# 18.5.2 Exponential Decay and Half Life

If a radioactive sample contains N0 number of undecayed nuclei at $\displaystyle t=0$, then number of undecayed nuclei N at time t is given by

$\displaystyle \displaystyle N={{N}_{0}}{{e}^{{-\lambda t}}}$

This is called an exponential decay. The larger the decay constant λ, the faster the sample is depleted.

The exponential relationship comes about because N decreases at a rate proportional to N. Ah, let’s just do the math.

We start from the fact that activity is proportional to the number of undecayed nuclei.

$\displaystyle A=\lambda N$

Since the activity is also equal to the rate at which the number of undecayed nuclei decreases,

\displaystyle \begin{aligned}-\frac{{dN}}{{dt}}&=\lambda N\\\frac{1}{N}dN&=-\lambda dt\end{aligned}

Integrating both sides,

\displaystyle \displaystyle \begin{aligned}\int\limits_{{{{N}_{0}}}}^{N}{{\frac{1}{N}dN}}&=-\int\limits_{0}^{t}{{\lambda dt}}\\ln\frac{N}{{{{N}_{0}}}}&=-\lambda t\\N&={{N}_{0}}{{e}^{{-\lambda t}}}\end{aligned}

N is not the only thing that decays exponentially. Since the activity A is proportional to the number of undecayed nuclei ($\displaystyle A=\lambda N$ and $\displaystyle {{A}_{0}}=\lambda {{N}_{0}}$), A should also decrease exponentially over time.

$\displaystyle \displaystyle A={{A}_{0}}{{e}^{{-\lambda t}}}$

In practice, the activity is measured using a Geiger-Muller (GM) counter. Since the count rate C is proportional to the activity, C should also decrease exponentially with time.

$\displaystyle \displaystyle C={{C}_{0}}{{e}^{{-\lambda t}}}$

Besides the decay constant, the half-life, T1/2, is the other parameter that is used to characterize the rate of decay of a nuclide. It is the (average) time taken for half of the nuclei in the population to decay. You can also think of it as the (average) time taken for a sample’s activity to be halved. The relationship between decay constant and half-life can be derived as below.

\displaystyle \displaystyle \begin{aligned}0.5&={{e}^{{-\lambda {{T}_{{1/2}}}}}}\\\ln 0.5&=-\lambda {{T}_{{1/2}}}\\{{T}_{{1/2}}}&=\frac{{\ln 2}}{\lambda }\end{aligned}

For example, Rn-222, which has a decay constant of $\displaystyle 2.09\times {{10}^{{-9}}}\text{ }{{\text{s}}^{{-1}}}$, has a half-life of

$\displaystyle {{T}_{{1/2}}}=\frac{{\ln 2}}{{2.09\times {{{10}}^{{-9}}}}}=3.31\times {{10}^{9}}\text{ s}=3.8\text{ days}$.

The cute thing about an exponential decay is that the quantity falls by the same percentage given the same amount of time. So it does not matter when we start counting. The number of undecayed nuclei N in a sample of Rn-222 number is halved after every 3.8 days.

Before we leave this section, let’s remind ourselves that the exponential formulae are accurate only if N is large. Radioactive decay is at its core a random event. In practice, there will always be random fluctuations in the measured count rate that result in deviations from the theoretical values.

Applet

Law of Decay (Walter Fendt)

Concept Test

Comics

Happy Half-Life Day

Killer Paper