Dallas wrote:
The origin of the question comes from the current edition of the
Jeppesen Private Pilot manual.
[...]

http://img.photobucket.com/albums/v1...allas/Jep2.jpg

Context helps. The kicker, and misleading, aspect is this:

"...assuming that all other variables remain constant...."

First, what the Jeppesen manual states is self-evident from the ideal gas
law:

P*V = n*R*T

So their text is technically correct, but really incorrect as far as
describing what *really* happens when some part of the atmosphere heats
up more than the surrounding atmosphere.

In the real atmosphere the volume V0 that is occupied by some n moles of
gas does not "remain constant". But neither does a volume V0 of gas
undergo "free expansion" as the temperature rises from T0 to T1, as I've
seen some people suggest. To be clear: "free expansion" means the total
energy of the n molecules in V0 remains the same as the volume goes to V1
- that is, as the gas expands it does no work, and no work is done on it.
(Or put mathematically, "free expansion" says that P0*V0 = P1*V1; that is
because the product of pressure and volume yields units of energy!) Of
course it is not the case that the energy remains constant because as the
gas expands it has to do work against gravity (i.e. it has to push
against the surrounding atmosphere). So in general what actually happens
is something _in between_ what happens when V0 = V1 (volume held
constant) and P0*V0 = P1*V1 (energy held constant). (And to determine P1
and V1 is why thermodynamics texts are filled with imposing looking
differential equations! :-))

BUT THE BEST FORMULA (and most relevant to aviators) DOESN'T DEAL WITH
VOLUME CHANGES! IT DEALS WITH DENSITY CHANGES! Here is how it is derived:

First divide both sides of the ideal gas law by V and rearrange
variables:

P = R*T*(n/V)

But n/V is just a density! So density = n/V and we get:

P = R*T*density

But R is "just" a conversion constant to make sure all the units work
out. If we only want to understand proportionalities we can discard the
R. Then dividing both sides by T yields:

density = P/T

*** SO IMHO THE BEST FORMULATION OF THE IDEAL GAS LAW FOR PILOTS IS: ***
**** ****
****** DENSITY = PRESSURE/TEMPERATURE ******

So when Jeppesen said "assuming all other variables remain constant" it
was basically saying "assuming the density remains constant." But air
density is the ultimate variable of interest to pilots! So Jeppesen was
effectively posing an example that said "assuming pressure and
temperature vary such that it doesn't affect the lift produced by your
wings!" Now *that* is a misleading (and useless) example IMHO!