How does ventilation dilute a passive tracer in a well-mixed room?

Ventilation can exchange indoor air with incoming air, but the concentration change depends on how that exchange occurs. A useful ideal model treats a room as perfectly mixed, so one concentration represents the entire volume and also the air leaving it.

Here the room volume V is fixed at 100 m³. Equal inflow and outflow Q keep that volume constant. The incoming air contains none of a passive tracer, and there is no ongoing tracer source, chemical reaction or exchange with surfaces. These assumptions make outflow the only removal process.

Tracer leaves at a rate proportional to the current room concentration C. The mass balance is dC/dt = −(Q/V)C. Its solution is C(t) = C(0) exp[−(Q/V)t]. The ratio Q/V is the air-change rate, expressed here per hour, rather than the fraction of original air guaranteed to leave in one step.

The initial example has Q = 50 m³/h, giving 0.5 air changes per hour. Starting from 100 concentration units, after two hours the model gives about 36.79 units. One nominal room-volume exchange has therefore not removed all the tracer: newly arrived air mixes with remaining air and can also leave.

Real rooms can contain poorly mixed zones, changing flows or continued sources. Different tracers can also react or interact with surfaces. The exponential curve is a conditional mass-balance result, not a statement that every location follows the same concentration or that a particular health outcome is assured.