2.  Atoms and Heat

 

 

 

 

Quandaries

 

When the asteroid hit the Earth 65 million years ago, it had a kinetic energy equivalent of 100 times its own weight in TNT. In the impact, virtually all that energy was turned into heat. The temperature of the rock (turned into vapor) was over a million Celsius, over a hundred times hotter than the surface of the sun. 

 

Why? How does kinetic energy turn into heat? What is heat? How did this lead to an explosion?

 

All objects sitting in a room should reach the same temperature. Yet if you pick up a cup made of glass, it feels cooler than a cup made of plastic. Many people unconsciously recognize plastic by its relative “warm feel.” 

 

How can two objects be the same temperature and yet one feels cooler? What mistaken assumption are we making?

 

Many scientists are worried that the Earth is warming. Some models predict that the continued dumping of carbon dioxide into the atmosphere (from the burning of fossil fuels) could warm the Earth by about 2.5 C, equal to 4.5 F. If this happens, we expect the oceans to rise by several feet--even if no ice melts. Many coastal cities, particularly those in Florida, would go underwater.

 

Why should the sea level rise if no ice melts? 

 

When we heat our homes by burning fuel, we are wasting energy. We could do a much better job by pumping heat from the cold outdoors into our homes.

 

Pump heat from the cold outdoors? This sounds like nonsense. Isn’t the burning of fuel 100% efficient, since all the energy goes into heat? How could we possibly do better than that?

 

 

 

Atoms and Molecules and the Meaning of Heat

 

Press your hands together hard and rub them vigorously for about 15 seconds. (It is actually a good idea to do this right now, before you read further, if nobody is watching.) Your hands feel warmer. The temperature of the skin has risen. You turned kinetic energy (energy of motion) into heat. 

      In fact, heat is kinetic energy, the kinetic energy of molecules.[1] Your hands feel warmer because, after rubbing, the molecules are shaking back and forth faster than they were prior to your rubbing. That’s what heat really is: the shaking of atoms and molecules, rapid in speed, but microscopic in distance.

      This is a good time to discuss the makeup of matter. All substances are made of atoms, and there are only about[2] 92 different kinds of these: hydrogen, oxygen, carbon, iron, etc. A complete list appears in a chart known as a periodic table, shown below:

 

      Each of the atoms in the periodic table has a number associated with it called the atomic number. It represents the number of protons in the atom; it is also the number of electrons in the atom (usually). The atomic number of hydrogen is 1, for helium the atomic number is 2, for carbon it’s 6, for oxygen it’s 8, and for uranium it’s 92.

      Molecules are combinations of atoms that stay clumped together. A molecule of water is written “H2O,” meaning it is made of two atoms of hydrogen (that’s the H2) and one atom of oxygen (that’s the O). Helium molecules contain only one atom (He), and hydrogen gas molecules contain only two attached atoms of hydrogen (H2).  But molecules can be very large. The molecule known as DNA, which carries our genetic information, can contain billions of atoms.[3] When molecules break apart or come together it’s called a chemical reaction. 

      In all materials, the molecules are constantly shaking. The more vigorously they shake, the hotter the material is. When you rub your hands together, you make the molecules in your hands shake faster. How fast do they shake? The answer is startling: the typical velocity of shaking is about the same as the speed of sound, about 700 mi/h, 1000 ft/s, or 330 m/s. That’s fast. Yet the particles (at least in a solid) can’t travel very far. They bump into their neighbors and bounce back. They move fast but, like a runner on a circular track, their average position doesn’t change.

      Atoms are too small to be observed with an ordinary microscope. If you move across the diameter of a human hair (typically 25 microns[4]), you will encounter 150,000 atoms from one side to the other. A red blood cell (8 microns across) has about 50,000 atoms spanning its diameter. Some molecules are so large (such as DNA) that they can be seen under a microscope, although the individual atoms in the molecules can’t be resolved.

