How do you use the exponential decay formula?

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Answer 1 · 2014-12-16T23:06:15

I'll use an example from radioactive decay:

$N_{t} = N_{0} e^{- λ t}$ $N_t=N_0e^(-lambdat)$

$N_{t}$ $N_t$ = number of undecayed atoms after time $t$ $t$

$N_{0}$ $N_0$ = initial number of atoms

$λ$ $lambda$ = decay constant.

$t$ $t$ = time elapsed.

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

$t_{\frac{1}{2}} = \frac{0.693}{λ}$ $t_((1)/(2)) =0.693/lambda$

So $λ = \frac{0.693}{t_{\frac{1}{2}}} = \frac{0.693}{8} = 0.0866 d^{- 1}$ $lambda=0.693/t_((1)/(2)) =0.693/8 = 0.0866d^(-1)$

Taking natural logs of the decay equation we get:

${ln N}_{t} = {ln N}_{0} - λ t$ $lnN_t=lnN_0-lambdat$

So ${ln N}_{t} = ln (5) - 0.0866 \times 10$ $lnN_t=ln(5) - 0.0866xx10$

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

From which $N_{t} = 2.1 g$ $N_t =2.1g$

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