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# More exponential decay examples

## Video transcript

let's do a couple more of these exponential decay problems because a lot of this is really is just practice and being very comfortable with the general formula and I'll write it again or the amount of the element that's decaying that we have at any period in time is equal to the amount that we started with times e to the minus KT where the K value is specific to any certain element with a certain half-life and sometimes they don't even give you the half-life so let's let's try an interesting situation let's say that I have a element let's let me just give you a formula let's say that I have some magic element here where it's formula is its K value I give to you K is equal to minus let me think of a excuse me I just had a lot of walnuts so my throat is dry let's say that K is equal to what K we're putting a minus for R so I'll say the K value is a positive 0.05 so it's exponential decay formula would be the amount that you start off with times e to the minus 0.05 T my question to you is given this what is the half-life of the compound that we're talking about what is the half-life so to figure that out we need to figure out what T value can we put here so that if we start off with whatever value here we end up with half of that value there so let's do that so we're starting off with n sub zero this is just some value our initial starting point we could put 100 there actually let's do that just to keep things just to keep things less abstract so let's say we start with 100 I'm just picking 100 out of air I could have left it abstract with n let's say I'm starting with 100 I take 100 times e to the minus 0.05 times T T is whatever our half-life so after our half-life we're going to have half of this stuff left so this should be equal to 50 we just solve for T divide both sides by 100 you get e to the minus 0.05 T is equal to 1/2 take the natural log of both sides of this natural log of this the natural log of that and then you get the natural log of e to anything I've said it before is just the anything so it is minus 0.05 T is equal to the natural log of 1/2 and then you get T is equal to the natural log of 1/2 divided by minus 0.05 so let's figure out what that is so if f 0.5 the natural log and actually someone just made a comment and I might as well do that I could just put this minus up here I could make this a plus and this a minus if I just multiply the numerator and the denominator by negative 1 and if I want to just to make the calculator math a little easier if you put a minus in front of a natural log or any logarithm that's the same thing as the log of the inverse of 2 over 0.05 makes a calculator math a little bit easier same thing so if I do to natural log divided by 0.05 it is equal to thirteen point eight six thirteen point eight six so when T is equal to T is equal to thirteen point eight six and I'm assuming that we're dealing with time in years that tends to be the convention although sometimes it could be something else and you would always have to convert two years but assuming that this original formula where they gave this key this K value of 0.05 that was with the assumption that T is and years then I've just solved at a flyff I've just solved that after thirteen point eight six years you can expect to have half of the substance left we started with one hundred wind it up with 50 I could have started with X and ended up with x over two let's do one more of these problems just so that we're really comfortable with the formula let's say that I have something with a half life half life of I don't know let's say I have it as one month half life of one month and after let's say that I let's say that I'm I love me just just for the sake of time let me make it a little bit simpler let's say I just have I have my K value is equal to I mean you can go from halflife to a K value we did that in the previous video let's say my K value is equal to point zero zero one so my general formula is the amount of product I have is equal to the amount that I started with times e to the minus point zero zero one times T and I gave you this if you had to figure it out from half-life I did that in the previous video with carbon-14 but let's say this is the formula and let's say that after after I don't know let's say after after 1,000 years I have I have 500 grams of whatever element is described the decay formula for whatever element is described by this formula how much did I start off with so how much did I start off with so I'm essentially I need to figure out n sub zero right I'm saying that after five thousand after a thousand years so n of a thousand which is equal to n sub naught times e to the minus point zero zero 1 times 1,000 right that's the N of a thousand that's that thousand and now I'm saying that that's equal to 500 grams that equals 500 grams so I just have to solve for n sub naught so what's the e value so if I have point zero zero zero 1 times 1,000 so this is n sub naught this is one thousandth of a thousand so times e to the minus one is equal to 500 grams or I could multiply both sides by E and I have n sub naught is equal to 500 e which is about two point seven one so that's 500 500 times two point seven one I don't actually have a on this calculator at least I don't see it is equal to 13 55 grams so it's equal to one thousand three hundred and fifty-five grams or one point three five five kilograms that's what I started with 1355 so hopefully you see now I mean we I think we've approached this pretty much almost any direction that that a chemistry test or teacher could throw the problem at at you but you really just need to remember this formula and this applies to a lot of things later you'll learn you know when you do compound interest in finance the K will just be a positive value but it's essentially the same formula and there's a lot of things that that this formula actually describes well beyond just radioactive decay but the simple idea is use the information they give you to solve for as many of these constants as you can and then whatever they're asking for solve for whatever leftover and hopefully you have I've given you enough examples of that but let me know I'm happy to do more