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# Statistical Mechanics Lecture 6

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#### Automatisierte Medienanalyse

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Steiger University no physical device ever
measured for its intended there mathematical thing medics intended to measure obsessed with usually truly is a thing measures were supposed to measure over a limited range of whatever happens to debate are 80 bringing the 1st what a political a spring back there just bringing where calibrated their measure along with you use to measure for your call on something so you connector spring balance to Italy upon it and you look at the the calibration and you say OK armor the number of ticks number fix but only the the stretch of the spring is a measure of the forest well that's not really true if you were my if you pull to larger break displaying and so forth so over some limited range of forest this spring of the spring constant point of this spring displacement measures the forest so how normally what the situation is is you'll have some quantity which you could imagine a laboratory easily it could be the length of the stretch Spring or it could be the height of the of the of mercury in May In a from armourer and you ask yourself what could be armed with big quantity in question what could depend on to fight the length of the spring displacement clearly depends on often depends on Bosnian sequel took it depends on the faucets being applied but it could depend on other things could depend on temperature it could depend on air pressure but she's that's not terribly important those variations of priests malls to northern New say what could depend on it can depend on the force the following spring is going to be some relationship between them my I work like that now was some crazy curve and then all some limited range over some limited range where the functions it is linear and of the function as smooth any function that smoothed the doesn't variants crazy way over a little of a limited range will be linear this linear then you can say all that limited range variations in the quantity that measuring a proportional to variations in New year are in whatever the independent variable happens be in cases bring balance it's just force say they would be Mercury in India column here what can the height of the column depend on wealth yet to depend on temperature where else could depend on depend on the air pressure in the room that much because sea air pressure and the rumors Edzard's isolated thereby be young might be closed right by by the glass of holds mercury and what else could depend on I can't think of anything can you think of anything might depend on besides the temperature Will I now maybe natural local gravity field yes right after OK so let me say it a different way it could depend on the horse and the direction that hold year of the thermometer manager of the local gravity field is pretty constant below the surface of the earth the only thing you could you could have a funny bad not sisterly bed but a thermometer which should behave differently if you held that horizontally you have the vertically and then you just say OK bye Ruelas behold the thermometer vertically and it depends very much then you just did you just say OK let's fix the direction of the thermometer make it part of the definition of the thermometer will hold a thermometer vertically anything else anybody can think of that the chemicals Noonan mercury-in-glass a pretty inert fell were pretty good that Wells the cause of a very long time Girardi during the product arms or make up beyond the mercury could the decay heat of long enough times in their 34 and 35 years 5 but the thing about it for while Gedo a very rare high approximation the only thing that behind her columns God to depend on is going to be the temperature or equivalently the energy of our energy that however we Mercury has absorbed from the 3rd most of the environment but at a quarter of that it still assuming that a thermometer is in equilibrium would be in line of course if thermometer was not in equilibrium and would be environment it is not measuring the temperature of the environment if only if there's been time established for the environment the company equilibrium of the mercury and then the only thing on is temperature but the height of column depends on temperature in some way let's suppose are this point here measured it we find out over some range over some range that it looks pretty linear a small range of range bigger range it's gonna go through the yard and also should make a temperature very hot the cart can't bigger if you make it hard right now drives a both so through the ceiling he so all the some Ranger looks pretty linear let's say you measure it at a freezing point of water the freezing point the water temperature the height Of these column and you find it's over here measure boiling water finds already if you were a lucky all over that particularly range this function will look pretty linear the site divided and 400 those segments and segment constitutes a new arbitrarily say this point you call 0 atomic yeah disintegrate scale this point to core 100 the violent or 100 steps is amount of measuring temperature down the amount measuring temperature tore high accuracy but 1st of all it's not measuring temperature in the absolutes and we've called this temperature 0 so it's not measuring absolute temperature when you when you go out in the wintertime and 0 degrees Fahrenheit among other it doesn't mean you going out and the absolute 0 so as a convention about where you put the 0 but smaller incremental changes in the temperature will be proportional to small incremental changes in the height a cop order times there which she really liked but boundary around how long it takes to establish normal equilibrium if depends on the thermal conductivity of the glacier the 3rd of the
