# Entropy

...the pieces don't tend to fall back together.

### The laws of thermodynamics

• What happens if you put a hot rock into the bottom of a cold glass of water?
• The zeroth law of thermodynamics: "any two objects that are in contact will eventually come to the same temperature."

• Will that temperature be hotter than the rock? colder than the water? Somewhere in-between?
• The first law says that energy is conserved if you take into account thermal energy.

• If there's no friction and you take a movie of some moving object, can you tell whether the movie is running forwards or backwards?
• The second law has to do with entropy and...

### The arrow of time

Newton's equations work just as well backwards as forwards.

(very little friction): In the middle, can you tell if the movie of these pendulums is being run forwards or backwards?

...and yet, even the one that you can tell is shot backwards doesn't violate any laws of physics that we know about so far:

Why couldn't a bunch of water molecules in a lake all of a sudden find themselves moving in the same direction, so as to eject the human swimmer?? And yet we know this rarely ~ never happens.

Coin-flipping exercise

If you are fascinated by the meaning of entropy, read The Library of Babel--a short story by Jorge Luis Borges.

### Second Law of Thermodynamics - in terms of heat

The law of heating is one way of stating the second law...

Thermal energy flows spontaneously from higher to lower temperature, but not from lower to higher temperature.

What does that mean for our hot rock and cold water situation?

## "Heat" Engines

A lot of modern engines work like this:

1. Some form of input energy (not thermal),
2. is converted to heat (thermal energy),
3. which is then converted to mechanical energy.

Many modern machines use heat to make some sort of mechanical motion happen. In a gas engine...

• The atoms in molecules re-arrange themselves into new molecules, and energy is released...as heat,
• The heat increases the temperature of the air in the cylinder.
• High temperature air has a much greater pressure than low temperature air, so the heated air pushes on the cylinder.

In a nuclear-powered submarine,

• The neutrons and protons in atomic nuclei re-arrange themselves into new nuclei, and energy is released...as heat,
• The heat is used to boil water into steam, which has a much higher pressure than the same amount of liquid water,
• the steam is forced through a turbine making a shaft turn...

### Steam engine

1. Heat is generated, for example:
• Coal / wood / natural gas / etc is burned (ChemicalE $\rightarrow$ ThermalE) to generate heat, or
• A nuclear reaction happens (NuclearE $\rightarrow$ Thermal E) to generate heat,
2. the heat is used to raise the temperature of water above the boiling point,
3. generating high pressure steam.

You can use the steam to push a rod back and forth (howstuffworks.com) to power a steam locomotive,

or more commonly:

The expanding steam pushes against the blades of a turbine to make a shaft turn,

If you attach the shaft to magnets, and rotate them past electric coils you can cause a current to flow--*electricity*.

A heat engine is a device that uses thermal energy in some form, converting some of it into mechanical energy (KineticE) in a cyclic process.

### Efficiency of heat engines

Just like any other energy transformation, a portion of the input energy goes to...ThermalE.

What's the efficiency of the cartoon heat engine shown?

energy efficiency = $\frac{W o r k_{out}}{ThermalE_{i n}}$

Example - A certain steam engine gets 50 kWh of heat from burning coal, and performs 12 kWh of mechanical work. What is its efficiency?

efficiency = 12 kWh / 50 kWh = 0.24 = 24%

### Second Law of Thermodynamics - Heat engines

The law of heat engines is another way of stating the second law...

Any cyclic process that uses thermal energy to do work must also have a thermal energy exhaust: Heat engines are always less than 100% efficient at using thermal energy to do work.

### Heat engine efficiencies

There is a pattern to be found here, for how the efficiency depends on $T_{in}$ and $T_{exhaust}$. Stare at the table, and see if you can see it...

• Remember absolute temperatures? Absolute zero in Kelvin 0K is -273 C.
• The size of a kelvin and celsius degree are the same.
• So 0 C = ____ K?
• Room temperature ~25 C = ______ K?
• 400 C = ______ K?
Using absolute temperatures (measured from absolute zero), it turns out

$$\text{ efficiency}_{max} =\frac{T_{Hot}-T_{Cold}}{T_{Hot}}= 1-T_{Cold}/T_{Hot} .$$

So the theoretical best (e.g. no engine friction...) you could do with a gasoline engine is... $$eff = 1-T_{cold}/T_{hot}=1-298/673=0.557 = 56\%.$$

So...how could you change a gas engine to be more efficient (and thus save lots of energy!)??

Efficiency of an engine with high temp of 500 C: $$1-298/773=.614$$

These days...

• almost all electricity is generated using heat engines.

Gas engines: Fuel and gas are compressed, and then ignited with a spark plug.

When you compress a gas it gets hotter. Problem: pre-ignition, so, you'd better not go above the temperature for spontaneous ignition

Diesel engines: No spark plug. Air is compressed to a greater pressure than in a gas engine, get's way hot, then diesel is sprayed in and ignites spontaneously.

