Copyright © Karl Dahlke, 2019
There is energy in motion. Consider a 19th century train. 🚂 Coal contains chemical energy in the arrangement of its carbon atoms. When coal is burned, the resulting CO2 has less energy, and the difference is released as heat, and a little bit of light. The heat boils water, which expands into steam, which drives a piston, which turns the wheels, which moves the train forward. Soon it is chugging along the track at 40 miles per hour. If you don't believe it has energy, try to stop it with your hand. Thus motion is a form of energy, sometimes called kinetic energy.
What is heat? There have been many theories over the centuries, including the theory of phlogiston, which was completely wrong, but stood for 1500 years. I'll skip past it, because it seems kinda silly now, but I guess it made sense at the time.
Heat is the motion or vibration of atoms and molecules. In other words, the energy of heat is really the energy of motion at a microscopic level. We have connected two things that seem unrelated: the energy of heat and the energy of motion.
The gasious phase is the most intuitive. Molecules are flying around the room, and when the gas gets hotter, the molecules fly faster. That's really all there is to it.
Liquids and solids are a bit less intuitive. Picture a block of ice, where the water molecules are locked in nplace. Each is connected to its neighbors, and together they give the ice cube its strength and its solid form. Yet these molecules, and even the atoms comprising each molecule, vibrate within their fixed locations. Perhaps the connections are not like hooks and fasteners, as Democritus thought, perhaps they are more like springs. The atoms form a solid, yet they vibrate back and forth within their framework. When you heat the ice cube, the molecules vibrate faster and faster. Eventually they move so fast that the connections are broken and the molecules slide past each other. The ice has melted and is now a liquid. Molecules move about, and collide with one another, but they still hold together in the liquid state. Heat the water further, and the molecules fly away from the surface of the liquid and into the air as steam. The water is boiling, and becoming a gas. If you think about this for a while it becomes intuitive. Atoms and molecules move and vibrate, and their motion is heat. More heat means more motion, and as the atoms move faster, solid turns into liquid, turns into gas.
Atoms can always move faster, thus there is no limit to how hot a gas might become, but at the other extreme, there is a lower limit to temperature. When the atoms of an object don't move at all, it can't get any colder than that. This is called absolute zero, the zero temperature, the ultimate in cold. Daniel Fahrenheit set 0 to the coldest temperature he could easily achieve in his lab, which was rather arbitrary. Anders Celsius puts 0 at the freezing point of water, a standard that could be replicated around the world. Lord Kelvin felt 0 should mean 0, so he established a new temperature scale, indicated by °K, that starts at absolute 0 and goes up from there. There is no "below zero" on Kelvin's scale. As a rough guide, 273 °K is the freezing point of water, 300 °K is a warm summer day, and 373 °K is the boiling point of water. This scale is only used in scientific circles, but it is very useful in that context. for example, double the temperature of an enclosed gas, in degrees Kelvin, and you double the pressure. That simple relationship doesn't work for celsius or fahrenheit. It only works when the temperature scale starts at absolute 0, where all molecular motion stops.
It's hard to imagine absolute zero. The coldest winter day you might ever experience is a dozen degrees below 0. The coldest day on earth, at the south pole, is -100 °f, -70 °C. Absolute 0 is -459 °F, -273 °C. Air condenses into a liquid, then freezes solid. Everything is locked in place and nothing moves. What would it be like to hold a block of air in your hand? (You better be wearing gloves.)
If you've ever blown up a balloon, or inflated a tire with a hand pump, then you have an intuitive understanding of pressure. A liquid or a gas exerts pressure on the walls of its container, the same pressure all the way around. With this in mind, the proper way to describe pressure is force per area, the force exerted on each little patch of wall. In English units, this is pounds per square inch, or psi. Every square inch of your car tire experiences 35 pounds of force due to the air inside. Imagine a 35 pound weight pressing against a square inch of rubber. The next square inch has another 35 pounds, so that 2 square inches experience 70 pounds of force pushing outward, and so on. Tires are thick, with steel bands, to keep from exploding.
The metric unit of pressure is grams per square centimeter, or gsc. That isn't really right because gram is mass and we want force, so think of a gram as a gram's weight on earth. 1 psi is about 340 gsc.
Fortunately there is another unit that is neither English nor metric, that we can all agree on, the pressure exerted on all of us by our own atmosphere. Miles of air press down upon us from above, just as miles of water would press down upon us if we stood on the sea floor. You don't feel this pressure because you have the same pressure inside and out. The air in your lungs, your ears, your sinuses, your stomach, your intestines, even the air dissolved in your blood, is all the same pressure. This is 1 atmosphere of pressure, and it's about 15 psi. Make a circle with your index finger and your thumb; that's about one square inch. A column of air, that big around, rising all the way up to space, weighs 15 pounds, and that is true for every square inch on earth. Ok, a little less air above the mountains, but you get the idea.
