Why is energy flux not equalized across an effusive hole between two gases?

Question

Consider two gases in a thermally insulated container separated by an insulated wall with a small hole in it (smaller than the mean free path).

We can claim that, in equilibrium, the net particle flux through this hole is zero and so $$ \int{d\Phi} \propto n\int{vf(v)dv} \propto nv_{th} $$ is equalized for each species, where $f(v)$ is the distribution of particle speeds, $n$ is the number density, and $v_{th}^2 = 2k_BT/m$.

Since for an ideal gas $n \propto p/T$, this implies that $$ \frac{p_1}{\sqrt T_1} = \frac{p_2}{\sqrt T_2} $$ i.e. $p/\sqrt T$ is equalized across the two species. You can look this up elsewhere online; I think it's called the Knudsen effect.

Here's the problem: we started from the claim that net particle flux must be zero. This is fair enough; if this weren't the case then the two species wouldn't be in equilibrium, right?

Ok, so what if we claim that net energy flux must be zero? An ideal gas has $U = U(T)$; a net energy flux would cause the temperature of the gases to change - meaning they wouldn't be in equilibrium! So let's try it. We claim as before that $$ \int{\frac{1}{2}mv^2 d\Phi} \propto n\int{v^3f(v)dv} \propto nv_{th}^3 $$ is equalized across the gases. Following the same steps as before, this leads to $$ p_1 \sqrt T_1 = p_2 \sqrt T_2 . $$ Divding this by our previous expression we find $$ T_1 = T_2 $$ if both net energy flux and particle flux are zero. Surely this can't be right; this is the same result for when the hole is large - it's like the size of the hole doesn't matter at all and effusion isn't relevant!

So where have I gone wrong?

• Is the assumption that the net energy flux is zero wrong? If so, why?
• In any case, why is it correct to assume that the net particle flux is zero?

Help with this is very much appreciated, thank you.

Answer 1

Those are two different concepts of equilibrium; and they occur in different circumstances.

The first concept is a dynamic equilibrium between two chambers with gas in Knudsen regime which are assumed to be maintained at two different temperatures $T_1,T_2$, or at least they are close to these two temperatures for some time interval of observation. In the Knudsen regime where the gas is rarified enough, there is no effective pressure equalization mechanism between the chambers, so in general $p_1 \neq p_2$, and a dynamic equilibrium with constant numbers of particles in both chambers may be reached without equalizing temperatures, pressures, or particle density. So this is not a thermodynamic equilibrium: while a steady state in both chambers can be maintained where they have different temperatures, in general there will be non-zero energy fluxes present, and thus net energy transfer from one chamber to the other. This transfer of energy is maintained by the heat reservoirs maintaining the different chamber temperatures.

The second concept based on zero net energy flux is requiring something that won't happen in above scenario. It is one of the characteristics of thermodynamic equilibrium: no net macroscopic energy fluxes in the system, thus no net energy transfer happens between the two chambers. Together with other requirements of thermodynamic equilibrium (such as constant number of particles already used above) this leads to the only possibility, equality of temperatures and particle densities, and thus also of pressures, which is correct result for a gas in state of thermodynamic equilibrium.

Answer 2

a net energy flux would cause the temperature of the gases to change - meaning they wouldn't be in equilibrium!

I disagree with this logic. By specifying an insulated wall, you've precluded temperature equilibration (and by specifying a very small hole, you've precluded pressure equilibration). So you can't then assume that the associated gradients will independently even out. The generalized driving force of $\Delta T$ and $\Delta P$ may exist, but you're not allowing the corresponding generalized displacements of entropy or volume transfer.

In contrast, you can reasonably propose that a constant amount of gas on each side constitutes equilibrium, as you are allowing single particles through the hole and it's possible for the net particle flux to be zero.