the following content is provided under a Creative Commons license your support will help MIT open courseware continue to offer highquality educational resources for free to make a donation or to view additional materials from hundreds of MIT courses visit MIT open courseware at ocw.mit.edu so radioactive decay is kind of a classic example of A first order process so we are doing one little tiny section of the chapter on nuclear chemistry and we're doing that all today and so all we're really covering is uh problems associated with first order processes um so uh there's a this just just a small introduction to this idea all right so radioactive decay has a lot of applications there are medical applications including um Imaging organs and Bones uh including the heart and so there is compound that you already saw um called cardiolite and so we talked about this in transition metals because you have a transition metal and what is the uh geometry of this compound what octahedral and we have cyanide ligans which are what kind of field strength strong yeah so um this uh compound was designed in part by an MIT Professor um Alan Davidson you could go talk to him about this incredible Discovery and invention um it's used about 7 million times a year to image various organs um and has been for a very long time it's off patent now um but this this uh this patent made Alan Davidson MIT and MIT chemistry Department in enormous amount of money and so you could go talk to him about it except he's happily retired living in one of his homes um so uh so you can't really do that but anyway so this uses an isotope of TN which is metastable isotope um and so it's 99 it's an isotope um of uh the uh the normal 98 uh Atomic uh mass and so the next challenge they're always looking for the next great thing um the next great Imaging agent so that's this is still a very active area of research um and there's actually a talk just this week on campus about uh work in this area so this is transition metals combined with radioactivity so it's a two topics uh here in in the class so another of course important use is the potential of nuclear energy and the current use of nuclear energy this has many challenges and I don't want to go on record of what I think about nuclear energy I think it's a complicated problem um there are a lot of challenges but one I'd like to bring up because I think it's particularly um interesting to me is what to do with the waste and so um one uh story that I heard about actually there was a documentary made about this Finland is uh had this idea to create this threemile long tunnel and they wanted to store um 12,000 metric tons of nuclear waste and they wanted the containers to store it for a 100,000 years and this documentary asked a number of questions about this idea such as what kind of container do you use and how do you know the material you design your container is going to last 100,000 years um as experimental science we like to test how long things last but you can't really do this experiment um also kind of brought up the idea do you guard this facility for 100,000 years um because you can make bombs out of a lot of this uh radioactive waste so you kind of need to protect it um but uh maybe you should just bury it and then you know no one knows it's there so you don't have to guard it so they can't find it and use it but then what if someone stumbles apart it and releases all of this uh radioactivity so that would be bad so do you put warning signs for people who will be around 100,000 years from now saying hey don't go in here it looks like a pretty tunnel but hey um you know the half-life of the things stored here are 100,000 years so this is this is pretty radioactive still don't go inside and if you write this sign what language do you put it in so um the documentary pointed out that neanderthals um existed like 40 thou 40,000 years ago so 100,000 years from now what's going to be going on how do you write a sign to people that long in the future anyway I just think that these are sort of interesting ideas and brings up the point that as scientists and Engineers we need to think not only about the science and engineering we're doing but the ramifications to society um and uh and sort of Sociology as well as politics involved in some of this this science so this is an interesting area for that intersection of the social sciences and the Natural Sciences and Engineering all right so radioactive decay definitely a useful thing dangerous and useful all at the same time oh look at that you know the clicker question is coming up at the bottom of the page okay we're not there yet it's okay we're not there yet all right um radio I just added that at the end and apparently didn't animate it well all right um so the decay of a nucleus is independent of how many nuclei are around it that's what makes it a first order process so because it's a first order process we can apply those first order integrated rate laws that we just dered so we had our rate law that the concentration of something a equals its original concentration to the e to the minus K which is our rate constant times time and also our Half-Life equation that we just used so instead of concentration of a though we're going to have a different thing to express of what we're interested in here which is n the number of nuclei so we can just write that same expression down but instead of concentration of a we're just going to use capital N so n the number of nuclei at some particular time equals how many nuclei were present originally times e to the minus K and here it is a rate constant still but it's a Decay constant in that the rate you're measuring is radioactive decay so it kind of has a special name although if you use rate constant for that it's that is what it is so that's okay T is still time and yes n um to the O is the original number of nuclei so we're just going to do a clicker question about how one goes about calculating the number of nuclei all right 10 more seconds all right so uh someone want to tell me for one of the Green Lantern t-shirts what is wrong with the other answers I I I think I saw your hand up first sorry folks let's see so uh answers one and two they have the wrong uh what was it the uh the the molar mass of uh Tech is that technum yeah uh and answer four does not multiply by avagadro's number so that's going to give you the number of moles of the particle right great job okay it's Thanksgiving I thought we needed a good prize today okay so right so one thing you also want to remember sort of your uh make sure that your units are good and it's really important in doing this you can