account in equals 3 and equal 7 and also in equal 7 M equals 3 that's the same pair we also don't want to count as a parent in in a we don't want the think that they're the 1st molecule in the 1st mark 2 forms of so we can does this by saying we only some In the big Aetna had some others them tour that have but they are sold In a great event GM that means you character 3 until but you don't count to Winfred how is it going to be it's going to be some potential energy you would now not further potential energy because he was saving for volume you on the distance between March in an hour kill em so let's write that as the distance between X In an ex and that means exit in X M means vector of separating be in position from the end and position the absolute value means the distance itself and the potential energy is a function of the distance between these 2 molecules we summit of any greater than a week they add up all of the potential energy so we can just call thing here but just call it you fool X but here's what I am now you Wizard it's what is going to be the small quantity the small quantity is is going to be you the potential energy is going to be assumed to be a good deal smaller kinetic energy between molecules roads we start with the ideal gas and then we simply turn bomb that turns on the potential buyer we keep that job very small so that at least at 1st potential energies stored by these pairs of molecules is much smaller than the kinetic energy of Tiger of better approximation of food begin by DRC is actually only 1 number regarding this potential which is important to define now question we're going to encounter in the following interval it's going to be let's take particle 1 in particle to doesn't matter which pair just 2 particles 1 and 2 D X 1 D X 2 0 now will mean by DX 1 1st of all particle 1 has 3 quarters to write x wines IDX I mean the interval over the volume element for particle 1 three-dimensional volume and I mean for the year here as I just write DX it means this kind of symbol he said no want right the potential energy as a function of the distance between them between them what I'm doing is I'm taking the particles end I did ratings on what all possible positions of the 2 particles the potential energy between them but sir suppose From moment that I hold the relative position the relative position x 1 minus X to fix and integrate over the position of X 1 only next 1 around dragging next to with how how could I do the interval overtook water 1 way is to integrate over 1 of the coordinates keeping the separation the other 1 fixed and then a the would integrate over the separation between 2 separate steps and well I guess if I hold at hold the distance between the 2 of these fixed but integrate the position of the pair of them all the space and going to get a factor of the volume was holding the 2 of them fixed and thinking of them as a unit and integrating them all the space I would get a factor of the volume of space now I've taking care of that de broke West holders point fixed because the other way because of holders . 6 and integrate over the position of the 2nd 1 Yao think that withstood the opposite order but cigarette 1st holding particle 1 fixed and integrating over the position of the particles what active fact is we think that will be scored X any acts you will love acts always doing is wholly 1 particle fixed Air warmly of the particle around an ailing up or degrading the total potential energy of volume element of a particle moving around got where the unit Ferber before work units of this quantity what units of the act of you energy what's the units of VX White right but this is really D 3 exits of volume integral so what have be 3 X volume the units of this thing Our volume and earnings energy this quantity is telling us how important energy is because but telling us how strong the potential energy use and held a how big volume does you act to helps spread out and let's get this thing and this thing is we are important numerical quantity that determines the strength of the potential of combination of how strong the potential is and how widespread its distributed but scholars you'll 0 see only program that will come into our calculation and you don't know if it's big or small big you don't know offhand words big because distributor over large volume but you was small or whether it's distributed over small volume was big 1 way or another you not resisting which will determine everything else right supposing we we held this particle fixed and integrated over the other 1 that would give us the goal here and it would be you not having them then let's integrate the position this molecule which means moved pay around that's going to give us another factor of volume of the total volume of the guests turned you not can everybody see that that victory in New York said you that to single yet picked up any pair particles any time see you How do you I were 1st we 1st hold this 1 fixed and integrate the other 1 everywhere now but of course we're not getting get much except when this is in the field of potential you might ask why doesn't holding his fund fixed in integrating the other 1 why we get a factor of volume for that be clause assuming pressure said resuming their whatever the function you ease a function which goes 0 zeroing the particles that far apart so what EU is it's a function which is only significant when the 2 particles Our within a certain distance beyond that distance across 0 also Freehold 1 particle fixed and integrate the the other 1 whether one-dimensional problem we would just say we in the rule would be area under the curve but not quite the area under the curve because it's a three-dimensional it has units of energy firms volume not energy kindly and Japan's volume but it is
