How long before you can smell perfume across a room?

Settled by Standard turbulent-mixing theory (as opposed to naive diffusion).

Kapitza's problem #210 (paraphrased on abitura.com: a perfumed student enters an auditorium, how soon does the professor across the room smell it) is a classic trick estimation problem: naive use of the diffusion equation with a molecular diffusion coefficient of roughly 0.1 cm^2/s predicts a wildly unrealistic time (hours to days) for scent to cross a room by pure diffusion, while everyday experience says it takes seconds to a couple of minutes. The resolution, standard in fluid mechanics pedagogy, is that ordinary air is never perfectly still; small convective currents and turbulence transport the scent orders of magnitude faster than molecular diffusion, and the true controlling timescale is set by room-scale air mixing, not the diffusion constant.

The 'obvious' physics approach (Fick's law of diffusion) gives an answer off by many orders of magnitude, and getting the right order of magnitude requires realizing you were solving the wrong problem.

Explanations that circulated

• Perfume reaches you purely by molecular diffusion (pure diffusion across a lecture-hall-sized room would take many hours to days for a scent molecule's mean-square displacement to cover meters; the actual timescale of tens of seconds to a few minutes is set by convective/turbulent air currents, not diffusion)

References

• Turbulent diffusion vs. molecular diffusion in indoor air (general physics of scent dispersal, standard textbook treatment), Standard fluid mechanics / indoor air quality literature, Various textbooks, 2020