Half-Life (Cambridge (CIE) A Level Physics): Revision Note

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:

$t_{1/2}=\frac{\ln 2}{\lambda }$

• 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

$\frac{1}{2}{N}_0=N_0{e}^{-\lambda t_{1/2}}$

• First, divide both sides by N₀:

$\frac{1}{2}=e^{-\lambda t_{1/2}}$

• Then, take the natural log of both sides:

$\ln \left(\frac{1}{2}\right)=-\lambda t_{1/2}$

• Finally, apply the properties of logarithms:

$\lambda t_{1/2}=\ln 2$

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

$t_{1/2}=\frac{\ln 2}{\lambda }\simeq \frac{0.693}{\lambda }$