particles being close now that the field cloak close enough to get feel each other's small Sony gasses die loot the particle spend most of the time being foreign a four-way that they simply don't feel richer some fraction of the time to particles would get close enough to get the feel of the within volume distant and that's when the forces between those 2 molecules become report that what fraction of the kind that really given molecules will be within the molecular distance of each other much much smaller the probability that the really did take the radio and ask What's a probability that all 3 of them will be within the molecular diameter of each other as vastly smaller depending on the volume of the guest of course is much more and the probability that 2 of them will be near each other so or if there was three-body forces the three-body forces would be less important not because a weaker but because just on the average the probability of finding 3 particles in the neighborhood of each other for a guy would get a square is even smaller that three-body forces there also were going to do an expansion and the expansion there are 3 body effects even if the forces only 2 body effect during the interview question is will follow the expansions far below or other terms they won't be important as long of particles are better Luke looked opaque set where we are but them now we get on autopilot where calculate a calculator partition function but the name Deborah oil which remember you integrate all with the momentum the momentum you integrate over the position now DPN PND X really stand for a very high dimensional multiple integrals how high is a dimension of the integral each 1 of them 3 times the number of particles they have detected the 3 components of momentum for the 1st article 3 components for a 2nd particle and so forth 3 in and that didn't bother writing that this is a multiple and the growth of power 3 and likewise DX is big multiple integral would've right persuade business factor of and factorial it makes no difference appalled we sometimes Korean makes no difference and then we have each other minus Ito minus what I need to remind the Beta Beta the inverse temperature times the energy of the whole system and the energy of the whole system is a term which is Connecticut Angie just write it as Peace squared off over to and capital and its status for this here had to put the summation starring in but this is the Scots enough that 1 terms and any other tournament is a potential energy so it's gonna be an exponential of the sum of 2 things and the exponential of the sum of 2 things is just the product of 2 exponential of a miners beta times such as call OAU of X and this stands you UXU stands a toddler potential energy as a function of all of the position and this is our job to try the computer now the glorious and simplifying feature of this Is that a fact arises a fact arises until a momentum integral and a position and to grow with the integrated itself being a product so fact Rifkin right it as the product of 2 integrals the 1st Centerbrook we in fact seen before I won't bother I will it actually tell you what it is just a destructive to remind you that needed the 1st integral momentum in the world while others armed guards that were wars are square root of voters tool in the hands of party and isn't an all-over Bader right here remember that raised the power 3 in over a 3 . 3 and remember that whose is Delcine interval flee each momentum until we used its exactly the interval that we did for the case of the ideal gas if you was 0 we would be doing the ideal guests have exactly the same thing to roll here when I didn't need it fortunately where are needed it but 1 of the intervals is exactly the way that we did the last time now the previous time when we did the integral over position before we had any potential energy remember what this interval position gave us they gave us the volume raised to the end Powell that's when this was just when you was 0 if you was 0 it's just the interval acts does the volume to the and power and remember that so let's actually multiply and divide by the volume of the and power let's put it in year volume and power In the end divide by volume for the power and the reasons for for doing that is because then we immediately recognize this integral as the old partition function for the ideal gets this goal here is just the partition function for the ideal gas so what we can call it z 0 0 now stands for the calculations when you was equal to 0 when there are no forces between the ideal gets us a partition function as a function of data depends on data of the ideal gas the other factor is the new thing here so this is a fact that we have to work on which they go with blackboard work on it and across I'm not telling you anything you want the than yourself she was 15 minutes where you realize very quickly the in Togo factorize is 1 of the factors if we put in the veto and would have been just a partition function of the ideal guests and the other factors a new thing that we wanna calculate areas and the the all over volume today in Etowah mind is Bader Of the potential energy know what's a small quantity of our calculation is that we're going to expand in terms of you the potential energy is a thing which is assumed to be small which starting with an ideal gas in a week turned on arm turns on the potential energy we turned on vary vary weekly of return on very weakly we can expand this exponential Taylor series In the strength of you but very easy it's just equal to 1 in minors data you of X those are the 1st 2 terms are my music is that each of mind something of course that is equal to 1 minus as plus a squared over 2 factorial blah blah blah blah blah but when is very small Of course when this is sufficiently small we ignore everything but the 1 boat
