The first-order rate law for the decay of carbon-14 nuclei,

is a differential equation. It tells us how the change in with time depends on . It would be desirable to have another expression that simply told how the varies with time in a first-order decay process:
  [C-14] = some function of t

Elementary calculus shows us how we can derive an expression for concentration versus time from a rate equation, using the process of integration. The general method is beyond the scope of this chapter, but we can give the result for a first-order decay:
  [C-14] = [C-14]oe-kt

Starting from an initial concentration at time t = O of [C-14]o the concentration of carbon-14 at some later time, t, decreases exponentially, as shown above. One property of exponential decay is that, if after a certain time interval the has fallen by half, then after another interval of equal length the concentration will have fallen by half again, or to one quarter its original value.