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)#