# Radioactive decay calculations for parent and daughter nuclides **URL:** \ **Category:** Factual Questions\ **Created:** [February 12, 2007, 4:59am UTC](https://boards.straightdope.com/t/radioactive-decay-calculations-for-parent-and-daughter-nuclides/391816 "2007-02-12T04:59:18Z")\ **Posts on this page:** 12\ **Page:** 1
**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\ **Post date:** [February 12, 2007, 4:59am UTC](https://boards.straightdope.com/t/radioactive-decay-calculations-for-parent-and-daughter-nuclides/391816/1 "2007-02-12T04:59:18Z")
To make things easier, I already know about exponential decay, half lives, etc. What I am unsure of is doing calculations when the parent nuclide decays into a daughter nuclide that is also radioactive. For example: Nuclide A has a half-life of 300 days Nuclide B 100 days Nuclide C stable Let’s suppose you are starting out with 100 grams of element A. How much of A, B and C will be present after 300 days? After 300 days, element A has undergone 1 half life and so there’s 50 grams remaining. Element A created 50 grams of element B which has undergone 3 half lives in 300 days. Does this mean what remains of element B is an eighth of the 50 grams or 6.25 grams? Is the calculation that simple ? ---
**Author:** ![Squink](https://avatars.discourse-cdn.com/v4/letter/s/b5e925/32.png) [@Squink](https://boards.straightdope.com/u/Squink)\ **Post date:** [February 12, 2007, 5:59am UTC](https://boards.straightdope.com/t/radioactive-decay-calculations-for-parent-and-daughter-nuclides/391816/2 "2007-02-12T05:59:04Z")
[QUOTE=wolf\_meister] Is the calculation that simple ? [/QUOTE] Nope. Ignore A for the moment, and try starting out with 50 grams of B. At day 300, you’ll have gone through 3 half lives and have 6.25 grams of B left. _That’s the same answer as you proposed for your A-\>B-\>C reaction_, but in that case you started out with _less than_ 50 grams of B. You started out with no B at all! You’re looking at two sequential reactions here, so the abundance curves will look something like [this](http://www.owlnet.rice.edu/~chem312/Handouts%20Folder/Sequential%20Reactions%201.jpg). ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [February 12, 2007, 6:01am UTC](https://boards.straightdope.com/t/radioactive-decay-calculations-for-parent-and-daughter-nuclides/391816/3 "2007-02-12T06:01:13Z")
> [@](#): > > After 300 days, element A has undergone 1 half life and so there’s 50 grams remaining. > Element A created 50 grams of element B which has undergone 3 half lives in 300 days. Does this mean what remains of element B is an eighth of the 50 grams or 6.25 grams? > Is the calculation that simple ? Certainly not that simple. That 50 grams of B was produced over a period of time, so most of it hasn’t been around for a full three B-half-lifes, and you’ll still have more B than that left. The proper way to approach this problem is with a system of coupled differential equations. Probably, though, someone else has already solved those differential equations and come up with a simple formula, since this is a fairly common situation. Unfortunately, I don’t know what it is off the top of my head. ---
**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\ **Post date:** [February 12, 2007, 6:13am UTC](https://boards.straightdope.com/t/radioactive-decay-calculations-for-parent-and-daughter-nuclides/391816/4 "2007-02-12T06:13:06Z")
Thanks **squink** and **chronos** Though I was hoping otherwise, I had a feeling that the calculations _had_ to be a little more complex than my example. Before asking a question in GQ, you can rest assured I have done a somewhat thorough job of web surfing. So, **Chronos** , any idea where I might locate those coupled differential equations? ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [February 12, 2007, 6:17am UTC](https://boards.straightdope.com/t/radioactive-decay-calculations-for-parent-and-daughter-nuclides/391816/5 "2007-02-12T06:17:14Z")
Suppose A decays at the rate a and B decays at the rate b. Then at time t the amount left is A(t) = A\_0\*exp(-at). At any time t, A is producing B at the rate aA(t), but B is decaying at the rate bB(t) so the differential equation for the amount of B is: dB/dt = aA(t) - bB(t) = aA\_0\*exp(-at) - bB. A particular solution to this equation is B(t) = aA\_0/(b-a)\*exp(-at) The homogeneous solution to the equation is B(t) = K\*exp(-bt). So the entire solution is B(t) = aA\_0/(b-a)_exp(-at) + K_exp(-bt). We must set K so that B(0) = 0. That is, K = -aA\_0/(b-a). Thus the solution is: B(t) = aA\_0/(b-a)\*[exp(-at) - exp(-bt)]. We could get the amount of C in a similar fashion, but it’s easier to note that A(t) + B(t) + C(t) = A\_0 ---
