Half-Life (Cambridge International A-Level Physics)

Half-Life Definition

• Half-life is defined as:

The time taken for the initial number of nuclei to reduce by half

• This means that when a time equal to the half-life has passed, the activity of the sample will also halve
• This is because the activity is proportional to the number of undecayed nuclei, A ∝ N

Determining half-life from a graph

When a time equal to the half-life passes, the activity falls by half, when two half-lives pass, the activity falls by another half (which is a quarter of the initial value)

Calculating Half-Life

• The half-life of a radioactive sample can be calculated using the equation:

$t_{1/2}=\frac{\mathrm{ln}2}{λ}$

• Where:

  • t_½ = half-life of the sample (s)
  • λ = decay constant of the sample (s^-1)

• This equation shows that:

  • the half-life and the decay constant of a sample are inversely proportional
  • the shorter the half-life, the larger the decay constant and the faster the rate of decay

Derivation of the half-life equation

• To find an expression for half-life, start with the equation for exponential decay:

N = N₀e–^λt

• Where:

  • N = number of nuclei remaining in a sample
  • N₀ = the initial number of undecayed nuclei (when t = 0)
  • λ = decay constant (s^-1)
  • t = time interval (s)

• When time t is equal to the half-life t_½, the activity N of the sample will be half of its original value, so N = ½ N₀

$\frac{1}{2}N_{0}=N_{0}e^{-λt_{1/2}}$

• First, divide both sides by N₀:

$\frac{1}{2}=e^{-λt_{1/2}}$

• Then, take the natural log of both sides:

$\ln ( 1 \frac{1}{2} ) = - \\lambda t_{1 \div 2}$

• Finally, apply the properties of logarithms:

$λt_{1/2}=\mathrm{ln}2$

• Therefore, half-life t_½ can be calculated using the equation:

$t_{1 \div 2} = \frac{\ln 2}{\\lambda } \text{asymptotically} \text{equal} \text{to} \frac{0.693}{\\lambda }$

Worked Example Strontium-90 is a radioactive isotope with a half-life of 28.0 years. A sample of strontium-90 has an activity of 6.4 × 10⁹ Bq.

Calculate the decay constant λ, in s^–1, of strontium-90.

Answer:

Step 1: Convert the half-life into seconds

28 years = 28 × 365 × 24 × 60 × 60 = 8.83 × 10⁸ s

Step 2: Write the equation for half-life

$t_{\frac{1}{2}}=\frac{\mathrm{ln}2}{λ}$

Step 3: Rearrange for λ and calculate

$λ=\frac{\mathrm{ln}2}{t_{\frac{1}{2}}}=\frac{\mathrm{ln}2}{8.83\times10^{8}}=7.85\times10^{-10}s^{-1}$