Radio Active Decay

Radioactive substances such as radium and caesium disintegrate over time — their mass decreases — and the RATE of disintegration is always proportional to the amount present at that time. If x(t) is the amount of the material present at time t, this translates to dx/dt = −k·x, where k is the (positive) constant of proportionality; the negative sign is essential here, recording the fact that x decreases as t increases (unlike the growth model, where no such sign is needed). Solving this by separation of variables in exactly the same way as the growth equation gives x = a·e^(kt) for some constant a — but since x is actually decreasing, this is more usefully written using the initial amount x₀ (the value of x at t=0): substituting t=0 gives x₀ = a, so a = x₀, and the formula becomes x = x₀·e^(−kt). This expression gives the amount of radioactive substance remaining at any time t.

The HALF-LIFE PERIOD of a radioactive substance is defined as the time it takes for half the original amount (or mass) of the substance to disintegrate — i.e., the value of t at which x = x₀/2. …