Question

How do you find the equation of exponential decay?

Answer 1

`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)`