So how many atmospheres of pressure are in your properly inflated car tire? Divide 35 by 15 and get 2.3. A car tire should be inflated to 2.3 atmospheres. This is net pressure, the difference between the internal air pressure and the ordinary atmospheric pressure outside. If you want to be technical, the tire has 3.3 atmospheres of pressure inside and 1 atmosphere of pressure outside. Thats a net force of 2.3 atm, or 35 psi.
In 1662, Robert Boyle published a wonderful relationship between pressure and volume, which is now known as Boyle's law. If gas is enclosed in a container, and you reduce the size of the container by half, squeezing the gas inside, the pressure doubles. This is the absolute pressure inside, ignoring the atmospheric pressure outside. If you divide the volumen by 3, you multiply the pressure by 3, and so on. This is an inverse relationhship.
An equivalent version of his law keeps the container the same size and pumps in more gas, which is what we are doing when we inflate a tire. If you double the amount of gas inside, the pressure doubles. This is really saying the same thing, because the bottom half of the container has the original amount of gas, and the top half of the container has the same amount of gas, so it has the same effect as keeping the gas constant and reducing the volume by half. The left is a container of green gas, perhaps chlorine. The middle has cut the volume in half, thus doubling the pressure. The right has pumped in an equal amount of blue gas, thus squeezing the green gas into the bottom and doubling the pressure. In reality, the blue gas and green gas would intermix; this is for illustrative purposes only.
The molecular explanation for Boyle's law is so beautiful, and so compelling, that it makes you believe in atoms, even if you didn't before. Gas molecules are flying around inside the container, and bouncing off the walls. All these collisions, trillions per second, exert a force on the walls, a force that we recognize as pressure. If you double the amount of gas in a container, there are twice as many molecules bouncing off the walls, thus doubling the pressure. Squashing the gas down to half its volume does the same thing; twice as many molecules are banging on each little patch of wall. The idea is intuitive, and it makes predictions that you can verify in the lab. Perhaps nobody has ever put 17 times as much gas into a container before, you're the first to do so. Boyle's law predicts the pressure will be 17 times as great, and it is. That is science!
In 1801, Jacques Charles published a wonderful relationship between pressure and temperature, which is now known as Charles' law. If a container holds a fixed amount of gas, and you double the temperature of the gas inside, that doubles the pressure. If you multiply the temperature by 3 you multiply the pressure by 3, and so on. This only works fore degrees Kelvin, where 0 means zero. there is no relationship on any other temperature scale. The molecular theory of heat explains Charles' law perfectly. When the gas is hotter, the molecules move faster. When the gas is twice as hot, the molecules bounce off the walls with twice as much energy, imparting twice as much pressure. The gas in the container on the left is cool, about room temperature; the same gas on the right is hot, twice as hot, thus twice the internal pressure.
|300 °K||600 °K|
This does not consider external pressure, the pressure outside the container. Perhaps you put a lid on a jar, trapping air inside. Pressure inside and out is 1 atm. Then you toss the jar in a fire, which is something you should never ever do! It heats up to 600 °K, and the pressure inside is 2 atm, while the pressure outside is still 1 atm. That's a difference of 1 atm, 15 pounds of pressure on every square inch of glass. The jar will probably explode, sending shards of red hot glass in all directions. If it doesn't explode, then in another 100 ° the glass will soften and melt, and release the trapped air.
If the container is flexible, the added pressure will press against the walls and expand the container. cool the gas, and the pressure drops, and the container shrinks. You can verify this at home. Blow up a balloon until it is full and round, but not so tight that it is ready to burst. Tie the end so it is sealed. Place it in the freezer for an hour and pull it out. It is noticeably smaller, perhaps shriveled. The gas has cooled, and exerts less pressure, yet the atmosphere and the rubber balloon present the same pressure from outside, hence the volume shrinks until the pressure inside equals the pressure outside. Rest the balloon on a table and watch it re-expand again as it warms up.