Gas in the UK currently costs ~2.5 $\times$ U.S. gasoline (> $7 / gallon) 35-50% of passenger cars sold in Europe are diesels. Should you always avoid the lowest efficiency heat engines? Must the exhausted heat *always* be wasted? Look up "co-generation"... Possible downsides of diesel engines?? ### Things run down Thermal energy has a lower "quality" in this sense: All other energy forms$\rightarrow$ThermalE: Easy to convert 100% to ThermalE. ThermalE$\rightarrow$any other energy form: Efficiency is often pretty low (much less than 100%) . ### Are heat engines the only kind of engines? DC electric motors can have efficiencies of more than 95%. ### Is the human body a heat engine? If the human body were a heat engine, what would its efficiency be?? Take$T_{in}$= body temperature = 98.6 F = 310.15 K, and$T_{ex}$= atmospheric temp = 70 F = 294.3 K$

The efficiency of a heat engine operating between those two temperatures would be... $$\frac{T_{hot}-T_{cold}}{T_{hot}} = \frac{T_{310}-T_{294}}{T_{310}} = 0.052 \approx 5.2\%$$

But we know its actual efficiency is closer to 25%.

This means our bodies are much more amazing than internal combustion engines (as if you didn't already know...): Our muscles are converting chemical energy directly to mechanical energy, without going through a stage where the chemical energy is turned into heat.

### Organized motion - water drop

What happens to the organized, collective motion of a water drop once it hits?

Consider two groups of 6 atoms.

• each atom has the same mass $m$
• each atom has the same speed $v$

So, the total kinetic energy of each collection is the same: $6*\frac{1}{2}mv^2$.

The only way they differ is in their degree of disorganization = entropy.

a) - organized motion might be the atoms in a drop of water all moving the same direction, or a bunch of electrons (in an electric current) all moving in the same direction.

b) - disorganized motion: "thermal energy".

If you want to use the kinetic energy from a group of atoms to push a car to make it start going, or lift a bucket of water from a well... Would you rather start with a) or b)??

### Organized motion - hot and cold

A heat engine could be used to connect a hot container of molecules, and a cool one and extract some work.

After simulating mixing hot and cold molecules .... Is there any way to get work out of the mixture? With a higher or lower efficiency than if we had kept the hot and cold separate?

• Why do they just mention thermal energy flowing from a hight-temp cup to a lower-temp hand and not mention the air temp around it? Wouldn't the thermal energy also being flowing into the cooler air?
• More examples of microscopic disorganization...?
• Do any other forms of energy [other than ThermalE] have this irreversible aspect?
• What about the process of freezing? Does entropy increase in that case?
and
What is meant by "the entropy of of all participants cannot decrease, but can increase". Can't water freeze and have its atoms slow down into a more uniform pattern?
• What does the law mean in terms of calculating physical processes?
• If energy can never be completely gone [lost], how does the energy used to break apart bonds or make things into steam get transferred to something else?
• Does entropy exist at all different temperatures of water?
• [Re: that example where the leaf gets "organized" with help from the sun...] Is it ever possible for something to do that without help from the sun?
• When we humans consciously organize items, are we the only occurances of organization happening in the universe?
and
When we heat something, is that the act of speeding entropy? Or is this usually just a natural process?
• Heat death in the future and lower entropy in the past..., Does entropy *cause* the end of the universe?

### 2nd Law as Law of Entropy

Entropy is a quantitative measure of a system's microscopic disorganization.

The total entropy of all the participants in any physical process cannot decrease during that process, but it can increase.

### Entropy violator?

[conceptual exercise #22]

A pan of water is set outside on a cold day.

Initially it is a liquid, but after a while, it freezes solid.

Molecules in a solid are more ordered than atoms in a liquid. So, is this a violation of the "Law of Entropy"?

[writing .... ]

1. What if the liquid water was cut off from any larger system: for example, put inside an insulated thermos bottle. Would it still freeze (just as fast)?
2. Back to the open pan of water: Is there a larger "system" that the pan of water is in contact with?
3. What happens to the larger system as the pan of water cools down? How would you talk about this in terms of entropy?

#### Selected responses

As the water cools, heat escapes in the form of atoms. These atoms are very random, which goes along with entropy...?

Heat is mostly a flow of energy, not atoms: What happens is that

1. the water atoms are initially moving very fast,
2. they collide with air molecules that are moving (on average) slowly.
3. At the same time, the warm atoms in the initally warm pan are bumping into the cold (slow-moving) atoms of the sidewalk that the pan was set down on.
4. After many, many collisions, the water molecules, and the pan molecules are moving, on average, slower than they were initially. The molecules in the air and in the sidewalk are moving, on average, just a little faster.
5. When you suck a small amount of energy out of a small system (the pan of water) its temperature will drop noticably. (Its entropy decreases.)
6. But when you add that small amount of energy to a big system (the air and sidewalk), that energy is shared around the whole system, and the temperature change might be too small to measure. But its entropy (as well as its energy) increases.
Some of the heat from the air would go into the water.

Well, no. On average the fast water molecules are losing energy, and the colder (slower) air molecules are gaining that energy.

The atoms in the water and air combine causing disorganization.

Well, no. Here it's mainly just energy that is exchanged between the pan of water and the environment. More slowly moving water molecules are more "organized" than faster moving ones.

But this change in temperature is not the only way for atomic systems to get disorganized. It is true that if you mix orange juice and pineapple juice, the resulting mixture, where orange molecules and pineapple molecules are evenly spread out in the juice pitcher is less organized that the separate juices that you started with.

### Suggested exercises

Conceptual exercises in Chapter 7: 1, 4, 6, 9, 10, 11, 16, 19, 22

Image credits