When we covered conservation of energywe found that the total mechanical energy of a closed system was constant.When this law is extended to include energy in the form of heat and thermodynamicwork it is called the first law of thermodynamics.
What is heat? Heat is defined to bethe energy flow from one object to another as a result of a temperaturedifference. In other words if we put two objects in contact with each otherand they have different temperatures, energy will flow from the hot oneto the cold one until the average kinetic energy of the molecules in eachis the same. According to the relationship between average kinetic energyand temperature, this means they end up with the same temperature (thisis sometimes referred to as the zeroth law of thermodynamics; twoobjects with different temperatures will, when put in contact, reach thesame temperature). Heat flow is measured in Joules and in Calories (4.184J = 1 cal; Note; a food calorie like on the back of a box of cookies isactually a kilocalorie or 1000 calories,)
Heat energy, Q, can be transferredby four main mechanisms: convection, conduction, radiation and evaporation.The first three are discussed in your book (read those sections); we talkedabout evaporation above.
An alternate concept for an integral is thatit is equivalent to the area under a the curve formed by the function beingintegrated. So for our new definition of thermodynamic work the work doneis also the area under the curve formed by PdV on a plot of P versus V.
Two important examples which we willuse below are work done by an isothermal process and work done in an adiabaticprocess.
The first law of thermodynamics saysthat the change of internal energy of a system has to equal the heatflow into the system minus the work done by the system. In other words,you have to account for all the energy coming into and out of the systemso that you have the same amount you started with (conservation of energyby another name); dU = Q -W. In other words we can account for all theenergy in a system if we monitor the internal energy, the heatflow into(or out of) the system and the work done by (or on) the system.
Comments
1) Note that a system doesn't contain orhave a certain Q; heat Q is energy flowing into or out of the system. Thereis no answer to the question 'How much heat does that mass have?'. Likewisea system doesn't contain a certain amount of work; you may or may not getwork out of a system where there is heat flow and internal energy change.
2) A system does have a certain internalenergy (we discussed the definition of internal energy in the section ontemperature, pressure and gasses). You can say 'this system has an internalenergy of 100J'. Another way to say this is that internal energy is a statevariable (as is temperature, pressure, volume). Work and heat are not statevariables.
3) The first law doesn't say anything abouthow much heat goes into changing the internal energy and how much goesinto work for a particular system. This is the provence of the second lawof thermodynamics.
Suppose we have three coins andwant to know how many different results we could get from tossing them.Here are all the possibilities:
| coin | Toss 1 | Toss 2 | Toss 3 | Toss 4 | Toss 5 | Toss 6 | Toss 7 | Toss 8 |
| 1 | H | H | H | H | T | T | T | T |
| 2 | H | H | T | T | H | H | T | T |
| 3 | H | T | T | H | T | H | H | T |
What does this have to do with thermodynamics?Suppose we have three molecules which can go randomly into two sides ofa container. Let's call the left side T and the right side H! Using thesame reasoning we can see that it will be three times more likely to findtwo molecules in the H side and one in the T side than it is to find allthree in the H side. In other words it is much more probable that moleculeswill spread out in roughly equal numbers between the two sides of the containerbecause there are more ways to have that happen.
Mathematicians have worked out probabilitytheory to tell us what results to expect in these kinds of situations.They can prove that the probable number of heads, P(H), in a coin tosswith N coins is given by P(H) = N/2 ± squareroot (N) with a 93%confidence limit. Lets calculate this probability for a few values of N.
| N | P(H) = N/2 | ±sqrt N | % error |
| 100 | 50 | 10 | 10% |
| 1000 | 500 | 31.6 | 3.2% |
| 10000 | 5000 | 100 | 1.0% |
| 100000 | 50000 | 316.2 | 0.3% |
| 1000000 | 500000 | 1000 | 0.1% |
| 10000000 | 5000000 | 3162.2 | 0.03% |
| 100000000 | 50000000 | 10000 | 0.01% |
| 1000000000 | 500000000 | 31622.8 | 0.003% |
Look what happens as the number of coinsincreases!! If you guess you will get 500,000 heads when tossing 1,000,000coins you only expect to have an error of 0.1%! In other words for largenumbers of coins you expect to be very close to exactly half headsand half tails as compared to small numbers of coins where you occasionallydo get all heads or all tails.
What does this have to do with thermodynamics?Let's think about putting molecules in a box again. Generally the numberof molecules in a container is quite large; chemists generally imagineworking with a mole of atoms which is 6.02x1023 atoms. Applyingthe discussion of the coin toss we see that the error in assuming the moleculesare equally divided between the two sides of the container (N/2 in each)is very small. Or stated a different way, it is very unlikely (very smallerror) that we find that the molecules are NOT divided equally betweenthe two sides. The likelihood of having precisely half in each side increasesdramatically with the number of molecules.
