hey and welcome so in this video we're going to show you how to calculate average atomic mass and then we're going to calculate it for three different elements carbon lead and uranium so before we show to calculation first I want to point out what number we're referring to when you look at a periodic table such as this one you'll see that for any element we'll discuss uh carbon first you can see that there's a number at top the atomic number number there's a symbol the name and then at the bottom we have this atomic mass all right and I like to call it an average atomic mass because we're taking into account that uh there may be more than one isotope for any given element and that we have to take an average of those masses as a result okay the techical term is not to say average it's to say a weighted average okay I don't always use the word weighted because sometimes that confuses or intimidates people but all we're getting uh all we're referring to when we say weighted is uh we're just taking into account stuff like uh for carbon for example most of carbon has a mass of about 12 atomic mass units whereas about just only 1% of carbon has a mass of about 13 so since most of the carbon is this Mass with about 12 99% of it is you can see that in that weighted average uh we have a average mass that is closer to 12 than it is like 12 and a half or 13 so all right so what's the formula um well the formula doesn't have a ton of symbols in it it's just to say atomic mass is equal to the sum of abundances times masses all right so for every isotope okay here's a little preview for like carbon there's three Isotopes really in reality we're only going to worry about carbon 12 and carbon 13 for each isotope you take its abundance and you multiply it by its mass and then once you get each of those answers you add up all the answers and boom there's your atomic mass all right and an alternate way of showing this if a fancy mathematical notation doesn't uh intimidate you or maybe you like it we have this capital Sigma symbol for total okay atomic mass is equal to the sum total for all isotopes of the abundances times the mass all right this is best shown through example though all right so let's first do carbon all right so carbon we have again we have this one and that one we're not going to worry so much about carbon 14 carbon 14 is is very important we get carbon 14 on Earth from cosmic radiation and once an organism passes away they stop ingesting carbon so the less of a radioactive signature you get from carbon 14 uh the older you know uh something is um so that's interesting but we're just going to stick with these two all right carbon 12 and carbon 13 these are the stable isotopes of carbon not radioactive uh or anything like that okay so the atomic mass I'm just going to write am for carbon is equal to we have to do this for each of the two isotopes so this is going to be the abundance I'll just write abund times the mass and that's going to be for carbon 12 and then we're going to add that to the abundance times the mass for carbon 13 if we had more Isotopes we would write them there we would keep that going but we don't have to be worried about that here okay now before we can do this there is something a little sneaky we have to deal with uh we have percent that uh we're going to have to deal with we can't can't use the abundance in a percentage form we're going to have to turn this into a standard number you may be wondering well why did I give as a percent that's because often in these problems or often when you look at the data it's given in a percent instead of a decimal okay so let's turn these abundances into standard numbers make sure I'm still on the screen good all right and this is just going to be a standard number [Music] standard number okay AKA a decimal AK a fraction okay in chemistry and geology Fields they'll often refer to this as um or the the standard number version as a fractional abundance even though they may not actually write out a fraction okay there got it on the screen so all right and to turn something out of percent to a standard number you move the decimal point twice right let me use orange here you go one two so we would have 98 93 and then and for the second number 1.07 we go 1 two now careful we have an empty space here so we'd have to put a zero there and we'd get 0107 okay and at this point if I have a percent column given I just kind of scratch it out I don't totally scratch it out so in case I do need to read it again I can but this is my reminder to myself I'm not going to use this anymore okay so now the only two columns we're going to care about is this column and that column all right so let's do this so put an equal sign here we want the abundance of carbon 12 12.0000 Z looks like there's four zeros there AMU got to always write out your units and then we're going to multiply that by the abundance 9893 and then we just set this up for our second one 13.33 atomic mass units don't forget your units and then we multiply that by 0107 okay let's let you catch up all right now we have to unfortunately or fortunately uh depending how you feel about sigfigs we do have to keep track of sigfigs here so right here 1 two 3 4 five six uh 1 2 3 4 so our answer here is going to be four sigfigs because that was six that was four we go with the lowest number in multiplication and then here we have 1 two 3 four five six sigfigs and then one two three remember zeros in front those leading zeros never count as SigFig so just one two three so we're going to have a three SigFig answer over there okay we're just making a note we're going to round at the end so [Music] 12.0000 time 9893 we get 11. 