      Even though you can’t see atoms, you can see the effect that their shaking has on small, visible particles. With a microscope, you can see the shaking of tiny bits of floating dust (1 micron in diameter). This phenomenon is known as Brownian motion.[5] The shaking comes from the dust being hit on all sides by air molecules, and if the dust is sufficiently small, this bombardment does not average out. For a wonderful simulation of this effect, see the Web site at

http://ephysics.physics.ucla.edu/ntnujava/gas2D/ebrownian_motion.htm

There is a similar site at

www.phys.virginia.edu/classes/109N/more_stuff/Applets/brownian/brownian.html

 

 

The Speed of Sound and the Speed of Light

 

Is it a coincidence that the speed of molecules is approximately the speed of sound? No-- sound travels through air by molecules bumping into each other. So the speed of sound is determined by the speed of molecular motion. Sound traveling through a gas cannot move faster than the velocity of the gas molecules.[6]

      You can easily measure the speed of sound yourself. One way is to watch someone hit a golf ball, chop wood, or hit a baseball. Notice that you see the event before you hear the noise. That’s because the light gets to you very quickly, and then you have to wait for the sound. Estimate your distance to the person, and estimate how long it takes for the sound to reach you. If the distance is 1000 feet, then the delay should be about 1 second. (If you do this at a baseball game, then it is helpful to sit as far from home plate as possible.) The velocity is the distance divided by the time.

      When I was young, and afraid of thunder and lightning, my parents taught me a way to tell how far away the sound and light was coming from. For every 5 seconds between the lightning flash and the thunder, they said, the lightning was 1 mile away. If there was a 10-second delay, then the lightning strike was 2 miles away. To me at that age, a mile was about the same as infinity, and so that put me at ease. The rule works because light travels so quickly that it covers a mile in a tiny fraction of a second. In other words, the light arrives virtually instantly.  But thunder, since it is a sound, travels at the slower speed of sound: 330 m/s = 1000 ft/s = 1 mi/5 s (approximately) = 700 mi/h.

      Knowing the speed of sound can be useful to measure distances. In 2003, I found myself on a boat some distance from a glacier that was dropping huge pieces of ice into the water. I measured that the sound took 12.5 s to reach me.  That way I knew that the edge of the glacier was 2.5 mi away (1 mi for every 5 s). 

      The speed of light is much greater: 186,000 mi/s, or 3 x108 m/s. Although that sounds super fast, we can express it in a way that makes it sound much slower.  Modern computers take about 1 billionth of a second (a nanosecond) to do a calculation. (Some go much faster, but you should know that a nanosecond is typical.) In that billionth of a second, light travels only about 1 ft (30 cm). That’s why computers must be small. Computers must often retrieve information to do a calculation, and if the information is too far away, it has to waste several cycles to get it.[7] If the computer speed is 3 GHz, then the light goes only 4 inches in one cycle.

Remember:

the speed of light is about 1 foot in 1 computer cycle (1 ns)

     

 

The Enormous Energy in Heat

 

The average speed of the molecules that compose this book is the speed of sound, but they are all moving in random directions. Suppose I made them all move in the same direction. Then the entire book would be moving at the speed of sound, 720 miles per hour. Yet the total energy would be exactly the same.

      This example illustrates the enormous energy that is contained in the heat of ordinary objects. Unfortunately, it is often not possible to extract that energy for useful work. We’ll discuss this further when we get to the section on heat engines.  There is no good way to change the directions of the shaking so that all the molecules move together. Yet we can do the opposite. When an asteroid hit the Earth 65 million years ago, all the molecules were initially moving at 30 km/s in the same direction. After the impact, the directions were all different.

      When kinetic energy is turned into heat, we can think of this process as coherent, regular motion becoming randomized. The molecular energy changes from being neatly “ordered” (all molecules moving in the same direction) to being “disordered.” The term disorder is very popular in physics. The amount of disorder can be quantified, and that value is given the name entropy. When an object is heated, its entropy (the randomness of its molecular motion) increases. I’ll discuss entropy further at the end of this chapter.

 

 

Hiss and Snow: Electronic Noise

 

Radios, when tuned in between stations, sometimes give a hissing sound. What is the origin of that hiss? Old TV sets, when there is no station present, show white spots jumping on the screen that reminded people of snow. What is the snow?

      The surprising answer is that the snow and the hiss are due to the same thing: electrons jumping around in the electronics of your set. They are in constant motion due to heat, and when there is no other signal present, you get to watch (or listen) to them move. Even though they are not molecules, they share the energy of shaking.

