Statistical Mechanics Lecture 6
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Title 
Statistical Mechanics Lecture 6

Title of Series  
Part Number 
6

Number of Parts 
10

Author 

License 
CC Attribution 3.0 Germany:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
Identifiers 

Publisher 
Stanford University

Release Date 
2013

Language 
English

Content Metadata
Subject Area  
Abstract 
Leonard Susskind derives the equations for the energy and pressure of a gas of weakly interacting particles, and develops the concepts of heat and work which lead to the first law of thermodynamics.

00:00
Weapon
Melting point
Oncotic pressure
Handwagen
Transmission line
Magnetic core
Spring scale
Decay heat
Scale (map)
Year
Measurement
Order and disorder (physics)
Force
Thermodynamic equilibrium
Bird vocalization
Brickyard
Lecture/Conference
Alcohol proof
Thermometer
Tire balance
Spring (device)
Limiter
Caliber
Stream bed
New Year
Spare part
Temperature
Water vapor
Line segment
Ship breaking
Decay chain
Vehicle armour
Particle displacement
Ford Mercury
Chemical substance
Glass
Direct current
Schwache Lokalisation
Atmospheric pressure
Atomism
Cartridge (firearms)
Absolute zero
Remotely operated underwater vehicle
Ground (electricity)
09:15
Weapon
Ideal gas
Bestrahlungsstärke
Photography
Order and disorder (physics)
Thermodynamic equilibrium
Cosmic distance ladder
Particle
Separation process
Packaging and labeling
Thermometer
Cardinal direction
Atmosphere of Earth
Magnetic moment
Water vapor
Distributor
Ship breaking
Combined cycle
Minute
FACTS (newspaper)
Excited state
Density
Series and parallel circuits
Mintmade errors
Frequenzumrichter
Narrow gauge railway
Tiger (cryptography)
Measurement
Abrasive
Ruler
Hot working
Bomb
Schubvektorsteuerung
Year
Force
Foot (unit)
Interval (mathematics)
Alcohol proof
Bahnelement
Radiation
Temperature
Modulation
Material
Typesetting
Pressure
Field strength
Electronic component
Ford Mercury
Gas
Twoway radio
White
March (territory)
Heat
Cogeneration
Finger protocol
Hour
Membrane potential
Remotely operated underwater vehicle
Optics
Week
27:24
Weapon
Capital ship
Ideal gas
Effects unit
Geokorona
Cosmic distance ladder
Multiplizität
Particle
Aerodynamics
Bird vocalization
Tool
Atmosphere of Earth
Schwellenspannung
Combined cycle
Minute
Power (physics)
Vertical integration
FACTS (newspaper)
Autopilot
Monday
Autumn
Series and parallel circuits
Centerbrook Architects
Cartridge (firearms)
Fahrgeschwindigkeit
Beta particle
Mintmade errors
Gas turbine
Measurement
Noise figure
Ruler
Hot working
Year
Boat
Force
Interval (mathematics)
Rootstype supercharger
Spare part
Musical ensemble
Temperature
Hose coupling
Typesetting
Collision
Cylinder head
Field strength
Flat engine
Electronic component
Hilt
Twoway radio
Gas
Computer animation
Miner
Amplitudeshift keying
Week
45:34
Weapon
Girl (band)
Spread spectrum
Ideal gas
Bestrahlungsstärke
Day
Boat
Year
Force
Order and disorder (physics)
Interval (mathematics)
Cosmic distance ladder
Thermodynamic equilibrium
Particle
Fieldeffect transistor
Apparent magnitude
Roll forming
Brickyard
Continuous track
Lowpressure area
Spare part
Tool
Temperature
Watercraft rowing
Lumber
Array data structure
Pressure
Club (weapon)
Ground station
Gas
Power (physics)
Twoway radio
Vertical integration
Source (album)
FACTS (newspaper)
Focus (optics)
Flight
Finger protocol
Density
Direct current
Computer animation
Beta particle
Flugbahn
Measurement
Gas turbine
Miner
1:03:43
Ideal gas
Effects unit
Scale (map)
Order and disorder (physics)
Radar
Plain bearing
Thermodynamic equilibrium
Aircraft carrier
Cosmic distance ladder
Light
Particle
Rope
Turning
Continuous track
Stellarator
Limiter
Magnetism
Jet (brand)
DVD player
Power (physics)
Electronics
Mixing (process engineering)
Density
Current density
Cartridge (firearms)
Atomism
Gas turbine
Measurement
Book design
Noise figure
Porcelain
Ruler
Hot working
Bomb
Short circuit
Year
Differential (mechanical device)
Measurement
Force
Towing
Mitsubishi A6M Zero
Fieldeffect transistor
Temperature
Mining
Modulation
Watercraft rowing
Hose coupling
Lumber
Pressure
Digital electronics
Field strength
Gas
Molekülphysik
Yacht
Cogeneration
Finger protocol
Metre
Wind
Rolling (metalworking)
Miner
Week
Natürliche Radioaktivität
Last
1:21:53
Girl (band)
Roman calendar
Submarine
Effects unit
Order and disorder (physics)
Cosmic distance ladder
Manipulator
Particle
Variable star
Aerodynamics
Roll forming
Continuous track
Water vapor
Mechanic
Space probe
Hydrogen atom
Power (physics)
Excited state
Density
Wood
Atomism
Cartridge (firearms)
Friction
Plane (tool)
Measurement
Gas turbine
Noise figure
Hauptsatz der Thermodynamik 2
Ruler
Hot working
Trajectory
Transmission line
Navigation
Schubvektorsteuerung
Differential (mechanical device)
Force
Cash register
Pen
Gas meter
Rotating radio transient
Fuel tank
Automobile
Spare part
Rail transport operations
Pressure
Typesetting
Electronic component
White
Gas
Heat
Fuel
Angle of attack
Remotely operated underwater vehicle
Miner
RWE Dea AG
1:39:20
Ruler
Prozessleittechnik
Liquid
Trajectory
Hot working
Effects unit
Quality (business)
Transmission line
Blow torch
Day
Differential (mechanical device)
Force
Relative articulation
Aircraft carrier
Thermodynamic equilibrium
Kette <Zugmittel>
Fuel tank
Volumetric flow rate
Lecture/Conference
Automobile
Musical ensemble
Spare part
Tool
Temperature
Water vapor
Drill bit
Weather
Energy conservation
Surfing
Pressure
Typesetting
Bending (metalworking)
Piston
Gas
White
Heat
Bridge (nautical)
Concentrator
Fuel
Excited state
Series and parallel circuits
Cartridge (firearms)
Thrust reversal
Frequenzumrichter
Gas turbine
Synthesizer
Last