8716 am all right and then I'm going to underline my sigs sigfigs okay so I got those four and then let's do the other one no glare okay 13.33 [Music] times 0. don't forget that zero it's there 107 and we get uh [Music] 139 and a bunch of other stuff okay uh so if you get a bunch of other stuff you have two options uh the first option is to just write it out completely the second option which is kind of a balance between uh not writing everything but minimizing rounding error decently would be to just write uh two more places beyond your sigfigs so um 0.139 are my sigfigs here I got those three sigfigs and to minimize rounding error I'm going to just write two more digits Beyond one4 I did 14 instead of 13 because everything after that three that 531 if I have to cut it off it's better to round up there okay all right so I'm going to add uh these two uh together all right so uh before we do that if you're like whoa we got all these Sig big things going on what's going on in any atomic mass problem like this you're going to go through two different stages of Sig THS the first stage as you can see here you went through multiply SigFig um multiplication SigFig which is all about going with the low the lowest number of sigfigs and here we're going to have addition sigfigs and addition sigfigs is all about um rounding your answer wherever your smallest common SigFig is so in this case 11.87 one6 our last SigFig our smallest SigFig is in the H hundredths place whereas in this value 13914 our smallest SigFig is in the thousand's place the hundredths is bigger than the thousandths which means we're going to stop it in the H hundredths place I'll show that here in a minute okay I like to underline my sig figs and we can see that's where the sigfigs end write out my units okay I can use this existing uh answer in my calculator or I can just start cleanly 11. 8716 +13 914 and we get 12.01 074 atomic mass units all right so we would write 12.01 we wouldn't write like 12.02 because everything after that one does not round it up so here's my final answer okay before we go to the next problem there's a great way there's a really good way to check your work if your you're working with Earth abundances if you're on some different planet or uh you know Celestial body or like some hypothetical planet or madeup element sort of problem you know those exist out there then you can't use this technique but if you're on Earth all you got to do is just check your periodic table and the data that you use may not be exactly the same as what was used for your version of the periodic table or maybe your periodic table rounded different ly but it should be very close to uh the answer on a periodic table sometimes you'll get lucky and you'll get the same thing so uh this periodic table I got from uh ptable.com says carbon has an atomic mass of 12.011 and we got something very close to that if we had more sigfigs on our Mass measurements we had maybe better Mass measurements some better um abundances then maybe we would have gotten a 12.011 but for a problem like this I'm I'm happy with getting something very close to the periodic table if we were working on Earth all right so similar problem with lead the reason why I chose to include this problem anyways is it gives us the additional practice of working with mul multip Isotopes not just two but here we actually have four all right and be before you stick around decide whether to stick around for the uranium part um you may be thinking this is a similar example to lead which it is but I threw in the extra um curve ball the little different thing in this problem I gave everything here the abundances I gave them as standard numbers and not as percents so if you're wondering how you would handle that then you can stick around for that last example all right but let's go ahead and work on the lead right so lead's one of those uh elements that has like more than just one or two stable isotopes on Earth lead has uh these four uh Isotopes where we can measure their abundances all right the first thing I like to do is if I see any percents I destroy them immediately and I will turn them into standard number numbers I'm going to get this other paper out so I don't leak ink to the next side all right abundance so just going to be a standard number and again a lot of people in the field like to call this a fractional abundance fraction or fractional um but I like to say standard number better your choice all right so slide that decimal point over twice we're going to get 0.14 these problems you may uh have seen it already I don't like to write the zero in front of the decimal point it confuses me sometimes 2 41 221 and. 524 all right so I don't need these anymore so we're just going to multiply across it's basically it's basically like this column times that column if these were standard abundances a standard numbers but they gave us a a percents again so we had to do that all right so um let's do it atomic mass of lead PB is equal to O that's a multiply .014 Plus 20597 74 AMU time. 241 plus 26.97 AMU times 221 stinks when the numbers like all look similar um but nature decided that 207. 