Van der Waals Equation: How Surface is Considered

[Chestermiller]

If I may, I'd like to bump up a question from my earlier post on this thread: How do we define the pressure in the middle of a volume of non-ideal (van der Waals) gas? Is it the pressure that would exist on an imaginary surface at that location? It there a definition that doesn't refer to such a surface?

That is the way I envision it. The normal force per unit area on the imaginary surface. Just like the tension in the middle of a string.

 

[Swamp Thing]

In that case one would have to assume a surface that screens nearby molecules against forces from molecules on the other side of the surface. Then the molecules hitting the surface would have reduced momentum due to attraction of molecules further away from the surface -- hence reduced pressure a la van der Waals.

In that sense the virial expansion picture (bulk effect) would be equivalent to the surface effect picture.

 

[Chestermiller]

In that case one would have to assume a surface that screens nearby molecules against forces from molecules on the other side of the surface. Then the molecules hitting the surface would have reduced momentum due to attraction of molecules further away from the surface -- hence reduced pressure a la van der Waals.

In that sense the virial expansion picture (bulk effect) would be equivalent to the surface effect picture.

It's an imaginary surface, not a real one. So it would do nothing. The pressure is the force per unit area exerted on the molecules on one side of a massless entity by the molecules on the other side of the massless entity. Are you not able to visualize a surface (massless) in space?

 

[Swamp Thing]

Are you not able to visualize a surface (massless) in space?

When I imagine a non-shielding surface, it seems to me that such a model might not capture the van der Waals deviation from ideal gas behavior. But let me have another think about this..

 

[Chestermiller]

When I imagine a non-shielding surface, it seems to me that such a model might not capture the van der Waals deviation from ideal gas behavior. But let me have another think about this..

In addition to the repulsive forces exerted by molecules from one side of the conceptual plane surface on those situated on the other side of the surface, in the case of a gas, there is actually flux of molecules (and accompanying momentum) from one side of the conceptual surface to the other other side. For an ideal gas, the latter is the only contributor to what we call the pressure.

The flux of molecules from one side of the conceptual surface to the other, in conjunction with the flux of molecules from the other side across in the opposite direction is equivalent to molecules at an actual rigid surface colliding with the rigid surface and bouncing back (i.e., exerting a force on the rigid surface).

 

[hutchphd]

So the surface analysis, where we think of molecules being pulled back into the gas (leaving less momentum to create pressure on the wall) is just one of those nice heuristic pictures that happen to give useful results? In some sense similar to the popular "waves between two ships" picture of the Casimir force...

Another question: what do we actually mean by pressure of gas in the middle of a container? Is it the momentum per unit time per unit area due to collisions on a hypothetical test surface? In that case, the bulk picture can be linked with the surface picture by introducing a hypothetical test surface which doesn't interact with the molecules, thus leading to a situation like the actual container walls. Maybe... or maybe there is a definition of internal bulk pressure that doesn't involve a special surface?

I believe what is bothering you is the subtlety of what actually happens at a real wall. As @Chestermiller says the internal pressure is defined according to real momentum flux across an imaginary surface deep in the bulk. This is unambiguous and involves only the usual stuff.
What happens at a real surface is complicated and depends upon the exact nature of the material. There could still be van der Waals forces and they might be large enough to adsorb the gas! So there will be a surface layer where the density of the the gas will be affected (either greater or less) . But this layer is very thin because the VDW forces are short range. And so the net contribution to the net free energy is de minimus and the net pressure on a real wall corresponds to the internal pressure. It just gets a little messy right at the surface .

 

[vanhees71]

In addition to the repulsive forces exerted by molecules from one side of the conceptual plane surface on those situated on the other side of the surface, in the case of a gas, there is actually flux of molecules (and accompanying momentum) from one side of the conceptual surface to the other other side. For an ideal gas, the latter is the only contributor to what we call the pressure.

The flux of molecules from one side of the conceptual surface to the other, in conjunction with the flux of molecules from the other side across in the opposite direction is equivalent to molecules at an actual rigid surface colliding with the rigid surface and bouncing back (i.e., exerting a force on the rigid surface).

