If I Blow smoke out my window, will any come inside my room?

Question

My girlfriend and I have got into a lively debate about whether or not my flatmates can smell the smoke I blow out my window. Can you help us? Here is the scenario.

1. The smoker, places himself perched upon the window, with his cigarette placed in his left hand, which remains outside permanently and remaining at least 15cm away from the window at all times.

2. It's winter in Edinburgh (it's 2 degrees celcius outside). The room temperature is about 21 degrees.

3. When the smoker exhales, he projects a thin stream of smoke directly outside (at 90 degrees). (so as, to move it as cleanly and as far away as possible)

4. There is very little wind outside.

Assuming that:

1. Hotter air has a higher air pressure than cooler air
2. The room is hotter than outside, so it has a higher air pressure.
3. High pressure air always flows to low pressure air

We can hope to assert:

1. That although the smoke is hot upon being exhaled, it will cool rapidly when exposed to the 2C air and thereby achieve an air pressure significantly low enough to be unable to cross ithe nside/outside threshold due to difference in air pressure between itself and the room.

Answer 1

If a house were filled with water to the top of the door and you opened the front door, water would run out the bottom part of the opening and air would come in from that outside through the top part of the opening. This is because the water is more dense than the air. In the case of your open window, the inside air is warmer (less dense) than the outside air. So outside air will come in through the bottom part of the window opening (from the outside), and inside air will leave through the upper part of the window opening. The air coming in through the lower part of the opening will carry smoke from the cigarette with it into the room.

The real proof of this is that the room gets cold with the window open. So cold air must be coming in from outside (and bringing smoke back in with it).

Answer 2

1. Hotter air has a higher air pressure than cooler air

This isn't the whole story. Air is generally well modeled as an ideal gas, which means pressure $p$, temperature $T$, density $\rho$, and molecular weight $m$ are related by $$ p = \frac{k_\mathrm{B}}{m} \rho T. $$ Hotter air has higher pressure if density and molecular weight are kept equal.

2. The room is hotter than outside, so it has a higher air pressure.

This won't be true at all. If you have an open window, the pressures will equilibrate rapidly. How rapidly? The characteristic timescale for this is the sound-crossing time of the room. Since the speed of sound is hundreds of meters per second, it only takes a fraction of a second for pressure differences to vanish. This is due to your third point, that air flows from high pressure to low pressure. Indeed it does, just much faster than you imagine.

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So what does happen? The air in all places is the same pressure. Differences in temperature are compensated by differences in density. (Recall how hot air rises -- this is really saying that pressures and molecular weights being equal, less-dense air rises.) The smoke itself might also have a different molecular weight. All of this though is missing an important point.

Your nose can pick up very trace amounts of some substances. One doesn't need to have a visible cloud of smoke to be able to smell smoke. In particular, the analysis so far has neglected diffusion. Even if the bulk of smoke is visibly moving in one direction, random motions of gas particles can cause a few stray smoke particles to be sent flying in all different directions.

Formally, diffusion spreads an (infinitesimal) amount of material infinitely far from the source in no time at all. In order to see if this really can cause the smoke to be smelled, one would have to calculate the diffusion coefficient (doable based on the molecular weights and temperatures of the substances), note the advection velocity of the bulk material (roughly measurable or guessable), and compare the amount diffusing against the bulk with the sensitivity of a typical person's nose (a pure biology question I'm not suited to answer).