of the original partition function for the ideal guests who never even really needed because her religious considering the corrections plus logarithm our 1 data and squared off over to be plans you not but why we expanding and were expanding in the magnitude a view of you want to be precise your orders are expansion parameter we're trying to make a Taylor series expansion or an expansion in powers of you naught but Healy have the logarithm of 1 minus a constant times you know what was our goal Our goal is to expand it In powers of you not so what is a Taylor series expansion of lager of warmer warmer and acts Lada Warren minus the small acts of Illinois pledges to fire his mind Largo 1 0 right so that those give us and we Taylor series expand 1 minus Texas is an exercise in I'll do a Taylor series expansion you differentiate huh so I don't work the mind is that and already the yup is so this is just equal minors X Largo born minus X is equal to minus exit from 1 of the right all out right the whole thing out minus X I think it's a plus X squared forward to minus think execute those 3 plus extra before World War not factorial despair 1 too former but would not interested in things which quadratic when you was small but 1 this is my ex actor when Taylor series explained that we don't have the right goal on getting more and we don't have the 1 year it's just a lot of disease year-old muddiness fears minus minus is a plus quite mightiest they squared all at twice the volume you not matters we want now when never really interested in Malaga the partition function where interest in what you can get from sorrow pick something interesting to get you get from the party for Malaga petitions of energy but calculating energy of the guests that's a equal from minus the derivative of the partition function of minus the derivative of the lobby of the partition function with respect to what direct bear I'll giving it the 1st firm over here that's just could be the original expression that we had the idea for the ideal gas on the calculator again and know what happens fight different log 0 with respect the logs and 0 with respect to beta I'd get the energy of the ideal gas at temperature beta so what's the 1st the number of molecules fires 3 halves forms of temperature 3 areas and arms treaty that's the 1st terms from here as just good although boats Maxwell-Boltzmann were warriors who realized that the thermal equilibrium temperature T each particle has energy 3 have stayed at a 1st and the 2nd terms Apple we want to differentiate with respect of data we have that we have to change what was a minus sign killers minus sign here will get a plaster differentiating this with respect to Bader just gives us In squared all over To be friends you not now of course we always expected a situation like this the total energy should be proportional to the number particles this term here is proportional to the number of particles this Karen here we're the quadratic in number particles proportional to the square of the number of particles that crazy if I have entered the 23rd particles are getting energy of of the 40 sex I don't care what units used and the 46 is debate but that's all right because of volume was also big so let's factor out in the West factor and in now what is in Oviedo a a a city entered this city's is whatever it is it's not a big no small but may be small because we're talking about a I would gas but he is our formula for the energy a term which doesn't depend on the density although of course it does depend on this depends on number particles depends on the number particles but only the number particles and is proportional to the Member particles and he is another the proportional to the number particles it that came on the potential energy between particles but also depends explicitly on the density 1 doesn't depend on a factor of the density yup because it's an energy per particle but this should really be in energy should proper particle this should be an energy per particle only if dozen of the particle nearby so that we focused on want particle family said What's the probability that is another particle body it would be proportional to the events the height of the density the more probable that there's in neighboring particles so the Karachi term here it's also contained the factor that isn't in square he and squared is the number of pairs of particles roughly so it in 1 part for each particle is an energy but the energy is proportional the density of particles as I say that because each particle density was very small you might have a particle OK but the chances that there was another particle nearby would be negligible as a density increases the chances that that particle is near a 2nd 1 start the increase and a factor of the density here OK so we could express this in terms of of 80 energy per particle the energy per particle as a 1st term which is just proportional to temperature it doesn't matter it doesn't care how many of the particles and body and a 2nd terms is proportional to a potential energy between pairs of particles and another factor of the density and that of course is to be confirmed for radio this related to 1 thousand easily measure the energy of a gas so this is not way conferment Watson interesting thing to concur that you could measure that is easy to measure whoever temperatures in input the temperatures input here told notices time doesn't depend on the temperature interesting fact doesn't depend on temperature OK what's another thing measures against the pressure depression easy to measure pressure so the question is what does this say about the pressure so let's calculate the pressure we have everything we need this is logs his long z remind you want
