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Lecture 18. Dynamic Effects in NMR Spectroscopy


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Title Lecture 18. Dynamic Effects in NMR Spectroscopy
Title of Series Chemistry 203: Organic Spectroscopy
Part Number 18
Number of Parts 29
Author Nowick, James
License CC Attribution - ShareAlike 3.0 USA:
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 and the work or content is shared also in adapted form only under the conditions of this license.
DOI 10.5446/19258
Publisher University of California Irvine (UCI)
Release Date 2012
Language English

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Subject Area Chemistry
Abstract This is a graduate course in organic spectroscopy, focusing on modern methods used in structure determination of organic molecules. Topics include mass spectrometry; ultraviolet, chiroptical, infrared, and nuclear magnetic resonance spectroscopy.
the Bond character is somewhere between that of a year of a single bond that of a double bond is in fact it's sort of about 30 per cent double bond and 70 % single bond that make up this picture so as a result you have slow rotation about this bond analysts say smaller which means if I put a start next to this Saffo group urologist but sort of a star here to remind us that special you do have a dynamic equilibrium where you slot positions right so in this case the start FO group consists of the carbon In this case the starter for group restraints of the Carbondale but this equilibrium is slow on what will save the NMR timescale isn't that I really want to do in today's talks is to give us more of a feeling of what slow what's fairness both in time and also in energy so when I say this is about 30 per cent double bond character I want you to get a feeling for what that means in kcal per mole and how I know that number and then I also want us to get a feeling for what happens as we crossed from the slow regime To the fascist regime so get a 1 equation a couple of calibration points on energy and time out of out of today's talk now if if you think about this let's take a simple a situation the case that people invariably use for didactic purposes is something with single it's so the case that sort of was the classic was dying manifold former managing just because it's easy to easy to think about museum to to simulate so here with methyl groups of course you have single it's and just like the Affeldt groups for 1 of them is more downfield the 1 sister the was more downfield when strains from the carbonyl it is more upfield you have the same with the methyl groups here and I just wonder will call these H a N H 3 and so you have some equilibrium hearing call this K and K B in the case of a perfectly symmetrical molecule the rate constants going to be the
same but if this were 2 groups Save disparity size like a church funeral group and a methyl group than the rate 1 way would be faster than the other because you would have an equilibrium constant that wasn't 1 another word you'd have an equilibrium constant work edible peer groups involved here would spend more time here in the west Balkan group would spend more time here voiced fears are excellent just imaginable a thought experiment for this situation so this situation we just saw it was 1 where you have to say so this is just my old drawing of an H 1 in a more spectrum In at some condition where your small you're going to see 2 sets of peaks for the same window if it were very fast you'd see 1 peak and just drawing the same spectral window and plotting its outside yes how can put these for the scientists on the scale on the same scale and so we can say this is fast and of course What's the best way to take something that's slow and make it faster laboratory heated up so this is called the decision are it's somewhere between that weight of cold and hot you hit a middle point which is called medium out medium OK you had a middle .period medium where your at what's called coalescence now at coalescence of what's happening is each of these peaks is getting broader and broader until they merged this is the uncertainty principle at work remember we talked about in line with and I said that if you were able to measure the velocity for infinitely long know the words if there were never any spin flip any relaxation in the swapping of population between Elfenbein states your lines would be infinitely shot but I said we're talking about the and certainty principal your lines a typically about a Hertz wider a little less than hurts wide because you're relaxation time is on the order of a couple of seconds in other words you cannot get your lines infinitely shocked because you're only literally measuring the velocity for a finite amount of time on this you heat things up what's happening is your flipping faster and faster and so your lines of broadening out so what's really happening at coalescence so let us go back to the equation I presented for it by the uncertainty principle which it is if you look at your line in theory there some exact the position of the law in other words in theory if you could make that measurement infinitely your line would have this position but what you're getting a signal loud here because you're not measuring that line with infinite time you're not able to because of relaxation and so you have a certain with enacts the value when we talked about the uncertainty principle we called Delta Nu right Delta Nu basically is the flying with at half height in other words it is the level where you're sort of within these error bars a your plus or minus Delta Nu of that theoretical central value and we said from the uncertainty principle Delta Nu times the time that's the lifetime it's not really a half-life it's almost like a half-life its and his wife 1 over 2 . 