The formula
N = N₀ · 2^(t / g)
| Symbol | Quantity | Unit |
|---|---|---|
| N | Final population | |
| N₀ | Initial population | |
| t | Time elapsed | min |
| g | Generation time | min |
What it means
In exponential phase a bacterial population doubles at regular intervals, and that interval is called the generation time. Under optimal conditions, Escherichia coli does it every twenty minutes: starting from a thousand cells, three hours later there are over half a million. That growth is what lets food poisoning assemble itself over a summer afternoon on a kitchen counter.
How to work it out by hand
- Divide the elapsed time by the generation time
- That number is the doublings
- Raise two to those doublings
- Multiply by the starting population
What is worth knowing
This formula only describes the **exponential phase**, one of four in a real culture. Before it comes a lag phase, where the bacteria adapt to the medium and barely grow; after it the stationary phase, when nutrients run short and death matches division; and finally the death phase. Extrapolating the exponential beyond a few hours gives absurd figures: a single bacterium doubling every twenty minutes would exceed the mass of the Earth in two days. Generation time is not a constant of the species either: it changes with temperature, medium and oxygen, and that is why refrigeration works — it does not kill bacteria, it stretches their generation time enormously.