Most intro texts state Boyle's, Charles' and Gay-Lussac's ideal gas laws
and then simply state the equivalent of "These laws can be combined into
one law: PV=CT". This is more than just another "it can be shown ..."
lacuna; it misses a great opportunity to introduce the student to a
powerful (and subtle) argument of mathematical reasoning which, once
clearly understood, will carry him through many future hand-waving
lecture assertions, because each lecturer will assume that someone else
has already explained it.
Given for a fixed quantity of ideal gas:
1) at constant T, PV = a constant
2) at constant P, V/T = a constant and
3) at constant V, P/T = a constant (Obviously, P is absolute
pressure and T is absolute temperature - all quantities are positive),
how does one deduce that PV = CT for all states, where C is independent
of the variables P,V,T?
There is more than one way. How do you do it? Above, I refer to a
particular argument which serves a wider pedagogical purpose. If it
doesn't surface in your posts, I shall reveal this (purely mathematical)
argument.