3 instead of 1 over 2 . 0 dealt New Times time is equal to 1 over routes to pride in other words if you're able to make that measurement for a 2nd then Delta Nu is going to be . 2 2 worth lying west at half time full line with is going to be . 4 4 per if on the other hand you're always be able to have that lifetime be say a 10th of a 2nd sourcing to equals .period 1 sack now we get to Delta Nu it is equal to 2 . 2 hearts in other words now that line has become 4 . 4 hertz wide at half time so what you were really saying coalescence we have missteps Of this broadened huh like this what you're really seeing is too fat Florencia that adding up underneath there so let me make this sort of with
dotted lines To show a fat Lorraine C. like so and if the separation of the lines here right of this separation if we follow this Delta Nu was longings for want of a better term in other words the separation of the lines in parts at this point at coalescence then and now each of these lines is fat and out so that its Delta Nu not the Delta Nu wines but this this distance here fist Delta Nu it is now so this is what we call Delta New World wines that this Delta Nu is now half Of the Delta Nu of lines does that make sense no the words each of the lines is broadened out so that its careful with at half height is halfway across and that is when your lines are going to be called last where you're no longer going to see at stage and the last line of the right if the bond anymore they're going to be merged together and eventually just have a single single peak in Europe this situation here but right now when they're broadened out there broadened out to a point where where they have merged together and so at that point Delta Nu of the lines the separation of the lines times town which is now going to be our life time at coalescence is able to show Over route to power right this is just the equation that we have over there except now because we have the difference in each of these is fat and out halfway if we have 2 of them it's going to nail being too and so what this boils down to this and is a simple equation that's how we knew just work this out is equal to 0 . 4 5 0 Over the Delta Nu Of the lines no other words the wife time and coalescence is equal to . 4 5 0 divided by the separation of the the lines at a lower temperature because I make sense all the . 5 4 is simply what happens if I'd taken my calculator to divided by route to divide by 3 . 1 4 1 5 and then I put that in the numerator and put the dealt life the company said that of well you mean the 201 say because here are Delta Nu it is here Of the separation of within what you can measure it's exactly have let me show you may be the best way is for me to show you how things look as you various here so basically if you go any coming into effect words more argued stocked appalling and you'd start to pull together if its last you'll see a dimple in the middle and let me show you exactly what this what this can best be be pictured as in rest just simulation this is from a chapter on dynamic in Amara spectroscopy I'm in a book and but see which but this may be a book on dynamic and more spectroscopy so this is an old Justin old drawings of a simulation of what you would expect and it's really based on dying therefore former admits actually prize sunlight idea that on a on a 60 megahertz spectrometers something like this so their simulation is as follows the reason I say it's a 60 megahertz spectrometry 4 lines in this simulation are Delta Nu lines it is equal to 20 hertz In other words at 16 megahertz anymore spectrometer that would be about three-tenths of a ppm which is pretty reasonable now 1 of 500 megahertz spectrometer that would be . 0 4 PPE so anyway for their role simulation they're saying and mentioned that you have 2 lines those 2 and they're separated by 20 and I mentioned that you have a team to that's a relaxation time 0 . 5 seconds no
the words of mentioned that the of lifetime for this molecule due to relaxation was half a 2nd in other words that your lines are now about . 9 hurts with full West at half height that halfway the fall forward and so that's how your normal spectrum of now a mention that you start to heat the sample off so that you have rotation between the 2 so you have the 2 flipping back and forth so a mention here for example that can work you know our equilibrium This is our lifetime methyl former Mayor Ed spectrum where we can call this a and B you're star a mansion now that are rate constant yeah mentioned that a rate constant was 5 . 0 2nd if you're rate constant is 5 per 2nd than your lifetime is 1 overcame their rights to your lifetime at this point is 200 milliseconds it's . 2 and so your lines had net because they're not
