Thus far in the math/science subjects:
- Math/Science Overview
- Basic Types of Numbers
- The Atom and Quantum Physics
- The Periodic Table
- Molecules and Ions
- Chemical Reactions
- The Basic Tools of Algebra
1. On a surface level, we know what heat is from our personal experience. Earlier this week I found myself outside in a windy cold without a jacket. When I got indoors, it was wonderful to feel "the heat." On the other hand, I've been to Greece and Israel in the summer when it was almost unbearable to be outside in the mid-day. I wanted to get out of "the heat."
In chemistry and physics, heat refers to the transfer of energy due to temperature differences. When I came inside from the cold, the warm building transferred energy to my skin, and my body liked it. In Greece, the energy transfer was almost too much, making me want to jump in a pool or go indoors.
Temperature is a snapshot of the level of this energy. Technically, it is the amount of kinetic energy of the molecules of a material, where kinectic energy has do to with how much those molecules are moving. The slower the movement of the molecules, the lower the temperature. The higher the movement, the higher the temperature.
2. There are different ways to measure temperature. In everyday life, we have thermometers. These often work on the principle that liquid expands when heat is added. So you put a scale of some sort on the side of a container with a liquid, then the liquid will take up more space in a hotter temperature and less space in a lower temperature. As the liquid moves up or down the scale, you have a measurement of temperature.
The actual scale itself is mostly arbitary. Scientists use the Celsius scale, which sets 0 at the freezing point of water and 100 as the boiling point of water. One degree Celsius is thus the difference in temperature that, if done 100 times, will make ice first become a liquid and then become a gas.
The freezing point is the temperature at which water turns from a liquid to a solid (at normal pressure). This is the point which can make the difference between rain and snow. The boiling point is the temperature at which water turns from a liquid to a gas. We call the transition from a gas to a liquid condensation, while the other direction is called evaporation. The transition from a solid to a liquid is called melting.
These are all concepts we encounter all the time in everyday life. The wind in our face is gas pushing on our faces. We shower in liquid water, and we sleep on a solid bed. If you add enough heat to anything, it will eventually become a liquid and then a gas. If you remove enough heat from something, it will eventually become a liquid and then a solid.
3. The United States is nearly alone in the world in still using the older Fahrenheit scale.  In it, the freezing point of water is at 32 degrees and the boiling point at 212 degrees. If you divide the 100 degrees Celsius between the freezing and boiling temperatures of water by the 180 degrees Fahrenheit between the two, you can see that a Celsius degree is 5/9 the size of a Fahrenheit degree.
We can thus come up with formulas to convert from one to the other.
C = 5/9(F-32)
F=9/5C + 32
F=9/5C + 32
So you take away 32 degrees from F to calibrate for the fact that the freezing point is 32 degrees different. Then you calibrate the size of the degree by using the 5/9 ratio. Going the other way is basically the same process in reverse.
4. We've already mentioned the three principal phases of matter: solid, liquid, and gas. Every element and molecule has a temperature at which it changes from one state to the next at normal pressure. Since temperature has to do with the movement of atoms and molecules, we can project down to a point where such molecules would have no energy whatsoever.
This point is called absolute zero, the absolute lowest temperature there could be. It turns out to be -273.15 degrees Celsius. Lord Kelvin (William Thomson, 1824-1907) had the idea that there would be such a point, so a third temperature scale takes the Celsius unit but approximately starts at absolute zero. So 0 degrees Celsius is 273° Kelvin, and the boiling point of water is 373° K.
5. The specific heat of a substance is the amount of heat (or calories) that it takes to raise the temperature of one gram by one degree Celsius.  Water is the standard. The specific heat of water is 1 calorie/gram. That means that 1 calorie is the amount of heat it takes to raise the temperature of water 1 degree Celsius.
The heat of fusion is the amount of calories that it takes to move a substance from a liquid to a solid. During this transition, heat is added but the temperature of the substance does not change. For water, it is 79.7 calories/gram to change the state of the ice to liquid. The heat of vaporization is the amount of calories to move a substance from a liquid to a gas. It turns out to take much more energy. For water, it takes 539 calories/gram to change water into steam vapor.
