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Lecture 08. The First Law.

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there's a quiz Friday at Queen's 3 it's gonna look just like with 1 and 2 very similar 1st couple problems will be pretty straightforward the last 2 or 3 will be a little more difficult far exceeding the roles open everything except computers including my parents and the like but they will have a scan France for you know Jean-Marc is going to give this lecture on Friday because I'm going to be gone so you have the quiz is usually will give a
lecture as usual I'm writing a lecture it's going to be about examples pertaining to Chapter 14 salts important stop examples are very helpful in terms of understanding the material OK so just to let you know will post the lecturers usual and I'll be back on
Monday OK so on Monday last Monday we were talking about this this is 1 of the examples we looked at and we were calculating the partition function Cielo that's this molecule right here it we just said all we have to do to calculate the overall vibrational partition function is calculated partition function for each 1 of these modes separately and multiply them together but the modes are independent of 1 another firmly in terms of whether the occupied a notch right whether the continuum of states associated with each 1 of these modes is occupied the act independently of 1 another so we can treat them independently and so we can calculate the partition function using the equation we'd arrived on Monday and here's what that looks like we got these numbers 1 . 1 to 1 . 0 1 1 . 0 0 all in these numbers makes sense to us mainly because of you we know the thermal energy at room temperatures 200 wave numbers and so on we don't expect these modes to be super
occupied right because 200 wave numbers is a lot
less than 450 year 945 1100 and so these numbers sort of qualitatively makes sense to us but we don't have a lot of intuition about this decimal place at this point but what should we expect these rotational part petitions rotational these vibrational partition functions to be but it's easier to have intuition about translation and rotation in some ways because in both of those cases the temperature is always much higher than the characteristic
temperature so do these numbers makes
sense it's a little hard to say I mean qualitatively we expect them to be wanting change but the changes the apart it's a little hard to know about so let's calculate these characteristic temperatures of 450 wave numbers 945 1100 by reads expected so excited vibrations feels like of the OK what all so are added that right I calculated that characteristic vibrational temperatures for each 1 of these modes and I got 647 Calvin 1300 1500 Calvin temperatures that are way higher than this temperature In other words we don't expect these modes to the we don't expect the occupation of excited states to be very high because these temperatures were this occupation is expected to occur are way higher than the temperature were paying attention to hear 298 Calvin we shouldn't get significant occupational 450 until 600 Kelvin we shouldn't get occupation of 945 until 1300 Calvin at temperatures that are way higher than the 1 that we care about in this problem now what partition function would we expect write
the characteristic vibrational temperature because we're going to be around that characteristic vibrational temperature lots of
what I did here is I made a lot of the partition function on this axis as a function of the temperature on this act it's just a plot of this equation right here right and I did that for for 3 different energies 3 different age newsstand wave numbers that's this red data right 100 wave numbers that this yellow data In a thousand wave numbers OK that's this green data right down your thousand wave numbers hundredweight numbers and 10 wave numbers honorable these up in just a 2nd OK so let's calculate what the characteristic vibrational temperature is for 10 wave number motor mean of course in real molecules we hardly ever see a vibration as this little energy doing this to be unusually low part of that do the calculation anyway the vibrational temperature would be only 14 degrees Kelvin and so far I look at this plot go to 14 degrees Kelvin and I follow it up to this red line here alright I find out that queue at 14 Greek categories Calvin is 1 . 6 so for writing at the
vibrational temperature but if the temperature of the environment is the vibrational temperature
we're going to expect the partition function to be 1 . 6 in other words were occupying the grounds state of course that's the wine white but were not that 1st excited state is not fully accessible by the system only 1 . 6 Total states including the ground state or accept thermally accessible this characteristic vibrational temperatures if we want that to be true indicating that that 1st excited state is fully accessible the system we have to go all the way up to 21 degrees Calvin right there is to know if I follow that down that turns out to be 21 degrees Kelvin right so I have to go to a temperature that about 30 per cent higher then the characteristic vibrational temperature to get full occupation of that state
but why my showing this to you because we don't really have any intuition about what we expect this partition function to be at these characteristic temperatures the characteristic rotation temperature the characteristic vibration temperature those of the 2 that we care about the most usually in rotation where