Thanks to Charles, we can calculate the effect. It isn't large, but it's noticeable. Room temperature is 295 °K. Water freezes at 273 °K, and a freezer is colder than that, perhaps 255 °K. Temperature drops by 13%, and pressure drops by 13%. The flexible balloon shrinks by 13% so that the pressure is equalized inside and out. Since the volume of a sphere is proportional to the radius cubed, the radius only drops by 4%. That's not much of a change, and maybe you won't notice it, but you can probably measure it. In reality the balloon might shrink more than these calculations would suggest. There is moisture in the air, and that moisture condenses out as ice. You might notice ice crystals on the inside of the balloon. The water vapor has been taken out of the equation. When the balloon warms up, ice turns back into water, and then moisture in the air. If you want the calculations to be spot on, you have to fill the balloon with perfectly dry air, and since you blew it up from your lungs, there isn't much chance of that. 😃 🎈
When a gas is heated in the free and open air, there is nothing to contain it. Its pressure increases by Charles' law, then it expands until its pressure is once again 1 atm. The heated gas is spread over a larger volume. It weighs the same, but it takes up more space. It is thus lighter, or less dense if you will, and it rises. Now we understand why smoke rises from a fire; it's hot. But the Montgolfier brothers didn't understand that principle, as they built the first working hot air balloon in 1780. They thought there was something magical about smoke - perhaps smoke was lighter than air - and that was a reasonable hypothesis at the time; Charles' law was still 20 years away. They sat in their basket with a roaring fire between them and an open canopy overhead to catch the smoke, and sure enough, they rose into the air. They tried burning different materials: grass, wood, paper, leather, almost anything, to see if some smoke had more lifting power than other smoke. The material didn't matter however, because the temperature of the rising smoke was the same, and that creates the lift. Nova
Jacques Charles rode in one of these early balloons, but he wasn't comfortable sitting next to a roaring fire, so he ascended in a hydrogen balloon, which acquires lift through a different principle - hydrogen is naturally lighter than air. Still, he was fascinated with the hot air balloon, and soon developed Charles' law, which explains the lift in precise and measurable terms, consistent with the atomic theory of gases.
A year from now someone at a party might casually ask you about Boyle's law and Charles' law. You'll think back to this book and try to remember which is which, but you're not sure. Mnemonics can help, but sometimes you have to reverse them. Like Jane Goodall and Dian Fossey - one studied chimpanzeees for 50 years and one studied gorillas for 50 years, but which one is which? Let's see, Goodall and gorilla both start with G, so naturally, Goodall studied chimps and Fossey studied gorillas. It's backwards.
Boyle makes you think of boiling water, and heat, so naturally, Boyle relates volume and pressure, and Charles relates temperature and pressure. It's backwards.
One comprehensive equation relates pressure, volume, and temperature for an ideal gas, i.e. a gas without moisture or other complicating factors. The nonsense word pivnert helps me remember it.
pv = nrt
P is for pressure, v is for volume, and t is for temperature in degrees Kelvin. n is the number of moles of gas, which is fancy jargon for how much gas you have. Finally, r is a constant that depends on the units, liters or gallons, psi or atmospheres, etc. We're not going to do any algebra here; let's just note the relationships. If temperature increases on the right, then either pressure or volume has to increase on the left. Tha gas is hotter, and it either expands, or if it can't expand then it is under more pressure. If pressure increases, it is because you heated the gas, or squashed it into a smaller volume. If volume increases, then either temperature increases, or pressure decreases. If n increases, i.e. you're shoving more gas into the system, like pumping air into a tire, then either pressure or volume increases, or maybe you have chilled the entire system down so that it can accommodate more gas.
The calory is a unit of heat, the amount of heat needed to raise 1 gram of water 1 degree celsius. Like anything else in the metric system, a kilocalory is 1000 calories. It is the amount of heat needed to raise a kilogram of water 1 degree celcius. Unlike everything else in the metric system, we have developed alternate notation for the kilocalory, the Calory, with a capital C. This shorthand is used on nutrition labels across the world. If a piece of pie has 84 Calories, it provides 84 Calories of energy to your body, which can help you stay warm in the winter, or maintain a steady heartbeat, or run a race, or think as you read this book, since your brain consumes a quarter of your food energy. If it's a cold winter day, and that pie is completely digested and metabolized, and if all that energy is turned into heat to keep you warm, and if you weigh exactly 84 kilograms (185 pounds), the pie can raise your body temperature 1 degree celsius or 1.8 degrees fahrenheit. If you don't need the 84 Calories today, that energy could be stored as fat, to carry you through a famine in the future. We evolved in a world where a stady food supply was not guaranteed. to further confuse things, nutrition labels sometimes use calory with a small c, when they mean Calory. In the context of foods and nutrition, calory always means Calory.