staying in the system Bertrand as long you can think of this as we started here were cold and here were starting the sample here here they actually have a very slow OK k is equal to in this case arm . 1 per 2nd in other words it's a 10 second life time they're not swapping at any appreciable as you heat up the sample in simulation they go decay equals 5 cables 10 per 2nd 0 lifetime is now . 1 seconds and finally you get to a point any you notice so here you are and now you're line which is still less than half the distance between you still see this temple here this is it and now you can
kind of see right at this point now it they are coalescing together so right it pays equal the 44 . 4 per 2nd there now coalesced together and then as you heat the sample lot more as you get hotter and hotter as you get faster and faster now we get decay equals 100 so now your lifetime is 10 milliseconds and then you get to take calls 500 0 lifetime is now to MS and finally you dedicate calls tend house so now your lifetime is attentive that's a relaxation time that's urinated line where so what they're saying of course is all lines have a native lined with let pretend that we had a native line with of about hurts this 1 you could just as well haven't Beatty 1 either T-1 where T 2 is going to contribute to line where I don't know why they chose to to it's it's completely it's completely arbitrary because whether your magnet dissertation is spreading out the XY .period so you're no longer able to get signal or your mechanization is returning to the z axis you still have lined with in the case of small molecules typically T 1 is the predominant relaxation pathway In the case of very large so strict 9 for example in the case of very large molecules like proteins T 2 is that predominant relaxation mechanism in this depends in part on how fast the molecules tumble and held consists of other thoughts and questions what was that means that people just it's Flatgap it's basically this is right this is perfect lesson all right so this is this is our time simulated data for sort of a textbook example and now I know what I wanted to show you were a real example show you how to get get 80 armed out of this and then we're going to translate that into a free energy of activation so OK so the case the case then I'll show you which is which is kind of holding has just 1 I've holed from From my only only experience it's a sort of neat molecule because we're going to see that there is actually 2 different things going on here so the molecule is were found but diaphanous lower tho tell you won't mind and I'll show you the spectrum of here we have another handout for you yeah 1 of the great things about being in graduate school is a lot of time to get to observe stuff that's cool and beautiful and relates to your classes and this just happened to be something I I noted when pack actually rise in graduate school it's like Oh this is cool this as an example and this is something I was doing on a synthetic methods project is working out and so the young so I had my my sample of Diana fold tell you what my Diana Fallaw Thell tell your mind and I started to warm it up by I noticed I was curious because there were some broadness of the peaks here so you had your 2 apple peach This is your Mathilde the writes for the 3rd year of the ch students the center ch theories and I just curious about what was going on said this was a sample India mess D 6 DMS so has a very high boiling points you can heated up to a high temperature Duro chloroform boils at 60 and 66 degrees so if you were to try to heated up to 160 and more to the M R 2 would act if you're lucky just blow the top off the 2 if you're not lucky explode the probe either way you have a very very angry department at you because you trashed the him spectrometer in series so as you were made up the CH chose coalesced in 1 hand is really perfect for the coalescence temperature Of the CH 2 now the sage chose a pretty far apart the CH 2 these are on this 1 said 3 . 4 5 is a 300 megahertz animal respect Ramadan and the other 1 is at 3 . 0 3 the metals are a little closer together so they're dealt new lines the separation of alliances smaller so they actually will coalesce even with small rotation so the further you are a part the faster you have the spin in order to have the 2 lines coalesce into 1 if 2 lines very close together you only have to spin it slowly only have their rotation slowly To get coalescence if 2 lines are very far apart you have to have rotation very quickly so we were already call all right call year to indicate that it's that it's in the past tense and here at 100 degrees were not yet coalesced so somewhere here at about 105 I would say it would have been would have been coalescence if I had bothered to demand that do that expert but let's firm moment focused on these 2 metals and I want us to figure out the EU rate here and then we're going to translate that raid into an energy so for this stage chose the Delta new lines the separation of the lines is equal to 3 . 4 5 minus 3 . 0 3 times it 300 megahertz spectrometer so that's 126 so now literally it's plugged in charge in this equation tell the lifetime at coalescence which is 110 degrees for this particular set of resonances the lifetime coalescence is just equal of . 4 5 0 divided by 126 which is equal to 3 . 6 times 10 To the negative 3 seconds I didn't
