80
votes
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Is there any way for a gas to pass through a solid metal?
Yes, some gases can diffuse into and through metal. It is the bane of the high-vacuum engineer's life. Hydrogen is the worst because it tends to dissociate into atoms at the surface and the nucleus, a ...
- 7,528
70
votes
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Why is dry soil hydrophobic? Bad gardener paradox
This effect appears like a paradox, as dry soil makes a very bad water conductor. Two effects prevent water from infiltrating:
Air in the soil pores cannot escape: dry soil includes lots of air ...
- 1,118
23
votes
Does diffusion cause the bottle to move to the left?
Yes, the bottle will move.
It will move in such a way, that center of mass of system "bottle and everything inside it" will remain at the same position.
This is because there are no external ...
- 4,232
20
votes
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What does it mean that a substance can be smelled from far away?
You are not missing anything. Rather I think you are placing too much emphasis on the scientific accuracy of something said for effect in a very chatty presentation.
The spoken words almost ...
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16
votes
Why is dry soil hydrophobic? Bad gardener paradox
Dry objects may tend to not break the surface tension, or "skin" of the water. With wet or damp objects the moisture in them will tend to merge with the water's skin.
Just as if you were to spill a ...
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14
votes
Does diffusion cause the bottle to move to the left?
Yes the bottle will move as there are no horizontal forces and hence the center of mass (CoM) of the system must remain fixed. It is a consequence of conservation of momentum. It is hard to really ...
- 4,894
14
votes
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Can a tea light really heat a greenhouse?
"a greenhouse in the winter does not retain heat well"
This is the crucial point: the rate of heat loss will be roughly proportional to the temperature difference between the inside of the ...
- 36.6k
12
votes
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How quickly do farts spread?
The kinetic energy of a particle of mass $m$ and velocity $v$ is related to the temperature by
$$
\frac{1}{2}mv^2~=~\mu kT,
$$
for $\mu$ a dimensionless constant. The velocity $v~=~dx/dt$ means that ...
- 12.9k
11
votes
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Diffusion coefficient for asymmetric (biased) random walk
PREFACE
After several edits, this answer provides a naive explanation of why your approach failed, how to fix it (naive-ish) and a completely different (but right) approach to solve the problem.
Intro
...
- 4,894
11
votes
Is diffusion a nonequlibrium process?
The question is far from being trivial. Basically, one has to distinguish between a macroscopic diffusion as the effect of a starting macroscopic gradient of density or concentration in a condensed ...
10
votes
What is the difference between solutions of the diffusion equation with an imaginary diffusion coefficent and the wave equation's?
It seems like one can transform the diffusion equation to an equation that can replace the wave equation since the solutions are the same.
I think replacing a real constant to an imaginary constant ...
- 5,270
10
votes
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The derivation of the advection-diffusion equation uses $\nabla\cdot(c\vec{v})=(\vec{v}\cdot\nabla)c$. Why doesn't the order of the derivative matter?
If $c$ and $\vec v$ is an arbitrary pair of functions, then the identity you wrote is false; instead it must read
$$
\nabla \cdot (c\vec v) = (\vec v\cdot \nabla) c + c (\nabla \cdot \vec v),
$$
which ...
- 135k
10
votes
Physical meaning of potential in heat equation
The heat equation, as you've written it, models the flow of energy via thermal conduction (heat) through some region with well defined boundary conditions. You have yet to provide the specifics of the ...
- 2,770
10
votes
Is there any way for a gas to pass through a solid metal?
Not really, but sort of.
Helium atoms do not form molecules, and the atoms are small. They fit between the spaces in iron, and can diffuse around inside.
This is not a fast process. That is to say, it ...
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10
votes
Why doesn't the Diffusion Equation with an initial condition of $u = x$ lead to an even distribution?
The key is the boundary conditions.
If you do have $u(0,t)=0$ and $u(L,t)=L$ (ignoring the incorrect units here) then indeed $u(x,t)=x$ solves this differential equation with the boundary conditions, ...
