## Determining half-life of a radioactive sample

If I know that a sample of X has a half-life of 270 years how do I confirm this experimentally? What data would I have to collect? How about if the half-life of X was much shorter i.e. in terms of days?
 Use some kind of device capable of measuring radiation quantitatively, like a geiger counter. With this, you can discover how much the sample is radiating at some point in time. Where k, c, and A are unknown constants, x is the amount of radioactive matter remaining, L is the half-life, and t is time, you can reason out what measurements you'd need this way: radiation (change in amount of radioactive material in the sample) = k multiplied by amount remaining dx / dt = k x dx / x = k dt ln x = k t + c (integrate both sides) x = A e^kt (exponentiate both sides, let A = e^c) for the half-life: 0.5 A = A e^kL (half of the amount at t=0 will remain at t=L) (ln 0.5)/L = k so from knowing L, you know k. measure dx / dt, divide by k, and you know a value for x. Repeat the same experiment at other times, recording x and t for each. If the values you measure satisfy x = A e^kt for the value of k you calculated, then you are right about L. If they do not work, you can use the equation to find the true value of k and L.