**Author:** ![Squink](https://avatars.discourse-cdn.com/v4/letter/s/b5e925/32.png) [@Squink](https://boards.straightdope.com/u/Squink)\ **Post date:** [February 12, 2007, 6:20am UTC](https://boards.straightdope.com/t/radioactive-decay-calculations-for-parent-and-daughter-nuclides/391816/6 "2007-02-12T06:20:24Z")
[QUOTE=Chronos] Probably, though, someone else has already solved those differential equations and come up with a simple formula [/QUOTE] [Sequential (Consecutive) Reactions](http://www.chem.neu.edu/Courses/1382Budil/ComplexChemicalKinetics.htm) ---
**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\ **Post date:** [February 14, 2007, 6:48am UTC](https://boards.straightdope.com/t/radioactive-decay-calculations-for-parent-and-daughter-nuclides/391816/7 "2007-02-14T06:48:46Z")
Let’s see if I have this right. The formula for the amount of the second element in a radioactive decay chain is: λ[sub]a[/sub] \* (Beginning amount of A) \* (e[sup]-λ[sub]a[/sub]\*t[/sup] - e[sup]-λ[sub]b[/sub]\*t[/sup])      (λ[sub]b[/sub] - λ[sub]a[/sub]) Where λ[sub]a[/sub] = the decay rate of A or (ln(2) / half-life) = (0.6931471805599 / 300 days) = 0.00231049060187 The decay rate of B (or λ[sub]b[/sub]) = 0.00693147180560 I tried writing the formula in Excel and got the following data: Starting with 128 grams of A and 0 grams of B After 0 days we have: 128 A and 0 grams of B (pretty good so far) 100 days 101.59 A 18.797 B 200 days 80.635 A 24.317 B 300 days 64 A 24 B 400 days 50.80 A 21.398 B The amount of B seems to peak at about 250 days which fits in with the type of “rate curve” in the chart that **squink** had referenced. So, do you folks agree with these results? ---
**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\ **Post date:** [February 16, 2007, 6:43am UTC](https://boards.straightdope.com/t/radioactive-decay-calculations-for-parent-and-daughter-nuclides/391816/8 "2007-02-16T06:43:34Z")
Sorry for bumping this thread but I’ll assume that this formula (based entirely on **OldGuy** ’s posting) is correct: λa \* (Beginning amount of A) \* (e[sup]-λa_t[/sup] - e[sup]-λb_t[/sup])         (λb - λa) Now I am wondering what is the formula if ‘C’ is radioactive and decays exponentially to ‘D’ ? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [February 16, 2007, 9:59pm UTC](https://boards.straightdope.com/t/radioactive-decay-calculations-for-parent-and-daughter-nuclides/391816/9 "2007-02-16T21:59:49Z")
Have you tried checking on [1728](http://www.1728.com/)? There are a lot of formulae available there. 😉 ---
**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\ **Post date:** [February 17, 2007, 5:41am UTC](https://boards.straightdope.com/t/radioactive-decay-calculations-for-parent-and-daughter-nuclides/391816/10 "2007-02-17T05:41:38Z")
**Chronos** Yes, I’ve tried looking there but those calculators are only applicable for a “mere” single decay reaction. (Still, a fine site nonetheless). 😃 If you are wondering if this information will end up as a calculator on my site - no. Why? If for no other reason, I’ve found that the complexity of a calculator is _inversely_ proportional to its popularity. A calculator for determining data for circles, spheres and cylinders? Wildly popular !!! An advanced calculator for Kepler’s 3rd Law? [crickets chirping] Anyway, this all came about because I was thinking about adding more information to the half-life calculator (explaining λ, τ, etc) but like most “Dopers” I got a little carried away and wondered about, daughter nuclides, etc. So, if a “Doper” wishes to offer a solution beyond a 2 decay chain that would be great but I appreciate the help already given. And that’s that’s name of that tune. ---
**Author:** ![yoyodyne](https://avatars.discourse-cdn.com/v4/letter/y/a9a28c/32.png) [@yoyodyne](https://boards.straightdope.com/u/yoyodyne)\ **Post date:** [February 17, 2007, 11:26pm UTC](https://boards.straightdope.com/t/radioactive-decay-calculations-for-parent-and-daughter-nuclides/391816/11 "2007-02-17T23:26:10Z")
What you’re looking for is known as the Bateman Equation. [Here](http://www.eas.asu.edu/~holbert/eee460/RadioactiveDecay.pdf) is a PDF with some good info. ---
**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\ **Post date:** [February 18, 2007, 5:00am UTC](https://boards.straightdope.com/t/radioactive-decay-calculations-for-parent-and-daughter-nuclides/391816/12 "2007-02-18T05:00:13Z")
**yoyodyne** Thanks for that link. Yes, it seems the Bateman Equation is _the_ formula for those calculations. And thanks to everyone for all your help. 🙂