00:05

so there wouldn't be going over some undetermined you questioned it's going to be a lot

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similar to how the discussion sections were continue over some questions that you guys have all seen before so this is the view so of the cover

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chapters 13 and 14 and say we're going to go over a good portion of the Chapter 13 and the that started so there's no need to problems on them 1 of them was going to be covering up the molecular their statistical mechanics of a particular system and the other one's going to be covering thermodynamics of gasses are busy with group petitioned there and there's going to be a some point it's about a problem which has extra credits

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100 evolves the details to

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and so here is a summary of all the partition functions in translation rotational vibrational states something you wanna keep in mind is that you want to remember that the mass that you including the translational partition function isn't our kilograms and also for the most part just remember to mind units if the units on matching up and counseling on whether they should go

01:18

back and rerunning the answer because all the numbers and sometimes the hard to get a hold of the other thing

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you as my 1 member and maybe go back and look at is the cemetery in lecturing it will we go over how to calculate the cemetery for any particular system and for the electronica of partition function usually no formally just add up the DeGeneres season the contributions from each state and Data General numbers and just 1 over Katie so anything you see that's 1 over Katie Candace translate into better so for the 1st example signal this problem we've seen this before arriving in discussion section 1 of them where we talk about an hour and it's doubly degenerate Telectronics stayed at 121 with numbers and that we do generate Grant stayed at 0 with function or 0 wait number the 1st thing and this is obviously going to be as Will it could be as on the us but we have to plot out of the partition function as a function of temperature considered all 1 thousand K are the next part of the question asked the term populations what the was is the population excited state with a population ground state and the electronica contribution to the internal energy at 300 so right here we can see these simple animal diagram depicting the system to the general ground states and cities chanted excited states and so the partition function for the levels as that has seen here is that the degeneracy and the contribution from the energy so as we can see in this partition function than have a contribution from the ground state which is that with the I will said Peter equals 0 and the contribution from the excited state where the energy is going to be equal to 121 with numbers so that those to gather this becomes a partition function and so once we multiplied out minding our own units this is the resulting "quotation mark as we report this out as a divided the function of temperature reconsidered 0 Calvin the only contribution were getting this from the ground state as intuitively followed by

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knowing that the status is not to be populated 0 Calvin and as we still the temperature

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of 2 thousand Calvin we can see how the contribution from the 2nd state contributes to the overall partition function so in order to calculate the term populations we will use this formula which has on where we take into account the number of number of particles in number of molecules in the particular state in question in this case they over all of the contributing partition function elements so in this case since we wanna calculate that the population of the ground state if we go back and we look at the way function the contribution from the ground state or not with function partition function contribution from the ground state is simply too so we leave that up here and we carry out the overall a partition function that we drive down there too to get better so I figured that 300 Calvin and the population of the ground state is about 64 per cent now the the way to get excited state is we can simply subtract 1 from this quantity and get 36 per cent or if you want to

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kind of like the chance yourself carry out the same calculation with the contribution from just excited state in an hour and now we have

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to go over the electronica contributions from smaller energy so which equations wind accusing and this will be the cause shit that you guys will see on on Friday and so the idea is that we throw a bunch of his on and see if you started well enough you should know which 1 specifically use and this is the moment when used notice looks like the average energy per molecule except there's no brackets on the EU and there's an depicting the actual loss number from all so as we use this equation of 300 Calvin we will denote the partition function contribution 300 Calvin is 3 . 1 1 9 and then we plug in a partition function to find the derivative with respect to beta and this turns out to be in a simplified version of this so then bonds mind all of our units and reuses equation we will carry out the 1 unit kind of approximation and the end result is going to be 519 jewels per mole is an election so far

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income Watson through this pretty quickly and remember any questions the leaders have pilfered Athens discussion sections of 1 today and tomorrow even if you're not technically registered and Fulford time and so here's another mid-term exam

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question from a couple of years ago Our we have 3 vibrational modes at 680 wave numbers 330 and 973 so the 1st question where asked is if the molecules school before Calvin how much vibrational energy bills return and will get the 2nd 2

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parts of the questions as we get so how do we figure this out while we