- 58.5k
10
votes
Is the Navier-Stokes equation valid in $d=2$ spatial dimensions?
There has actually been a fair amount of activity in this area recently, see, for example, this set of lecture notes.
I would argue that the answer to your question is, "yes", if properly ...
- 18.9k
9
votes
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What is the difference between solutions of the diffusion equation with an imaginary diffusion coefficent and the wave equation's?
Both Schrödinger and Wave Equation have plane wave solutions, that's right. The difference is the dispersion relation, which is quadratic for the Schrödinger equation and linear for the wave equation. ...
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votes
Why is dry soil hydrophobic? Bad gardener paradox
Diffusion and adhesion are different phenomena. Diffusion happens due to a concentration gradient, and this is how the water is slowly absorbed by the soil. This happens faster in dry soil per unit ...
- 1,132
9
votes
Is there any way for a gas to pass through a solid metal?
Permeation of atomic hydrogen through metal has been performed in a study(ref.1):
[...] atomic hydrogen is supplied to the metal surface by reaction
with an acid, by electrolysis, or by ionization, ...
- 1,270
9
votes
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Why doesn't the Diffusion Equation with an initial condition of $u = x$ lead to an even distribution?
To complement BioPhysicist's nice discussion of the mathematical resolution to your problem, I will revise my answer to provide a physical resolution.
The diffusion equation can be thought of in two ...
- 71.4k
9
votes
Inverting the heat equation
You are right, this is a classically ill-posed question, and here is why.
If you are measuring the temperature $k$ along the wire at position $x$ and time $t$, there are any number of different ...
- 97.8k
9
votes
Accepted
How are far from equilibrium systems studied analytically?
If the answer you are seeking was known, there would surely be a lot of Nobel Prizes associated with it! But I can make some relatively useful comments.
When you say: "It would be interesting to ...
- 1,748
8
votes
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What is the proper way to model diffusion in inhomogeneous media (Fokker-Planck or Fick's law) and why?
It is a sticky question, and as van Kampen puts it, " no universal form of the diffusion equation exists, but each system has to be studied individually." https://link.springer.com/article/10.1007/...
- 666
7
votes
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Why is the mean free path divided by $\sqrt{2}$?
The calculation done above assumes that one molecule is moving, while all of the others are standing still. This is a much easier situation to handle; you basically just count up how much volume is ...
- 105k
7
votes
Accepted
Physical meaning of potential in heat equation
As discussed here under "Incorporating lateral heat transfer" (disclaimer: my site), if you're considering a 1-D transient heat transfer problem as suggested by the variables $x$ and $t$, then the ...
- 28k
7
votes
Why diffusion happens?
In a nutshell: probability! (In the jargon of thermodynamics we say the system will move from a state of low entropy $S_1$, to one of higher entropy $S_2$)
Imagine a partitioned container. To the left ...
- 35.5k
6
votes
What is the difference between solutions of the diffusion equation with an imaginary diffusion coefficent and the wave equation's?
I don't think your understanding is fundamentally flawed. There are many ways to see the resemblance between the heat/diffusion equation and Schrödinger equation, one of which being the stochastic ...
- 708
5
votes
Do particle velocities in liquid follow the Maxwell-Boltzmann velocity distribution?
The velocity distribution in a liquid does follow MB. This is routinelly seen in molecular dynamics (MD) simulations in liquids such as water (and in more complex systems!!, proteins in liquids, ...
5
votes
Accepted
Why do lighter atoms and molecules diffuse upwards?
It's a statistical walk, but its biased. Consider the energy in the system containing all of the molecules. One factor is the potential energy due to gravity for all molecules, and the other factor ...
- 51.7k
5
votes
Accepted
Characteristic time for heat diffusion
Since the heat equation for 1D time-dependent conduction is $$k\frac{\partial^2 T}{\partial x^2}=\rho c\frac{\partial T}{\partial t},$$
the hand-wavy way to derive the characteristic time is to ...
- 28k
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