6.9.6 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 = N₀ e^–λt

• 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 = A₀ e^–λt

• 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 = C₀ e^–λt

• 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)

Step 1: Write out the known quantities

• • Decay constant, λ = 0.025 year ^-1
  • Time interval, t = 5.0 years
  • Both quantities have the same unit, so there is no need for conversion

Step 2: Write the equation for activity in exponential form

A = A₀ e^–λt 

Step 3: Rearrange the equation for the ratio between A and A₀

Step 4: Calculate the ratio A/A₀

Therefore, the activity of Strontium-90 decreases by a factor of 0.88, or 12%, after 5 years