Hauptsatz der Thermodynamik 2
1:56:03
Pressure
Ideal gas
Computer animation
Lecture/Conference
Tire balance
Mechanical fan
Differential (mechanical device)
Gas turbine
Gas
Switch
2:00:01
Pressure
Cylinder head
Warren and Wetmore
Combined cycle
Bomb
Order and disorder (physics)
Heat
Excited state
Fuel
Weight
Cartridge (firearms)
Plane (tool)
Temperature
00:05
Steiger University no physical device ever
00:12
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 mercuryinglass 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
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insulate it will take too much time for the heat the past of the energy to pass between the mercury and the environment but the entered 3 simple experimentally when the thermometer stop changing that's wants an equilibrium so how long does it take to change a temperature from 0 to 100 degrees or something like that How long does it take the thermometer Tula to equilibrate me into his come half a minute and some of the monograph but experimentally the answer is when the thermometer stops changing its in equilibrium With B minor he a lot of the best sense if you also depends on the environment Yocum have a high temperature and have very very few molecules if the density is very low but the temperatures high and there's no reason why that can happen it just means kinetic energy of the molecules time kinetic energy of molecules high temperatures high but it may be very few molecules you bring the command into the problem is not so equilibrate very rapidly because very very few molecules in hid so depending on the density of material equal abrasion time could be very long or to be a lot shorter if you would've been so that we air density was what they won 100 0 1 onethousandth of the air density in the room here today but you would keep the temperature saying the thermometer would take a longer time to equilibrate but nevertheless experiment with the rule would be its equilibrated such changing a afoot also fluke but that means the air temperature be Air and Radiation mutton equilibrium right all drive hard to bring radiation and a room in thermal equilibrium with the error there was a radio air molecules of mutual they radiate much assure you in this room here radiation optical radiation is not always nowhere near being equilibrium with an hour on the other hand is very very few photo arms in this room compared with a year with euro air molecules we really don't count for for much this I can see is a photo show was don't kill happened that s why don't I just off the top of my hair do a calculation of how the ideal guest breaks them when narrow forces between molecules just purely off the top of my head and throw away my lecture notes but because I'm looking at them staring at them don't get the wrong idea and that using OK against of weakly interacting particles why weakly interacting well we've done the problem of the 9 interacting guest molecules will limit I it's easy to make a little step little steps I usually where you change problems known solve problems it's all offer and easy to change the problem a little bit numerically change some parameters just a little bit and then try to expand the small changes that's the usual trick your have some problems but you solve 4 some quantity epsilon equal 0 epsilon being some small number they you we formulate the problem with epsilon not being 0 it's still hard to solve but if you're lucky you might be able to expand the power series of and then check whether the succeeding orders are as big as the thing you start with how you tell when I wonder what series but a series like that they breaks down you Utah tell it down when the 2nd term is as big as the 1st so we don't do that problem today with the problem of a weakly interacting gas and of course weakly interacting because I looked because the molecules army average rather far from each other will assumed the range of the forces is small by comparison with the distance between the particles will assume that the potential energy between particles of forces are small will do our calculations and then we'll look at the resulting say where break down where does the correction term become as large as the uncorrected project will find out if that's could depend on the city the bigger the density the more likely are crash into each other's aggression to each other they never experienced the fact that they were interacting with you make density to through all jumbled and squeeze on top of each other and the ideal gasses a very bad approximation our on the other hand you could keep the density loyal old but of the range of the forces long if particles were to interact with each other strongly when they were 6 feet apart the molecules in this room even if that even if the interaction was pretty weak that would still be a very very serious modification of the ideal gas law and so we want to win we start by assuming that we can make an expansion of a small parameter we make the the expansion mathematically explained and then check whether the corrections the 1st term our big or small as big or small or big smaller than me the term correct thing OK so at Phillips set up that problem or if there was a calculates the partition function for a system of molecules for which state energy is not just the some of these Connecticut energies where there are forces forces means a potential energy so let's start formula formula is that the energy of a set of molecules is the sum of all modules some or all of them in stands for which marked you were talking about peace where he stands for the sons of the squares of the components of the moment the P squared twice the last at the usual kinetic energy of a of a molecule plus now going to assumed that every molecule experiences a forest From what every other molecules that means that there's a potential energy for each a pair of molecules and the public potential energy is sum of all that Perez of molecules sums over pairs not sums repairs like apples and pears but sums over pairs of molecules OK so heavily right some Paris we say a some not all that if you can and now and then and label to particles of particle the 190 2nd bowl well I can't quite so we don't
