Exponential Decay Equations

This lesson covers: 

  1. The equation linking undecayed nuclei, original nuclei, decay constant, and time
  2. Calculating remaining activity using initial activity, decay constant, and time
  3. Using radioactive carbon-14 for archaeological dating

Mathematical model for radioactive decay

The relationship between the number of undecayed radioactive nuclei (N) remaining from an original number (N0) can be determined by:


N = N0​ e−λt


Where:

  • N = number of nuclei remaining
  • N0 = initial number of nuclei
  • λ = decay constant (s-1)
  • t = time (s)


As the nuclei decay over time, their activity (A) also changes. The activity at any given time t is:


A = A0​ e−λt


Where:

  • A = activity at time t (Bq)
  • A0 = initial activity at t = 0 (Bq)

Worked example - Calculating the number of remaining nuclei

Calculate the number of undecayed nuclei remaining after 10,000 years given the initial number of nuclei is 1.0 x 106 and the decay constant for the substance is 1.209 x 10-4 per year.


Step 1: Formula

N=N0​e−λt


Step 2: Substitution and correct evaluation

N=1.0×106 e−1.209×10−4×10,000=2.99×103

Radioactive dating using carbon-14

The radioactive isotope carbon-14 is extensively used in archaeological dating because:

  • Living organisms absorb carbon-14 from the atmosphere.
  • Upon death, the decay of carbon-14 in the organisms begins.
  • The comparison of the remaining carbon-14 enables the dating of archaeological samples.


Key Points:

  • Carbon-14 has a half-life of approximately 5,730 years.
  • The measurement of the carbon-14 percentage facilitates the calculation of the sample's age.
  • This method is applicable to materials that were once living, such as wood or bone.

Worked example: - calculating the age of an archaeological artefact.

Estimate the age of an archaeological sample if the initial activity of carbon-14 was 0.25 Bq and has now decayed to 0.1 Bq, with the decay constant for carbon-14 being 1.21 x 10-4 per year.


Step 1: Formula

A=A0​×e−λt


Step 2: Rearranged formula

t = −λln(A/A0​)​


Step 3: Substitution and correct evaluation

t=−1.21×10−4ln(0.1/0.25)​≈7,573 years