a formula for the pressures rotate it intends olives the Helmholtz free energy that's a freeway right Rick Sira Carrera ranked the losers quantity which was a ritual Miners TU blogs and the formula of this remind you of a formula for the pressure was the derivative With respect the volume Minor sorry at fixed temperature was a formula we worked out last time the pressure is a derivative of AT or just might miss become plus temperatures times a derivative on the logic of Zee With respect volume S fixed temperature this was a formula worked out last time for pressure so now we have to do is apply if there are 2 terms when we differentiate logs equanimity get turned from here was 2nd as the pressure of the ideal gas price a get the pressure of the ideal gas P E V equals in T P E equals in Cecil key equals V was a T and T divided by V you're right that's the pressure of the ideal guess NOV it is wrote the 1st is the density current density of particles times the temperature that's exactly what we had last time from differentiating logs not now what about this term he lets differentiate that but 1st we have to men multiplied by temperature so let's multiply despite temperature we multiply this by temperatures in the correction term works correction year multiplied by temperature by T was times data so by the time we have finished multiplying by temperature this just becomes minus In square 3rd only to movie times you not minus sign actually want minus tee times log ZERO deter the mightiest signs of flying all over the place as only a question of whether it is an order an even number of them right I'll tell you a bit theirs the number is such that the answer is we wanted differentiate was see what we wanted differentiate this would minus signs all over the place the number of an a up to an even number OK but what happens only differentiate this with respect to the volume its Columbia plus while others differentiate this with respect to volume we square In the denominator so With a get in squared over 2 0 volume squared end you not know what it is not the not except as vendor but it's not you go through it please check out at what is clear his squared roh squared 1 half rose squalor plans you not to see among other things we're expanding and powers of the density when the density is a low density is very low is linear in the density of the pressure as if the density is 0 there's no pressure doesn't matter what the temperature is there's no particles there is no pressure so is proportional to the density and also proportional to the temperature because the temperature 0 the molecules are moving at all and they don't bounce off the walls of all this journal is sensible this terms proportional Perot squares so when was more this is all while smaller than there was a small role in Rome if you may grow small enough for this is going to be much more than this and um that's good because returned to expand in this variable risk and we expect what we expect is when the density is lower this is less than that and it's true that we get smaller and smaller rose square overall goes to 0 as to you not it doesn't seem to care about the temperature and eventually it to turn the pressure that does not depend on the temperature but it does depend on energy the on the on on this 1 parameter which governs the strength of the interaction of stellar row square so cost can be measures that can be measured you take a gas grill you simply start thirdly lately measure its temperature you measure pressure you measured density and you change the density and keep the temperature fixed by keeping it in equilibrium you could change the density by changing the volume price it is changing the volume forget the body just keep track of the density and you calculate the pressure as a function of the density and factors see that it's not just a linear function but starts to increase with a quadratic terms for a of lot of them this is about but there is no known medical helping may be he followed that up by Otto the beyond OK so that the players film that the energy scale from three-body interactions is about the same as for the tow body interactions bomb than in the next term would be a border row cube three-body defined 3 bodies at the same place will have a probability proportional broke cube Dubai at the same place where squared so the next turn in the expansion of whether it's coming from three-body forces or anything else which involves 3 bodies which will be of order a rope Q yuan or typically squared divided by the temperature Best of the mixture French or be bigamy other can't kinds of that other kinds of terms also army sorry I respect him if they are 3 body forces and three-body forces have the same order of magnitude as you lot this is not you know what here not you squared it's just you want and then or have the figure the temperature in the right place from which
you what with the potential energy is between a pair of hydrogen molecules of hydrogen atoms helium Haley was always a clear Queensland might reactive site what Louisiana has a animals forces or more to to restrict them some legal the 612 potential 1 was a Lorillard 12th forms of 6 compete may leave some of which looks like there's this is not the point of point there given giving the dynamics given the formula for the energy how you proceed getting the formula for the energy is not a problem statistical mechanics getting the formula for the potential energy is not a problem mystical mechanics of problems require mechanics problem molecular structure it's a problem of atomic physics whatever happens to be that starting point for thinking about a statistical mechanics opaque bets not topple this is very straightforward as I said it's blind navigation just right down the energy formula start going in rules here and there of course there are threats but the tracks tricks of experience knowing knowing that I said multiply divide by volume and power that's just a matter of experience so it doesn't take a hell of a lot of inside it's a blind manipulation of the symbols our it's now at the end of the calculation it takes some insight to say Look am I getting sensible crime that what me is of of the rights sunny does it make sense to say that the forces are repulsive that it increases the pressure between the effect is it reasonable and proportional the Rose squared the factor of the half well we know that came from came from me in times and postwar who they might threaten in hindsight say you have should be be one-half there you go over account counted every parent of particles and you can't 1 2 was