How do you find the equation of exponential decay?

Question

6911 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2014-12-09T00:54:03

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

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:

$\frac{- N}{t} \propto N$ $(-N)/(t)propN$

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

$- \frac{d N}{d t} = λ N$ $-(dN)/(dt)=lambda N$

Rearranging gives:

$\frac{d N}{N} = - λ d t$ $(dN)/(N)=-lambdadt$

Integrating both sides:

$\int \frac{d N}{N} = - λ \int d t$ $int(dN)/(N)=-lambdaintdt$

So

$ln N = - λ t + c$ $lnN=-lambda t+c$

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

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

So

$ln (\frac{N_{t}}{N_{0}}) = - λ t$ $ln(N_t/N_0)=-lambda t$

So

$(\frac{N_{t}}{N_{0}}) = e^{- λ t}$ $(N_t/N_0)=e^(-lambdat)$

So

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