take this back um to remember to use use the number that is here this atomic mass number not the one from the periodic table um in uh in calculating the problem oh actually the periodic table disappeared from that oh well okay so it's it's uh if you use the periodic table it's a close answer but sometimes sometimes it won't be so close um but remember when it tells you about the isotope it always has the um atomic mass that you should be using in the problem as part of the question so keep that in mind all right and yeah you definitely want to remember avagadro's number and the answers are such that it's hard to tell that you messed up with the wavelength it's really easy to tell you messed up if you didn't use avagadro's number because it doesn't make any sense with these it's a little harder all right so um remember to use the Isotopes atomic mass and also remember to use alagad number when doing this and then uh then you should be fine all right so this is really similar it's really similar depending on whether you're talking about chemical kinetics or nuclear kinetics um in doing these problems in terms of the equations um but uh in in chemical kinetics you're measuring the concentration whereas with nuclear kinetics you're measuring Decay event and so usually how do you measure Decay events and the most common way is here our uh Geer counter so just want to It's always important every once in a while at MIT to double check that uh the rooms that you're teaching in have not been contaminated by some wonderful experiments so so far we're good so here this uh is working you can hear the chips I think this is this is pretty good you don't have to be concerned about this there's always some background level it's fine so so um there are um gases in here that uh will get ionized by radiation which gives off then that's translated into that clicking noise so that's what has happening so it's measuring with our thing um whether there are any radioactive events going on and this is called a guer counter and we use x-rays in my lab so I went and uh and stole this from our x-ray facility before I came here luckily it's almost Thanksgiving so no one was collecting any data so no one will get in trouble for taking this right now um and Hans Gyer is the person who came up with this idea and this device does anyone remember where we heard that name before think back class two so he did that amazing uh gold foil experiment um and uh so our pingpong balls that we were throwing were duplicating the experiment that he did as a graduate student and luckily I think he was smart enough to realize that when you're working with he was working with a lot of radioactivity at that point they want should know exactly how much radioactivity you're working with and so uh you know these are very early day experiments and he came up with this device um that helped him know how safe he was and this is still you know the sort of the standard thing to have these around and double check that radio that there is no radiation leaks in places like that so the guer counter all right so uh also a couple of more uh terminology things Decay rate is also called activity or specific activity so you're talking about sort of how active your substance is that's really you know how how radioactive is it um and activity also has the letter A so we talking about the concentration of a and now we have a again there's a lot of A's in this unit um so that's the change now in the number of nuclei over time um that's the rate expression or the rate law K the Decay constant times the number of nuclei and because activity is proportional to the number of nuclei we can also take this expression that we had before that had the n's in it and rewrite it with a so now it's really just like that first order expression we had but without the concentration term um so we have the activity at some time equals the original activity of the material um times e to minus KT and all of these equations are going to be on your equation sheet um but you know if you mess up and use the wrong equation for this it doesn't matter as long as you're using the first order equation um whether it's concentration or activity it's the same idea that you can you can determine if you know the rate constant or the Decay constant how much material is left how much activity is left how many nuclei are left after a given amount of time all right so I know what you're all thinking now you're thinking this is is fantastic but what about the units um we must hear about the units so the SI units for uh for activity are the Beckel BQ and one Beckel is one radioactive disintegration per second and this is the newer unit the older unit was called a curee and sometimes you will still see this in the literature and a curee one curee was 3.7 * 10 10th disintegrations per second so was a much larger number than the current SI unit um so this was what one gram of a radium uh specific activity was so they Ed this big number but it's not was not really practical because you want to tell people like how much radiation would be safe for them to have in a year or something like that and you didn't want to use this this giant number for that so we've uh moved to here so anyone know or want to guess uh who uh who the older unit was uh named for of radioactivity one might think Marie cury but a lot of the uh evidence suggests it was actually her husband Pierre cury um who it was named after it's a bit controversial um but they both worked together and they worked with um HRI Beckel and they all won the 1903 Nobel Prize for discovering radioactivity 3 years later um Pierre C was killed crossing the street um he slipped when it was raining and a horse and wagon I guess ran over him and killed him so this is I think an example of you know someone who's so brilliant but you say they're so brilliant but do they look both ways before they cross the street so you are all very brilliant and I encourage you look both ways before you cross the street anyway so he he died and some of the stories are that they Nam the unit after him um as a tribute others say well it was really for both of them but in any case now it's named after the third person um hre Beckel okay so radioactivity um I'm going to tell you a little bit about radioactivity this chart is not in your handout because you're not responsible for knowing all this information so I just didn't put it in there um but you can look this up there's a couple of points I did make in your handout which is there are different types of radioactive of nuclear radiation we have Al um we have alpha particles alpha decay beta Decay gamma Decay