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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 threedimensional 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 onedimensional 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 threedimensional it has units of energy firms volume not energy kindly and Japan's volume but it is
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just integral did you get if you held 1 of them fixed and integrate the other 1 allrounder so speak having done that we can now integrate the 2nd particle 1 that we held fixed normally do that it would just give us another factor of the overall volume of the system it's not they person use your head all in the that attract a threat that's that's right this Utah is equal let's say it's qualitatively equal qualitatively equal the height of the function what colleges got you kind of volume over which the function is significant this volume is roughly the volume of a couple of times a volume of a molecule basically be forces between molecules a potential energy between molecules is only significant when the 2 molecules all within a couple of molecular banned the diameters of each other so his volume is roughly the volume of a couple molecules the other volume is a volume of the whole sample of the whole lot boxer gets the small volume here is absorbed into you we have to worry about anymore it's absorbed into you know what you know was a combination of the small molecular side volume Kansas strength of the potential energy between molecules and they we have to take into account other integral the other integral the total volume of yes I just thought I II or is it part but what you're saying the distance between particles with assuming it depends on the distance between particles of removed to put it to use it all In case of that result that is fine he said but as a ploy might get no no no no no no backed him saying it's not be cause it all depends on the distance between that's right of renowned for all the molecule fixed over here and take the other 1 and integrated allrounder we get EU not would put the 1st molecule only here integrate around it will get the same number but we're saying that part that was very clear about what that role that once and that sort of thing that that the ways In all of this always boundary effects the boundary effects and were always using is the fact that for a large volume of volume is much bigger than the area of surfaced of volume ratios of budget everywhere right this is not exactly true just because when you take the molecule and you move it near the walls of the year of the box you made a little bit of a mistake by assuming the other molecule can be anywhere around it it can only be anyways around it but not served by means of walls who mistake there the mistake is not important as long as the box is much bigger than volume and the volume here words if the boxers much bigger than any of the molecule then error made it is very small negligible but here we are we have this integral I would say that because we use it but I just want to get up there he is a result of their growth is the son of the death possible figures article that that's exactly what it is but some proportionable 1 volume to volumes for the reasons we set as you its other units Our volume square turns energy but it's only proportional to 1 volume as it already young question you said that so far ratio has such that bother it is reasonable to say that because that is because of all of the services so of the shape that a series of smaller set it does but as a sphere is big it's still true that surfaced volume was very small on the answer is doesn't depend on the shape of the subject except more surface area effects it ends don't dynamics in general and statistical mechanics we often ignore surface surface things a so the exception mark to the surface is the minimum but put it this way the number of molecules which are within our molecular diameter of the surface is much much smaller the number of molecules which are further their molecular banner biographer research so there's a small fraction of fractionally speaking the number of molecules for which were making a mistake by by saying the potential energy only depends on distance between them and ignoring the boundaries the percentage of the system a very very small wicket estimated for a box but we don't have after we know that surfaced volume is is very small but it would take about small at some point we're going to start to experience the importance of surface about small a known as the creator that's what of course we are the existence of a potential energy the derivative of potential energies of forest and of course creates collegians we we not have left the a thinker collisions whenever they have the think about following the particles are win after think about you say this is the beauty of statistical mechanics you lose all intuition about what's really going on but you have she said mechanical rules you gone the autopilot you'll follow the rules and