In addition a real gas also implies dissipation, i.e., shear and bulk viscosity, which are nothing else than contributions to the stress tensor (which also includes the perfect-fluid pressure).

It's generally clear that you get the meaning of such quantities from the balance equations for conservation laws for energy, momentum, and angular momentum, which can be formulated for a finite volume, involving corresponding volume and surface integrals, as well as local differential forms. To each such quantity from these considerations you get a density and a current and an equation of continuity. These derivations lead to the equations of motion like the Euler equation (perfect fluid) or the Navier Stokes equation.

All this has also to be complemented by thermodynamical considerations, i.e., an equation of state for the fluid, which brings particularly entropy into the game, for which in the non-perfect-fluid case you have of course no conservation law but the increase of entropy due to dissipation leading to irreversibility.

Another point of view is more microscopic: You an derive all these fluid-dynamical equations from the Boltzmann(-Uehling-Uhlenbeck) equations in a systematic expansion of the phase-space distribution function in terms of moments leading to a hierarchy of fluid equations starting from Euler's perfect fluid equations to the Navier-Stokes's equation and higher-order dissipative-fluid dynamics equations (which are particularly needed in realtivistic fluid dynamics, where the first-order Navier-Stokes equation leads to problems with causality), leading to a microscopic definition of transport coefficients via statistics ("coarse graining" to get effective descriptions for collective macroscopic observables by averaging over microscopic degrees of freedom).

 

[Chestermiller]

In addition a real gas also implies dissipation, i.e., shear and bulk viscosity, which are nothing else than contributions to the stress tensor (which also includes the perfect-fluid pressure).

The VDW equation applies to thermodynamic equilibrium situations, where dissipation is not present.

 

[vanhees71]

Sure, I was referring to the question about the meaning of pressure in the interior of a ideal or real gas.

 

[Swamp Thing]

To refresh my memory, I reviewed this... https://en.wikipedia.org/wiki/Van_der_Waals_equation#Conventional_derivation .

In the section "Conventional Derivation" it says that van der Waals assumed:

1. An attractive force between the particles

2. The inter-particle force is fairly short range; the majority of particles are surrounded by particles on all sides, hence no net force.

3. No attractive force between particles and wall (which I interpret as a purely repulsive force that is much shorter in range than the inter-molecular attractive force, but grows very steeply at "ultrashort" range).

Due to the above, particles "within range" of the wall have a net force pulling them away from the wall, due to the unbalanced pull from particles further inside the container, since there are no particles in the outer side of the wall's surface (and the wall itself exerts no attractive force).

These assumptions lead to a reduction in pressure a/Vm^2:

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QUESTION:-
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Now to get back to my question about pressure in the middle of the container, let's assume that the above container itself is surrounded by gas at the same temperature and pressure, and that the wall is extremely thin -- so that molecules on opposite sides of the wall are close enough to potentially attract each other.

But we can split this into two different thought experiments :

(a) The wall, although of nearly zero thickness, screens the intermolecular forces trying to act through it.

versus...

(b) The wall allows intermolecular forces to act through it

In both cases , the wall bounces the interior molecules back inwards and the exterior molecules back outwards.

It seems to me that model (a) corresponds closely to the three assumptions cited above, and will correctly reproduce the negative correction term in pressure. On the other hand, in model (b) a molecule very close to the wall is still effectively surrounded by other molecules pulling it every which way (including some acting through the wall), hence no inward force, hence no van der Waals correction to pressure.

If the above paragraph is valid, then a zero-thickness but screening surface in the middle of the bulk of a gas would be the correct imaginary surface to "measure" or define internal pressure.

This might seem like nit picking, but the nature of the test surface, i.e. (a) or (b), seems to be a key to defining internal bulk pressure correctly.

It's not very important whether the test surface actually bounces molecules back physically, or merely "watches" passively and totals up the momentum from all the molecules passing through it in one direction -- but it seems to be important to define whether intermolecular forces can act through it.

And I'm suspecting that model (a) is hidden somewhere implicitly in the virial expansion approach (although I must admit I know nothing about that derivation).

Edit:
Or... we can just assume a small imaginary inner container that has a vacuum inside it, and use that as a test surface for gas in the outer container