977 AMU time 524 okay sometimes I kind of like to do my math more uh vertically when I have a bunch of abundances especially all right so uh before I do anything we'll keep track of sigfigs once again all right so 6 six six six everything's six sigfigs here so the sigfigs are going to depend on these abundances so here we have uh two sigfigs leading zeros never count then we have three sigfigs three sigfigs three Sig fig all right so let's do these one by one all right let's do the top one all right [Music] 2 85 five 6 22 I guess I didn't have to write out the whole thing um I could have just two sigfigs I could have just had round that up to a six there I'm going just call that good so I don't waste so much time writing unnecessary stuff that is a six all right uh 25974 4times 241 49.63 sigfigs and then I'll write two more digits so I can't write 3 n if I'm just going to write two more digits because the seven rounds it up to a 40 so I'm going to write a 4 Z all right next 206 n76 times. 221 we got 45 Point see if we can keep it on the screen 7 and then 42 okay that one gets rounded up to a two there and then lastly here's our most prominent contribution at a whopping 50s something percent so we got 207. 977 multiplied by there we go 0524 all right so we get 108 and our sig figs will end right there we don't even get to U past a decimal point 98 all right so we don't even get past a decimal point so maybe not the most precise data in the world but whatever okay label my units all the way down that's how I do it sometimes um just draw an arrow it's the same information am all right so right if you're a fan of drawing the line or underlines had the most common spot of right here didn't go past the decimal point okay all right so let's type these into the calculator sweet so all this sums up we added these together they all sum up to 207 218 atomic mass units we're only reporting to the ones place so we get 200 7 atomic mass units okay um some chemistry teachers or some chemistry like homework packages they would have you write a DOT there saying that you measure to the one's place um in my view that's not necessary to do unless let's say we had like 200 if you want to say that zero and that zero are significant then you would put a period there so that that zero you're telling someone hey I actually have measurement data over there um I would leave it like that but some people will put a period there anyway so I'm not going to choose sides you you do whatever you like or whatever your teacher likes all right how do we do let's check the periodic table 207.200 we did it all right last example and then a little bit of commentary on what the heck an AMU is and then we'll get we'll get out of here all right so all right so I wrote a note here for Uranium this is just a standard number abundance it's fractional abundance not percent so that means we can use these just as written that makes our life definitely easier cuz now we can just go like that now if your teacher like show all work don't just work off a table I guess you would rewrite all that out but um I'm okay with that if it's already kind of set up for you all right so we'll just do uh each of those um let me make some notes about sigfigs first so this uranium 234 isotope we have six sigfigs here oh and same with 235 and 238 I'm not going to worry about uranium 233 in this problem because it is Trace so they don't even give us a number to work with anyways um Trace elements they're often going to be like in parts per you know billion or something parts per trillion um really really uh small numbers like uh that carbon 14 we talked about earlier that Trace I think I think I remember reading somewhere was like about one in a million atoms or something of carbon all right and then as for the rest of these uh abundances uh that actually have numbers if you were to add them all up it would add to exactly one or uh AKA 100% okay so six sigfigs there and then so this one has just two sigfigs this one has 1 2 3 4 and this one 1 2 3 4 5 six all right and I think this is ready for calculation let's do 234 first so we have 234. 041 times 0.0000 0 54 and again these were not in percent so I can just use them straight like that all right there's my sigfigs for the first calculation 012 underline those two Sig and then 64 all right let's do the next one uranium 235 so we get 23544 times oh got to stay in the screen 0.7 204 1 6 9 three those are my sig figs and then I'll carry two more 2 six atomic mass units all right and then we got uranium 238 the dominant isotope 23805 [Music] times 99 2742 and we get 236 323 and then we have two three afterwards atomic mass units all right so let's sum these up it looks like our sigfigs for each of these ended in the thousandths place each time right and then we'll put our decimal point right there all right and we're adding these up all right so we get 238 02 913 and then that's not going to round up so we would write 238.22 38.3 238.22 so we were actually more prec prise than uh whatever was written here but we basically got it um right on the dot so very nice okay so closing thought we were working with with atomic mass unit you may have been wondering well Josh what on Earth is a atomic mass unit so I've talked about that a little bit in a previous video but in case you just needed a refresher um one atomic mass unit is 112th the mass of a carbon 12 isotope a single atom of that isotope all right you take one single atom of a carbon 12 and you take one 12th of its mass and that is one atomic mass unit so that is to say if you took the mass of um six protons and the six neutrons that make up carbon 12 that is equal to a total of 12 nucleons nucleons just a fancy word for something you have in a nucleus um importantly we we have a equal number of protons and neutrons here all right so you divide by 12 you have your one atomic