      Lowering the temperature can reduce such noise, and high-sensitivity electronics often have to be cooled to reduce the hiss and the snow. In the Chapter 9 “Invisible Light,” I’ll talk about a device for seeing in very low light that had such a cooling system attached. But too much cooling can cause the device to cease operation, since a transistor (discussed in Chapter 10 “Quantum Physics”) actually depends on the fact that room-temperature electrons have some kinetic energy. Without that kinetic energy, the electrons become trapped and electricity doesn’t flow. If you cool a transistor, and remove that energy, the transistor no longer functions. 

      Now that we have described heat as the kinetic energy of the molecules (and sometimes of the electrons too), we can address a trickier question: what is temperature?

 

 

Temperature

 

      Temperature is closely related to heat. Stop for a moment and think about it. When it is 100 F outside, it is hot. When it is below 32 F, water freezes. But it is very tricky to state exactly what temperature is. It is what you read with thermometers. But what does it measure? The answer is surprisingly simple:

      Temperature is a measure of the hidden kinetic energy of the molecules.

      By the “hidden kinetic energy” I mean the usually unobserved energy of fast (speed of sound) but microscopic (in distance moved) shaking. When we get to the section on temperature scales, I’ll give the equation that allows you to calculate the kinetic energy from the temperature.

      The temperature increases when the average shaking energy of its molecules is greater. (We use the word average because at any given instant, some of the molecules may be moving faster than others, and some slower, just like dancers on a dance floor.) If two objects have the same temperature, then their molecules have the same kinetic energy of vibration.   

      Here is a surprising consequence of what I just said. Suppose two bars, one made of iron and the other of copper, have the same temperature. Then their molecules must have the same kinetic energy, on average. Will the iron molecules and the copper molecules have the same average speed? The surprising answer is no. The copper molecules will be shaking faster, on average. 

      Remember that kinetic energy is given by KE = ½ mv2. Copper and iron have different masses m. So the heavier iron molecule must have a smaller velocity v in order to have the same kinetic energy KE. See why temperature was once even more of a mystery than heat?

      Remember this:

At the same temperature, lighter molecules

move faster (on average) than heavier ones.

     

The Zeroth Law of Thermodynamics

 

The key discovery that makes temperature a really useful idea is the simple fact that two things that touch each other tend to reach the same temperature. That is why a thermometer gives you the temperature of the air--because it is in contact with the air, so it gets to the same temperature. The fact that objects in contact tend to reach the same temperature was such an important observation that it was given a fancy name: the zeroth law of thermodynamics.[8] 

      Touch a hot iron object to a cold copper one. Because they are touching, the fast molecules in iron now bang into the slower ones in the copper. The iron molecules lose energy and the copper ones gain. The temperature of the iron will drop and that of the copper will rise. Only when the temperatures are the same does the transfer of energy stop. The “flow” of heat is actually the sharing of kinetic energy.  Heat (kinetic energy) is given up by the hot material to the cold one. The flow stops only when both materials have the same temperature.

      This means that if you put a bunch of things in the same room and wait, that eventually they will all reach the same temperature. Of course, that doesn’t work if one of the objects is a source of energy, such as a burning log. But if no energy is going in or out of the room, all objects will eventually reach the same temperature. 

 

Where is our hydrogen?

 

The element hydrogen is, by far, the most abundant element in the Universe.  Hydrogen atoms make up 90% of the atoms in the Sun. The same is true for the large planets of Jupiter and Saturn. Yet in the atmosphere of the Earth, hydrogen gas is virtually absent. Why? Where is our hydrogen?

      There is a remarkably simple answer, and it comes from the zeroth law of thermodynamics. The Earth once had lots of hydrogen, but we lost it to space.  Hydrogen in the atmosphere of the Earth would have the same temperature as the nitrogen and oxygen. Therefore, the molecules of hydrogen have the same kinetic energy, on average. But since hydrogen is the lightest element (only 1/16 the weight of oxygen), it must have a higher velocity. Since energy depends on the square of the velocity, the velocity must be a factor of 4 larger (so the square is 16). This high average velocity turns out to be enough for the hydrogen to escape from the Earth like a rocket![9] The Sun and Jupiter have much stronger gravity than the Earth, so they kept their hydrogen. We’ll discuss escape velocity in more detail in Chapter 3 “Force and Gravity.” The Earth lost its hydrogen gas because our gravity is too weak.

