## Temperature and Heat

If you push a box along the surface of a table and let go, the box slows down and stops.  The box started with some kinetic energy, but where did it go?  A large proportion probably went to increase the temperature of the table and the box.  But why should it take energy to do this.  The reason is that the atoms and molecules in the box and the table are in motion.  Even though the atoms are held in position by spring-like bonds, they vibrate back and forth on a microscopic level and have an average kinetic energy because of this motion.  (Because the "springs" have potential energy when stretched or compressed, there is a potential energy associated with the motions too.)  The absolute temperature of a material is proportional to the average kinetic energy per atom or molecule in the material; so we call this microscopic motion thermal motion.  (Technically it's proportional to the average translational kinetic energy per atom or molecule.)  The absolute temperature scale most people use is the Kelvin scale.  The temperature in Kelvin, TK, can be related to the temperature in centigrade, TC, or Fahrenheit, TF, by

• TK  = TC +273 = (5/9)(TF -32) +273
• TK is proportional to the average KE per atom or molecule.

Therefore it requires energy to increase the temperature of something.  Since this energy is associate with the microscopic motion of the atoms and molecules, we can't see it, but we can feel its effect, e.g.. something feels hot when the average KE per atom is large.  The molecules of nitrogen and oxygen in this room have an average speed of about 500m/s, or about 1100 mi/hr, due to this thermal motion.

If we are going to raise the temperature of something, we have to add thermal energy to it.  If we want to lower its temperature, we have to remove some of its thermal energy.  The thermal energy transferred from on object to another is called heat.  (Heat is a "type" of energy and it's measured in Joules.)

### Specific Heat

If you want to increase the temperature of a piece of aluminum, or any other material, you have to increase the average kinetic energy per atom, i.e. you have to add energy to increase the thermal motion.

The amount of energy you must add to increase the temperature of 1kg of material 1K is called the specific heat of the material. For instance it takes about 4186J of energy, or heat, to raise the temperature of liquid water 1oC or 1K. The specific heat is

4186J/(kg· deg.C)

To raise temperature of a mass, M, of a material T degrees requires an amount of heat

Heat = Specific Heat´ M´ T

Therefore to raise the temperature of 2kg of water 10oC requires about (rounding 4186 to 4200)

4200J(/kg deg.C) ´ 2kg´ 10oC = 84,000J

The specific heat of water is relatively large compared to many other materials. (Lead’s is about 130J/(kg deg.C) and Aluminum’s is 900J/(kg deg.C).)

### Heat flow

If we have two objects at different temperatures, heat will naturally flow from the warmer object (higher temperature) to the cooler object (lower temperature).  This is something like water, it naturally flows downhill.  This will continue until they reach the same temperature.  When they are at the same temperature, we sat they are in thermal equilibrium.  {However, we can intervene and make heat flow from the cooler object to the warmer one, but we have to do work to make it flow backwards, just like we can make water go up the hill by putting it in a bucket and carrying it up or using a pump to force it uphill.  In both cases we have to intervene and do work to make it go "backwards".}

There are three basic mechanisms for transferring heat from one object to another.  They are

• Conduction
• Convection

Conduction

In conduction the heat is transferred from one atom to the next as they collide with each other. In the figure below, the atom on the left initially has more kinetic energy than the others. It collides (pushes and pulls) on the atom next to it and transfers energy to it. The second atom in turn transfers energy to the one to its right as it collides with it and so on until the "extra" energy is shared by all the atoms. (Red represents a lot of KE, pink less, orange even less and green represents the least KE.) Note that the atoms do not move from one end of the chain to the other as the heat is transferred down the chain.

Many of the best conductors of heat are metals (the "free" electrons conduct heat efficiently.) They conduct heat easily and rapidly. However, diamond is a surprisingly good thermal conductor, pure diamond is four or five times more conductive than silver even though diamond has almost no free electrons.  Materials that do not conduct heat very well are called insulators. Materials like glass, most plastics, and wood are thermal insulators. Gasses are usually poor conductors (thought they can convect heat quickly).  As a result, materials with lots of small air pockets like Styrofoam and down are very good insulators because the small air spaces inhibit convection so heat must flow by conduction and air is a very poor conductor.

The rate of heat transfer or conduction from one point to another depends on several things.

1. The temperature difference - the greater the temperature difference the faster the transfer
2. The type of material - the better the conductor the faster the transfer.
3. The cross sectional area - the larger the area the faster the transfer.
4. The thickness or distance - the greater the distance, the slower the transfer.

Convection

This is a mixing process, and as a result it is confined to liquids and gasses. The atoms move from one location to another and carry the heat with them. This mixing motion of atoms is due to buoyancy. If I heat a gas, it will expand and become less dense. If it is less dense than the gas surrounding it, it will rise because the buoyant force pushing it up is greater than its weight. This motion produces the mixing and the consequent transfer of thermal energy (flow of heat).

It should be noted that if the air above is warmer than the air below, there will be no mixing due to convection.  If the surface water in a swimming pool is warmer than the water below, convection will not mix the water and it may take a long time for conduction to equalize the temperatures.

There is one interesting exception to the rule that objects expand when warmed.  In the temperature range of 0oC to 4oC water contracts when heated.  Outside this range is behaves "normally", but between 0oC to 4oC it is "backwards".  Water is most dense at 4oC.