you get the answers in a completely rigorous way and you don't have to worry about whether some assumptions that you may have made such as whether particles have the same average velocity if knew the walls on Monday of the walls of their knew each other and I knew each other it afterward so a Tillie's good to go back and say You know what really all we are doing and we did the end so that we would have expected from her from naive thinking but if we want to really know whether we have the right here best set out the rules God Pierette fall equations air at end look at say does is make steps so we're raising his father's is part of a full using not high on the list 1st of all there could be terms in the energy which depend on 3 particles of time the energy between 2 particles conceivably could depend on the presence of the nearby 3rd particle so we Baynham assumption when we've written that energy can be written as a sum of pale now the point is why is it a good approximation 1st of all to use the ideal gas and the answer is it's a good approximation when the probability of the
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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 fourway 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 threebody forces the threebody 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 threebody 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 allover 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
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interested in keeping track of the correction as squared will be much smaller than SFS a small so that keep the 1st correction but 1 miners beta you of X source club ready in the rid of the exponential and replaced by 1 miners beta you of acts but what does the integral role 1 arrest anybody guests would be in the goal of the 1st the 1st 2 rows veto the Enright mutiny and abided by weakening and that's 1 of flight to euros not flow not adjusted the 1st Church here is just 1 and me the role would be Exs is end right so the 1st and the road is just the number 1 Vitaly and Vitaly and then minus Bader times into will be acts over Vitaly end but your Rex OK I think I have everything yet selectively induced integral here looks terrible yours terrible some but that's OK because this the integral over son was just as some of the girls and just some of go pairs of particles every pair of particles gives exactly the same answers every of the pair particles so up then he we have the sort being greater than any of the us matter in greater and greater than in you'll of the distance between them OK this is a sum of integrals let's focus on Wall and every 1 of them is exactly the same answers every other 1 because every pair of particles is a stiff nothing distinguished about particle choose particle 7 noted for particle 50 in particle 494 give the same answer and the fact they will give exactly the same answer as particle 1 and particle till just not started just particle to and particle 1 but how many such terms so I have family such pairs are which is what said a little in times and this 1 or 2 so we can just do this as DX 1 X 2 Oh and NDX all the other DX is all the other 1 is how many of them are their primary Exs here 3 and now we worry we get to it here so day in list To well yes 3 times and still but in my right arm but what's the next step is the next step is exactly that the Sadie arm but but as a factor of 8 times in 1 or 2 it was just explained to us and choose To my data times and times in minus 1 divided by 2 at once we put that factor where we don't have to some anymore all the areas particles adjusted particle 1 particle to nothing special about and I thought a lot of times and might swore on followed thanks no no no no it doesn't there are many in the goes here it's a sum of integrals that every integral has exactly the same form I was exactly the same form each into is an integral where tool of the axes Our insider you and all the other ones of martensite you Your honor Idaho thing now the see Buraydah was to fly a about that but that's a 1st turn the 2nd terms exactly the same except where I see 1 and 2 0 I put 2 with 3 dogs exactly the same here and still right each 1 of these intervals as a sane and array NYTimes in minus 1 or 2 of them and sold by just about OK what about these intervals over here what they give me the ones that that aren't in you here each 1 is me of volume was a factor of volume powers and my tour but now what about this in over you we raced into growth X 1 BX to of view of X 1 minus X 2 is equal volume kind you know what that was the definition of you not solved when I do not think the growth I wish I were taller but I'm not could you remember this remind but but exactly the Inter we have here so let's replace it by that it's 1 might best data now in times in mind this 1 over to we can approximate that bike In square making mistake yes we are making mistake very small compared to the peace we keep we could keep me here and undermined this 1 our let's just approximated by and squared done worse things than that when we did Sterling's formula In times and minus 1 yard thousand tons of thousands or millions minus a thousand wealth of scores of 4 million divided by a tool spread now we have Vitaly and minus 2 over Vitaly A. that what would be squared sorry but the outwardly square right and then we have this in the role this interval gives us another power of volume up upstairs so it becomes find you 0 so now we have the partition function the partition function is equal to the partition function of the free gas without interactions tying this factor which is 1 minus data In square over to divided by the volume times you where we would do it this well again with normal pirate everybody knows that what you do with a partition function is you take its logarithm only interesting me are about 1 the of the petition so let's take its logarithm next Price forces tires Z equals so a lot of disease or 1st of all equal a lot