different into warn of abstract OK if they get tough for a it said woes the right back road cut there expect 1 more power of Rome this has to do with the probability that 2 particles come together the other terms only required 1 particle particle is either there it's not there it doesn't matter whether another particles nearby so the probability for finding the probability of fine particle little volume it is proportional to the higher the density the more likely it is to find a particle what's the probability of time finding 2 particles small volume is proportional twirl square it's a probability to find 1 part is if you to particles if you only have to particles let's start with 1 particle of you only have 1 particle density is rolled what's the probability of finding the particles and their unit volume throughout her in her None proportional to roll were a threat to particles and your anyone them both to be in the same volume is proportional to the square that salt garnered serve straightforward OK now something that's much less straightforward end I remember it really confusing me when I when I learned thermodynamics unfortunately the professor was more confused than I was could straightening out it had to do with heat and work and something called exact differentials for many people have heard of such things and have been frustrated in a found themselves are warned kill a textbook because exactly that only 2 people 3 4 you may want to only 2 but but old but cycle mathematical a interlude talk about exact and inexact now exacting inexact in this context has nothing to do with precise vs. in precise exact is a technical mathematical term which probably predates home ology theory the fancy things like that but it may have 1st turned up and thermodynamics so let's talk about the notion of exact differentials on exact differential that sort of thing and the function of 2 variables physical variables as the y plane text and why and the function is solemn figure was the altitude just the altitude as a function of report at altitude altitude that was the altitude and it's a function of X and Y OK let's talk about what happens when you move from 1 point to another if taking a little walk in this altitude during this altitude landscape and taking a little walking take a step that corresponds to a excellent BY all we we assume we know the function the White House Woods's our altitude change I the opportunity altitude changes BAF equals partial of f with respect that X at the rate of change and if you moved along the x-axis can't be amounted to move lost the rate of change of F along the y-axis pens you don't have to be moving along the x-axis of the axis you could moving at a angle this nevertheless is the formula and let's try this form F sub XBX plus ethos of why want this is called a differential B.S Zurich he restrictions on what half of extra why can't be in the water that this really be the small change in some warning function Mr. are basically 1 restriction but take the following quantity these 2nd F by D X by Y this is the 2nd derivative of F you 1st differentiate with respect or why continued differentiate the result with respect to act it's a few of calculators that and it's easy fear the probe we won't do it tonight remorse not that doesn't matter in which order you do derivatives so that translates In the statement over here the by DYE that's and now differentiating it with respect that acts of rats thee a derivative with respect that acts of FY 8 assists termed the right hand side we have exactly the opposite we have derivative of F X with respect wife or fast if I give you the
exchange but in that case you're right it's also called reversible reversals another word for change and therefore no change of entropy the Nikkei's bottom line if the entropy doesn't change how was that I was the energy change minus pressure times volume that's a change in the energy of the gas if you don't change the entropy actually went through last time definition of pressure it's essentially a definition of pressure How does the energy change if change the volume slowly and without allowing any heat the flow and reversibly if you like with no change entropy the formula now where about the change in energy if you keep the the volume fixed but you a little bit of heat Delaware you got some energy talk from somewhere else and you dump it into the system however you'd if by you could do it but if but flame here and heating there was a change in energy in that case without changing the volume of its equal for that temperatures times the the change in interplay that is basically a definition of entropy all temperature of sorts the chain definition of temperature change of energy is by definition the temperature times a change in entropy interpret we defined independently we defined probabilistically visitors there was no surpassing you change both the volume and the entropy this me you do a process you could think of it as little incremental process sees 1st do a change of volume keeping the entropy fixed than you change the entropy little bit keeping the volume fixed but at the end of the day you make a small change of both volume and entropy and how do you do if you change the volume a little bit and you put a little bit of heat do you do a general profits the evolving both kinds of things in that case D E he s D E is evil Tom minus DVD plus PBS that's call the 1st law of thermodynamics but it just expressed his energy conservation incidentally who's doing work misses work this is work it's called D W D work and this is called D Q Q scans that he is an obvious accused the honestly where to came from I don't qualities something I don't know where it came from Q. spans the heat and as the definition of heat This is the definition Of the heat that you put in PBS Cheney energy by changing the entropy that's an irreversible change it could correspond to dumping some heat in this corresponds to changing energy cozily work on whom incidentally is