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.