FZX: Personal Lecture Notes from Daniel W. Koon St. Lawrence University Physics Department CHAPTER 13-14

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1 FZX: Personal Lecture Notes from Daniel W. Koon St. Lawrence University Physics Department CHAPTER Please report any glitches, bugs or errors to the author: dkoon at stlawu.edu. 13. Heat and Temperature Temperature Scales Fahrenheit, Celsius, Kelvin Linear expansion Equivalence of Heat and Energy Calorimetry I: change in temperature Calorimetry II: change in phase Calorimetry problems Digging deeper: conduction, convection and radiation 14. Thermodynamics State Variables First Law of Thermo Entropy page 1

2 FZX, Chapter 13: HEAT and TEMPERATURE We have said that mechanical energy is conserved when only conservative forces act, and that conservative forces are forces that don t behave like friction. So what happens to the mechanical energy lost in a system when friction is at work? The energy goes into heating up the objects that participate in the friction. Rub your hands together quickly to convince yourself of this. We produce heat, but what is this heat? Can we understand it in terms of physical concepts we already know? But of course. We can think of heat as the local mechanical kinetic energy of the molecules in a material. A related term that we often confuse with heat is temperature. We can think of temperature as measuring the mechanical kinetic energy per molecule, or simply the average heat per molecule. The English language reinforces the confusion between heat and temperature since we talk about how hot it is when what we mean is that the temperature is high. Temperature is a very handy quantity, since it tells us which way heat energy will flow if two objects of dissimilar temperatures come into contact. Since the molecules of one object will line up with the molecules of the other, transfer of energy takes place between individual molecules. Thus it is the kinetic energy PER molecule -- the temperature -- which determines which direction any energy would flow. It is not necessarily the object with more heat which gives off its heat energy. An ant sitting on a large block of ice will give up some of its heat to the much larger block, even though the ice actually has more heat energy itself. This is like with foreign aid. It is generally the country with a larger per capita income that gives aid to another country, rather than the country with the larger total income. If the total national income of Switzerland is smaller than the total national income of China, it is still much more likely for aid money to flow from Switzerland to China than vice versa. If two objects have the same thermal energy per molecule, no heat flows between them. They are said to be in thermal equilibrium. We can rephrase this into what is called the Zero th Law of Thermodynamics : Two objects in thermal equilibrium with a third object are in thermal equilibrium with each other. If T a = T b and T a = T c, then T b = T c. [ Zero th Law of Thermodynamics] [Mathematical equivalent of Zero th Law] TEMPERATURE SCALES Our interpretation of temperature as the local kinetic energy per molecule leads to two conclusions about temperature scales. First, there must be a zero temperature at which all motion of the molecule ceases. Secondly, there is no such thing as negative temperature, since that would suggest negative kinetic energy. Unfortunately, the two principal temperature scales which we are most likely to use were invented before people understood temperature to be related to energy. They invented scales simply as a way of classifying objects in terms of their relative willingness to either soak up or give off heat. Consequently these scales allow for negative values of temperature. And that s okay, as long as we don t try to attach any physical significance to the negative sign in these negative values. On the other hand, it is useful to define a temperature scale which does call the point at which molecular motion stops zero. Such a scale is called an absolute temperature scale. FAHRENHEIT, CELSIUS, KELVIN The oldest temperature scale still used is the Fahrenheit scale. We are all familiar that water boils at 212 o and freezes at 32 o. Why were the numbers chosen as they were for this scale? The Fahrenheit scale has two basic reference points -- 0 o is the point where the saltiest saltwater freezes and 100 o is body temperature, more or less. Body temperature is not such a good absolute reference point, but the freezing and boiling points of water under controlled conditions are. The Celsius scale uses the freezing point of [unsalted] water as its 0 o point, and the boiling point as its 100 o point. Because of the way that temperature is defined, a change of nine Fahrenheit degrees equals a change of five Celsius degrees, regardless of whether the change occurs near 0 o C, -273 o C, or wherever. Because of this, the conversion between these scales is linear and straightforward: o T ( C) = 9 Rearranging terms, we get o o [ T ( F) 32 F] 5 page 2

