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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.
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    $\begingroup$ I would simply answer this question experimentally. $\endgroup$ – CuriousOne Feb 18 '16 at 21:44
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    $\begingroup$ No matter what precautions you take, some smoke molecules will get inside your room (on a related subject, air molecules bounce back and forth between the walls of your room around 100 times per second, so diffusion is incredibly fast). The question is whether it's a noticeable amount - and I suspect the only way to answer that is to test it. Sorry. $\endgroup$ – lemon Feb 18 '16 at 21:44
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    $\begingroup$ @lemon, careful when you say that air molecules bounce back and forth between the walls ~100 times per second. While it is true that the typical speed of a molecule is ~1km/s, the mean free path is very tiny, of the order of tens of nanometers! In other words, a molecule doesn't keep going in the same direction for very long! The (self) diffusion coefficient is of order cm^2/s. In one second, you can expect a molecule to diffuse something like a centimeter. I agree with you that some smoke will always get in the room, but mostly this is because of flows of air, not just diffusion. $\endgroup$ – Greg P Feb 19 '16 at 1:17
  • $\begingroup$ keep the cigarette on the high part of the window (even few cm inside the room), exhale towards the top as well, all the smoke goes out. Like smoking close to a fireplace. $\endgroup$ – scrx2 Feb 19 '16 at 11:34
  • $\begingroup$ Could you elaborate on smoking close to a fireplace please? ^^ $\endgroup$ – Baconbeastnz Feb 19 '16 at 13:02
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  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.

  1. 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.


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).

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  • $\begingroup$ The statement that the pressures will equilibrate rapidly on a sonic time scale is not accurate. In a flowing fluid, like the natural convective flow that gets set up, viscous turbulent stresses in the gas will support the pressure gradients that develop, particularly horizontal gradients. These pressure gradients drive the fluid flow. The flow will consist primarily of cold flow into the window at the bottom and hot flow out the window at the top. $\endgroup$ – Chet Miller Feb 18 '16 at 23:57
  • $\begingroup$ Indeed I took some shortcuts and assumed static conditions. In my defense, the pressure gradients that do appear are small compared to the iso-density case I was comparing against. Certainly my windows in winter aren't holding back 1-2 pounds per square inch of pressure difference. $\endgroup$ – user10851 Feb 19 '16 at 1:03
  • $\begingroup$ I agree. The pressure gradients driving the natural convection flow, in my judgment, would be pretty low. $\endgroup$ – Chet Miller Feb 19 '16 at 1:48
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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).

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