     

The cold death

 

Stars are very hot, and molecules in space are very cold. Eventually the stars will stop burning, and eventually everything in the Universe may reach the same temperature. By keeping track of everything, we can calculate what that temperature is. If we ignore the expansion of the Universe (see Chapter 12) then the average temperature of the Universe turns out to be –270 C.[10] Because the Universe is expanding, the eventual temperature may be even lower. Philosophers have called this the “cold death” of the Universe, and the thought of it gets some people depressed. But being cold doesn’t necessarily mean life will be uninteresting. A detailed analysis made by physicist Freeman Dyson showed that even as the Universe gets very cold, life can continue, and the complexity of organized thoughts could get greater and greater. That might take additional evolution, but we have hundreds of billions of years for that.

      What would life be like in such a universe? What would the descendants of humans look like? Some people estimate that, because of the extreme cold, in order to remain complex, active creatures, they would have to be very large, perhaps as large as planets are now; maybe even bigger.

 

 

Temperature Scales

 

The concept of temperature was invented long before it was understood. It was measured using devices called thermometers. People could make thermometers that would always agree, more or less because (as stated in the zeroth law) it doesn’t matter what material the thermometer is made out of.  So temperature became a standard idea. We’ll talk about how thermometers work later in the chapter.

      There are two common temperature scales, the Fahrenheit Scale, and the Centigrade scale. Centigrade has recently been renamed “Celsius.”[11] Celsius is also abbreviated C, just like Centigrade, and Fahrenheit is abbreviated F. The scales are defined such that the freezing point of water is 32 F and 0 C, and the boiling point of water is 212 F and 100 C.[12] 

      We can convert between Fahrenheit and Celsius by the following rules. Let TC be the temperature expressed in the Celsius scale, and TF be the temperature in the Fahrenheit scale. Then

            TC = (TF –32)(5/9)

           

            TF = (9/5)TC + 32

 

      Examples (try the equations yourself):

 

            Freezing of water: TF = 32 gives TC = 0

            Boiling of water: TC = 100 gives TF = 212

            “room temperature” TC = 20 gives TF = 68

     

 

Degrees

 

Until recently, it was common to refer to temperature in degrees. A temperature of TF = 65 was read as “65 degrees Fahrenheit” and written 65o F. However, the word degree doesn’t add any meaning, and some people were confused by it. (It has nothing to do with angles, which are also measured in degrees.) So scientists are now adopting a new convention of dropping the degree symbol. Thus 32o F is usually shortened to 32 F. You’ll see it both ways. There is no physics in this; it is just notation. I’ll sometimes use the traditional terminology, just because of the fact that that is how you will hear it used most, and because it sometimes makes it clear that we are talking about temperature.

      Note that Celsius degrees are bigger than Fahrenheit degrees. A change of 1 C is a change of 9/5 F = 1.8 F ≈ 2F. As an approximation for changes in temperature, remember:

1oC  ≈  2oF

 

Digression: Which is metric, C or F? The original Fahrenheit scale was designed to make 0 F the coldest temperature that could easily be reached in a laboratory.  That was done by mixing ice and salt, and that is what is called 0 F. The temperature of 100 F was originally chosen to be body temperature. (They made a slight mistake, and average body temperature is actually about 98.6 F.) On this scale, water freezes at 32 F and boils at 212 F. When the centigrade scale was officially adopted (by the French, under Napoleon) they decided that the two standard points should be the freezing and boiling points of water. So on the Centigrade scale, water freezes at 0 C and boils at 100 C. Some people think the Centigrade scale was more “metric” than the Fahrenheit scale, and that is nonsense. Both scales were based on standard points 100 degrees apart; they just chose different standard points.