temperatures are much higher than the characteristic rotational temperature but in vibration not that were often really close to it but we don't really have any intuition about this that's why I'm doing this now what about here is that I blew this up now but that's 500 degrees back here was 50 so I made at 500 now this is about 100 wave number modes and all that yellow line that was down here that's blown up but there it is OK now the vibrational temperature for that mode is 144 Calvin bright but notice by good 144 and I got to this plot I get 1 . 6 again but the partition function at the vibrational temperature stays the same right you know we change the energy the mode by a factor of 10 if I want to fully access the 1st excited state and have looked at 280 qualitatively exactly what we saw with the lower energy mode we have to go to 30 per cent higher temperature of roughly In order to get full occupation of that excited states and of course if I do this with the thousand wave numbers the same thing happens 1 . 6 knowledge characteristic vibrational temperatures in order magnitude higher but the partition function at that temperature there's still 1 . 6 all yes so Fogel 30 per cent higher than 1400 I get 2018 and that's what it would take to get to OK so that's the intuition we want
have right if the temperature equals the characteristic vibrational temperature were going to
expect a partition function to be 1 . 6 1 . 5 but OK that's just just the way the partition function works it's it's counterintuitive your intuition would tell you if you get to this characteristic vibrational temperature than does not mean that that 1st excited
state should be fully accessible destination the partition function be true what that's my intuition tells me right but what I'm telling you it is that's not the way it
actually works you have to go to a temperature that's higher than its characteristic vibrational temperature to get to that partition function of 2 but ah yes that's what the show's atrociously on gear this year this OK now we're going to start talking more specifically about thermodynamics instead of the statistical mechanics some of you may be happy about that
somebody made said I'm impossible for these subjects are difficult for me proving to be perfectly honest and I feel much the same way as
Arnold Sommerfeld focus the limits is a funny subject the 1st time you go through it you don't understand it all if we can all relate to that the 2nd time to go through it you think you understand it except for 1 or 2 small points not many of us have arrived at that point the 1st time to go through it you know you don't understand it but at that time you're used to it doesn't bother you anymore this is what he said to a he said the summer
Alaska White was a writing a textbook about thermodynamics so this is a
pretty famous guy he made a lot of contributions to quantum mechanics worked out X-rayed wave theory was nominated for the Nobel Prize 81 times according to Wikipedia's but I don't think he ever won nominated 81 times and all you can be nominated 81 times is that you were nominated like 10 times a year fur 8 Harris Wikipedia could be wrong on this point OK so assuming this is your 1st time fuse material the bar is set low you don't understand it that doesn't make you any different than this guy right here is very famous and I personally have had tremendous difficulty with this material through time
and maybe that's why I'm a good person to teach this to you because I appreciate how difficult thermodynamics it's not an easy OK so you're ready for this so when
we think about the universe the total energy is assumed to be a fixed quantity but the total energy in the whole universe now not takin a planter is assumed to be a fixed quantity as a function of time right that's a hypothesis this fixed amount of energy has been redistributed without being added to or subtracted from as far as we can tell for almost 14 billion years any physical quantity that I cannot change with time is a conserved quantities so energy would be an example of that but it's a conserved quantity in the universe right the 1st lot anemic postulates the energy is conserved in every process no there is no proof but this is true right it's like shorting the equation there's no proof that the shorting the
equation shorting proposed destroying the equation and we compared with experimental observations that we make in quantum mechanics and all systems really work we keep finding out that the shorting the equation predicts the right thing over and over again no matter how the system changes right so we have more and more confidence that its correct but there is no proof that it's correct
1 of the most confusing things Is the sign convention right we have to pay careful
attention to it right if he falls out of the system any process that transfer the transfers heat from assistant to the surroundings is text ferment if he flows into the system in other words the heat total heat of the system gets higher as a function of that process it's an end-all Thurman process Exel thorough make indoor Thurman write more thermodynamic jargon and the system has an internal energy which were just gonna call you we've encountered you already but now we're assigned you to the internal energy of this system OK case the change in you as a consequence of any process work heat right the changing you is going to be just given by the final internal energy the system subtracted by the initial internal energy the system this is the property of any state function on the internal energy is a state function right this difference can always just be assessed by subtracting the final state from the initial state and this is true for any type of state function will talk about others later on yes it's the final use the system of the process of yes that's the initials and since the changing you can be affected by work or be it follows that the change in you is equal to 2 plus W since that's the only way energy can enter or
exit the system is in terms of your work but those are 2 options all right if Q was
positive and W is positive in the internal energy the system is going to get bigger now we know intuitively what he did within Canada