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sum over all the contributions of the vibrational weight function vibrational modes inside and so we put them in his way numbers and or end result is of course going to being weighed numbers which is the equal 991 wave numbers or if you wanted carry out the calculation jewels you can convert all these weird numbers to jewels using new and then we'll get our concern jewels so the end result jewels for this particular question is going to be 1 . 9 7 times tend the negative 20 duel so for the 2nd part of the equation were asked that if a leader of this system is warmed up to thousand Calvin what fraction of these molecules in the 600 sir among

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other Eurasia has yet to accept the rules this part of what we're

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trying to explain that the system is yet basically the ground said there's no contribution of

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translational rotational vibrational energy in the accepts it that this will work

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so for what fraction these molecules is the 680 wave numbers state vibration excited so what what part of the year how

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many of the molecules are really excited 680 vibrational wave number so much

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equations that we use for that no there's this 1 right here which is the partition function contribution for vibrational partition function and the calculation for which is what I doubt will think of and this is the equation we use from problem on where we noted to calculate the population of which set the stage for each state in question now if we use the vibrational partition function plug it into V overall partition function in the denominator do we use the vibration partition function from simply 680 wave numbers or do we use from all 3 all the answer for this 1 specifically it is we use the 680 wave number of vibrational mode because that's specifically will rest however if at all it is a

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question its new noted that 330 and 973 wave number modes are equal to 0 0 then we include them but I'll get to 2nd

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as actually get him and so in order to calculate this we use the vibrational partition function or yet we use the vibrational partition function for 680 with numbers on it's just east of the negative or at the into the negative EDI over Katie or BAD if you use that over the vibrational partition function contribution from 680 where numbers so this is a simplified formula because if we take the denominator and carry out the simplification the sons of appear and this is a result in part now if you ask Our specifically where the vibrational contribution from our 330 wave numbers and 973 wave numbers is equal to 0 then we include them in the denominator so back to the question of what fraction of these molecules is a 680 wave number of vibrational mode excited so we calculate our term population from using well used by now the generously Times said the contribution over the vibrational partition function the end result that the simplification works like this and so once we carry out our calculations of course minding units we will have negative . 4 8 9 that's what yet is a negative . 4 8 9 for the expanded appear I once to carry that over and calculated we notice that the contributions . 2 3 7 does this

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actually makes sense so if we carry out the

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calculation for a temperature of the vibrational mode we find that the vibrational temperature is at 978 Calvin which is well below thousand Calvin so we expect the vibrational partition function to be appreciable it turns out that using this the overall partition function equal to 2 . 5 8 specifically in this example for months effect that covers the entirety of this question of others apart destiny questions

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those problems and that wasn't reason have sometime at the end for questions 1 so here's the

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partition equation that we've seen that you guys remember the discussion section we want think it was in Week 3 we are exhausted by the Katie over 2 contribution for any quadratic terms in the Hamilton onions for the translational reservation on vibrational energy so if were to think about this in the air and the Hambletonian operator for princess news concerning his translational energy we have a quadratic term here moments and scored over to and the potential quadratic term half Explorer so frightening quadratic terms over which and considered a form of aid he squared or be squared we will get a contribution of Katie over 2 but from all of these guys we use authority over to which

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has all member that aura and just the same thing but offers or 1st the I'm all of these things while Carrefour's to 1 specific molecules so far to

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get that he constant which is the amount of energy and molecule can store per-unit temperature all we have to do is take the Katie over 2 or specifically on entire contribution from Katie over 2 from in a vibrational rotational and translate yet translational states and in terms of temperature and we find that capacity for 1 quadratic contribution is going to be equal the conservative and that so if we take the classical Hambletonian for 3 D translation where we have the required under terms for momentum in the acts the y and z you will notice that all are equal partition contribution is going to be equal to 3 Katie operative which makes sense because the story for and so if we take into account are complete energy and drive via the capacity for the energy will see that it is equal to 3 or over 2 4 or more so as we can see on the graph this is just the translational energy contribution that we see and we're

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going over the other contributions they can happen here and here's all when we include the rotational and vibrational way

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so this also turns out to be the heat capacity for all monotonic gasses because if it's a single molecule it can't were not a single molecule single Adam

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attend rotation there's nothing in vibration something because it has nothing to vibrate off so for

