# How do you use the exponential decay formula?

Dec 16, 2014

I'll use an example from radioactive decay:

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

${N}_{t}$ = number of undecayed atoms after time $t$

${N}_{0}$ = initial number of atoms

$\lambda$ = decay constant.

$t$ = time elapsed.

" ${I}^{131}$ is a radioactive isotope with a half - life of 8 days. Starting with a mass of 5g, what mass will remain after 10 days? "

${t}_{\frac{1}{2}} = \frac{0.693}{\lambda}$

So $\lambda = \frac{0.693}{t} _ \left(\frac{1}{2}\right) = \frac{0.693}{8} = 0.0866 {d}^{- 1}$

Taking natural logs of the decay equation we get:

$\ln {N}_{t} = \ln {N}_{0} - \lambda t$

So $\ln {N}_{t} = \ln \left(5\right) - 0.0866 \times 10$

$\ln {N}_{t} = 1.61 - 0.866 = 0.744$

From which ${N}_{t} = 2.1 g$

I can use grams instead of number of atoms as they are proportional so the constant will cancel out.