3 T o ( 5 9 o F) = T ( C) + 32 o F The Kelvin scale, which is an absolute temperature scale, is simply related to the Celsius scale by o T ( K) = T ( C) Note that there is no o sign associated with the Kelvin scale. This is a convention you should get acquainted with. The reason that Kelvin is an absolute scale is that 0 Kelvin represents the lowest temperature possible -- the point at which the heat energy per molecule is zero. Any time we want temperature to stand for heat energy per molecule, we will use the Kelvin scale. LINEAR EXPANSION: A number of quantities change with the temperature. One of these is the length of an object. The more kinetic energy the molecules in an object have, the more they push against their neighbors. The more they push against those neighbors, the further apart is the equilibrium distance between molecules and the longer the object becomes. If this is the only process occurring, then, for small changes in temperature, the relative change in the length, L/L, is proportional to the temperature change. We can quantify how willing such an object is to change its length with temperature by the quantity α = Δ L 1, [Linear thermal expansion of an object] L ΔT where α is called the linear coefficient of expansion, a measure of the relative change in length of a material per change in temperature, L is the object s original length, αl is the amount by which it changes, and T is the change in temperature. All that this equation says is that if the temperature changes a little, most objects will change their lengths a little. Usually α is positive, indicating that warm things tend to expand. An important counterexample is water near its freezing point. Ice floats, which means that it is less dense than water -- and it s not just because of trapped air bubbles. Between about 4 o C and 0 o C, the density of water decreases by about 0.04%, and it decreases even further as it freezes into ice. It s a good thing that water displays this anomalous expansion, since it allows the top of a lake to freeze first, insulating the rest of the lake, and keeping it from freezing solid and killing much of the aquatic life. By the way, when some object expands or contracts, the new length at the new temperature can be written -- by using algebra on the equation above -- as ( 1 + ΔT ) L + ΔL = L α page 3

4 EQUIVALENCE OF HEAT AND ENERGY Heat can be made to change the temperature of things. This observation was made long before it was realized that heat was a form of energy. Consequently, units of heat were developed, even though they are redundant, since we could use units of energy -- Joules -- to measure heat. These original units of heat are called calories, and one calorie is the amount of heat required to raise the temperature of one gram of water by 1 o C. Since heat is a form of energy, there must be a conversion factor between calories and joules: 1 cal = 4.19J [Conversion between units of heat and energy] A kilocalorie (10 3 cal) is the amount of heat required to raise 1 kilogram of water by 1 o C. Unfortunately, this is the unit often called calorie when it is used to measure the energy equivalent of food. To avoid confusion, let s call it a food calorie when we are measuring the energy content of food, and use the conversion 1 food calorie = 1 kcal = 4190J [ food calorie ] page 4

5 CALORIMETRY I: Change in temperature In addition to changing an object s size, heat can do other things, most importantly changing the temperature of the object or converting it from one phase -- solid, liquid, or gas -- to another. The study of these changes is called calorimetry. How do we change the temperature of something? We HEAT it: that is, we add heat energy to the thing which we want to raise the temperature of. When we say that we are going to heat things up, we mean that we are going to make the temperature rise. If we put a kettle on a stove, heat is being supplied to the water in the kettle at very close to a constant rate, and if we measured the temperature, we would find it rising at a constant rate too. We can therefore say that Q = mcδt [Heat required to change temperature] where Q is the heat supplied, m is the mass of the thing being heated, and Δ T is the amount by which we change the temperature. Of course, we hedge our bets by saying that c -- the specific heat, which characterizes how hard it is to raise the temperature of some material -- is not necessarily constant, and may change some as T changes. r r Think of this equation as being like Newton s Second Law -- F = ma. The left side in each equation represents the cause -- the thing that we can adjust -- and the right side represents the effect -- the result of twiddling with the left side. Repeat after me: Cause, effect. Cause, effect. Cause... CALORIMETRY II: Change in phase Once the kettle comes to a boil, however, the temperature stops changing. The water is now at 100 o C (212 o F for all you non-metric fogies), and the heat goes into boiling off the liquid water into steam. Obviously, the more water there is to boil, the more heat is needed. The amount of water being boiled, however, is the only variable. We can thus write Q = ml [Heat required to change phase] where L is called the latent heat -- the amount of heat per kilogram needed to change the phase of the water from liquid to gas. There are two flavors of latent heat that we will deal with: the latent heat of vaporization,, for either boiling or its reverse process, condensation; and the latent heat of fusion,, for either freezing, or its reverse process, melting. For water, the latent heats are 22.6x10 5 J/kg = 540cal/g for vaporization and 3.34x10 5 J/kg = 80cal/g for fusion. The tremendous amount of heat that goes into boiling water times as much as needed to raise it from its freezing temperature to the boiling point -- means that when steam condenses, it releases a lot of heat energy. This is why it is so easy to get serious burns on your skin from steam. Never stick your face right above a kettle of boiling water to see how the spaghetti s doing! To summarize the four different phase changes we most commonly work with -- we will ignore sublimation, the process by which solids evaporate directly into gas -- L f L v CHANGE IN HEAT +ml v -ml v +ml f -ml f PROCESS Boiling Condensation Melting Freezing page 5