measure it at 105 but I think it's about 105 as the coalescence temperature so for the CH freeze the CH 3 days were separated by a 60 hertz and so in that case towel so the lifetime at their coalescence 105 degrees for them what say is going to be about 7 . 5 times 10 to the negative 3 seconds in other words about 7 . 5 milliseconds to put it to put it in terms of that other example at room temperature rotation about this and that bond is slow on the inner timescale in other words it's on the order of let's say seconds or hundreds of milliseconds so we see 1 Affeldt peak we see another Rafael peak as we warm it up it rotates faster and faster and faster as it gets warmer and warmer and warmer by 105 degrees it's spinning around with a lifetime of 7 . 5 ms the metals being close together have coalesced and we heated up a little more to 110 degrees the methylene is being further apart now have spinning at 3 . 6 millisecond lifetime the methylene sieve coalesced and and by the time I needed up to to 150 or 160 the peaks are now relatively sharp and usually by that point it's pretty hard to get the GM so we probably would see a quartet and a trip there if I could check the spectrum but I wanted translate this life time into a free energy sold so 1 of the take-home messages here from this example it is below a millisecond let's say is fast on the end of timescale and you know above the 10 milliseconds or 100 milliseconds is slow on the animal times let's now see how that relates to free energy of activation so what I want to do is translate arcade the delta G double dagger In 1 can from transition state theory you have the Irene equation which basically deals with the amount deals with the Boltzmann population of molecules that are able to cross an energy barrier and the hiring equation is at the rate constant is equal to cap which is the transmission coefficient which is generally taken as 1 of triumphs of the Boltzmann can't times the temperature Over plunks constant time speed to the negative delta G double dagger Over when you have the gas Constance temperature what this is really the way the Eyring equation is derived is basically you're setting up an equilibrium between molecules in the ground state molecules in the transition state and assuming that have the molecules go over the transition state at that point and of course Of course Kerry here at coalescence K is equal to to 1 over towel so there were going to use our 110 degrees K is equal to 1 over Tower 1 over a lifetime so what I wanna do is figure out Our free energy here so at 110 degrees Celsius if I now just plug into this thing I get 1 over 3 . 6 times 10 to the negative 3 is equal to 1 . 3 5 times 10 to the 20 3rd into the negative 20 3rd and at 110 worth 383 Calvin we divided by wonks constant 6 . 6 3 times 10 to the negative for the 4th and for the sake of taking on a map for a moment I'll just keep this has lead to the negative Delta chief delta G double dagger all 4 are too Baron if I just continue to work through my equation I get 3 . 4 8 4 times 10 to the negative 11th people's feet to the negative delta G double dagger Over and if I work through that some debts but delta G double dagger is equal to a negative 1 . 9 8 7 times 10 to the negative 3rd times 383 times the natural log of 3 . 4 8 4 times 10 to the negative 11 and when all is said and done the number I get is 18 . 3 you can provide so what is
that saying that same coming back to this point I raised at the beginning of class what's the degree of double bond character and an well you've got 8 18 kcal per mole barrier to rotation if that were a single bond you'd have essentially no debt barrier to rotation if that were applied behind you could say maybe you know 60 70 kcal per mole in other words like an ethylene the pie bond is 60 65 kcal per mole so I look at that and say Oh that's about 30 per cent 25 30 per cent of a pipe no other words if I had ethylene qualify had if I had an assist to beauty and no matter how hard I heeded Indiana Morris spectrometer I'd never see summarization between system trains to bewteen the energy barrier to rotate about a real pie bond is so high you just don't get that thermal G but with this partial pie bond I've added 18 kcal per mole derriere part there's something else that's really cool that's embedded in this in this spectrum so take a walk take a look at the Spectrum at 30 degrees Celsius and you'll notice even at this point whatever methylene is a world brought you see that there is actually this is a call molecule there are actually 2 dynamic processes that are going on here inside I figured at the time I was just curious but no 1st and using this as an example and is a fine example cause it actually ties into simple concepts in stereo sold anywhere I figured I'd want a cool the sample down and take a look at idea myself freezes just a little below room temperature so you can't do super high temperature and in chloroform you can't do super low temperature an Aymara Indian myself there chlorinated solvents you can use like 1 1 1 I to chew dike or can go up to very high temperatures and down to very low temperatures but DMS 0 is common so I use that and chloroform form is coming you can use math going on if you need to get a lot so took the animal or spectrum in poor form to to see what the heck is going on it is really really beautiful self so at room temperature which happened to be that day 22 Celsius now we were just not quite almost a coalescence and you notice