6. As a substance absorbs energy, its molecules become more excited. They vibrate more. Their kinetic energy, the energy of movement, increases. The substance expands. This is the general rule. As temperature increases, the substance expands. As temperature decreases, the substance contracts. 
Bridges usually have metal expansion joints because the steel of the bridge expands and contracts significantly. Steel is a good conductor of heat and so expands more than other materials. Other materials are good insulators with high specific heats (like the materials we use in our homes to "weather proof" them). So the joints allow the steel to give and take without destroying the bridge.
7. What we are saying is that volume and temperature are directly proportional. As the temperature goes up, the volume goes up.  The same is true of pressure. As the pressure goes up, the temperature goes up. Meanwhile, volume and pressure are inversely proportional. As the pressure goes up, the volume goes down. As the volume goes up, the pressure goes down. 
When you put all these elements together for gases, we get the ideal gas law: PV = nRT. P is the pressure. V is the volume. T is the temperature. n is the amount of substance.  R is a universal gas constant.
8. Earlier, we mentioned that water froze and boiled at a certain temperature "at normal pressure," What we mean here is the pressure of the air, the atmosphere, at sea level. Accordingly, this standard is called one "atmosphere" of pressure.
There is a point when the pressure, volume, and temperature are such that all three phases (solid, liquid, gas) are in equilibrium, called the triple point. If the pressure on water at 0.01° C is lowered to 0.00603659 atm, then water can go straight from gas to ice or to liquid with only minor variations in temperature and pressure. All substances have a triple point of this sort.
Plasma is a fourth state that matter can take when it reaches outrageously high temperatures and strong electromagnetic fields, causing molecular bonds to break and the atoms to become highly ionized. The sun is mostly plasma as is the majority of the universe.
9. There are four laws of thermodynamics. The zeroth law is fundamental to all of them, but was added after the others (thus zeroth). The zeroth law is that the temperature of two bodies or systems, when they are in contact, will eventually reach thermal equilibrium--they will end up having the same temperature. This is what happens when you have two rooms, one of which is hot and one of which is cold. If you open the door between them, eventually they will both have the same temperature.
The first law basically states that energy cannot be created or destroyed, the conservation of energy. It changes form, but it doesn't disappear or come out of nothing.
Many have heard of the second law of thermodynamics. It states that the entropy or dissipated heat that cannot be recovered from a system tends toward a maximum. From a broader perspective, you might say that the universe tends toward overall disorder rather than order. There will never be a "perpetual motion machine" that never stops without adding more energy. You can spin something really fast, but it will eventually slow down and stop.
Finally, the third law states that entropy finally comes to zero at absolute zero. So when the universe finally has dissipated its heat in entropy until the universe finally reaches equilibrium at absolute zero, entropy will finally reach zero as well.
10. The principles of thermodynamics, extensively developed in the 1800s, made possible inventions like the automobile and the refrigerator. The internal combustion engine converts the expansion of gas as it is heated into mechanical energy.  A refrigerator cools a space by a liquid that evaporates at a low temperature (freon usually). As it expands in coils inside the fridge, the substance pulls heat from the space inside the fridge.
Next Week: Math/Science 9: Basic Geometry and Trigonometry
 I personally think this fits with a general resistance to science among Americans on a popular level.
 The joule is 4.184 calories.
 Water is highly unusual in that it actually expands from about 4° to 0°, This is why pipes break during a freeze. The water expands as it freezes in the pipes, breaking the pipes. Then when the water melts, the pipe no longer is intact.
 Called Charles' Law
 Called Boyle's Law.
 The amount is measured in "moles." A "mole" of a substance is an amount first defined by Amedeo Avagadro (1776-1856), also called Avagadro's number. It is defined as the number of carbon atoms in 12 grams of carbon 12 (see an earlier post for what an atomic number is). Carbon 12 is a carbon atom with 6 protons and 6 neutrons. The number of carbon atoms in 12 grams of carbon turns out to be 6.02 x 1023 atoms.
 Called the Otto cycle.