6 CALORIMETRY PROBLEMS: There is a class of calorimetry problems in which a number of items of different temperatures are all placed together in some insulated container, or else the system comes to an equilibrium temperature so quickly that we can ignore losses to the surroundings. For such problems, we start with an equation that says Q1 + Q = 0, [Insulated system] where Qi is the heat gained or lost by each thing in the mix. If Qi is positive, the item picked up energy, if it s negative, it lost energy. If there are only two items exchanging heat, we can alternately write this equation like this: Q gain = Q loss, [Two-item system] where I ve included the absolute magnitude signs to remind you that you can t have a positive thing equal to a negative thing. These absolute magnitude signs make the math cumbersome if you are solving for either an initial or final temperature. Whichever equation you start with, the next important part is to determine what kind of heat processes are occurring for each item. If, for example, a small ice cube is dropped into a large vat of molten lead, the ice will probably melt into liquid water, increase its temperature to the boiling part, and boil completely away. If there is much less lead and much more ice, it may not go through all these steps. In any event, you need to replace the Q i which stands in for the ice cube with a term -- ml,, or -- for each change in the object. Notice that if the ice starts off at -20 o f mcδt ml v C, you have to include an mc ΔT term for the ice heating up to 0 o C before it can melt, and that c will be different for ice -- or steam -- than it is for liquid water. page 6

7 ADVICE: In doing calorimetry problems, you will often have to follow one of the items through a few different temperature and phase changes. You should sketch a separate drawing for each change, labeling each drawing with what s known about that process. This helps a lot in keeping track of all the terms. Q i page 7

8 DIGGING DEEPER: Conduction, Convection, and Radiation Okay, HOW does heat energy move from one object to another? We can discuss three processes that allow this transfer. First, conduction is the transfer of heat energy which occurs when two objects touch. If we consider conduction along an object with a uniform cross-section, A, but with a temperature gradient along its length -- that is, a difference in temperature between its two ends -- then the rate at which heat flows can be expressed as P = ΔQ / Δt = ka( ΔT / Δx) [Heat conduction] where Δ T / Δx is the temperature difference per length along the direction that heat flows, and k, the thermal conductivity, is a parameter that varies from one material to another. For insulation, we choose a material with small k, so that the rate of heat flow will be smaller for the same Δ T / Δx. Air is a very good insulator, as long as we don t let it move. Some of the best insulation -- styrofoam and fiberglass, for example -- is mostly air, surrounded by some material that keeps the air localized in pockets, so that it doesn t move. If air -- or any other material -- moves, it can carry heat with it. This is called convection. We use air as a convective material when we blow on something hot to cool it. Convection is harder to describe mathematically than conduction, and we won t even try in these notes. How does the Sun heat the Earth? It s not by either of the two processes we ve just mentioned, since there s hardly any material between the Sun and the Earth. Heat is transferred instead as radiant energy -- visible sunlight, as well as infrared and ultraviolet light -- which is converted to heat when it gets to the Earth. We can imagine the Sun to be an ideal radiator, which, for historical reasons is called a blackbody. For such an object, the rate at which radiant energy is given off is given by P = ΔQ / Δt = σεat 4, [Radiative heat transfer] where σ=5.67x10-8 W/m 2 /K 4 is a universal constant, ε, called the emissivity, is a number between 0 and 1 which tells how efficient a radiator we have, A is the surface area of the radiating body, and T is its temperature in Kelvin. Looking at this equation, we see that any object that is not at absolute zero temperature must radiate energy. However, it may actually absorb more energy than it gives off if its surroundings are warmer than it is. Often, more than one of these processes is taking place at the same time. If we sit around a campfire, we feel the glow of the fire -- radiation -- and feel warm air coming our way from the fire -- convection -- and sometimes a hot cinder flies out from the fire, hitting our skin -- Ouch! conduction!. Generally, however, physicists focus on problems where the heat transfer is mostly just of one type. page 8

