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Re: Text omission = lost pedagogical opportunity



David Bowman has revealed the mathematical argument which I was after.
There is of course the preliminary lesson which comes from turning "@
constant T, PV=a constant " into "PV=f(T)". The lesson concerning the
several meanings of "constant" should not be quickly dismissed, as a
valuable lesson in itself.

I simplify the main argument by using only two of the three gas laws (2
suffice): eg solving each of the two equations PV=f(T) and P/T=g(V) for
P, equating the results and re-arranging yields: f(T)/T = V*g(V). I
write this statement on the board, explicitly noting that the LHS depends
only on T, while the RHS depends only on V. I then draw beside it a
sealed rigid cylinder of gas, and label its state as P,V,&T.

Then I draw a candle flame beneath it and observe that this will surely
heat the gas and raise its temperature, but the volume cannot change. I
then remark that such possibilities limit the unknown functions f(T) and
g(V) to a very narrow and simple class. Getting no response, I prod with
questions like "since the RHS does not change, what must be the behavior
of the LHS, . .? ". Invariably, I am met with a bevy of frozen, blank
stares.

At this point, the solution requires absolutely no mathematical
calculational skill, only a small step in (almost forced) logic. One can
almost see three possibilities wrestling in the students' heads:1) the
equation itself is an impossible statement, 2) this is a clever trick, a
paradox or 3) this is so esoteric and mysterious that I will never
understand this math stuff. It does happen, but it is rare that I have
been able to prod a beginning class to a successful resolution.

Even after the epiphany, one must brood with the students over the
solution for some time before the sense and power of the argument takes
hold. They have then drunk the first draft of the deepest and broadest
argument in physics, the argument from symmetry (in its broadest sense).

Queries / Comments:
Is there a "Piaget stage" before which such reasoning is not available?
What is it that impedes the imagination from allowing "The LHS must also
remain constant" - just the appearance of the changed variable T on the
LHS?
Would this be a valid IQ test question (perhaps written in a more common
context)?
There is usually a genuine sense of awe at the power of this reasoning,
at the end. It is easy to presume that the general equation goes far
beyond (at least any two of) the input equations and adds new
information.
This exercise is free enough from math that it could be presented in a
general public (Rotary club?) talk on "The power of quantitative
reasoning".
It is worth giving the students this PVT introduction to the "separation
of variables" argument. Pressure and temperature are more vividly imaged
by students than are the coordinate arguments of an electric field or a
wave function.

-Bob

Bob Sciamanda sciamanda@edinboro.edu
Dept of Physics trebor@velocity.net
Edinboro Univ of PA http://www.edinboro.edu/~sciamanda/home.html
Edinboro, PA (814)838-7185