Here's a continuation of
-- discussion of PV=NRT
-- the question "what happens if V is doubled?"
-- the question of what "Other Things Being Equal" means
The short answer is that (in the absence of considerable
additional information) you have no idea what happens
when V is doubled.
It is useful to contrast this thermo situation with
ordinary geometry. In geometry, there is a notion
of moving "due north". It is taken for granted that
whatever the "due north" direction may be, it is
perpendicular to the east/west direction, so that
by moving due north you don't change your eastness
or westness.
This notion of "perpendicular" is spectacularly lacking
in thermodynamics. The PV=NRT equation has four unknowns.
If you are changing V,
-- you must not assume that P is constant
-- you must not assume that N is constant
-- you must not assume that T is constant
*) maybe some of them remain constant, but
*) maybe none of them do.
To say the same thing in fancier words: in geometry
there is usually a natural way to choose a basis that
is orthonormal, but thermo has a superabundance of
non-unique bases, many of which have equal claim to
being "natural".
Thermo revolves around partial derivatives and directional
derivatives. There are, alas, many variables and many
directions of interest. The directions include:
-- entropy
-- enthalpy
-- energy
-- pressure
-- volume
-- temperature
-- density
-- etc. etc. etc.
Returning specifically to the equation PV=NRT, that is
one equation in four variables. If you specify what
happens to one variable (V for instance), the other three
remain totally undetermined. Generally you need !!two!!
more equations before you can infer a simple relationship
between P and V.
Most people learn geometry before they learn thermo, so
they start with the unrealistic expectation that moving
in the "volume" direction is like moving in the "north"
direction, with some other variables that naturally and
implicitly remain constant. This expectation is absolutely
wrong physics and must be unlearned.
This difficult process of (un)learning is made even
more difficult by the traditional notation for partial
derivatives, which is clumsy, ambiguous, and misleading.
Possibly constructive suggestion: If I were teaching
this, I'd probably adopt the functional-programming
formalism used by Sussman and Wisdom http://mitpress.mit.edu/SICM/book-Z-H-5.html
Learning that formalism is a burden, but at least you
understand something at the end -- as opposed to most
students for whom learning the traditional formalism
is a waste because they achieve no real understanding.
Some (but not all) geometric intuition successfully
transfers to thermodynamics. The equation PV=NRT is
a constraint, which you can visualize as a surface
(three-dimensional hypersurface) in the four-dimensional
space. This space has coordinates, but
-- it lacks a dot product, therfore
--- it lacks a notion of distance, and
--- it lacks a notion of angles, and
---- most particularly it lacks right angles and any
notion of an orthonormal or even orthogonal basis.
-- even the choice of coordinates is non-unique. You
can label the points using (P,V,N,T) but multitudinous
other labelling schemes are just as valid.
The typical "phase diagram" is a lower-dimensional slice
through such a hypersurface. By taking slices in different
directions you get the various types of phase diagram:
you can slice it with a plane of constant energy, or a
plane of constant entropy, or whatever.