this who worked on the gas to resist a work done by the guest on the coast and fly there was opposite of course the work done there's a change in the energy of the guests so this should be thought of as the work done by the piston on the gas if the pressure as is positive and you change the volume you lower the energy of the guests white because you do work on the piston pressure positive and you push against the Pistons you do work on the stand and therefore you lower the energy of the gas arrange the energy of the Pistons part of day work here is the work done on the gas and this says the change in energy is just definition is a definition this is work and this is the question is whether Q. What is really of function is DQ really an exact differential Is there such a thing as the heat of the system is not a problem areas you could go from weather back if you normal volume and the entropy you know everything about the system volume and entropy is thermodynamically enough to know everything positive temperature Tauziat energy because everything OK so you think of going from 1 value of volume and entropy toe and not taking a walk on our a volume entropy space the question is whether he but you have to put in the go from 1 point to another depends on the path or doesn't depend on another way of saying it is you've got to warm the closed-loop you put little bits of you put that's a little bit to working it take little bits workout to take little bits of them over the heat out at the end of the day if you come back to the same point is the total amount of heat that you've put in 0 as it would be if he were real function and came back from a point or does it depend on the path a path for your answers it depends on the path of prove that we prove that the DEQE is in not an exact differential leaders W what is true is if you take a system tourists cycle or a series of steps and come back to the same state that the total energy that you put into it is 0 you bring in around the back to the same point be energy will be the same but the heat the 2 put in all the work could you take out or whatever will depend on a route that you that you eases is also has filed a we do them slowly so that they stay equilibrium that's will figure out music curl type formulation check and see what they know about your heart attack actually leaders 1 of 2 steps OK so wet let's just right with what this docket leaders it lets I remember are independent variables are entropy and volume so let's right now that our we got a right to demand for now here it is Q 0 a system of busy quoted D E minus a plus PDV fire changed independent variables From S & V energy and volume energy and volume of this right now energy
however I moved from 1 point to another DQ will be equal to the change in energy plus the pressure times change in Bali OK so now let's right let's assume that Qiu was a real function to exact differential 1st of all the switch say that D Q will buy D E is Eagle don't want curiosity of it Q. by D E the change in Q if you change the energy keeping the the volume fix is equal to 1 1 thing was another thing that tells you that the change in Q when you change the volume of keeping the energy fixed is the pressure that's what it wants said if there really was a function Q that this was a derivative of is a true properly cast doubt whether there really is a function that satisfies the we test the Cairo this is CQ By he lets
differentiator with respect the volume were we get so
yet the left inside these 2nd fueled by EU by devaluing volume is equal to 0 balance differentiate the right inside with respect to energy where we get we get the derivative under pressure with respect to the energy if there really was a functions cube awards of function that did depend on how you move from 1 after another filet have to say that by the energy would be equal to 0 as curl test differentiate this 1 with respect of volume you 0 differentiate this 1 with respect to energy this would give you be pressured by energy will you leave that if you have a box of gas and you change the energy in the box of gas that the pressure was changed not of course it does but let's check for an ideal gas with jacket for an ideal gas from the calculation of ideal guests the pressure of an ideal guests Peavey yes physical till PVD Jose aimed teacher sales P equals PC or if you leave it that way that's 1 equation good and other corrasion I want is energy is equal to to 3 fans and TD and he is equal to two-thirds says that In is equal to two-thirds EU which with this is equal to 0 2 Arabs yield volume glad they're right volume pressure two-thirds it volume OK
Miller comeback that 5 over here only ask when you change the price when you change the energy does the pressure change figs volume at 6 volume and the answer is yes and fix volume you change the energy the pressure changes so it is simply not true that BP by D is simply not equal to 0 . 8 0 the head of the curled past fail the getting tired of the curl testers failed indicating that there's no such thing as QB being function all of where you are of what the state of the system depends on how you move from 1 place toward our another word on the amount of heat that you have to put into a system in order to change it from 1 state for another state depends on the path to take that's why he is not a good bomb the description how much he can put into the system is not a good description of the state of the system it's as good as asking how tired you are and whether that determines your rapporteur not how tired you are depends on how you went from 1 point to another and not just a change in how to sell his like that he is like that and workers like that only the combination of the 2 of them is really are functions of the state of the system function the state of the system means a function of the temperature and volume or energy and why are you OK on fuel 0 any questions this is very subtle stuff is very tricky I know everybody has so many