some of those involve a mass change so like an alpha particle is the same as a helium 4 nucleus two protons two neutrons um beta Decay involves an electron gamma is a photon um so they're they're definitely different types some Mass change some not there also um really dramatic differences in Half-Life so again halflife depends in the material in question it depends on that decay constant that rate constant um and if we look at this table we can see things from milliseconds um and if we look at some of these D um is for day A is for year Y is also for year so sometimes you'll see y for year sometimes you'll see a I think most people guess that Y is for year that A is for year I don't really know um but anyway in this table it's a so if you see that don't be confused and GA that's Giga years so 10 10 to the 9th that's where the Finland uh 100,000 years comes from that we need to keep the stuff safe for a very very very very very very long amount of time for Giga years okay so in some Decay processes such as uranium 238 you have more than one type of uh nuclear radiation going on and it can involve a very long and complicated series of different events so here at MIT we spend most of our time talking about science and engineering but I feel like every once in a while we should throw in some poetry um into our science classes so once a year I like to read a chemistry poem to enrich our lives and today is that day in 2014 and the poem I'm going to read to you is called the days of our half- lives and it is by Professor Mala radak Krishna she got her PhD here at MIT uh in the chemistry department and she wrote this book which she wanted me to point out is available on on Amazon if you're looking for a Christmas present for a very very geeky friend of yours um and so and it's illustrated by another MIT chemistry PhD Mar Riley who actually did the illustrations for the videos that I've been showing you in class so um I might chemist just really multi-talented individuals okay so I will read you this poem now and I as I read it to you I will point out what is happening in this uh decay process because all of mala's uh uh poetry is scientifically correct all right so Days of our half lives my dearest love I am writing you to tell you all that I've been through I've changed my whole identity but loved I can't pretend to be when I was uranium 238 you were on my case to start losing weight for five billion years I'd hoped and I'd prayed and finally I had an alpha decay two protons and two neutrons went right out the door and now I was thorium 234 but my nucleus was still unfit for your eyes not positive enough for its large size but this time my halflife was not very long cuz my will to change was really quite strong it took just a month not even a millennium to Beta Decay into protactinium but still rejected me right off the bat protactinium who's heard of that so baited Decay I did once more to become uranium 234 myself again but a new isotope you still weren't satisfied but I still had hope three Alpha decays T was hard but I stayed on through thorium then radium and then raid on I thought that I would finally please you my mask was a healthy 222 but you said although I like your mass I don't want to be with a noble gas they dress so well you had a point though I wasn't reactive so in order to please you I stayed proactive a few days later I found you and said two more Alpha decays and now I am LED but you shook your head you were not too keen on my mass number of 214 I had a bad experience with that Mass before and an unstable actine walked right out the door so in order to change I went away but all I could do was just baited Decay my hopes and my dreams started to go under cuz baited Decay does not change a mass number to bismuth and polonium I hoped and I beckoned my halflife was 164 microseconds and then finally I Alpha decayed and then I was led with a prew worthy mass of 210 got to admit I was getting quite tired and my patience with you had nearly expired you were more demanding than any I dated and much of my energy had been liberated but you still weren't happy but you had a fix I really like the number 206 so I waited for years until the day which began with another beta Decay and then one more and finally in the end I Alpha to Le 206 my friend to change any further I wouldn't be able no longer active but happily stable it took me a million years to do but look how I've changed and all just for you wait what did you say I've gotten so old that you'd rather be with a young lass of gold well I give up we're through my pumpkin shouldn't all my effort be counting for something well you won't be able to rule me anymore cuz I'm leaving you not for one Adam but four that's right while you were away diffusing I found some chlorines that I found quite amusing and we're going to form LED cl4 and you won't be hearing from me anymore see over the years I've grown quite wise I've learned that Love's about compromise you still have half of your half lives to live so go out there it's your turn to give thank you there's a whole book of them on Amazon okay so that is first order first order is pretty exciting it has nuclear Decay second order not quite as exciting but we should talk about it anyway all right so second order integrated rate laws we're not going to go through a derivation it's in your book but here is the equation if you do do the derivation so now we have one over the concentration of a at time t equals rate constant K * t plus 1 over the original concentration of a and we could plot this one over concentration of T versus time and if we did that you would have the opportunity for another clicker question all right 10 more seconds all right 90s yeah it's kind of hard to come up with clicker questions in this unit so uh but but it's fun for Thanksgiving we'll have lots of 90s okay um so right so we can just look and this is actually an expression for a straight line again so we're plotting on the Y AIS one over our concentration of a any at all the various different times um and versus time over here and so our intercept is going to be one over the initial concentration of a and our slope is going to be what K right so again you can measure your concentration as it changes with time how the concentration changes plot it and just determine your rate constant for that particular material all right so second order halflife we can do another derivation but um in this case I will just give you the equation so halflife equals 1 over K times your original concentration of a and so this is different from first order there is a concentration term in the equation so in for second