If a cup of water is exactly 32 °F, or 0 °C, is it water or ice? Answer: it could be either. Start with an ice cube that is several degrees below freezing, and gradually warm it up, one calory at a time. When it reaches the freezing point, calories continue to pour in, but the temperature does not change. Instead, the ice melts. It takes 80 calories to melt a gram of ice, without changing its temperature one iota. Once the ice is melted, the heat raises the temperature of the water, as you would expect. The heat that melts a solid, such as ice, without changing its temperature, is called the latent heat of fusion, and it can be substantial. It takes as much heat to melt a block of ice, as it does to raise the same amount of water from room temperature all the way up to the boiling point. the heat breaks the molecules apart and lets them flow freely as a liquid. In the other direction, water gives up its latent heat when it freezes into ice. Here is a graph of temperature versus time, assuming we are adding heat to the system at a steady rate. The ice starts at -10 °c, then rises to 0 °C, then melts as you add 80 calories per gram, then rises to 10 °c. The flat line is heat pouring in and melting the ice, without changing the temperature.
did you ever wonder how 3 or 4 ice cubes can chill an entire glass of lemonade? This wouldn't work without the latent heat of fusion. Here is a demonstration you can do at home. Find a box of marbles, or rocks, or other self-contained objects that won't dissolve in water. Take some ice out of the freezer and put it in a measuring cup, 3 or 4 cubes. Put the marbles in another measuring cup, until the two weigh the same. It doesn't have to be exact. If you like, let the marbles be heavier; give them the advantage. Quickly put the ice and the marbles back in the freezer. Wait several hours, until the ice and the marbles are the same temperature, well below freezing. Run the tap water until it maintains a steady temperature, then fill two large glasses with water. Put the cold marbles in one glass, and the ice in the other. Wait for most of the ice to melt, then measure the temperature of the water in the two glasses, with your hand or with a thermometer. The marbles have barely cooled the water at all. One marble can chill one marble's worth of water, but thanks to its latent heat of fusion, one ice cube can chill 5 or 6 ice cubes worth of water. You should notice the difference just by putting your finger in the two glasses, or take a drink from each glass, but please don't accidentally swallow a marble. For best effect, use insulated glasses, so that heat from the outside does not confound the experiment.
For a more direct experience, hold an ice cube in your right hand, and an equivalent weight of chilled marbles in your left hand, and wait for the ice to melt, and see which hand feels colder.
As you might guess, water has a latent heat of vaporization when it turns to steam. Add heat to the system at a steady rate, and the water reaches the boiling point, then it stays at that temperature as it transforms from liquid to gas, then the temperature of the steam rises again. The latent heat is a whopping 540 calories per gram. It takes almost 7 times as much heat to convert water into steam, as it took to raise that water from room temperature up to the boiling point. this latent heat makes a steam burn particularly nasty. All that latent heat is dumped into your skin as the steam turns back into water. Here is another temperature graph, as heat is added to the system at a steady rate. Ice rises to the freezing point, then melts. Then the water rises to the boiling point, then turns to steam. Finally the steam, as a gas, is free to rise above the boiling point. The scale is the same as the previous graph.
Why do we sweat, and why do dogs pant? As sweat evaporates into the air, it changes from liquid to gas, which requires heat - heat that is carried away from the body. It is thus an effective cooling mechanism.
Most animals have too much fur, so they pant instead. Air moves rapidly across the throat and tongue, causing the saliva to evaporate away, taking its latent heat with it. Sweating is more efficient than panting, but both processes expend a lot of water, and a person, or animal, could become dehydrated if that water is not replenished.
The kangaroo, well adapted to the desert heat, employs a special trick. She licks a particular spot on her forearm that is crisscrossed with veins and capillaries. The water evaporates, and cools the blood beneath, which cools the rest of her body.
As a demonstration, run water until it is warm, the same temperature as your skin. Place one hand in the running water and keep the other dry. Wave both hands in the air, or place them in front of a fan, and note the cooling effect as the water evaporates into the air. 540 calories per gram - there it goes.
When a liquid becomes a gas, its volume expands a thousand fold. In the case of water to steam, the expansion is 1700, but 1000 is a good approximation for most liquids. If a pan of water is boiling on the stove, and it completely boils away, then 1700 pans of steam fill your kitchen. The air feels humid, and the walls may even experience condensation as the excess steam turns back into water. This expansion accounts for some of the latent heat of vaporization. It takes energy to expand something. The steam seems unconstrained, but it actually pushes the air around it; it lifts the air above it, 15 pounds per square inch. The overall height of the atmosphere is just a hair higher after the water is boiled into steam, and that lift takes energy. The heat of vaporization is less if the ambient pressure is less, e.g. on a mountain top, where there is less atmosphere to lift.
In 1770, James Watt developed the first practical steam engine, which turned this expansion into useful work. Coal burns, and water boils, and expands 1700 times, and pushes a piston, which turns a wheel, which moves a train, or runs a factory. It was a simple idea, that started the industrial revolution and changed the world.