54:41
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 yearold 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 MaxwellBoltzmann 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
1:03:46
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 threebody interactions is about the same as for the tow body interactions bomb than in the next term would be a border row cube threebody 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 threebody 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 threebody 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
1:12:51
doesn't work right temperature of after think about it can't think about Modula2 mix corrections are our radar many cases this is a this is a good approximation for Di Luca gas the 1st correction or would get said his new observation wrote in his fastball it yeah but you really should close and calculate the next and ask when it's bigger smaller OK so when is it a good approximation to just keep history such a good approximation the ideal gas Wind Rose squared human off is much less than rolled times temperature survived by road from other words when the temperature is bigger than wrote you not temperature as a measure of the kinetic energy of particles your or times role was aware the units of world are of you not rebuffed I want to units of Rome so what's the units of you Norton rural energy so comparing energy and left with energy on the right but see our wrote times you is basically a potential energy per particle wrote times you Is the potential energy per particle when the potential energy Papa is much less than the kinetic energy that is a good approximation that's a would do Inge were really I am studying the gas in the range where the potential energy is much smaller than the kinetic energy so that I did this mostly to show you how the rules work to show you do not limited to be on the idea of guests and show you could you have guessed this Formula One now you might have guessed that proportional to Rose squared because most pairs of particles in my guess that involves the potential energy would not be so obvious that involves exactly that integral but it's plausible that EU terms of volume was there would you have guessed the 1 half they if you really really smart you might just 1 hair on were show you'll write probably not mechanical calculation of calculating the partition function Our that's the ticket that euro a way to do these things and that's a way to be sure you're right OK so we got a couple of problems is that all there is pressure on all of us who oversaw the there's a new Harper get role the temperature is much much bigger very very much midair book a total exercise to work out Harlem B potential energy between a pair of molecules it is roughly Alexei electron volts or something like that so could convert that would have you as you like you who is for electron volts the density of particles as just a number of particles the room divided by a very volume of the room are are you need something else you need volume of a molecule are busy energy scale which is a couple of electron volts times the volume of a molecule or a couple molecules couple molecular volumes problem is a molecule molecule and the mines 7 centimeters or something like that Adams America's 8 centimeters Amour list in the minds the Senate leaders so you can take the density volume of a molecule times the electron volts per unit convert everything from meters and everything else or whatever unit you light electron volts was fined and compare that with the temperatures are measured in units of energy you would find in this room this is vastly smaller and that you have to start getting into the range where modules a sore almost butting up against each other it's when they start to have appreciable probability of being on top of each other that this becomes violated show how much would you have to shrink the rule of law I have to think about but you have to shrink it can quite a lot before that became Porto is using his father there's a smuggler talk about that but that's OK so you 0 the potential energy between the pair molecules could look like this conceivably could look like this now normally award but Our God what does represent does a which represents attraction which represents repulsion misrepresent repulsion the energy goes opposite bring molecules together this represents attraction will you think offhand would be the effect of repulsive force on the pressure young repulsion between the molecules clearly that the report was really really strong and you try to squeeze the molecules of the way they're really saying I wanna get close to you buddy pressure they go up when the pressure when you not as positive as repulsive situation it corresponds to an end and extra extra bit of pressure so the pressure stronger if the molecules repel modules attract that means that you just choose you not being negative OK that's the best be near ideal gas the almost ideal guests of weakly interacting garish the other limit yes that also got off a gravitational yacht to know what the forces between 2 molecules are you have to know something from molecular physics bimolecular atomic physics from there repulsive the average aren't they are army averaged a repulsive maybe however tend to look like this theory porcelain at short distances and then there's an attractive tales attractive an attractive tell weeks off slowly but the average tends to be repulsed the averages repulsive and for most molecular situations positive but we're not doing molecular physics on that carrier calculators you need a new leader Handbook over physics and chemistry or something you look up some lawyers and that they would tell
