Half-Life (AQA A Level Physics): Revision Note

Half-life

• Half-life is defined as:

The average time taken for a given number of nuclei of a particular isotope to halve

• Since activity $A$ is proportional to the number of undecayed nuclei $N$, the activity of the sample will also halve

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

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

• Where:

  • $N$ = number of nuclei remaining in a sample

  • $N_0$ = the initial number of undecayed nuclei (when t = 0)

  • $\lambda$ = decay constant (s^-1)

  • $t$ = time interval (s)

• When time $t$ is equal to the half-life $t_{1/2}$, the activity $N$ of the sample will be half of its original value, so $N=\frac{1}{2}{N}_0$

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

• The formula can then be derived by first, dividing both sides by $N_0$:

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

• Then, taking the natural log of both sides:

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

• Finally, applying properties of logarithms:

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

• Therefore, half-life $t_{1/2}$ can be calculated using the equation:

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

• This equation shows that half-life $t_{1/2}$ and the radioactive decay rate constant $\lambda$ are inversely proportional

  • Therefore, the shorter the half-life, the larger the decay constant and the faster the decay

Half-life from decay curves

• The half-life of a radioactive substance can be determined from decay curves and log graphs

• Since half-life is the time taken for the initial number of nuclei (or activity) to reduce by half, it can be found by:

  • drawing a line to the curve at the point where the activity has dropped to half of its original value

  • drawing a line from the curve to the time axis, this is the half-life

Log graphs

• Straight-line graphs tend to be more useful than curves for interpreting data

  • Nuclei decay exponentially, therefore, to achieve a straight line plot, logarithms can be used

• Take the exponential decay equation for the number of nuclei

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

• Taking the natural logs of both sides

$\ln N=\ln \left(N_0\right)-\lambda t$

• In this form, this equation can be compared to the equation of a straight line

$y=mx+\hspace{0.17em}c$

• Where:

  • y-axis variable, $y=\ln N$

  • x-axis variable, $x=t$

  • gradient, $m=-\lambda$

  • y-intercept, $c=\ln \left(N_0\right)$

• Half-lives can be found in a similar way to the decay curve but the intervals will be regular as shown below: