# Ideal gas with sticky wall

A collection of $N$ point particles is held inside a cubic box of side $L$. The walls of the box are sticky. This means that when a particle collides with a wall, it stays attached for a short period of time $\tau$, and then it is released. When it is released, it goes out with the same velocity that it would have if the collision had been elastic in the first place.

What is the spatial distribution of particles? For simplicity, you can assume that only one of the walls is sticky, and the rest are normal. In this case there is only one gradient to be concerned with.

If there were no sticky wall, the distribution would be uniform. I have the intuition that the presence of the sticky wall will increase the concentration of particles near the wall. Think of ergodicity: since the particles spend more time near the wall ($\tau$ extra units), that means that the probability of finding particles near the wall should also be larger. But I am not sure how to prove this, nor how to find the quantitive distribution.

Added: What is the velocity distribution of particles in the bulk of the box? And in the wall?

• Show effort and identify a specific aspect of the problem where you are stuck. – DanielSank Oct 2 '17 at 18:14
• @DanielSank I added some comments. This is not a homework question. – becko Oct 2 '17 at 18:25
• The name of the "homework policy" is unfortunately very misleading. Whether or not a question is literally homework is completely irrelevant to the fact that questions like this one should show effort and ask something specific. See this meta post. – DanielSank Oct 2 '17 at 18:30
• @DanielSank I honestly do not see why the question is trivial. – becko Oct 2 '17 at 18:34
• Nobody said it was trivial. – DanielSank Oct 2 '17 at 18:34

## 2 Answers

Consider the set of particles with a velocity $v_i$ towards the "sticky" wall.

Then replace the sticky wall with a normal perfectly elastic wall, but locate it a distance ${(v_1 \times \tau)/}{2}$ behind the location of the sticky wall. Note that the location of this wall is a function of the velocity under consideration

This new, enlarged vessel will mimic exactly the behaviour of the sticky walled one, at least for particles with normal velocity $v_i$.

The sticky box has volume $L^3$; the enhanced box has volume $L^2\times (L+{(v_1 \times \tau)/}{2})$

Can you see how the distribution of particles with different velocities will be affected by the sticky wall?

Edit:

All the particles will be uniformly distributed spatially.

However, the energy distribution will be skewed.

The presence of free-flying particles with a particular energy/velocity will be diminished. A fraction of them can be visualized as being in the extra space; the fraction in the real, $L^3$ space is just the ratio of that real volume to the volume including the moved wall. And that ratio changes for each different velocity.

Consider: a fast, high energy particle will spend $\tau$ stuck to the wall, perhaps $0.001 \tau$ on a round trip down the box and back, another $\tau$ stuck, and so on. Almost never part of the free distribution.

But a slow particle might spend $\tau$ stuck in the wall; then spend $1000 \tau$ driftinnnng down the length and back, another $\tau$ stuck... Almost always part of the free population.

• I was thinking about this exact way of analyzing the problem. Then I realized that a slower particle would be "returned to the gas" relatively sooner than a faster particle (in a situation where the MFP is long compared to $v\times \tau$, a cold particle would stay longer). Does this cause a cooling of the gas? – Floris Oct 2 '17 at 17:53
• The box has effective volumes that are different for each particle, depending on the particle's velocity. I don't see how that's helfpful, sorry. – becko Oct 2 '17 at 18:22
• I think that @DJohnM has centered qualitatively the point: if you fix a population with one velocity you obtain that particles that lives in the extra volume are the one glue to the wall. This is true for all velocity, so you have that in the true box the density is still uniform, and there are some particles stick on the wall. – Annibale Oct 2 '17 at 18:39
• @becko : if you want to find what is the fraction of particles stick on the wall you have to calculate the velocity distribution (that is unrelated to the glue on the wall in your formulation), and so it is the Maxwell distribution for an ideal gas, and integrate it multiplied for the fraction of particles stick on the wall with fixed velocity. – Annibale Oct 2 '17 at 18:43
• @AlessandroCandido I see. So that means that the distribution in the bulk of the box is still uniform. However, the density of particles stuck in the wall is a Dirac Delta, correct? In fact, using the extended box argument, the probability that a particle with velocity $v_i$ is stuck in the wall is $L^2 \times (v_i \tau)/2$ – becko Oct 2 '17 at 18:43

I believe the question is similar to phase equilibrium. Suppose you consider equilibrium between a liquid and its vapor. When a vapor molecule "hits" the liquid it does not rebound, but becomes part of the liquid and spends some time inside the liquid, before by pure chance it is ejected out again. Of course the time spent by a molecule inside the liquid is not fixed but a stochastic variable.

Larger stick-time $\tau$ is equivalent to the situation of low vapor pressure liquid. In other words, increasing $\tau$ will result in smaller fraction of molecules freely moving inside the box and the rest of them stuck to the walls when system reaches steady state. The molecule number concentration will be a step function, with change occurring at the sticky wall.

In your question the molecule is ejected from the sticky wall as if it were reflected (after time delay $\tau$). In the case of liquid-vapor system, molecule escaping the liquid has probably no memory of its velocity when it was last captured by the liquid. But if statistically the velocity distribution remains the same for molecules ejected from the sticky wall and for molecules that hit the sticky wall, the resulting velocity distribution inside the box (excluding walls) must be identical to that in the absence of sticky-walls.