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10 FZX, Chapter 14: THERMODYNAMICS STATE VARIABLES: We described fluids in terms of a couple of variables -- pressure, the speed of flow, the height of the fluid. We can think of these variables as state variables, because they describe the state of the fluid. We are about to consider a class of gases for which the state variables we are interested in are: P = pressure V = volume T = temperature (in Kelvin, of course!) n = the amount of gas, measured in moles = the number.of.molecules/avogadro s.number = N/N A [State variables for a gas] Avogadro s number is the number of particles x in a mole. The particular simple model we shall use to describe gases is called the ideal gas. Ideal gases can be defined as gases that obey the Ideal Gas Law, which says that PV = nrt [Ideal Gas Law] where R is a constant -- Rankine s constant -- equal to R = 8.31J/mole. K The Ideal Gas Law is a culmination of a number of laws -- Boyle s Law, Charles Law, and the Law of Gay-Lussac -- which each tell a part of the story. Consider what happens if we have a sealed container of gas. If we heat it, it can expand -- as a balloon does -- or the pressure increases if the volume is held fixed. One can also ask what happens to the temperature if we compress the container or make it expand. If we have some quantity of gas in a sealed container, such that n remains constant, then we can vary any of the other three variables, P, V, and T, and see what happens to the others. Generally, we keep one of the other two variables fixed, or else we adhere to some other constraint. Some of the most common such constraints are defined as follows: Adiabatic process: No heat flows Q=0 Isochoric process: constant volume V=0 Isobaric process: constant pressure P=0 Isothermal process: constant temperature T=0 There are a few limits to the Ideal Gas Law. It assumes that the gas consists of pointlike molecules which bounce off each other, but otherwise do not interact. Phenomena like liquefaction of a gas, which require some sort of intermolecular interaction, cannot be explained with the Ideal Gas Law, because, well, if the molecules don t interact, how do they attract each other enough to become a liquid? If we consider the individual molecules that make up a gas, we can describe how much material there is by the number of molecules, N, N = nn A The Ideal Gas Law can thus also be written as PV = NkT, where k = R/N A = 1.38x10-23 J/K [Definition of Boltzmann s constant, k] Now we are getting somewhere. Since temperature is the kinetic energy per molecule, NkT must be proportional to the total kinetic energy of those zillions of dancing gas molecules. We can define the total internal energy of the gas, which is just the total kinetic energy of the molecules, and it turns out to be -- for an ideal, monatomic gas -- U = 3 nrt = PV [Internal energy of ideal gas] We can do one more thing with our ideal gas model. If we put it in a cylinder, and push down on the lid of the container, so that we decrease its volume, we need to exert a force on the gas to move the cylinder -- provided we have a tight enough seal that we don t lose any of the gas. The gas, meanwhile, is doing negative work, since it exerts a force -- page 10

11 proportional to its pressure -- on the lid in the direction opposite the way the lid moves. The work done by a gas when we change the volume it sits in is W = PΔV [Work done BY gas] If the pressure the gas exerts is not uniform, then we need to use the average pressure. If there is no change in volume, then no work is done: we DO NOT replace P V with V P. Remember, we have to move the walls of the container in order for there to be any work done. FIRST LAW OF THERMO Now, if we supply heat to a gas, that heat is just energy. It can be used to do work -- by expanding the gas -- or to increase the internal energy -- raising the temperature. It is like with your income: some of it goes into savings, and part goes into spending. We can summarize this common sense with what is known as the First Law of Thermodynamics: Q = ΔU + W (Heat can either do work or raise the internal energy of an object.) [First Law of Thermo] where Q is the heat entering the gas, U is the change in the internal energy of the gas, and W is the work done by the gas. The lefthand side tells us where that energy came from, the righthand side tells us where it went. ENTROPY We mentioned, when we introduced temperature scales, that heat flows from warmer things to colder things. This is not always true, or you wouldn t be able to make ice cubes in your freezer. What IS always true is that heat energy, when left on its own, flows from warmer to colder objects. We can analyze this in terms of entropy, which we will first define loosely as the amount of disorder in a system. (We will come up with a more exact definition very soon.) Imagine taking a snapshot of the molecules in a block of ice. They don t travel far, but as long as the ice is warmer than absolute zero, they do have kinetic energy, and they will travel some. The warmer the ice, the more they ll travel, and the more random their positions will be when you take the snapshot. When the ice gets warm enough, the disorder is so great that the molecules no longer stay near their neighbors, and the ice becomes liquid. As we supply heat energy to the ice, the disorder increases, and a constant amount of heat will create more disorder in very orderly, super-cold ice than in relatively warmer ice. We can define the change in entropy as ΔS = Q / T [Change in entropy] where S is the entropy of some material, Q is the heat added to the material (negative if heat is removed), and T is the temperature, measured in Kelvin. Since the temperature may change as heat is added or removed, the T above should be the average temperature for that object. Using this definition, we can state the Second Law of Thermodynamics: Δ S Universe = ΔS1 + ΔS , [Second Law of Thermo] where S Universe, the entropy of the Universe is the total entropy of a system of objects that may be in contact with each other, but which do not give off or absorb any heat from their surroundings. page 11

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