If either the rate constant or the half-life is known, the other can be calculated. Half-lives usually are used because they have an immediate physical meaning.

The exponential decay curve (fig 4 opposite) can be used to give the rate law a physical meaning. The rate of change of concentration with time, d[C-14]/dt, is simply the slope of the first-order decay curve at any time, t.

Because carbon-14 is disappearing, the slope is negative. By drawing a tangent line to the decay curve at several points and examining the slope, you should be able to verify that the slope of the curve at any time t is proportional to the remaining concentration of carbon-14, measured in the vertical direction. This is what the rate law for a first-order process means:

Thus far we have established that the rates of chemical reactions depend on the concentrations of reactant species, and that this dependence often has a complicated form that cannot be predicted directly from the coefficients of the balanced chemical equation (as the equilibrium-constant expression can).