right we have an intuitive feel for what it's wants work it's
a little more abstract than he all right it is technically speaking the force with which we act on the system multiplied by the displacement in the direction of this force and as different kinds of forces that we can think about but let's just think now about a physical pressure right so technically it's the force which is a vector quantities the integrated over some path right and so in general this work is going to be dependent on the path that we take now that's a very abstract notion would make a concrete here in a minute OK forced times displacement now let's think about mechanical work real called mechanical work has to do With pressure mechanical equilibrium has to do with pressure pressure is the same inside and outside of the system now at assist system is going to be the inside this container here which has a volume the survived that is controlled by the position of this piston along this axis here which is the x axis this axis that extends from X equals 0 all the way up to x 0 that's the initial starting point of the Pistons but the piston also has an area a that I'm not
indicating in this diagram
OK so there bit a pressure inside the president of 180 and there's a temperature that we're going to maintain constant because we're going to maintain the temperature constant using this water back In initial volume is the survive OK now equilibrium here so the external pressure on the internal pressure they have to be equal 1 another that's mechanical equilibrium now here's a over here on the move this man passed over to the Palestinians
what's going to happen yes In beautiful
now we've got a final pressure of final volume that's different from the initial pressure the initial volume let's work through that simply can figure this out all right the external pressure is the the final pressure that the initial pressure that we started with "quotation mark the pressure that's being applied by this mass right here that math is math and that's the gravitational constant met the area the Pistons OK and we started out with 180 and so that's over a plus 180 him but the external force is the force From the pressure that's outside of the all right which is the same as the initial pressure acting on the area the piston that's the area that that this did not hear that the area OK and that's the mass times the gravitational constant so that the total force right that the force from external gas pressure which hasn't changed its name same as was initially and that the force of the mass acting with acceleration of gravity of the president OK so we said earlier workers forced times distance and so I can perform in role right here right now is that the substitute for the force there's the put that into inside the integral now it's 90 yachting lauds the actors access the direction of Pistons moving right now now we're going from X X minus H 1 of my talking about were going from x 0 but when we put the mass America the piston goes down the x 0 minus those of the limits of the integration OK so when I evaluate that integral groups I didn't say why then a minus sign Simitis sign because this forces directed in the negative direction it's directed down where the positive
X direction is up right so the force is negative yes
it is in this area OK so I can that I can do this integral and I reason it's so easy to do because the bosses constant it doesn't depend on access so I can pull it out of this integration and I can just plug these limits and Bloomberg and I get a H OK so the work is just given by this really simple expression here by and multiplying proportional to age OK with me so far now 1 adds to the volume while the changing the volume is just the final volume minus the initial volume but here we are at the final volume all right I can figure out what this volume is because it's this displacement multiplied by the area right area the piston times a displacement that it's moving through insult that's the client that the final volume right there in the and 0 minus age and that is the initial volume until the difference between them is this minus 8 times age that's a change in volume but OK so now .period take that don't have the people's minus I can make substitutions this work expression of here for example a equals don't be overrated a because Delta the over age and so put Delta the over age here age consulate at minus the Delta me that 1st term right when I make a substitution here I'm going to get Ng Delta the over a I'm just substituting from this expression right here OK so this is the survive by way a lot of the subscript OK so I can factor around the delta the right is still to be here the year and so this is workers at the mine is the survived last energy over a lot that's just the external pressure on the way we define it earlier but just the total external pressure right that the initial pressure and the pressure imparted by the mass that we stuck on the piston title the and so were equals mine is P external I'm still the maybe yeah we can use this equation to calculate the mechanical work right this is a derivation of animal thinkers derivation is in your book the turns up to be helpful way
