## Radioactive Decay Equations

When atomic nuclei become *unstable*, they must emit either an α or β particle, or energy in the form of a γ ray. They become more stable in the process.

*A*= activity of a sample, Bq (Becquerels) = the rate of decay = the number of nuclei that will decay in any given second*N*= the number of unstable nuclei remaining in a sample*λ*= the probability that any given nucleus will decay in any given second- = the half-life of a sample, s (seconds)

If *λ* is the probability that any given nucleus will decay in a second, and there are *N* nuclei remaining, then it stands to reason that *λN* gives us the number of nuclei that will decay in a given second;

The number of nuclei remaining after some time *Δt* is *reduced* (hence the minus sign) by *ΔN*. Then, the activity of the sample is:

Or,

This is a *linear differential equation* whose solution is:

where

= the number of undecayed nuclei at time zero

*N* = the number of undecayed nuclei at time t

Thus, radioactive decay follows an exponential decay curve.

Now prove that

and

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