Question

I'm looking for the physical description of a certain diffusion process, but I don't know how to precisely express it, making the search fruitless. I'd like to have some help formulating, or rather enunciating the problem, whose solution is probably very simple; though, obviously a complete solution is also welcome.

The problem: suppose one starts with a source of some gas, coming out from a hole, which can be approximated by a point, at a constant rate, while its concentration elsewhere starts at zero. How can the concentration at a given point, distinct from the source, be computed after a given amount of time? It can be assumed that this happens within Euclidean 3-dimensional space.

Edit: (10.13.2023) Thanks to everyone who answered, this helped me envisage better where my original concern, which gave rise to this question, fits in the theory. If I understood correctly, all the presented solutions share a common feature: the concentration c(t,r⃗ )
, which is zero everywhere, except at the origin, at t=0
, becomes nonzero everywhere instantly after the play button is pressed. Thus, it turns out this doesn't completely solve my original concern, which was: after I open my hermetically closed window, how long does it take for a mosquito flying nearby to realise I'm willing to feed it? (Of course, I'm trying to think about this in an abstract way.)

Assuming this happens after the first molecule of gas arrives at the mosquito nose, is it enough to compute the time when, at a given point in space, the concentration attains a value somehow corresponding to a molecule of gas? Or does it take more complicated considerations about quantum phenomena and randomness?

Answer 1

To compute the concentration at a given point after a certain time, we can use the diffusion equation which relates the rate of diffusion to the concentration gradient, surface area, and distance traveled by the gas particles.

Step-by-step explanation:

In this problem, we are interested in computing the concentration at a given point in space after a certain amount of time, considering a gas diffusing from a point source in Euclidean 3-dimensional space.

To solve this problem, we need to consider the rate of diffusion and the factors that affect it.

The concentration at a given point, distinct from the source, can be computed using the diffusion equation.

The diffusion equation relates the rate of diffusion to the concentration gradient, surface area, and distance traveled by the gas particles.

By solving this equation, we can determine the concentration at a specific point after a given time.

The time required for diffusion to occur is inversely proportional to the rate of diffusion, so as the rate of diffusion increases, the time required for the gas to disperse and reach a given point decreases.