we can use this equation we complied in charge let's do that currently work required to compressed 1 mole definitely a 180 m pressure against the 280 external pressure constant temperature from 2 leaders the while later so many gas is ideal work calculate the works when you work it was minus the external times the I complied in these change in volume From 2 leaders to 1 liter 1 leaders there final volume to lose initial volume we only need the external 200 AT external pressure my goodness OK so what I do this calculation I get 200 leader atmospheres now these are energy units but they're not energy units that we want used too often right we need to convert them into energy units that we care about like jewels OK case of IT leader atmospheres and I multiplied by here is just too here's the gas constant right 8 . 3 1 4 5 1 that's 1 guest constant .period 206 that another guest constant to Calvin's from consul Igor jewels a leader atmosphere "quotation mark that's pretty convenient because we want converter Jewell's rights of a multiplier leader atmospheres by this hour over this hour I get rid of them I work to jewels get 20 . 2 6 killer jewels months I did that so that's the work 20 . 2 6 killer jewels were done it's plus does that make sense what is the plus
but so every single time you do 1 of these thermodynamic calculations you gotta ask yourself the sign makes sense because the signs gonna mess you up all right the site is very important we gotta get a ride Frank it's a
bigger and bigger dealt minus versus plus it tells us everything about what's going on so we can it's more "quotation mark air you know in the past not such a big deals just sign air but here the sign is everything by tells us for the energy is going into it is going out OK so what is the plus I mean it means that the volume got smaller but we did work on the system now that's not intuitive right there's nothing intuitive about that because work is not an intuitive concept alright right but impede work if the wine gets smaller you did work on the system and that work should be positive like you're putting work into the system is
that intuitive now not to me it's not let's look at this graphically here's the
initial state of the system we got 2 liter containers were 180 and now we're going to comply after this final stage here 280 m and 1 later that's what we're doing this problem right but where is the work on this diagram which area the area this rectangle isn't no what we imagine that this process occurs in 2 steps 1st the pressures increased 280 m there we compressor 280 and that's the external pressure there were acting against this is the total work are you might think it's just that I know of where does that mean that it makes no difference what pressure we started at the all right
because no matter where we start we're going up in the 1st step of our imaginary process we're going to increase the pressure to the external pressure that were acting against and then we're going to compress it does amenities .period here here here here for anywhere between that's what I did McCain but that's the
word or it just happens that we started at 180 this means in principle we can arrive at the same finals day doing less work if we divide this compression process into 2 steps my goodness it might take less work to get there alright work is not a conserved quantities it's not like energy right it's not a conserved quantities use your imagination you can do less of it many of you know that already so here's what we did before we use 1 Matthew compress the Pistons that's tantamount to doing that right 1 man was used to accomplish this change and displacement of this piston from 2 leaders to 1 leaders right so instead of doing that with state 2 masses I will put 1 marathon at a time and after we would bolt masses I would be at the same place using final state of the system but now we did it in 2 steps we took the same manner as we signed in half and we
used to it in 2 steps it's going to save us work surprising intuitively that makes no sense it seems like more work are you saw the brick and haven't transfer to
parts of the work to the president here's what we did we started here we did this compression 1st and then we did this comparison I can calculate the work separately for each 1 of these 2 processes because I'm in gold 100 1 . 5 liters and 158 Tiempo Bay OK and there's a word for this guy right the 1st guy that was the the work was that area never work in this area by can call these processes 1 to but what I am together I get 17 . 7 3 killer deals that's less than we calculated we calculated 20 . 2 6 member that 17 . 7 3 that's last we had less work and we got the same final status OK the total work depends upon the path that we choose between 2 equilibrium states this is a very important point right the amount of work
that you have to do To get the 2 to traverse From an initial equilibrium status some final equilibrium state but that's not a conserved quantity it depends on the path that you take path dependent quantity that's a very important message of
thermodynamics so In principle more steps means less work right this is an example so let me just make a few general comments or talk money is an example of expansion or compression against a constant pressure yet this is mechanical work they were talking about mechanical equilibrium His other possibilities that reversible expansion could isothermal reversible expansion all right in thermodynamics of reversible process here is more jargon for you reversible what a reimbursable process a reversible process is a process that is easily reversed all right what is that need it means that if I'm applying perhaps the same tells the president direction influenced his approach to the so if I'm compressing a piston where
can I apply a tiny extra increment of four-second move that pistol a tiny out OK if I take that course Waibel move back by the same tiny amount all right and
so that was an infinitesimal change in pressure that I applied to that history all right in the process reversed in other words I compress the Gasol bit about the pressure weighed this came right back right by the same amount right only under these conditions ,comma system remained continuously from anemic equilibrium in other words if I if I do of if I compressed the piston at any macroscopic Lee observable race even if he tried to it really slow you're not doing it low enough you're introducing losses right dissipation energy getting lost why there's turbulence in the gas that you're exciting when you do this compression for right there's friction terms