Can we quantify this (I want an equation!)?Yes! Let us call the number of different ways a particular outcomecan occur (three ways to get two heads and one tail in our three coin tossfor example) the number of microstates and use the symbol omega, W.So for the case of three coins omega = 1 for all heads or all tails (thereis only one way to have this occur so only one microstate available) andomega = 3 for two heads and one tail (there are three ways for this tooccur so there are three microstates). Likewise omega = 3 for two tailsand one head. Now we define something called entropy; S = kB ln (omega) where kB is Boltzman's constant (kB = 1.38x10-23 J/K) and ln is the natural logarithm. Why choosethe logarithm and Boltzman's constant? We'll get into that below but basicallyit is so that this definition is compatible with other definitions plusthe fact that logarithms make working with very large numbers easier. Noticethe units for entropy will be Joules per Kelvin
Now we see something interesting. Noticethat the entropy for all heads is lower than the entropy for the case oftwo heads and one tail. In our three coin toss the entropy for all headsis kB ln 1 = 0 while the entropy for two heads and onetail is kB ln 3 = 1.5 x 10-23J/K which is largerthan 0. So another way to state the fact that getting half heads in a cointoss of N coins is more likely than some other distribution (two headsand one tail for example) is to say that the outcome with the highest entropyis more likely. States with higher entropy are more likely than stateswith low entropy. This is a statement purely based on the laws of probability.
We can also see that, as the numbers of coins(or molecules in a container) gets larger the state with the highest entropybecomes almost certain (the error gets VERY small). For a mole of moleculeswe can say with near certainty that the system will be in the stateof highest entropy. This is a result purely due to probability appliedto large numbers of molecules.
Version one of the second law of thermodynamics:Isolatedsystems tend go towards a state of maximum entropy.
Some comments. It is possible to makea loose connection between entropy and disorder (the connection is notexact but works most of the time). In our example of molecules in a boxwe see that we have more information about the molecules if they are allin the left side of the box than if they are spread all through the box.Somehow there is more order if the molecules are all in the left side becausewe have limited the possible locations to one side. But from our definition,having all the molecules on one side would be a lower entropy state thanhaving them spread over the entire container. From our reasoning abovewe recognize that the system would tend to be found (with very small error)in a more disordered state (high entropy) with the molecules spread halfin each side. So another way to state the second law is to say that closedsystems tend to go towards a state of maximum disorder (maximum entropy).
This can't be right can it? Don'twe see counter examples every day? What about living organisms which seemto be able to create order at the cellular level, at least until they die?Or ants building an organized nest. There is an important criterion mentionedin the definition: Isolated systems! Living organisms cannot live as isolatedsystems. They create order locally at the expense of disorder elsewhere.We take in energy in the form of food which has highly ordered chemicalbonds and expel waste which is in a very disordered state. So the totalenvironment of organisms plus food sources experiences a net entropy gainoverall. The second law applies to everything, including all living organisms.Another conclusion of this line of thinking is that the universe, a closedsystem, has to be experiencing a net entropy increase overall. The endstate of the universe (fortunately for us, billions of years in the future)will be total disorder (this is sometimes referred to as the 'heat death'of the universe).
Example: From our earlier discussionof the Maxwell Boltzman distribution of velocities we can predict anotherresult of the second law. Notice that the distribution for a low temperaturedoes not include as many choices of velocities as the distribution fora high temperature. So the same gas at a low temperature is more orderedthan at a higher temperature. This means a cool gas will tend to gain heat(if it has the chance to) so as to increase its entropy. This is also trueof non-ideal gasses where the heat flow goes into internal energy; heatflow into a gas at constant temperature increases the entropy; heat flowout decreases the entropy.
Example: Still one more example isfriction. When we slide a box across a table we do work on the box. Thisresults in the box gaining energy. Initially the energy is very organized;all the atoms in the box go in the same direction. But as energy flowsinto the surface the molecules of the surface gain kinetic energy thatis randomly oriented (the temperature increases). Now the energy is ina disordered state. We can't get this energy back for doing work (movingthe box for example) because it is in a more random state and entropy (disorder)does not decrease spontaneously.
Suppose we are an engineer and we wantto build the most efficient engine possible (and our name is Sadi Carnot!).We want to burn some fuel or use some other heat source and convert thisenergy to mechanical work. From the first law of thermodynamics we alreadyknow that the maximum mechanical energy output cannot be more than theenergy input from the heat source (you don't get something for nothing!).So the best possible result we might expect would be to burn fuelto get 100J of heat for example and get 100J of mechanical energy out.(NOTE: this turns out not to be possible! Read on.) Let's see whatelse is involved.
