Basically I started thinking this question in regard to farts. I thought to myself, say Alice and Bob are in a room. If Alice farts, how long would it take for Bob to pick up on the smell? After a bit of thinking, I realised that this could apply to all gases.

Is there a formula for gases spreading? Or some way of deriving a formula?


The kinetic energy of a particle of mass $m$ and velocity $v$ is related to the temperature by $$ \frac{1}{2}mv^2~=~\mu kT, $$ for $\mu$ a dimensionless constant. The velocity $v~=~dx/dt$ means that the rate of diffusion is related to the mass of the particle $$ R_\textrm{diff}~\propto~\frac{1}{\sqrt m}. $$ For a molecule of a gram molecular weight, this determines the rate of diffusion by this formula.

I have a bull terrier dog, aka a pitbull, who is a big teddy bear of a dog. He is very affectionate and with my other dog, a Brittany pointer female, the two are perfect frick and frack. However, this bull terrier lets the farts fly with a smell that should never be smelled. They are by far and away the most horrific farts ever.


The velocity of a molecular species of mass m is $$ v(T)~=~\sqrt{\frac{8kT}{\pi m}}. $$ Putrecine is a molecule with the shape

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with an atomic weight of $88$ this means it has mass $m~=~1.47\times 10^{-27}kg$ at a standard temperature of $T~=~300K$ we then have velocity $v~=~268m/s$. This is the average speed of a molecule.

Clearly the gas does not diffuse that quickly. However, I can assume that the molecules moves a distance $d~\simeq~\ell\sqrt{n}$ after $n$ random collisions. I am assuming here a random walk of sorts. The means free path is $$ \ell~=~\frac{kT}{\sqrt{2}D^2 P} $$ for $D$ the diameter of the molecule and $P$ the pressure. Assuming $P~=~10^5Pa$ and the molecule has a diameter of about $10^{-9}$m this gives a mean free path of $5\times 10^{-4}$m. Given the above velocity there is a collision about every $2\times 10^{-6}$ seconds or about $5\times 10^5$ collisions per second. So in a second the average molecule has migrated about $7$cm.

That sounds about right with standard experience. A standard smell, whether a nice cooking smell or a foul odor, fills a room within a few $10$s of seconds.

  • $\begingroup$ So solving for v, then integrating with respect to time would give me the particle location? Then how would I apply this to the mass of the entire gas body? $\endgroup$ – drunkBrain Oct 25 '16 at 10:58
  • $\begingroup$ To do an explicit calculation one might have to do a Boltzmann calculation. In general the lighter the molecule the faster it will diffuse. Methane has a gram molecular weight of 16. This makes it bouyant and readily diffusable. This is though not the smell of flatulence. These are due to putrecine and related alkanamine molecules that are a carbon chain with amine $NH_2$ ends. These are heavier and tend to hug the ground. $\endgroup$ – Lawrence B. Crowell Oct 25 '16 at 11:48
  • $\begingroup$ Assuming ideal gas, for example, how would I calculate how fast, say, 1 kilogram of gas diffuses at STP? $\endgroup$ – drunkBrain Oct 25 '16 at 13:05
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    $\begingroup$ youtube.com/watch?v=ADiI_-3TA3I $\endgroup$ – Lawrence B. Crowell Oct 26 '16 at 0:49

There is more than one process by which gases can be transported around a room. Besides diffusion (which spreads out radially from the source) there is also convection. This could be natural convection - ie expansion/rising and contraction/falling of warm/cold packets of air, causing circulation. Or it could be forced convection - ie draughts or fans causing movement of air. If the room is large, an aroma is more likely to spread by convection than by diffusion. Farts are warm and likely to rise.

Diffusion is much easier to calculate than natural convection (aka advection).

Whether or not an aroma is detected also depends on the sensitivity of the nose. An aroma which spreads by diffusion may be so dilute when it reaches Alice that it is below her threshold for detection, whereas if the fart is transported in a packet of air it may still be strong enough when it reaches Alice.

  • $\begingroup$ Great thing to consider. The natural convection of the hot gas and the forced convection of the fart itself probably give a very nice initial boost to the diffusion process (which explains why it can take like 1s to smell farts like 1m away). $\endgroup$ – JMac Apr 3 '17 at 13:27

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