 

 

Absolute Zero

 

What happens if the molecules actually come to a stop, and have zero kinetic energy? When all motion of the molecules stop, we say the temperature of the material is at “absolute zero.”[13] Such cessation of motion occurs at –273o C = -459o F.[14]

      Using this fact, we can define a new temperature scale called the “Absolute” or “Kelvin” scale (named after William Thompson[15]). Physicists find the Kelvin scale to be very convenient, because it simplifies equations. In the Kelvin scale, the average kinetic energy E per molecule is given by a very simple equation:

     

E = 2 x10–23 TK

        

where TK is the temperature in Kelvin (or degrees Kelvin). The constant, given in the equation as 2 x10–23, is very small only because atoms are so small. Don’t bother learning this number.  It is not important to know the numerical value for the kinetic energy of the particles. It is important to know their velocity (1000 ft/s, about the speed of sound) and that if you double the temperature (on the Kelvin scale) then you double the kinetic energy.

      The most remarkable fact about this equation is that it doesn’t depend on the kind of material. That’s just the zeroth law again. I find that to be an amazing and surprisingly simple law of physics. Ponder it for a few moments. Temperature is just the hidden kinetic energy. At room temperature, the kinetic energy of the atoms in the air is identical to the kinetic energy of the atoms in this book. That fact eluded scientists for hundreds of years. The only really tricky part is that the energy must be measured per molecule. This equation begins to illustrate what physicists sometimes refer to the “beauty” of physics. It isn’t really beauty in the traditional sense. It is just an insight, a simplicity that is missed by people who don’t study physics.

      You can convert from the Kelvin scale to the Celsius scale by subtracting 273:

 

TC = TK – 273

     

Thus, for example, TK = 273 is the same as TC = 0.  Put another way, 273 K = 0 C.

     

 

The Columbia space shuttle tragedy

 

On February 1, 2003, the Columbia space shuttle broke apart in flames as it reentered the atmosphere, killing all seven astronauts on board.

      The space shuttle always generates enormous heat when it reenters the thicker parts of the Earth’s atmosphere. That’s because it has very large kinetic energy, and to slow down (so it can land) it must get rid of that energy. 

      To calculate the energy per gram, we need to know the velocity. When the space shuttle orbits, it travels the Earth’s circumference of 24000 mi in 1.5 hours, so its velocity is 24000/1.5 mi/h = 16000 mi/h = 7000 m/s = 22 times the speed of sound. At the time that it began to fall apart, the Shuttle had slowed to 18.3 times the speed of sound. That is known as “Mach 18.3.” We’ll show why it has to move so fast in Chapter 3.

      In the optional calculation below, we show that if the kinetic energy of the space shuttle were all turned to heating it, its temperature would rise to

 

The Mach Rule:

 

T = 300 M2

        

where M is the Mach number. This is a useful equation that you will not find in any other textbook. For M = 18.3, this gives T = 100,000 K. That is 17 times as hot as the surface of the Sun. This is why the pieces of the Shuttle glowed so brightly, friction with the air made them very, very hot.

      There is no way to avoid this turning of kinetic energy to heat on reentry.[16] The Space Shuttle is designed to have heat-resistant ceramic “tiles” on the bottom surface. During reentry, these tiles face the onrushing air, and glow with a temperature of thousands of degrees. They can lose this heat by conduction with the air and by radiation. They cool off by the time that the shuttle lands.

      The shuttle contains little fuel and no explosives. It was the kinetic energy of motion, turned into heat, which destroyed the vehicle.

 

High Temperatures. Here’s a little trick you might find helpful. Suppose an object (such as a meteor, or the interior of the sun) has a temperature of 100,000 K.  How hot is it in C? The answer is 100,273. They differ by only 0.27%.  So here is a useful rule: when temperatures are really high, then the temperature in C is approximately the temperature in K. 

 

Optional calculation:

Let’s derive the Mach number equation. Now here is a trick that can allow you to get the answer very quickly. We know that at room temperature (300 K) the molecules in the Shuttle are moving at about the speed of sound, i.e. at Mach 1. Suppose all the kinetic energy of the orbiting shuttle was randomized, i.e. turned into heat. Then the molecules would be moving at Mach 18.3 (since that is how fast the Shuttle was moving). So as the energy of orbit turns into the energy of heat, the molecules hidden motion speeds up by a factor of 18.3.

      What does that do to their hidden kinetic energy, i.e. to the temperature?  Remember that the kinetic energy is E = (1/2) m v2. So if you increase v by a factor of 18.3, you increase the kinetic energy by a factor of (18.3)2 = 335.