people seen it and confused by are you probably still confused by bar why should you be why should it be the case that beat the heat that you have to put in versus to work to get from 1 place to another should not depend on how you get there and you go OK I'm tired of profound off the it's related to the efficiency of right so that exactly was set its connected with all of these they connect Europe things like history says armored are memory systems but here it's just a statement that he is not a yell function of of the properties of the property of the system he is not a property of a system now you know your state of tiredness it is a property of yield but it's not property of the landscape by a property of weighty Warren EXY plane not a property of the landscape is state of tiredness depends on how you went there from more Nextel 6 say what he'd say in the matter enough that that official who after normal for
pleads visit us up steam for down ET
Stoff <Textilien>
Erder
Messung
Gefrierpunkt
Maßstab <Messtechnik>
Auslenkung
Begrenzerschaltung
Schwache Lokalisation
Leitungstheorie
Atomistik
Vorlesung/Konferenz
Absoluter Nullpunkt
Abwrackwerft
Thermometer
Nachwärmeabfuhr
Eisenkern
Waffentechnik
Entfernung
Unwucht
Bett
Proof <Graphische Technik>
Glasherstellung
Druckkraft
Übungsmunition
Panzerung
Thermalisierung
Jahresende
Nassdampfturbine
Kaliber <Walzwerk>
Ziegelherstellung
Gleichstrom
Ford Mercury
Onkotischer Druck
Handwagen
Zerfallsreihe
Gummifeder
Jahr
Ersatzteil
Luftdruck
Federwaage
Unterwasserfahrzeug
Temperatur
Greiffinger
Magnetisches Dipolmoment
Bombe
Parallelschaltung
Intervall
TIGER <Kampfhubschrauber>
Fehlprägung
Spannungsabhängigkeit
Satz <Drucktechnik>
Minute
Woche
Trenntechnik
Schleifwerkzeug
Abwrackwerft
Array
Thermometer
Kaltumformen
Modulation
Feldstärke
Elektronisches Bauelement
Längenmessung
Angeregtes Atom
Weiß
Atmosphäre
Druckfeld
Bestrahlungsstärke
Ford Mercury
Jahr
Lineal
Frequenzumrichter
Wärmequelle
Temperatur
Direkte Messung
Verteiler
Teilchen
Phototechnik
Fuß <Maßeinheit>
Bahnelement
Stunde
Verpackung
Gasdichte
Kraft-Wärme-Kopplung
Waffentechnik
Windrose
Mark <Maßeinheit>
Kombinationskraftwerk
Optik
Proof <Graphische Technik>
Schmalspurlokomotive
Gas
Druckkraft
Thermalisierung
Nassdampfturbine
Strahlung
Ideales Gas
Sprechfunkgerät
Material
Unterwasserfahrzeug
Parallelschaltung
Intervall
Fehlprägung
Geokorona
Impulsübertragung
Satz <Drucktechnik>
Minute
Leistungssteuerung
Computeranimation
Woche
Schlauchkupplung
Schwellenspannung
Boot
Montag
Fahrgeschwindigkeit
Multiplizität
Großkampfschiff
Kaltumformen
Feldstärke
Längenmessung
Elektronisches Bauelement
Autopilot
Rootsgebläse
Übungsmunition
Werkzeug
Atmosphäre
Jahr
Luftstrom
Lineal
Betazerfall
Temperatur
Direkte Messung
Bergmann
Dreidimensionale Integration
Teilchen
Gasturbine
Herbst
Klangeffekt
Centerbrook Architects
Waffentechnik
Kombinationskraftwerk
Rauschzahl
Gas
Druckkraft
Band <Textilien>
Amplitudenumtastung
Heft
Ideales Gas
Zylinderkopf
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Ersatzteil
Boxermotor
Greiffinger
Ruderboot
Direkte Messung
Höhentief
Intervall
Bergmann
Erdefunkstelle
Dreidimensionale Integration
Leistungssteuerung
Feldeffekttransistor
Computeranimation
Teilchen
Boot
Gasturbine
Flugbahn
GIRL <Weltraumteleskop>
Array
Gasdichte
Kaltumformen
Brennpunkt <Optik>
Waffentechnik
Längenmessung
Tag
Gas
Gleiskette
Druckkraft
Thermalisierung
Werkzeug
Gleichstrom
Ziegelherstellung
Vollholz
Ideales Gas
Bestrahlungsstärke
Druckfeld
Knüppel <Halbzeug>
Scheinbare Helligkeit
Jahr
Source <Elektronik>
Sprechfunkgerät
Betazerfall
Ersatzteil
Frequenzsprungverfahren
Temperatur
Fliegen
Gesteinsabbau
Greiffinger
Messung
Drehen
Bombe
Leisten
A6M Zero-Sen
Leistungssteuerung
Woche
Schlauchkupplung
DVD-Player
Elektrische Stromdichte
Ausgleichsgetriebe
Porzellan
Modulation
Kaltumformen
Feldstärke
Seil
Längenmessung
Licht
Gleiskette
Übungsmunition
Magnetische Kraft
Satzspiegel
Vollholz
Druckfeld
Jahr
Schleppen
Lineal
Temperatur
Ruderboot
Direkte Messung
Gleitlager
Bergmann
Maßstab <Messtechnik>
Stoffvereinigen
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Jacht
Nanometerbereich
Feldeffekttransistor
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Atomistik
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Rauschzahl
Elektronik
Gas
Druckkraft
Thermalisierung
Kurzschlussstrom
Stellarator
Ideales Gas
Wind
Mechanikerin
Hauptsatz der Thermodynamik 2
Satz <Drucktechnik>
Leistungssteuerung
Leitungstheorie
Raumsonde
GIRL <Weltraumteleskop>
Ausgleichsgetriebe
Array
Kaltumformen
Pulsationsveränderlicher
Längenmessung
Elektronisches Bauelement
Diesellokomotive Baureihe 219
Übungsmunition
Gleiskette
Angeregtes Atom
Weiß
Römischer Kalender
Druckfeld
Flüssiger Brennstoff
Luftstrom
Lineal
RWE Dea AG
Gaszähler
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Hobel
Teilchen
Holz
Atomistik
Gasturbine
Klangeffekt
Gasdichte
Eisenbahnbetrieb
Rauschzahl
Gas
Druckkraft
Trajektorie <Meteorologie>
Nassdampfturbine
Innere Reibung
Ersatzteil
Schreibstift
Wasserstoffatom
Kraftstofftank
Unterwasserfahrzeug
Schubumkehr
Parallelschaltung
Kraftfahrzeugexport
Lötlampe
Wetter
Hauptsatz der Thermodynamik 2
Leisten
Bildqualität
Satz <Drucktechnik>
Schwächung
Förderleistung
Leitungstheorie
Flugzeugträger
Spiralbohrer
Prozessleittechnik
Kolben
Biegen
Kette <Zugmittel>
Gasturbine
Vorlesung/Konferenz
Kommandobrücke
Ausgleichsgetriebe
Klangeffekt
Wellenreiter <Aerodynamik>
Kaltumformen
Konzentrator <Nachrichtentechnik>
Tag
Energieeinsparung
Gas
Band <Textilien>
Übungsmunition
Trajektorie <Meteorologie>
Druckkraft
Thermalisierung
Werkzeug
Angeregtes Atom
Flüssigkeit
Weiß
Nassdampfturbine
Druckfeld
Flüssiger Brennstoff
Ersatzteil
Lineal
Frequenzumrichter
Kraftstofftank
Synthesizer
Wärmequelle
Temperatur
Ideales Gas
Druckfeld
Unwucht
Gasturbine
Vorlesung/Konferenz
Gas
Ausgleichsgetriebe
Schalter
Computeranimation
Gewicht
Angeregtes Atom
Bombe
Kombinationskraftwerk
Druckfeld
Flüssiger Brennstoff
Zylinderkopf
Hobel
Warren and Wetmore
Übungsmunition
Temperatur
Wärmequelle