The batch reactor

A batch reactor is a tank you fill, stir while the reaction runs, then empty. Nothing flows in or out while it's reacting, so the balance on A is just consumption = accumulation: the concentration falls over time.

For a first-order reaction at constant volume, the concentration falls exponentially: C_A = C_A0 × e^(−kt). Rearranged, the time to reach a conversion X is t = ln(1/(1 − X)) ÷ k. With k = 0.1 per minute, 90% conversion takes ln 10 ÷ 0.1 = 23.0 minutes, and 99% takes ln 100 ÷ 0.1 = 46.1: the last bit always takes the longest.

A batch cycle also needs time to fill, heat up, cool down, empty and clean, often longer than the reaction itself. So a plant's output depends on the whole cycle: a 23-minute reaction with 37 minutes of turnaround gives one batch an hour.

Batch reactors suit small volumes, many different products on the same equipment (medicines, dyes, specialty chemicals), and slow reactions. Each batch can be tracked and tested on its own, which regulators of medicines like. Large-volume chemicals are made continuously instead.

The same equations describe any first-order process that runs down over time: a batch of drug breaking down on the shelf, or a pollutant decaying in a closed lake.