as you call it down now were made at 10 degrees warmer Xerox and come out well by the time you're it's 0 degrees you notice that CH 2 is resolving itself they include 2 respect pinks and by the time you've gone down to negative 42 you can say these things happened to be double that of cortex and by the time the down to negative 40 degrees Celsius you can see that even even the other methylene which started as a quartet now has a more complex wedding so this is this is really kind of cool so there's a 2nd dynamic process With the coalescence temperature that just a hair above 22 in core from remember to different solvents have slightly different rate constants so will will say that I didn't measure it exactly but will say that coalescence is probably at about 25 and chloroform 98 Calif so so what I want to do now is to play with this process figure out what's going on and then look at look at the energies federer involved and get us dead calibrated on energy that self this system happens to be way cool so because you have these 2 substituent and the Mathilda at your methyl group is never going to be cold cleaner in other words you have a situation where the tell you what made rain is rotated out of play nowadays From the math from the the and if they are orthogonal were close to if I could to each other and this is a situation where you have what's called axial Carolyn it's very chemistry is cold it's the same thing you have an alley if you have 1 3 dimensional Ali there to an tumors of that office so if you have the simplest example you can come up
with this molecule is are so we can have an equilibrium here of true ain't trope ice American road emeritus and the Institute for a tried and true Barham trope ice American road are in future which means you're CH TRU are dire topic now the sage students that's next to the carbon meal as vertical magnetic anisotropy see you see that at very low temperatures you do see something that other than a simple but this 1 of the 2 protons the pro-war the pro-West have a high degree of magnetic and the 1 that's that's on the same site and so we actually have a separation there and in this case the Delta do as I called it down remember we could see the 2 2 different lines here and so here 1 of these alliances at 3 . 9 4 4 In the other line is it's . 3 0 and so the dealt the new alliance that is going to now now these
for those too 2 lines for the methylene it's going to be 3 . 9 4 minus 3 . 3 0 times 300 is 192 hurts In so odds then the lifetime medical licenses 0 . 4 5 0 and that's what divide by 119 it's 2 . 3 at times 10 to the negative stories seconds back I don't know if it's 298 A at 295 K I said I don't know I'd I'd say that's close to coalescence if it's at 295 K plug-in delta G double their is to 13 points calories so take a minute to think about this we've got to process season this molecule 1 that's almost invisible at room temperature because well that's invisible slightly above room temperature which is the rotation about this and the other that is the rotation about this spot in the rotation about this bond has an 18 kcal per mole 18 . 1 see 3 kcal per mole area so it doesn't become fast until you the thing up not like 110 degrees 105 degrees the rotation about this bond it is this kind of medium scale at room temperature you cool it down it becomes law you heated up it becomes fast and is invincible so that's kind of cool now I wanna give you 1 caveat because every every student I hate to say it in every every new persons arms 1st excuse when they see something the Indiana mark that they don't understand where they see to test 2 sets of pink is a hindered rotation by I'm making a very sweeping generalization the hindered rotations that you say are only going to involve things where you have an S P 2 Adam connected to an S P 2 of chances are if it's nasty to Adam connected to an B 3 out sp 3 Adam sp 3 out of it's going to be fast so hindered rotation and generally only 4 S P 2 the S P so here we have an sp to hybridize benzene connected to NSP to hybridize Papandreou when you have some extra steric hindrance it's slow to rotate here you haven't sp to connected with partial double bond character you have slow rotation and the rest of the "quotation mark it's it's pretty got Dawn flat it's actually that nitrogen really has asked me to what I what I mean specifically is I can think of no simple bonding situation where you have S P 3 Adams connected single sp 3 where anything is slow without some nasty to intervention cycle hexane ring flap where you have 2 sets of eclipsing 10 kcal per mole which is still fast funny and timescale at room temperature or cycle hacks another couple together that 1 if you cool to negative 7 negative 80 degrees does become can become slow actually let me use this as a chance would do that sort of answer your question other questions about men will what you rotate spend so basically the benzene ring is like this here with the methyl group pointing out and spins back and forth but it has the backing In doing so the methyl group has the bank past the carbon and there's enough steric hindrance there that it can't do it rapidly although it's OK it's not going to be perfectly perpendicular it'll be at about about a 60 degree angle and all nicely rocked back and forth but to cross to the other right trope summer that's where it's hard