1:21:54
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 onehalf 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 xaxis can't be amounted to move lost the rate of change of F along the yaxis pens you don't have to be moving along the xaxis 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
1:31:03
effects of the water and don't tell you anything else except effects my wife is clearly no guarantee you just stroke given any old 2 functions of Rexon worry it will not be true that I'll do some examples later and the condition the condition that affects in FY really the derivatives of a unique functions are exactly this over here now they get FY they form the components of the vector was that called gently but grainy and the gradient Internet for the other gradient of F and these took things are the components of a vector of a gradient was a statement over here just like worry Miners this is equal to 0 what's left inside the equations BY FY by VX his DEA effects but you Socorro it's Socorro efforts In this case it's only twodimensional threedimensional there would be other components of the girl is the Cairo and give end an object a set of components to components of that you could hat on postal components of vector really the gradient of some function is really a function which ethics in FY other components of the Grady and that test is this corrosive equal to 0 visit on uh implications of the curled being equal to 0 or tell you what it is it's intuitively obvious that a the following okays opposing it take a walk Mr. and you come back to to the sea and point Summers's change my alright let's say Alaska slightly different questions opposing it take a walk From 1 point toward another how much has changed or more answers just the difference of here and here but no answer is they end up all of the little incremental changes in as you move from 1 point to another and that is the course Delta have been moving from 1 to 2 it is just the integral Justice some of all little incremental changes of DF will change in F which is death XDX FY Y that integral along this curfew is a change in F happen if you chose another curve you went from 1 to 2 we did the same fanciful change of at him because of his is the altitude here and there you go from wanted to move on 1 path for another path we're changing the altitude society so that's says that is Hawaii integral doesn't depend on what path to take is that truly in general for any X in FY better pay and the integral like this doesn't depend on the path so let me give you an example of just calm would've example saucily example but a real example of that but that operation by driving your car from 1 point toward other along to different trajectories and I'm interested in is not a change in the altitude I'm interested in the change in the reading of my guests meter alone or how much gas you sell does the change in the gas meter because the amount of gas city you lose the quantity of gas left in the car were forced to the guest and those that depend on the on the rickety take now of course it does on for example if I take a route from 1 point to another that keeps you at the same altitude at all times will use a different amount of gas that if you go if you go from 1 point to another over until these changing going over the hill will be or larger change and change in the gas tank by another way to say this incidentally if you go around in a closed loop stunning workplace coming back to the same place how much altitude change 0 how about the change in euro senior I guess that I'm not only does it depend on how you go but it's certainly won't come back the same value afterward now last nevertheless the changing your fuel now stands for fueled the change in the fuel can written in this former how much fuel you used to go although X distance and how much fuel used to go a little white distance on the turning the change in the fuel could also be written as a differential but it's not a differential of some functions it was a differential of a welldefined function if you roll around the rule the change has to be 0 it's quite clear that we're talking about a on another example could be helped hired a where are you will shift around but these are the approximation of simplicity that you don't get tired at all if you don't if you don't change out to and walking sliding longer friction with a surface or something normal effort what about that climbing over hill coming back to the same place well if you got tired by going down the hill that maybe you could say that tiredness ratio would change but that's not true you 20 tired during downhill so whatever these efforts are they are such that if you around in a closed loop you do get a change there is a complete mathematical equivalent to the question of where the line integrals around closed hands give 0 0 and whether after all Of the function is equal to 0 mathematical state the curl has to do with going around a little corner little and asking how much things change is curl itself going around the big loop is warranted to grow so the main point is not every pair of functions FX in FY what defined in this way they can always be defined a differential but that differential is not necessarily the differential function so only give you some examples till example specifically so very simple examples but coincidentally when this is for the vector field F for the FX in FY out cold exact when it's not true the called inexact exactly it does not refer to accuracy it refers
