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More exponential decay examples

A few more examples of exponential decay. Practice calculating k from half-life, and calculating initial mass.  Created by Sal Khan.

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  • blobby green style avatar for user Yuichi Kitagawa
    What is "e" for? why the alphabet becomes 2.71?
    (32 votes)
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  • starky seed style avatar for user sreeni
    Sal says that after one half life, you could expect 50% of the substance to decay. Since the particles have only a 50% chance of decaying, Is it possible for the substance to not decay at all after one half life?

    Thanks in advance :)
    (1 vote)
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  • blobby green style avatar for user Aaron jacobson
    at what's the log inverse.
    (5 votes)
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    • piceratops tree style avatar for user Jayden Tan
      10 to the power of X

      i.e log (100) = 2
      Therefore 10 ^ 2 = 100 which is true :)

      As a more general rule Log a ( b ) = Log ( a ^ b )
      In chemistry log base 10 is used almost all the time so: a will always = 10

      edit: Just saw that in this video Log base e is used and not log base 10.

      i.e the inverse of Log base 10 ( b ) = 10 ^ b
      and the inverse of Log base e ( b ) = e ^ b

      Hope that helps :)
      (5 votes)
  • blobby green style avatar for user JenniferOTimmins
    Can you show me how to calculate a 1/2 life using disentegrations per minute. For example, I start with a compound that is 9.7 x 10^5 dpm, then several days later I end up with 2.2dpm, how do I figure out the half life?
    (3 votes)
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    • blobby green style avatar for user drpatin
      Disintegrations per minute are handled the same way as number of atoms. DPM are considered an activity A(t), so instead of the equation N(t)=No*exp(-kt) you have A(t)=Ao*exp(-kt). From the talk, he mentions that half-life = ln(2)/k, so use the above equation for A(t) above, solve for k, and then convert to half-life.
      (3 votes)
  • leaf green style avatar for user Jess Carrion
    What doe kt stand for?
    (3 votes)
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  • leaf yellow style avatar for user Boaz
    Does anyone know of somewhere I can practice problems like these?
    (2 votes)
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  • aqualine seed style avatar for user Kayli Nicole Champneys
    I'm having a hard time understanding what K is. Help!
    (3 votes)
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  • aqualine ultimate style avatar for user claemiller
    this question might seem a little off topic but if atoms can decay is there something opposite to that?
    (3 votes)
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    • spunky sam blue style avatar for user Ernest Zinck
      No. Nuclei don't spontaneously add protons or neutrons.
      You might call electron capture the opposite of beta decay.
      In beta decay, a nucleus spontaneously emits an electron. In electron capture, a nucleus captures an inner electron.
      However, the same nucleus doesn't undergo both processes.
      (2 votes)
  • duskpin seed style avatar for user School Learn
    Can I use the formula N = N_0 (0.5) ^ (t / n), where N is the final amount, N_0 is the starting amount, t is the time and n is the half life to solve these problems as well?
    (3 votes)
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  • starky tree style avatar for user Veronica Faust
    I have a question about the time units. Do I always have to convert any unit time given to years? Or can I use another unit as well? Thank you.
    (1 vote)
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    • female robot grace style avatar for user tyersome
      The time units will depend on the process being modeled, but must be used consistently.

      For example, if you determined k using t in seconds, then the half-life would also be in seconds.

      Similarly, if you are given a k then you need to know what time units were used to derive that value.

      Does that help?
      (4 votes)

Video transcript

SAL: Let's do a couple more of these exponential decay problems, because a lot of this really is just practice and being very comfortable with the general formula, and I'll write it again. Where 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 try this interesting situation. Let's say that I have an element. Let me just give you a formula. Let's say that I have some magic element here, where its formula is, its k value I give to you, k is equal to minus, let me think of a-- [coughs] Excuse me, I just had a lot of walnuts and my throat is dry. Let's say that k is equal to, well k, we're putting a minus in front of it, so I'll say the k value is a positive 0.05. So its exponential decay formula would be the amount that you start off with, times e to the minus 0.05t. 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 1/2 of that value there. So let's do that. So we're starting off with N sub 0 This is just some value, our initial starting point. We could put 100 there. Actually, let's do that, 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. And I take the 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 1/2 of this stuff left. So this should be equal to 50. We just solved for t. Divide both sides by 100. You get e to the minus 0.05t, is equal to 1/2. You take the natural log of both sides of this. The 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 0.05t 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. 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. It makes the calculator math a little bit easier. The same thing. So if I do 2 natural log, divided by 0.05, it is equal to 13.86. So when t is equal to 13.86. 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'd always have to convert to years. But assuming that this original formula, where they gave this k value 0.05, that was with the assumption that t is in years, and I've just solved its half-life. I just solved that after 13.86 years, you can expect to have 1/2 of the substance left. We started with 100, we ended up with 50. I could have started with x and ended up with x over 2. 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 of, I don't know, let's say I have it as one month. Half-life of one month. And after, well let's say that I-- well let me just for the sake of time, let me make it a little bit simpler. Let's say I just have my k value is equal to-- I mean you can go from half-life to a k value, we did that in the previous video. Let's say my k value is equal to 0.001. 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 0.001 times t. And I gave you this, if you have 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, I don't know, let's say after 1000 years 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 essentially I need to figure out N sub 0, right? I'm saying that after 1000 years, so N of 1000, which is equal to N sub naught times e to the minus 0.001, times 1000. Right? That's the N of 1000. And 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 0.0001 times 1000, so this is N sub naught. This is 1/1000 of a 1000-- so times e to the minus 1 is equal to 500 grams. Or I could multiply both sides by e, and I have N sub naught is equal to 500e, which is about 2.71. So it's 500 times 2.71. I don't actually have e on this calculator or at least I don't see it. So we'll have 1,355 grams. So it's equal to 1,355 grams, or 1.355 kilograms. That's what I started with. So hopefully you see now. I mean, I think we've approached this pretty much at almost any direction that a chemistry test or teacher could throw the problem 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 this formula actually describes well beyond just radioactive decay. But the simple idea is, use 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's left over. And hopefully I've given you enough examples of that. But let me know, I'm happy to do more.