed.

But the car engine can most definitely not be reversed. Think of it: you shove the piston down and the CO₂ and water produced in combustion spontaneously turn back into gasoline and air ... no way could that happen!

We also know that the Carnot engine is very efficient, whereas the car engine is not. This is not a coincidence. It is almost as if there is a "right way" and a "wrong way" to do things. Slow and reversible is right, fast and irreversible is wrong. (In no way should these "right" and "wrong" designations be construed as normative judgements.)

It is, in fact, possible to prove that the Carnot is better than any other engine between the same two temperatures. But since car engines are the most likely kind of engine you will ever see, let's just analyze that one.

The description of the four stages above is accurate, but we'll have to change the way we phrase things to determine the efficiency as compared to a Carnot engine. The cycle, called the Otto cycle (that shouldn't be too hard to remember!) is in every important way the same as this:

1. Some gas in a piston is compressed from to with no heat from the outside.
2. The gas is held at constant volume while it is put into contact with a heat bath at temperature until its temp is
3. The piston expands, doing work, with no heat from the outside.
4. The gas is put into contact with a cold bath at and allowed to cool to .

So, we get around the combustion and exhaust by substituting our hot bath and cold bath. In part 1 we do work, in 2 heat is absorbed, in 3 the engine does work, and in 4 heat is expelled. We can now calculate the efficiency

The efficiency of the Otto cycle (last equation in the aside above) does not depend on the two temperatures, but rather on the compression ratio, or the volume of the piston when it's expanded divided by the volume when it's compressed. In modern engines this can be as high as 14 or so. Nevertheless, even for a very large compression ratio the efficiency is lower than the Carnot engine operating between the same two temperatures. Note that this is for an ideal Otto cycle, without friction in the piston etc., so car engines will never realize this.
Considering the temperatures most often used, with the hot temperature at about 1000 K and the cold temperature at about 300 K, you see the efficiency vs. compression ratio for the two engines displayed below.

(Of course, you can make a Carnot engine with low efficiency by operating it at two temperatures that aren't very different. I'm only addressing practical temperatures here. It is not the case that every Otto engine is worse than every Carnot engine, but it is the case that with only two heat reservoirs available, the Carnot is more efficient.)

But why is the Carnot always better? Why is reversibility always the best for efficiency? We must be doing something fundamentally different. But what could it be?

So far I've kind of just talked about heat moving from one system to another, without much regard to how that transfer actually happens. The way that heat actually gets around is hugely important, so we can't keep avoiding the issue.

Of course, we know that heating is the transfer of energy from one set of atoms, which I'll call object 1, to another, object 2. Aside from exchanging particles, what are our options for how to effect this? There are 3:

Firstly, we could put the two in physical contact, so that the atoms of 1 can hit the atoms in 2. For a solid this is easy. For a liquid or gas to do this we have to put some kind of membrane between the two objects to keep the particles from mixing. No big deal. This is called conduction. When you put your steak right on the grill, the heat is conducted from the hot grill into your steak because the jiggling iron atoms in the grill knock into the atoms in the steak and start them moving faster. This relies on there being a significant amount of area in contact.
Secondly, we could use some medium, like air, to transfer the heat. That is, we could let object 1 heat some air, and then blow that hot air over object 2. The net effect is that heat was transferred from 1 to 2. This is called convection.

Thirdly, we could use the fact that all objects emit electromagnetic radiation to transfer the heat. Object 1 radiates light, and the light carries energy. When that light hits object 2, it can absorb it. This is called radiation.
That's basically it. Conduction, convection, and radiation are the ways to get thermal energy from one thing onto another. I suppose you add in "an engine", but the engine itself is really some combination of these 3, so it's superfluous.

There are actually grills and ovens that do each of these things. Grills and pans use conduction, by putting the food in contact with them. This heats the side in contact very quickly. Since the food doesn't do a good job conducting the heat through itself, this mainly cooks the meat near the surface, allowing for searing. Convection ovens offer more even heating while still being relatively fast because they heat all sides of the meat which are accessible by the air flow. A normal oven uses radiation and heats the air inside it so that heat goes into the food from all sides. Is that conduction or convection? I have no opinion. Recently, some grills offer infrared heating, which uses radiation that isn't visible to cook.

What accounts for the rate of heating? If you put two things at very different temperatures together, heat flows quickly from the hot one to the cold one. If the two objects are not too different, the the heating is slower. The heating rate is usually proportional to the difference in temperature

This isn't very surprising. Now, if we want an equality we just need a constant out front. In this case we call the constant . doesn't have a name, so people just call it , or R-value. Then

Large R-value is a very good thing if you're trying to insulate something like your house. Then, even if it's very cold outside and very warm inside, the heating rate won't be too big, making you have to run your furnace all the time.

In the US, the R-value given for insulation is in ft² F h/BTU. Supposing that it's 75°F inside and 30°F outside, if you have R-13 insulation, the house will lose 3.5 BTU per square foot of wall every hour. To maintain the temperature, the furnace has to therefore put this much heat into your house every hour. Windows have a much lower R-value, with a regular air-filled window having a value of 2, so the heat needed for every square foot of window is higher by a factor of about 6.

Now here's a relevant question we can answer: when you leave the house on a cold day, should you leave the heat turned on, or turn it off until you get back? After all, when you get home the house will be really cold, so the heater will have to run for a long time to restore the temperature. If you leave it on, it only has to run a little bit from time to time. On the other hand, what if you left for 2 months? Surely you wouldn't argue that the heat should be left on.

The answer is that you should turn off the heat. Because the heating rate (or loss of heat by the house) is bigger when the temperature difference is bigger, you're always better off letting the temperature fall.