How do you find the equation of exponential decay?

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Answer 1 · 2014-12-09T00:54:03

$N_t=N_0e^(-lambdat)$

Exponential decay and growth occurs widely in nature so I will use radioactive decay as an example.

When an atom decays it is a random, chance event. The number of atoms decaying per second depends only on the number of undecayed atoms N.

So we can write:

Rate of decay:

$(-N)/(t)propN$

We can replace the $prop$ sign with an = sign and the constant $lambda$ . We can also use calculus notation:

$-(dN)/(dt)=lambda N$

Rearranging gives:

$(dN)/(N)=-lambdadt$

Integrating both sides:

$int(dN)/(N)=-lambdaintdt$

So

$lnN=-lambda t+c$

If we apply the limits of integration such that when $t=0$ , $N= N _0$ and when $t=t, N = N_t$ we get:

$lnN_t-lnN_0=-lambdat$

So

$ln(N_t/N_0)=-lambda t$

So

$(N_t/N_0)=e^(-lambdat)$

So

$N_t =N_0e^(-lambdat)$