Radioactive Decay Equations

Posted in A2 Unit 5: Radioactivity, AQA A2 Unit 5 by Mr A on 11 Jan 2010

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
  • T_{\frac{1}{2}} = 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;

 

A = \lambda N

 

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

 

A = -\frac{\Delta N}{\Delta t}

 

Or,

 

\frac{\Delta N}{\Delta t} = -\lambda N

 

This is a linear differential equation whose solution is:

 

N = N_{0} e^{- \lambda t}

 

where

 

N_{0} = 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

 

T_{\frac{1}{2}} = \frac{ln 2}{\lambda}

 

and

 

A = A_{0} e^{- \lambda t}

 

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