order halflife it does depend on the starting concentration so that's really the big difference in first order it doesn't depend on the starting concentration just depends on the rate constant or the Decay constant which depends on the material in question with second order you do need to know how much you had originally so again how do you know if it's a first or a second order process and here it you really have to determine it experimentally so one thing you could do is measure how a changes over time and then plot your data using the equation for first order and you may see that uh yeah that does not form a straight line so when you're plotting with natural log of the concentration of a but then if you try plotting it one over the concentration of a may get a beautiful uh straight line with your data and so you'd say that's a second order process so again you're determining these things experimentally collecting data plotting the data determining rate constants determining the order of the reaction all right now this is very exciting what we're going to talk about is the relationship between rate constants and equilibrium constants so I love this I love when we come back to stuff that we've talked about before and see it in a slightly different way so at equilibrium we talked about how it's a dynamic process and you have the rate of the forward reaction equal the rate of the reverse reaction the reactions are still still going they haven't stopped but the rates are equal in both directions so we've talked about how to write an equilibrium constant for a reaction so if we have a reaction of a plus b going to C plus d we can write our equilibrium constant K and it's products over reactants unless one of our products of reactants is a solid or a very dilute solution it's the solvent and I heard from your Tas that in the last problem set some people had forgotten what goes into Q or k expression so that's good to review that for this next unit in exam four in the final um so products over reactants now suppose I tell you that it's a second order process and uh the rate of the forward reaction here A plus b we can write the rate law for that forward reaction being second order first order in a second or and first order in B so the rate constant for the forward direction is k1 and then times the concentration of a times the concentration of B for the reverse reaction the rate constant is K minus one and this is generally true in all the problems if it's the first step you have plus one K1 on the top K minus one on the bottom so we have K minus1 times the concentration of c and d so that's the backward Direction all right so at equilibrium these rates are equal we just talked about that we've seen that before the rate of the forward reaction so K1 * a * B is equal to K minus1 * C * D when you're at equilibrium so we can rearrange this equation now and say C and D over here over divide by A and B it's going to be equal to K1 over K minus1 and we also just saw that c * D over a * B was equal to K so therefore our equilibrium constant k equals K1 the rate constant for the forward Direction Over K minus one the rate constant for the reverse Direction so here we're relating equilibrium constants and rate constants so we thought a lot about what's true if you have a big equilibrium constant if you have a big equilibrium constant if you have an equilibrium constant much greater than one what's the ratio to products and reactants at equilibrium there're more or less products at equilibrium than reactants more so we thought about that and now we can think about the relationship of the rate constants so if K is greater than one is k1 greater and less than k minus one greater and so that would then be the case where you have more products than reactants at equilibrium if K is less than one a case where there's more reactants than products at equilibrium then you have K1 is less than K minus one so again we can think about this in terms of thermodynamics we can also now think about it in terms of rates okay so one more thing that we need to uh cover before we end today and that is about elementary steps and molecularity which I just love saying that word so on Monday we're going to talk about mechanism of reactions most reactions do not occur in one step and we need to think about mechanisms I said it was Wednesday but it's actually Monday so it's coming up it's very exciting and we're going to talk a lot about elementary steps when we talk about mechanisms so an elementary um step is um one of the steps in the reaction so reactions usually don't occur in One Step they have many steps and each step is called an elementary reaction so we talked about last time that for the overall order of the reaction you can't just look at the stochiometry and say what the order of the reaction is so you can't predict it from stochiometry for an overall reaction but if it's an elementary reaction if it's a step that Elementary reaction is written exactly as it occurs so in that case the order and the rate law can be predicted so this is you breaking it down into sort of the smallest unit the smallest step this Elementary reaction so you can just look at the stochiometry for a single step for an elementary reaction and predict the order and the rate law so Elementary reactions occur exactly as written so that's what we're going to do on Monday we're going to break down our elementary our our mechanisms into Elementary steps write out the rate laws and then figure out um what kind of mechanism we might have all right so finally molecularity so molecularity is just the number of things that come together um to form a product and um here we have three names unimolecular bimolecular and termolecular unimolecular process what do you guess how many rea are coming together to form product one B molecular what do you guess two ter molecular is a little harder but just give it a whirl three yes so bimolecular is very common um so and termolecular is not so I have uh three uh three molecules to come together and if you try to think about how you get three things to come together all at the same time that's kind of rare usually when there three things reacting there multiple steps involved but two is good now finally we'll end with a clicker question um think about which of these would be examples of unimolecular processes 10 seconds yeah so it actually is one and two so uh most people got uh the two yes that's radioactive decay but the other you can have a decomposition so here we have decomposition into its elements is also uh A first order process all right Happy Thanksgiving everybody see you you next Monday