mass unit and so this is kind of the um average mass the average between a proton and a neutron in carbon okay and this was chosen to be the atomic mass unit not just for carbon but for all of chemistry the entire periodic table all Isotopes so why did we choose a carbon based uh system for making atomic mass unit for all of matter well carbon made a convenient Choice carbon 12 in particular because there's a lot of it around like we showed earlier 99% of carbon is carbon 12 this stuff and it gives this nice mathematical even ratio between protons and neutrons so the atomic mass unit therefore gives a good approximation for the mass of a proton or a neutron a nucleon and the mass of our proton and neutron is already about the same the neutron is just slightly slightly more massive than the proton but basically the same okay and so that allowed us to work with masses very conveniently when we're talking about single atoms or single Isotopes rather than writing out this uh long sort of cumbersome number all right so that's that's the idea between atomic mass units um and then one more parting thought maybe you noticed in the earlier problems that the mass number was very close to the actual measured Mass 12 here we have a 12 well for carbon 12 it's going to be exactly 12 because well that's the AMU definition 112th okay 12 * 112 um equals one whole carbon 12 atom hopefully that made sense all right um but now look look this is interesting carbon 13 has a mass it's not it's not exactly 12 it is 13.33 and you may be think whoa how come it's not exactly 13 okay well maybe one explanation is we have six protons and we have seven neutrons okay but we're using an average then maybe that shouldn't matter okay so then you have a point there so I'm just going to cut to the Chase and I'm going to introduce a weird word weird phrase Mass defect okay Mass def effect is the term that we actually don't have to worry about a lot okay we don't have to worry about this too much Mass Effect means that um a Proton or Neutron has I'm going to put this in quotes slightly different Mass between different isotopes okay Mass defect a Proton or Neutron has slightly different Mass between different isotopes that may sound a little weird you may be like well a proton is the same everywhere right no matter which element it's in no matter which isotope it's in and same for the neutron it's the same no matter what right and you are correct okay so what's this idea of mass defect um and defect maybe not the best word uh it's maybe the word adjustment is a better word or mass difference or maybe Surplus or deficit okay but yeah effectively protons and neutrons seem to have a slightly different mass and we bust out this nice equation here okay we have energy is equal to M is for mass and then times we have some constant which is squared now that constant happens to be the speed of light okay but we'll focus right here there is an equivalence between energy and mass that is to say Mass you can think of even as a stored form of energy for most of chemistry uh we don't worry about that a ton but if you're doing radio nuclear chemistry or you're Nuclear Physics that might interest you um a little bit but Mass being a stored form of energy okay so if we're talking about like a carbon nucleus we have six protons and we have six neutrons in its nucleus okay boink boink boink we can keep drawing these okay and then Neutron Neutron Neutron Neutron Neutron Neutron you can start making up these pictures all right and that's like carbon 12 and then for carbon 13 we would have seven protons oh excuse me six protons and then we' have seven neutrons okay now in the nucleus if you have all this positive charge uh in the nucleus you would think that the nucleus should just explode okay but there is something holding the nucleus together called the strong force or the strong nuclear force and the strong nuclear force is so strong that it can hold all this together even with those positive charges trying to repel each other okay at these sort of um subatomic distances the strong nuclear force is somewhere around like a hundred times or so stronger than the electromagnetic uh Force okay now depending on the stability of the nucleus sometimes you need more of this strong nuclear force sometimes you need less of this strong nuclear force okay so because you're going to have sometimes differences in strong nuclear force sometimes you need more of it sometimes you need less of it comparing different isotopes together since this is the form of energy it's going to have a measurable effect on Mass okay and that's where you get these slight Mass differences and that's what we call Mass defect okay for your typical chem 101 student this is of no practical consequence okay here I'll just write that in case you need that for Kem 101 thankfully not practic thankfully not practical all right thankfully not practical but what you do need to do is use those actual masses okay the long story short is because it's not going to be exactly the same to the mass number it's like 99% to Almost 100% the same often but because it's not uh exactly the same you have to use these actual masses instead of the mass numbers like for this carbon example I can't use 12 13 14 um well we didn't do 14 we can't use 12 and 13 we had to use this 12.0000 and the 13.33 okay all right I know that was a bit of a longer explanation but a long explanation just to tell you that everything is okay so I hope this helps and uh yeah any questions comments suggestions um let me know all right have a good day