Radioactive Decay Equations (OCR A Level Physics): Revision Note

Radioactive Decay Equations

• In radioactive decay, the number of undecayed nuclei falls very rapidly, without ever reaching zero

  • Such a model is known as exponential decay

• The graph of number of undecayed nuclei against time has a very distinctive shape:

Radioactive decay follows an exponential pattern. The graph shows three different isotopes each with a different rate of decay

• The key features of this graph are:

  • The steeper the slope, the larger the decay constant λ (and vice versa)

  • The decay curves always start on the y-axis at the initial number of undecayed nuclei (N₀)

Equations for Radioactive Decay

• The number of undecayed nuclei N can be represented in exponential form by the equation:

$N=N_0{e}^{-\lambda t}$

• Where:

  • N₀ = the initial number of undecayed nuclei (when t = 0)

  • N = number of undecayed nuclei at a certain time t

  • λ = decay constant (s^-1)

  • t = time interval (s)

• The number of nuclei can be substituted for other quantities.

• For example, the activity A is directly proportional to N, so it can also be represented in exponential form by the equation:

$A=A_0{e}^{-\lambda t}$

• Where:

  • A = activity at a certain time t (Bq)

  • A₀ = initial activity (Bq)

• The received count rate C is related to the activity of the sample, hence it can also be represented in exponential form by the equation:

$C=C_0{e}^{-\lambda t}$

• Where:

  • C = count rate at a certain time t (counts per minute or cpm)

  • C₀ = initial count rate (counts per minute or cpm)

The exponential function e

• The symbol $e$ represents the exponential constant

  • It is approximately equal to $e$ = 2.718

• On a calculator it is shown by the button $e^x$

• The inverse function of $e^x$ is $\ln y$, known as the natural logarithmic function

  • This is because, if $e^x=y$, then $x=\ln y$