Try not to use the expression "heat rises".  Heat is a transfer of energy, not a substance.  Hot air will rise if it's surrounded by cooler air because the hot air is less dense than the cooler air.  As a consequence it will produce a heat transfer upward.  However, water at 4oC will sink if surrounded by water at 0oC, because the warmer water is more dense.  This will tend to transfer heat downward. Also, conduction and radiation can transfer heat in any direction.

Many liquids are poor conductors of heat and most gasses are insulators, i.e. thermal insulators, and heat transfers over large distances in these materials usually involve convection. This is especially true in gasses like air. We depend on convection to transfer heat in our houses.

Convection takes place more readily in large spaces.  If you can break up a large pocket of air into many small pockets, it will slow down the rate of heat transfer due to convection.  Since air is a good insulator, it will provide good insulation if you can prevent convection.  Therefore many insulation techniques involve breaking up a large pocket of air, e.g. in the walls of a house, into many small pockets, e.g. with fiberglass insulation, so you restrict convection and convert the space into an insulating layer.  This also makes down a good insulator.

This refers to electromagnetic radiation, traveling electric and magnetic fields that can move through empty space carrying energy. This energy comes in bundles called photons. Electromagnetic radiation is often classified by how energetic the photons are.  (Two other ways of characterizing it is by the frequency and by the wavelength.  The energy per photon is proportional to the frequency and inversely proportional to the wavelength.)

Note that visible light is only a small part of the electromagnetic spectrum.

All objects above 0K (absolute zero) emit electromagnetic radiation because of the thermal motion of their atoms. Thus they are emitting energy in the form of electromagnetic radiation. The higher the temperature, the more energy they emit per second (i.e. power). Also the energy per photon increases with temperature. We are not too hot (about 300 – 310 K) so we emit infrared radiation. The sun is much hotter (the surface is around 5800K) and emits visible light along with some ultraviolet and infrared. The energy we receive from the sun comes from radiation.

Radiation emitted because of the thermal motion of the atoms in a material is called thermal radiation.  It's also called blackbody radiation, because black objects are good emitters of this thermal radiation.

For visible light, the color of an object tells us how good it is at absorbing radiation.  Objects that are white reflect light rather than absorb it, therefore they do no heat up rapidly when light shines on them.  Black object absorb light and reflect very little.  Therefore they heat up more quickly when light shines on them.  We say white is a poor absorber and black is a good absorber.  Objects that are good absorbers are also good emitters of thermal radiation.  Black objects are perfect emitters (and absorbers) of thermal radiation.  A black object that is emitting radiation but not receiving radiation will cool down more quickly than a white one, because the black object is better at emitting thermal radiation and therefore will emit more energy per second than the white one of the same geometry and temperature.

We also adsorb radiation emitted by other objects near us. If we are warmer than the walls of the room around us, we transfer more heat to the walls by the radiation than we receive from the walls, providing a net transfer of heat to the walls. If we are cooler than the walls we will receive more than we emit, thus gaining energy.

Two things determine the average temperature of the earth. The first is the rate at which we receive energy from the sun. The second is the rate we radiate energy away. The balance between these two determines the temperature of the earth.  On the daytime side of the earth, we typically  receive more energy from the sun than we radiate away and the daytime side of the earth warms up.  On the nighttime side, we radiate more energy away than we receive and we cool down.

### Newton's Law of Cooling (or Warming)

For all of the mechanism of heat transfer, the greater the temperature difference, the greater the rate of heat transfer.  If two objects differ in temperature by 10oC and the rate of heat transfer is 100J/s, an increase in the temperature difference to 20oC will mean that more than 100J/s of heat are transferred.  If the temperature differences are not too large, the rate of heat transfer is approximately proportional to the difference in temperature.  For the above example, doubling the temperature difference will double the rate of heat transfer, from 100J/s to 200J/s.  This is why turning down the thermostat in the winter will save money on your heating bills.  The furnace runs to replace the heat you lose to the outside.  If you lose less heat per second, the furnace has to replace less heat over the course of a day.  If the temperature difference is less, you lose less heat per second.  It is similar in the summer when you set the thermostat up to make the temperature inside closer to the outside temperature.  Then less heat is transferred into the house every second and you have to pump less heat out through the air conditioner.

### Sensations

Why do some objects feel cooler than others even though they are at the same temperature?  For instance, if you get out of bed on a cold morning and step on a carpet, it does not feel too cold on your feet.  However if you step onto a tile floor, it feels much colder than the rug, even though the rug and tile were originally at the same temperature.  If you stepped onto a large piece of aluminum (a metal), it would feel even colder than the tile.  The reason is that we sense the temperature of our own skin.  For something to feel cool, it must take heat away from our skin, and do so fairly quickly or our body will transfer heat to the skin to keep it warm.  The carpet is not a good conductor, but rather an insulator.  Therefore it does not conduct heat away from our skin very rapidly and so our skin cools slowly and the carpet does not feel cold because it doesn't cool our skin much.  The tile is a poor conductor, but a much better one than the carpet, so it conducts heat away from our foot more rapidly and cools it down so it feels cold.  The aluminum is the best conductor of the three and cools our skin rapidly and therefore it feels the coldest.

Why can you pull a piece of aluminum foil from a hot oven and not get burned, but if you do grab a glass casserole dish you will probably be burned?  Here the aluminum is a good conductor and the glass is a poor conductor.  However, the mass of the foil is small and it does not have to lose much heat to cool down to the temperature of your fingers.  Also, it is thin so its geometry makes it hard to transfer heat by rapidly by conduction.  Therefore when you grab the foil, it does not transfer a lot of heat to your hand in a short time.