(g) graphical methods and spreadsheet modelling of the equation t N N T T…

The idea

Rather than using the analytic exponential solution, spreadsheet modelling involves a step-by-step numerical method. Starting with an initial number of undecayed nuclei $N_0$ and a chosen time interval $\Delta t$, the change in nuclei is calculated as $\Delta N = -\lambda N \Delta t$. The new number of nuclei after $\Delta t$ is then $N_{new} = N_{old} + \Delta N$. This process is repeated iteratively across rows in a spreadsheet to generate a decay curve. For the model to be accurate, the time interval $\Delta t$ must be significantly smaller than the half-life. Graphical methods involve plotting the modelled values of $N$ against $t$ to visualise the exponential decay curve and determine the half-life.

What earns the marks

• State the iterative equation for the change in the number of nuclei: $\Delta N = -\lambda N \Delta t$.
• Calculate the new number of nuclei after a time interval $\Delta t$ using $N_{new} = N_{old} + \Delta N$.
• Explain that a spreadsheet can model decay by applying this iterative calculation row by row for successive time intervals.
• Identify that the time interval $\Delta t$ must be small compared to the half-life for the numerical model to be a good approximation.
• Describe how plotting the iteratively generated values of $N$ against $t$ produces a decay curve from which half-life can be found.

Common mistakes and how to put them right

• Using the analytic equation $N = N_0 e^{-\lambda t}$ instead of the iterative numerical method.

  Put it right: Spreadsheet modelling for this specification point specifically requires the step-by-step use of $\Delta N = -\lambda N \Delta t$.

• Choosing a time interval $\Delta t$ that is too large.

  Put it right: If $\Delta t$ is comparable to or larger than the half-life, the model becomes inaccurate; $\Delta t$ must be small.

• Forgetting the negative sign in the rate equation.

  Put it right: Ensure $\Delta N$ is calculated as a negative value because the number of undecayed nuclei decreases over time.

Exam tips

• When asked to model decay, explicitly state the formulas you would enter into the spreadsheet, such as `=-lambda * N * delta_t`.
• Be prepared to explain why the iterative model deviates from the true exponential curve if $\Delta t$ is too large.
• Remember that $\lambda$ must be in the same time units as $\Delta t$ (e.g., both in seconds or both in hours).