After asking my own question on half-lives and radioactive decay despite having read the library article on Radioactive Decay, I felt that I should post a less formal, more in-depth explanation of how to actually solve the equations.(adsbygoogle = window.adsbygoogle || []).push({});

There are two primary equations that I use when dealing with standard radioactive decay.

1. N=N_{0}e^{k*t}

2. r=ln2/k

N represents the final amount remaining.

N_{0}represents the initial amount.

e is the standard variable for ~2.718

k is the constant exponent for that sample.

t represents the amount of time that passed.

r is the half-life of the sample.

ln stands for natural logarithm.

Depending on which variable you're solving for, you'll need to set up your work in one of several different ways. I've posted each different type of problem, as well as examples, below.

You must usually first determine k before solving a problem (often that will be your problem). Set up your equation r=ln2/k, or switch it around to k=(-ln2)/r if that is more convenient and works for you. From there, you can plug in r (which must be given) and solve for k. Sometimes you may have simply been required to find r, in which case you must be given k (unless you're insanely intelligent and know of some way that I don't).

From there, you simply plug in your values to the other equation, N=N_{0}e^{k*t}. You should now be able to solve for any sort of problem without hassle, as you have the k value in addition to other already provided values by your instructor (remember, you can only solve for one variable at a time unless you have a system of equations).

Example:

The half-life of radium-226 is 1600 years. Suppose you have a 22mg sample. After how long will only 18mg of the sample remain?

k=(-ln2)/r

1600=(-ln(2))/k)

k=(-ln2)/1600

k=~-0.000433216988

N=N_{0}e^{k*t}

18=22*e^{-0.000433216988*t}

t=~463.210587512

463.211 years

As always, when you have your answer, remember to round to correct number of significant digits in chemistry, or typically three decimal places in mathematics. Make sure to include units.

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# Homework Help: Tutorial: Half-Lives and Radioactive Decay

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