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molecules in more than 1 atom vibration rotation can also contribute but vibrational contribution doesn't turn on until the vibrational temperature approaches that of the temperature of the the system in question so for a linear molecule where the temperature is significantly less than the vibrational temperature where we assume that the vibrational contributions insignificant but only the rotational significant because of the temperature of the higher the number the vibrational Temple rotational temperature sorry but we take into account for specifically linear molecule to quadratic terms 1 for rotation the x-axis among 4 rotation and the wife because rotation the axis of this not change the system since we have 2 terms our contribution is to Katie over to or forum all of these guys it'll be too artsy over and again the heat capacity for when we calculated out is just over to where so once

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we have a linear molecule where the rotational states heartache we can approximate the heat capacity of the system for translation rotational along States by 3 or over 2 plus 2

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operative which will be 5 or to so this is a Gandhi noting that there is no vibrational contribution to the heat capacity at these temperatures so once we take into account rotation we can see that the heat capacity contribution will be 5 too as we have just shown now for a nonlinear molecule which can exhibit 3 orders were 3 of quadratic terms for the rotational contributions are all we have to do is simply add another Katie reverted to it to make equal 3 Katie over 2 and similarly again for a mall these guys it's going to be 3 artsy over and once we take the heat capacity of these guys together for a nonlinear molecule of total contribution is going to be equal to 3 . again this is

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not taken into account vibrational state security at vibrational states because the vibration temperature is to lower the temperature in the room was too long comparison the vibrational temperature so this is the

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contribution from the translational and this is a contribution from the rotational functions so what about cases where the temperature in the room is high enough to have the vibrational states contribute so we go back to the equation for the vibrational Hambletonian which only really contains 2 terms of again have parents and apiece "quotation mark momentum over the reduced so following the rules of the petition theorem we will get 2 Katie over 2 or T but we have to take into account that any system can have for a linear molecule for ,comma are at 3 uninspired vibrational modes or 3 M 1 6 of its not time so this takes no account diatonic vibrational contribution which will equal our contribution to be 5 or over 2 instances diatonic system this contributions of equaling 6 minus 5 1 0 or and so as we can see all of these things but together it's 5 or plus 3 and minus-5 so that we use some of us together and we get 7 or over to and we can see the full heat capacity contribution from translational rotational vibrational states so let's use this to solve of for the constant volume more capacity of specifically we're going to go over 1 of these fights to by using the same rules and principles in In the words would consult procedure for and 6 8 6 so the vibrational mode of ideas is 200 14 with numbers and while this vibration contributes

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to Our constant volume he capacity at 25 degrees Celsius so let's calculate the

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vibrational temperature where we have a trio overcame we plug-in our vibrational mode Minder units and we calculate that at 308 Calvin we expect this month to be populated so because 25 degrees Celsius which is 298 Calvin is pretty close to this will take into account the contributions from the vibrational partition function so a quick approximation to determine whether vibrational modes will contribute is if we think this equation where the vibrational temperature if it's less than or equal to 2 times the temperature in question you want to include it under the guise of played around some of these functions I 5 . 1 woman my discussion sections where somebody's asked why or when do we take into account the vibrations and play around with

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the mouth I've noticed somewhat of a one-to-one relationship meaning if the temperature is close to that of the quantity and wave numbers irrespective of the units it will contribute however it was something about half like so if you're talking about assistance of 300 Calvin for sorry 150 Calvin and the vibrational notice something like 300 among contribute very much like something 2004 the significant figure on that again and play around these things get comfortable with them it's going to make

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it a lot easier for it but this is a simple rule of thumb that makes it easier to follow on so once we calculate our our own inequality for this we noticed that the temperature is going to be much greater than the vibrational wave function or vibrational partition function additional temperature answer yes we decide to include so we have to degrees of rotational to vocational degrees of freedom because it's diatonic species and may include 1 vibrational mode because the only way this can vibrant sense of the diatonic is itself so once we calculate the Saudis constraints into account we get 5 or over two-plus or which a 7 over 2 but the real value is it's pretty close its 3 . 4 0 note that if we leave out the vibrational mode but our contribution becomes 2 . 5 0 are which is significantly less than the heat capacity if we don't include the vibrational mode and my Lotus problems I think this is the together

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many questions so it was that it was pretty quickly chauffeured ask questions in any of the discussion sections on anything as far as stuff we've gone over if you don't feel comfortable the materials fine just ask these questions will be able to help you and I have office hours on Friday an hour before the exam at 10 o'clock at Natural Sciences to Room 1 1 1 5