# Why does it take so long for a gas to reach a thermal equilibrium?

Imagine a 1m cube of air in an insulated container, where you have heated the air in one half but not in the other. Intuitively, given molecules are moving so fast, you would expect the energetic particles from the hot side to distribute themselves throughout the box within milliseconds, equalising the temperature. Why does this instead take much longer?

I thought of this question while waiting for a warm room to cool down after opening a window. Clearly my mental model of temperature is wrong because it seems to me that heat should move through a gas way faster than it actually does.

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• In the case of a warm room cooling down, there is a substantial amount of heat energy in the walls, the floor, the furnishings, etc. that will continue to warm the room even if all the heat in the air was lost. – Michael Seifert May 4 at 11:59
• @MichaelSeifert: I understand what you're saying, and I've made the same point in my answer. Heat isn't a state function, though, and it's not correct to talk about "heat energy in the walls" or "heat in the air". Sorry for being nitpicky, but this is a common misconception in thermodynamics, and this question is on HNQ. – Eric Duminil May 4 at 13:37
• @EricDuminil: Fair enough; "thermal energy" is better terminology. I blame lack of caffeine and an early-morning answer. – Michael Seifert May 4 at 15:18

The mean-free path of a nitrogen molecule in air at STP is about 60 nm, so the molecules may travel quite quickly, but they do not go very far before the run into another molecule and get stopped.

• Ah I see. I was imagining the air to be 'almost ideal', i.e. particles bouncing around in the container only occasionally hitting each other. Does that mean that a very low density gas behaves the way I described, where it hits thermal equilibrium almost instantly? – sebzim4500 May 3 at 17:35
• @sebzim4500 that's a bit tricky to answer, because in gases of such low density the very concept of temperature becomes problematic. You can't really measure it in a thermodynamic way at all anymore, but instead need to directly measure the velocities of individual particles (can be done with Doppler spectroscopy). In outer space, the thin gases actually tend to be so hot (!) that the particles are ionised (plasma), which again prevents quick long-scale thermalisation because the particles are forced on gyration paths. – leftaroundabout May 4 at 9:01
• But when they hit another molecule, they transfer some of their energy. It's how fast energy is moving, not individual molecules, that matters. – Acccumulation May 4 at 17:43
• @Acccumulation well, they do hold in a collisionless-ideal gas that you somehow magicked to have Maxwell-Boltzmann distribution all along! But in reality, too-thin gases indeed don't have this. Space plasmas tend to follow a kappa distribution rather than Maxwell-Boltzmann (but they're anyway much more complicated than ideal gases, because of long-scale MHD effects). — Pragmatically speaking, it makes more sense to define an ideal gas as one where the mean free path is much longer than the mean particle distance, but still much smaller than the total system. – leftaroundabout May 4 at 20:18
• @Acccumulation makes a point, but the energy actually moves according to the heat equation, and therefore proportional to the gradient of temperature; the end result is that the energy spreads proportional to sqrt(t). – NoLongerBreathedIn May 5 at 1:55

I thought of this question while waiting for a warm room to cool down after opening a window.

A warm room is not just composed of warm air. The whole building has a huge thermal mass (basically, the heat capacity of walls, furniture, basement, ...) in comparison to the air inside it.

Your building is not perfectly sealed. Even when the windows are closed, the air change rate is close to $$\frac{1}{\mathrm{h}}$$. Which means that after 1 hour, the whole indoor air volume has been replaced by outside air. It does not mean that the indoor temperature is equal to the outside temperature after an hour, though.

If you open the windows, you might achieve an air change rate of $$\approx \frac{20}{\mathrm{h}}$$. Which means that after 3 minutes, you have replaced all the indoor air with fresh air. If you close the windows after 3 minutes, the indoor air temperature will almost revert back to the previous temperature, because the walls act as a thermal reservoir.

If walls have different temperatures (e.g. due to solar irradiance), convection will transfer heat much more efficiently than diffusion (the process you were asking for).

By the way, the best way to cool down a room in summer is:

• to avoid getting solar gains during the day, with external shading
• closing the windows during the day
• opening them wide all night long, in order to cool down the walls
• removing the external shading, if possible, during the night, in order to allow radiative cooling
• Re your hints about cooling down a room in summer I've had quite a few arguments from people who think "an open window on the shady side of the house lets in cool air". I ask them to imagine smoke travelling around the house in a matter of seconds but they just don't get it. – Arthur Kalliokoski May 4 at 9:51
• @ArthurKalliokoski: Same here. But I don't argue with them any more. I just enjoy the fact that my flat is 5 K cooler than theirs, without air conditioning. – Eric Duminil May 4 at 9:54
• Downvoter : constructive criticism is welcome! – Eric Duminil May 5 at 8:18

In addition to what @mike stone said, keep in mind that heat transfer from the higher temperature side, say left side of the container, to the lower temperature side, say right side of the container is the result of a progression of the transfer of kinetic energy between the more energetic molecules of the left side to the less energetic molecules on the right side by direct collisions, beginning at the interface.

For a given temperature, the kinetic energies of individual molecules are distributed about an average. Consequently, some molecules of the higher temperature portion of the gas may have lower kinetic energy than some molecules of the lower temperature portion of the gas. So collisions between such molecules can actually transfer energy from the cold side to hot side, slowing down the rate of heat transfer from hot to cold, particularly the lower the initial temperature difference.

Hope this helps.

• Not to mention some of the kinetic energy transferred to the colder side can transfer back to the warm side (since parts of the warmer side now have "colder" parts of them having transferred their energy to the colder side) – Michael May 4 at 16:22

Imagine that you're in a large room - say, 50 m x 50 m - which is so filled with people that on average you only travel about 1 meter before colliding with somebody else. Everyone on the left side of the room has a red hat and everyone on the right side of the room has a blue hat. The only way people move is by frantically sprinting into each other and bouncing off in random directions - that is, there is no coherent motion or intent behind their individual trajectories. How long will it take until the different colored hats are more or less uniformly distributed throughout the room?

The key question is not how fast any individual is moving at any one time, but rather how rapidly the group of red hats will spread out under the influence of chaotic, random jostling. The same is true of the atoms in your question. This kind of motion is called diffusion.

I should also add that your question body asks a somewhat different thing than the question title. When the gas comes to thermodynamic equilibrium, it isn't because the energetic molecules diffuse throughout the system, but rather that their energy does. Energy is transferred between particles in every collision, so even if a particular collection of molecules remains localized in one area for a while, their energy can spread throughout the gas.

• This analogy concerns rather two different gases which are at the same temperature. – leftaroundabout May 4 at 8:53
• @leftaroundabout Yes, but the body of the OP described a misunderstanding of particle transport. I've added a few sentences clarifying the distinction between that and the question title. – J. Murray May 4 at 13:41

The other answers have explained why the warm air in your room doesn't diffuse out the window as quickly as you would like.

A strategy that overcomes this issue is to open multiple windows and/or doors on different sides of the room, to allow a draft or breeze, such that there is a bulk transfer of cooler air in one window and of warmer air out another. This is a much faster and more thorough process. This strategy has played a major role in various architectural styles, such as that of the dogtrot houses that used to be common throughout the Southeastern United States.

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