in the process right there suck energy out of it right if you want the process to conserve energy you've gotta do it infinitely slowly it turns out
that's that's an odd notion but that's the way it works so How do we do that here right at the break and have already right and I I know I can do less work with with 2 halves the break when the dues undertake these 2 pieces of work now and put them any more parcels people know what that is right I'm inclined about make a powder all right there it's pounded the breadth and now I'm going to transfer 1 grain of this powder break on the piston the it moved so that the volume is now smaller by DVD right I was careful not to lose any of the breadth mind you but which is easy to do in this process but I was careful finds and on taking
the tweezers and putting
flake by play all of the break now on this piston and sure enough and that the same final location all right there is no way to do this more slowly than that but it takes months for right now my diagram looks like this at 1 tiny flakes on the history and a move right here and then to move right there and 3 right there to thousands to get all the way up to the final pressure you
would think but there is no
way believe it or not to do less work assignments now what we understand about this process let's see if we can figure out mathematically what we just did because this is kind of important to understand this here's a we did so as an example if external forces continuously just those infinitesimally higher than initial pressure that's what we did we just made it infinitesimally higher by putting 1 grain at a time on the Pistons but a pleasing to the ears of his visit that we can achieve limiting only work where the call that fracture that were adjusting Flake
grain by grain were in a call that PCs right
that's just to give it a name right now we know work is minus pieces DVD all work for each work associated with the addition of each grain I can organ increase that wording to decrease the volume rather buy DVD all right so we immigrated right this is the approach this is the pressure that depends on the volume that's a band that's what I mean by this notation times DVD what he says well that's an ideal gas is just energy over the right he's just entity over the NRT don't depend on the right so they can go out front and center girls I just got in a groove from the initial lined the final volume of DDE over me I can do that in a role that's pretty easy this is logs the final over the initial all right so that's the work like there use then all I get is that right the work as a differential and recognize the volume changes I had each grain is going to be super tiny alright and then I do this in a girl and I remember that about the ideal gas equation here to think about the party over the boil that work is equal to insult now I can't figure out how much area is under this that's the total work that I did in this case right but in so I do the calculation they got 1 mole times 8 . 3 1 4 5 1 0 times 2004 137 degrees Kelvin where did that come from well by golly that's a temperature I never told it to you but if I play again all of these constants using ideal gas equation that's a temperature again so we haven't said anything about what the temperature was here but by
golly it's pretty warm right it's
2004 and 37 degrees Kelvin insolvent .period and I the volume went down by a factor of 2 14 . 0 4 killer jewels of the modern energy involved by 14 . 0 4 4 killer deals that the area under the curve right here and by golly that's lower than the either 1 of these we didn't 1 step but was 20 we did it 2 steps it was 17 and now it's 14 right so this is reversible I felt thermal compression reversible means you
crush the break-up into a million flakes and add 1 at a time you with me there's no way to
do it in smaller steps than that in smaller steps wins In terms of minimizing work smaller steps wins let's work with more steps
counterintuitive absolutely I completely backwards right from a normal way of thinking right but now we get this equation for
work we got this equation for work do they all agree in terms of this time because of better pay attention to the side wiring mentioned but if the volume gets smaller the work is positive as well we can see that that's true here because if the 2 is smaller than the 1 in this is going to be negative that cancel that negative sign up to get a positive work right if V 2 is smaller than the 1 that's negative that onto that negative signs get positive work the want the final is less than that the initial the net ratio is less than 1 and a lot of less than 1 is less than 0 so again we cancel let negative side and work positive for on both
cases if the 2 is smaller than the 1 work is positive try to keep
that in the back of your mind because a check on your answer right the last thing you need to check on any answers is that sign right there make a sign near the sign is everything now that mechanical where is all the types of work but we're not going to talk about
because we've only got 10 weeks to do everything in physical chemistry except quantum mechanics we were not going to talk about
surface tensions this is very interesting subject were not going to talk about the extension of lines where there is an energy analyst Volodymyr electrical work and 100 other types of work that we could talk about we don't have time to really just a focus on this mechanical work right here see if we can understand that pretty well but OK I think I'm in a letter
GM Mark tell you about the rest of this thing is
130 slide in the letter I'm tired it of his
OK so on Friday quiz 3 fights same drill every other city and then restore electric engine more
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