if you wanna make a model of it this is a great 1 to use primal you can easily make a model and primal and you'll see how they they said put it to you still have axial carnality as long as you have a barrier all own did I O War waiters on my goodness and all thank you yes no I'm meant to I meant to have this going back here and there you go that male chaos yeah Mathilde back battle for the 2 entities set up by the just 0 1 1 2 and it so OK my films the methyl group methyl out flat metal bat rotate of that all right thank you thank you thank you all right I want to show you might the last thing I want to do is give you to take-home messages and let me start with the message and then and then I'll go on then I'll show you my thought on this are at my thought is the take-home message is is the animal timescale you know I like to have simple things in my in my mind as ways to keep things in his let's say less than 1 millisecond is fast about 1 to 10 milliseconds intermediate and greater than 10 milliseconds this these are obviously sweeping generalizations because they're going to depend on separation of wines and they're going to depend on field strength of the spectrometer but let me show you my my thinking on this if we imagine a Delta Nu of lines and it's 50 herds are going to give us mercenary Ariel's let's start with a scenario where over 50 hertz and let's say what that isn't ppm at 500 other state 500 megahertz because that sort of a typical modern spectrometer so that's going to be . 1 ppm C to alliance P at . 1 ppm separation and the towel the lifetime at coalescence Is . 0 0 19 seconds in other words it's what 10 milliseconds work a is equal to 111 perceptive so in other words with 2 lines that are close together if you're processes occurring on the order of 1 millisecond it's going to be fast if it's occurring on the order of 100 milliseconds it's slot if our separation of lines was 500 hertz that's pretty far apart that's 1 ppm but we saw half of ppm overnight example of it's 1 ppm the lifetime a coalescence would be . 0 0 9 seconds in other words 111 about 1 thousand 111 per 2nd in other words if the lines are further apart if it's spinning around here many times for millisecond it's fast if it's spinning around once every 10 ms that slows that's how I I sort of calibrate myself let me give you might other calibration that I like to keep keep in mind said the other calibration I like to keep in my hand is typical and Amara energies he is going to be say 10 to 20 kcal per mole
no words a process that's 15 kcal per mole kind of teeters between slow and fast at room temperature a process that's 20 kcal per mole is small at room temperature but it's going to be fast said Avery hot temperature a process it's 10 kcal per mole will be slow and very low temperature but fast at room temperature and so let me let me just show you my thinking on disulfide take if I go ahead and look at 1 over here and I'm just going to make a low table of 1 over Kennedy in seconds as a function I love delta G double the Aguirre and temperature and so I think this is me windowing myself there's your take-home message on top but let me show you show you my windowing myself so I mentioned we consider energies of 10 15 20 and 25 kcal per mole delta G double deck and then we consider from the the I'm From a rate equation we consider temperatures and I'm just gonna window at 3 temperatures negative 50 degrees C which is kind of cold to 98 take a 25 city which is kind of room temperature and 373 k 100 C which is kind of hot orange and then if I simply calculate From the calculates the rate that applies the life time in seconds is 1 ms for a process where the 10 kcal barrier at negative 50 that process has a lifetime of 3 . 5 times 10 to the negative 6 seconds at room temperature and 9 . 3 times 10 to the negative 8th seconds at high temperature in other words for a 10 kcal barrier process and negative 50 we teeter between slow and fast so that's sort of a intermediate or coalescence temperature by this point were fast and by this point were very fast said 10 kcal barrier cycle hexane ring fled slow when called OK if we go 15 108 seconds as the lifetime and negative 50 in other words it's slow it negative 15 but that 15 kcal barrier process becomes Ms 16 ms at room temperature in other words 15 kcal becomes intermediate at room temperature that was like that rotation about the benzene and it's fast by the time were hot it's 7 . 9 times 10 to the negative 50 seconds lifetime 20 kcal per mole 8 . 6 times 10 to the 6 lifetime that's for ever 75 at room temperature 75 seconds is slow but you get a hot in 68 MS and now we get into the intermediate regime in in high temperature that was like a M bond rotation 18 . 3 kcal had heated up to 110 and finally by the time we get to 25 kcal 6 . 9 times tend to be 11 seconds 3 . 5 times 10 to the 5th 2nd in 57 seconds in other words by the time we have a 25 kcal per mole area even at high temperatures you are still small alright so that's where I window myself and I say 10 degrees slow at low temperature 15 degrees no intermediate assays intermediate at room temperature 20 degrees intermediate and high temperatures are I am midterm will be the next time we have in class part on Friday and an open book part on Saturday a close but part of


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