Re: [Phys-L] Figuring Physics solution Jan 2018

[John Denker <jsd@av8n.com>]

On 01/23/2018 05:04 AM, Robert Cohen wrote:

> I'll start with the simple situation of a gas in a one-dimensional
> container (so the particles only travel along that one dimension), 
> made up of particles of identical speeds under no attraction and
> totally elastic collisions.   That way the speeds remain identical.
>
> I then add a small attraction so that if I remove the "top" of the
> container, the fluid doesn't necessarily spit out the top.  This can 
> then be considered a liquid.  The speeds won't always be identical at
> any given time but on average they are.
>
> At any given time, only the particle at the top is exposed to the
> vacuum above and thus only that particle can "leave" the liquid state
> and fly away as a free particle.  That particular particle need not
> be the "fastest" of them all, does it?  Does it even need to be going
> faster than the "average" of them in order to leave?
>
> I'd argue it doesn't.  It just needs to be going fast enough to
> overcome the attraction.  And, while undergoing the escape process, 
> it slows down AND the particle it was attracted to slows down.  This
> leads to a cooling of both, independent of whether it was initially
> going faster than those it "left behind" or not.
>
> In other words, the real physics is on the cooling that results from
> the "breaking" of the bonds (so to speak).  Focusing on the speed of
> those left behind (what JD calls "cancelling the sixes") allows
> students to ignore the real physics.

I agree with all that.

Let me tell you /why/ I agree.  It gets back to something
they teach in kindergarten:  Check the work.

One useful check is to see whether the model given above
is a dead end, or whether it lays a good foundation for
thinking about things in 3D.  This is not a check I expect
students to make in the introductory course, but it is
the sort of check I would make before teaching the 1D
model.

Let's split the difference and look at 2D.  Here is a
snapshot of some molecules:

             A         B        C      D            E 

1            x                                      x
2          x  x        x        x                 x  x
3        x  x  x     x  x        x     x        x  x  x
4      x  x  x  x  x  x  x  x  x  x  x  x  x  x  x  x  x
5    x  x  x  x  x  x  x  x  x  x  x  x  x  x  x  x  x  x
6  x  x  x  x  x  x  x  x  x  x  x  x  x  x  x  x  x  x  x

The particle most likely to evaporate next is at location
C2.  That's the one with the fewest bonds.  At this site,
it's a race to see what happens next:
 -- C2 might form a second bond, or
 -- C2 might break the bond it has, and fly off into
  the vapor.

As RC pointed out, being /at the surface/ is a key
prerequisite for being eligible to leave the liquid.

In 1D being at the top of the stack is an absolute
requirement, but in 2D the idea of being "at" the
surface is slightly more subtle.  C2 is not the only
particle at the surface, and indeed others such as
A1 and E1 are farther from the center of the bulk
liquid, but they each have two bonds.  Being at the
surface guarantees having "somewhat" fewer bonds than
in the bulk, but within the surface layer there are
further distinctions to be made.

Overall, I conclude that the 1D model generalizes
just fine.

=============

A lawyer might argue that some subset of what Hewitt
said was true, since the evaporation process is not
completely independent of speed.

However, main points:
  1) His main conclusion is not correct.  It is not
   only the faster ones that can break free.
  2) Students are not lawyers.  If you talk about
   speed and nothing else, they will assume that
   all that matters is speed and nothing else.


Additional lesser points:
  -- Essentially /nothing/ is completely independent
   of speed, so my lawyer says that Hewitt is asking
   about properties of the null set.  This set
   contains immovable objects, irresistible forces,
   and things that don't conserve energy, such as
   Hewitt's liquid that evaporates without cooling.
   You can make any predictions you like about such
   things, confident that no experiment will ever
   prove you wrong.

  -- The usual example of a simple thermodynamic
   system is an ideal gas.  That's fine as a starting
   point, but (a) that's not the only feasible
   starting point, and (b) it's not a very nice
   ending point, because ideal gases have a great
   many peculiarities -- such as having no energy
   other than kinetic energy -- that are not shared
   by other thermodynamic systems, including liquids
   and solids and molecules et cetera.

   The most interesting thing about particle C2 is
   /not/ its speed.

   When you are in check-the-work mode, when you see
   a dubious thermodynamic argument, the first thing
   to check is whether it wrongly imputes ideal-gas
   behavior to something else.

  -- In my experience, all-too-often Hewitt understands
   things at the starting-point level only, and generalizes
   naïve ideas to more complicated situations, without
   bothering to check the work.