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I have a curiosity to know how long it will take for a water molecule to move a certain distance in pure water. I can put it simply as how much time for a water molecule to travel the whole length of a typical glass of water if no collective motion is present (therefore very still water with no convective currents).


To simplify things, this is the model:

Suppose I have a completely still and ideally insulated glass of water where the temperature in each part of the fluid is exactly the same (therefore no convective motion).     In addition the top of the glass container is closed and no water air interface is present to avoid evaporation that if present changes the dynamic.    In addition the glass of water is big enough so that the dynamic of the water in the core is not influenced by the the limited range of motion of the water near the borders of the glass.

Since it is a liquid, the water molecules move around each others until the original water molecule is at distance r from where it started(A). In absence of gravity the movement is equally probable in all directions, while with gravity I suppose it may be more likely a layer layout of the water molecules making horizontal movements more likely (see elipse shape of the trace).

The questions are (no need to reply to both):
1) How much time to be at distance r in both situations (gravity or not)? (it will depend on temperature, so let's say at room temperature)
2) How much time roughly will it take for a water molecule to reach reach almost the top of a typical glass of water starting from the bottom?    (in absence of convective motion and evaporation)

enter image description here

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    $\begingroup$ Just keep in mind that the gravitational influence on a single water molecule must be negligible as compared to thermal motion. $\endgroup$ – M. Farooq Jun 13 at 18:40
  • $\begingroup$ I am aware that gravity force on the single isolated molecule is negligible compared to its thermal motion but if applied on many molecules it may cause water molecules to form layers sliding on top of each others and in an enough long distance to reach(like bottom to top of a glass) this may be relevant $\endgroup$ – C.X.F. Jun 13 at 18:51
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    $\begingroup$ Eukaryotic cells have transport systems because they can't wait for materials to get from one to the other side by diffusion. From a certain size up, animals have circulatory systems because oxygen would not diffuse quickly enough to reach the cells far away from the surface. The time of random walks scales with distance square. This calculator says 60 days: physiologyweb.com/calculators/diffusion_time_calculator.html $\endgroup$ – Karsten Theis Jun 13 at 18:53
  • $\begingroup$ I was expecting something that is many hours but actually seems days. $\endgroup$ – C.X.F. Jun 13 at 19:03
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I am posting here the answer since it has been quickly found thanks to the useful comments. To give an estimate the best way is to look at the self-diffusion coefficient.

This is defined as the diffusion coefficient when the chemical potential gradient equals zero, or more simply, it is the diffusion coefficient in a medium of the same molecules.

According to Ref.1, the self-diffusion coefficient of water at $\pu{25^\circ C}$ is $D = \pu{2.3 \times 10^{−9} m^2 s^{−1}}$.
The diffusion time can be calculated as: $$t \approx \frac{x^2}{2D} $$

As pointed out by @Karsten Theis, you can use the calculator and insert the water diffusion coefficient D.

For example, if the distance x is $\pu{8cm}$ like a typical glass of water and we are at $\pu{25 ^\circ C}$, then $t \approx \pu{16 days}$! Instead if the temperature is $\pu{90 ^\circ C}$ then it's around $\pu{5 days}$.

I have drawn a graph that shows how much time to reach the top of the glass depending on temperature:

Diffusion rate vs Temperature

References:

  1. Manfred Holz, Stefan R. Heila, Antonio Sacco, "Temperature-dependent self-diffusion coefficients of water and six selected molecular liquids for calibration in accurate $\ce{^1H}$ NMRPFG measurements," Phys. Chem. Chem. Phys. 2000, 2(20), 4740-4742 (https://doi.org/10.1039/B005319H).
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  • $\begingroup$ It's actually a bit longer because this is the time to go in any direction, but you have to go up. Also, it is a bit more complicated because any point on the surface will do, and they have slightly different distances from a given point on the bottom of the glass. Finally, it would be great to put error bars because this is just the average in a stochastic process, as anyone going on a random search to find their wedding band on a sandy beach will know. $\endgroup$ – Karsten Theis Jun 14 at 17:20
  • $\begingroup$ Also, random walks, varying path lengths,...not exactly a precise answer, more like an approximate statistical expected average (see, for example, the illustration here and also next page mit.edu/~kardar/teaching/projects/chemotaxis(AndreaSchmidt)/… ). $\endgroup$ – AJKOER Jun 14 at 17:38
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    $\begingroup$ Yes, I know it's not a precise answer but a very rough estimate just to have the sense of the timescale. However if someone wants to write a detailed answer this is very welcomed. $\endgroup$ – C.X.F. Jun 14 at 18:46

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