IA idea · Differential equations & dynamics

How long would a lake take to flush out a pollutant?

AA HLAI HL Solid Also in: Environment

Research question

Using a mixing differential equation with a lake's published volume and outflow, how long would it take for a pollutant's concentration to fall to 5% of its initial level, and how sensitive is that to the well-mixed assumption?

Adapt it: change the place, the data or the comparison until the question is yours.

Why it makes a good exploration

Environmental agencies use exactly this model to estimate recovery times after spills. Applying it to a real lake links a standard HL differential equation to a real decision.

The mathematics you'll need

Rate in − rate out modelling
First-order linear DE and its exponential solution
Half-life and time to 5%
Comparing continuous and discrete (Euler) solutions

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Find a lake's volume and outflow rate from a published source (national water agency or encyclopaedia) and cite it.

OpenStreetMap — Free map with exact coordinates of schools, hospitals, shops and stations; export or read off coordinates.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Derive dC/dt = −(Q/V)C for a clean inflow.
Solve and interpret Q/V as a flushing rate.
Apply to two lakes with different volumes.
Extend to a continuing pollutant inflow.
Reflect on stratification and incomplete mixing.

Pitfalls that cost marks

Unit errors between m³/s and km³.
Treating a huge lake as well mixed without comment.
Unsourced volume and flow figures.

Showing personal engagement

Choose a lake near you or one with a real pollution history.
Compare the model with a real recovery reported in the news.
Design a two-basin model and compare.

See Criterion C: personal engagement for what examiners look for.

Taking it further

Model a lake that receives a seasonal inflow (sinusoidal Q) and solve numerically.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).