How do you calculate the half-life of cesium-135 if seven-eighths of a sample decays in 6106 years?

Answer 1

As is asked, the half-life of the cesium-135 is 2035.3333... years (or 2035 1/3 years). We'll walk through the process to show you how to calculate it using an "easy" method. But before we launch, the half-life of cesium-135 is actually on the order of 2.3 million years, just so you know. Now let's look at the math. To make the calculations, we'll use a 2-step process to avoid some sophisticated "nastiness" and mathematical hair-pulling. First we'll need to find the number of half-lives, and we know that 7/8ths are gone in 6106 years. That means 1/8th is left. Since we're dealing with half-lives (or 1/2-lives) we'll need an expression to get from the "start" to end up with 1/8th of the substance remaining. Here's the equation: 1/2n = 1/8 [the n is the number of 1/2-lives it takes for a given amount of decay to occur] The 1/2 is the rate of decay, and that's because half-life means half goes away at every interval of time. This is for all applications involving radioactive decay. (We might have other applications in something like business where things like sales or production decay by 1/3rd or 1/4th or something else.) If you solve for n here, you'll get 3, 'cause it takes 3 half-lives for 7/8th of the material to decay and for 1/8th to be left. (The sequence is 1/2, 1/4th, 1/8th, 1/16th, 1/32nd, ..., and you knew that already.) We now take the number of half-lives and divide it into the time it took for the decay of 7/8th of the material to occur and 1/8th to be left, and here's that expression: t1/2 = t / n t1/2 = half-life of the substance under investigation t = time elapsed for a given amount of decay to occur n = number of half-lives that elapse in a given period of decay t1/2 = t / nt1/2 = 6106 / 3 = 2035.3333... years Simple and easy. Use a scientific calculator to solve the first equation or just do some repetitions to run down the number of half-lives. We have some other expressions we could use to pull everything together, but they're a little more complex. Links can be found below to citations and explanations, and you're a mouse click away from those higher order mathematical expressions to make the calculations. The object here was to make the math accessable to almost any student. No, we're not selling you short, but giving you a stepping stone to the "bigs" with this avenue to a solution. (If you're really adept, you can take the information here and derive your own equation! Never said you weren't challenged in addition to being informed!)

Answer 2

the half-life of cesium-135 if seven-eighths of a sample decays in 6*10^6 years.