1:39:20
to curl of effect if you lose your job I am as it did they want fled the exact it can be applied to system was so it is no man observers systems have to do with force if effects in FY correspond to the force sealed their poem exactness corresponds to lower concentration roads because talking right which is talking about mathematical properties and his days as a man I have and then back but not everybody is familiar with the concept of for now a version will be less is to quiet has a flood that has a great deal to do with curvature calm for example are tour surfers going around workers may change angles but prepared I go there now it has a great deal to do with curvature but done but I is it is some VfD differentials of functions others urges random expressions involving small changes on things that you carry with you don't about the guests fuel to get the gas in the car is a welldefined quantity of can't it just doesn't depend on someone you go from 1 extra will go from 1 next to know from 1 point to another into different groups and change the amount of gas in 2 different ways so the gas in the car is a thing you carry with you along a path and it definitely has a definite meaning along the Pan it's just not a function of where you are it's a function of the whole path of how you got there it is but differentials usually have to do with things changes that depend not only on the end points trajectory but depend on the whole trajectory like getting tired or the amount of fuel a new gas tank OK well we know what exact in exactly the gruesome examples about trying and selects physical Hawaii and have Hawaii is equal that acts that test exactness is exactly as it is this excellent calculate D F X by DYA ways that warned about the FY by DX what is this exactly yes this is exactly must mean that FX enough whitey on derivatives of some function could you guess what function fx in FY of the gradient of exwife take a function equals XYE is derivative with respect to X is why it's derivative with respect warriors acts of salt this is really the gradient of some function or about this 1 here as VAXes eagled the white FY physical to minus this is our possible rules for how the in your gas tank might work if you take away is perfectly good rule about the small differentials of the amount of gas in your car but does not define a function and are now that because this case DFX party wise 1 but DFY by Exs you go on right here want 1 minus 1 now where'd of 0 we're meant meant that the difference between them was 0 excuse me leave the rivers were the same the 1st time have an this these are simply not the derivatives of a real function of any function and if I want to take a little excursion calculating the changes half as I move around according to their D there's a move from 1 point to another I came back to the same point I would find that best in combat the same argue this case ballots talk about heat and work 8 more have to do with energy and they have to do with changes in energy so it's defined With thinking about a guest now doesn't have to be a guest Kabila liquid it could be anything but was thinking about energy volume and those kinds of things let's imagine so what the let's take all independent variable 2 independent variables the system as a box of guests and it has a job plunger allows us to a of in which allows us to change the the volume of the guests preceding bridge about what are the independent variables tool independent variables when could take them to be the temperature in the volume or you could take them to be the energy of box and the volume what taken to be the energy in the water energy and volume are independent variables energy and let's think about the chick energy in volume nonstarter carrier Bryant its were site entropy them volume entropy and law you if you not only entropy any 2 things will build bigger things will do it only depends on 2 things it depends if you like on the temperature and the volume but you could substitute if you like the entropy and what that's enough to determine the state of the gas the thermodynamic state of forget how much entropy and hemorrhage volume completely determined let's think about the energy let's take the change in the energy if we change the volume a little bit without changing entropy what's it what's a process called change entropy AT back right idiomatic process works a change in the energy changing the value this seems to be a disagreement about the terminology quantum physicists mean by adiabatic slow 5 I will mean by adiabatic both slowly and no heat
1:47:43
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 closedloop 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
1:56:12
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
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differentiator with respect the volume were we get so
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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 twothirds says that In is equal to twothirds EU which with this is equal to 0 2 Arabs yield volume glad they're right volume pressure twothirds it volume OK
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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
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pleads visit us up steam for down ET