What is a cyclic process (does this haveanything to do with Harley Davidson)? First off, to have an enginewe must be talking about a cyclic process. Whatever happens in theengine we have to make it to come back to the same state periodically sowe can keep getting work out of it. So for example if we had an expandinggas pushing against a piston we could get mechanical work out of the piston.But the only way to keep doing this (without having an infinitely longpiston!) is to eventually return the piston back to its starting point.In other words the internal state of the engine has to periodically goback to the original state. Another way to say this is the change ininternal energy is zero for a cyclic process. For the cycle dU = 0where U is the internal energy; we put everything back the way it was whenwe started. This means that the first law of thermodynamics for a cyclicprocess is W = dQ; the work done in a cycle is equal to the net heat flowinto the cycle. (Watch out for the word net here, remember what a problemit was in Newton's first law, F = ma where F is the net force.)
When we talked about conservation of energywe learned that some processes are reversible and some are not.If we raise a mass, m, up to a height h we store up gravitational potentialenergy mgh. We can get all of this stored energy back by dropping the massby the same height, h (in which case we end up with an amount of kineticenergy equal to the amount of gravitational energy we had stored). Thegravitational force is conservative; we can change potential energy toother forms and back again without losing any energy. Other forces suchas friction are non- conservative. Processes such as sliding a block acrossthe table against a friction force are not reversible; we cannot get theenergy back.
A more precise definition of reversible.Aprocess is reversible if it occurs quasistatically and does no work againstfriction. What do we mean quasistaically? Quasistatic means we do theprocess so slowly and in such a way that we could, if we wanted, at anystep reverse the process and go back to the previous state. In our exampleof lifting a mass to store potential energy we could imagine doing it veryslowly so we can get the energy that we just stored back at any step. Noticethat these two conditions also mean that any heat flow has to be such thatwe can always take a small step backwards and go back to the previous state.(This rules out work done by friction because we cannot get the energyback if it is lost to friction.)
Why do we want a reversible process forour 'perfect' engine? Well certainly if any energy is lost due to frictionit won't be as efficient as if we have an engine which does not loose energyto friction. Reversible process don't lose energy to friction so to getthe best possible engine we want the processes which make up the cycleto all be reversible.
What thermodynamic processes are reversible?Isothermal and adiabatic expansion are two examples of reversible thermodynamicprocesses. Isothermal means the process occurs at a constant temperature.Adiabatic means the process occurs with no heat flow at all.
So lets build an engine. Lets start at pointa in our diagram below (pressure Pa volume Va) and go isothermally (constanttemperature T1) along path (1) to point b (pressure Pb, volumeVb still at temperature T1). What is the physical process thatwould do this? Well we could put a piston of gas in contact with a heatreservoir of constant temperature T1 and let the gas absorbheat Q1 and expand. Then we could isolate the piston (superinsulate it so no heat enters or leaves for example) and let it keep expanding(slowly so it is reversible) to a new temperature T2 at pointc (with new pressure Pc and volume Vc). To complete the cycle lets go fromc to point d and let the piston cool isothermally (in contact with a heatreservoir at temperature T2 which absorbs heat Q2)and then further contract adiabatically (isolated) from d back to our startingpoint a. This has to be the most efficient cycle possible for an engineoperating between temperatures T1 and T2 since weused only reversible processes.
Ok so how much work is done during thiscycle? We can get this two ways. From the definition of work we couldfind the area under the curve for each step. Notice that for the coolingpaths (c to d and d back to a) we must subtract the area under the curve(we do negative work). This is the price we pay for getting back to thestarting point so that the process is cyclic but at least since the processesare still reversible we know we loose no heat to friction. So for thecyclic process the total work done is the area inside the cycle. Infact this is always the case; the area inside a cyclic process diagramedon a PV diagram equals the work done in the cycle. So one way to find thework done is to find the area inside the cycle.
For this cycle however, the easy way to findthe work done is use the first law. We know that for the process a to bto c to d to a we have W = dQ since from the first law the change in internalenergy is zero for a cyclic process. But the only heat flow occurs duringthe isothermal parts a to b and c to d (adiabatic means no heat flow!).So the work done in this cycle will be W = Q1 - Q2(Q2 is negative because heat flows out of the cycle duringthe isothermal process c to d).
So why can't we get a bigger area (andtherefor more work) by following a different path on the PV diagram?We could do that but the paths would not be isothermal or adiabatic andso would not be reversible which are the most efficient choices. Rememberwe are looking for the most efficient engine, not the one that does themost work.
