csc@watmath.UUCP (Computer Sci Club) (09/20/83)
Jim Stekas original submission on trapezoidal holes did not reach me.
I will endevour to use a classical treatment (the second
law holds even in ideal models!) to show that trapezoidal holes (or those of
any geometry ) pass equal numbers of molecules in both directions. I hope
I do not repeat Mr. Stekas argument.
Consider the usual ideal gas model. In any given volume we expect (on
average) to have an equal density of particles (could be very low), with the
particles moving in random directions (i.e. no direction preferred on average).
Now consider the situation near any wall (planar and perfectly reflecting).
We note that density of particles and distribution of directions of motion
is exactly the same. The particles not coming from behind the wall are
replaced by those bouncing off the wall.
. | .
. | .
r | v
. | .
.|.
. | .
. | .
. | .
. | .
|
wall
real space virtual space
The easiest way to see this is to imagine a "virtual space" behind
the wall, with "virtual particles" which are the mirror images of the
real particles. Note above that the reflection of a real particle can be
seen as the real particle leaving and a virtual particle entering. Now
the virtual space has the same density of particles as the real space
and the same random distribution of directions (naturally as it is just
the mirror image of real space). Therefore a volume of real space near
the wall has just as many "virtual" particles approaching from the right
as real ones from the left. Volumes of space near walls are not different
(as far as density and directional distribution of particles) from volumes
not near walls. Note this argument does not depend on such things as mean
free path, interparticle collision, etc.
Now consider the hole. Particles in this local are moving randomly with
NO PREFERRED DIRECTION. THE PRESENCE OR ABSENCE OF WALLS DOES NOT EFFECT THIS!
Clearly just as many particles will pass one way as another.
As for trajectories. Consider the point at the center of the narrow end
of the funnel. If we "look" toward the one box we have an entire unobstructed
view through 180 degrees. If we look toward the other box we see directly into
the box through part of the arc. Through the rest of the arc we see into the
box indirectly (by one or more "reflections" off the sides of the funnel).
We see into the box THROUGH 180 DEGREES! There are just as many trajectories
going each way. (or use symmetry. Any trajectory from box 1 to box 2 is also
a trajectory from box 2 to box 1. Assuming random particle motion there is
no reason to expect a preferred direction down any trajectory).
The pressure of a fluid is constant throughout any container, Pascals Law.
With equal pressures on both sides we will certainly have no net flow of gas.
William Hughes
PS The above was meant to try to give intuitive insight, not formal proof.
Hence such nonsense as looking along and counting trajectories.