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[Phys-L] isothermal "pressure"



Hi --

Part 1:

Question: For a gas in a cylinder with volume V, pressure P, and
energy E, would you say

P = - partial E / partial V ??? [1]

Answer: The RHS is horribly ambiguous. In the case of a monatomic
ideal gas, possibilities include:

- partial E / partial V | = 1.5 P [2]
| P

- partial E / partial V | = P [3]
| S

- partial E / partial V | = 0 [4]
| T

Lesson: A partial derivative is not much more or less than a
directional derivative. When writing a partial derivative,
always indicate the direction.

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

Part 2:

Suppose our "system" consists of some gas in a cylinder with
a piston. External to the system, a spring pushes on the
cylinder. In the familiar adiabatic case, we can use the
principle of virtual work to understand how the gas pressure
opposes the force of the spring. Equation [3] is a straight-
forward expression of PVW.

Recently I was asked about equation [4]. It was alleged that:
In analogy to equation [3], if the gas was in sufficiently
intimate contact with a heat sink, if the piston moved
inward a little bit, the energy of the gas would be
unchanged. By PVW, that would mean the gas was exerting
no force on the piston. The spring would crush the gas to
very high density. The rate of crushing would be limited
mainly by how fast the heat sink could do its job.

That seemed like a paradox. Thermodynamic equations were
allegedly saying something that common sense said couldn't
possibly be true.

Disclaimer: Something in the indented paragraph isn't true.
Normally I abhor saying things that aren't true. I think
discussing paradoxes is a poor pedagogical technique, especially
in introductory courses. (Yes, this includes all those infamous
relativity paradoxes.) I especially abhor _contrived_ paradoxes
... but this one wasn't contrived; it came to me fair and square.

Therefore: I suggest you not strain your brain too much thinking
about this paradox. The best way to proceed is to learn to
formulate the question in such clear terms that paradoxes cannot
arise.

My analysis can be found at
http://www.av8n.com/physics/isothermal-pressure.htm

I go on to milk this example for some lessons about
-- the importance of _locality_ in physical laws, and
-- the shiftiness of the concept of "work" in thermo and PVW.
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