Version two of the second law of thermodynamics:Notice that in order to make the process cyclic (get back to our startingpoint) we have to expel some heat (subtract the work done by the lowercurve). Evidently it isn't possible to build a cyclic engine which doesnot give off heat. This is why your car engine gets hot and has tohave a radiator, nuclear power plants have cooling towers and computershave fans. All processes which exchange one form of energy for another(heat into work in the case of engines) have to generate waste heat.This is not just energy lost to friction; as we have shown above,even for the most efficient engine possible with no friction there willbe waste heat expelled to he environment. Real engines also have heat lossesdue to friction but even if we could get rid of the friction all engineswould expel heat because of the second law.
Version three of the second law of thermodynamics:TheCarnot cycle is the most efficient heat engine possible. The abovecycle (using two isothermal and two adiabatic paths) is called the Carnotcycle after Sadi Carnot who first proved it was the most efficient cycle.One implication is that there is an upper maximum theoretical efficiencyof any engine. We can only make engines as efficient as the Carnot enginebut no better.
Thus for our 'perfect' engine the efficiency= (1-T2/T1) x 100% whereT1 is the temperature of the hot reservoir and T2is the temperature of the low reservoir.
What does this efficiency equation mean?An important thing to realize here is that the efficiency of our perfectengine can never be 100% because 1 - T2/T1 is alwaysless than one. How about raising the temperature of the upper reservoirto get better efficiency? Fine, that does raise the efficiency but if youdesign an engine which runs at too high a temperature you have a problemfinding materials that won't melt or burn up. How about lowering the lowtemperature reservoir? Cars use the atmosphere as the low reservoir butyou could use a trunk load of ice as the low reservoir instead. This wouldwork but it also has practical limitations.
But what about my solution to the worldenergy problem!? My idea is to extract heat from the ocean! Hey thereis lots of internal energy out there why can't we use that? Notice thereis nothing in the first law which says we can't do that. The first lawsays we can't get more energy than is there but it doesn't say wecan't get whatis there. So can we do this? From the above argumentswe see that there has to be a cool reservoir in order to have somewhereto expel heat, according to the second law. What would this reservoir be?Unfortunately the earth and atmosphere have about the same temperatureas the ocean on average. So there is no cool reservoir for our waste heat.So we can't do it. Oh I suppose we could drag an iceberg around as a coolreservoir but this doesn't seem real practical.
Version four of the second law: Ina closed system heat does not spontaneously flow from a cool reservoirto a hot reservoir. Notice that if this occurred we could separatetwo parts of the ocean, have one part spontaneously give up heat to theother so we have a hot and cool reservoir and then run an engine to dowork between the two reservoirs. But this would violate our second versionof the second law because we'd get work out with no hot and cool reservoirsto start off with.
How do these versions of the second lawof thermodynamics connect with the version based on entropy, disorder andprobability? Recall the ratio Q/T plays an important role in findingthe efficiency of the Carnot cycle; the hot reservoir decreases by thismuch and the cool reservoir increases by this much. For this reason itwas given a special name; entropy! S = Q/T (well actually it is usuallywritten for very small heat flows dQ so dS = dQ/T). This definition ofentropy was actually made long before the one based on probability.
Let's revisit the 'get energy from the ocean'idea again from the perspective of our first definition of the second law.First off, reversible cyclic processes have to result in no change of entropyfor the engine. We end up in the same place we started from so the disorderhas neither increased nor decreased for the engine. So far so good.Now lets imagine using this cyclic engine and only a hot reservoir (theocean for example) with no cool reservoir. Well the hot reservoir wouldlose entropy (become more ordered) in this process since heat flowsout of it (S = dQ/T is a negative quantity). But according to the first(disorder) version of the second law the overall, total entropy has toincrease. The only way to get entropy to increase overall for thisprocess (or even to remain constant) is to raise the entropy somewhereother than the hot reservoir or the engine. In other words you have toraise the entropy (increase disorder) of a cool reservoir somewhere else.But this is saying the same thing as version three of the second law: ifthe process is cyclic we've got to have waste heat exhausted to a coolreservoir somewhere in the process. So all versions of the second law givethe same results.
How about some non-obvious applicationsof the second law?
We know computers organize data. Physicallythis occurs as ordering of the current flow in the computer circuitry orordering of magnetic particles on a storage device such as a hard drive(we do NOT want the physical representation of ones and zeros in the computerto be random!). So from the first version of the second law we can seethat computers are a type of heat engine- they use electrical energy tocreate order. We can conclude that computers will always give off heat.
Refrigerators are heat engines run in reverse:we put work into the refrigerator, take heat from the inside and expelit to the outside. By reversing the above arguments for heat engines youcan show that there is no way to build a refrigerator which only coolssomething. Because of the second law there will always be waste heat givenoff, the coils on the back of your refrigerator will get hot. This is alsowhy air conditioning units have to have part of their apparatus outside;there has to be a hot reservoir where you can expel heat. A Carnot refrigerator(the Carnot cycle run backwards) is the most efficient refrigerator possible.