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Re: [Phys-l] gas laws



Regarding John Denker's post:

[apologies to those who have already received this via the chemistry
list]

Hi --

I collected some thoughts on "gas laws" into this document:
http://www.av8n.com/physics/gas-laws.htm

I tried to make it accessible to a fairly wide audience.
I had it vetted by a couple of 14-year-olds who have never
taken HS chemistry or physics; they didn't follow every
detail but they got the main points.

It's about 5000 words, with 12 illustrations.

===========

Here's an interesting exercise: Choose the best completion:
Other things being equal, for an ideal gas, ...
__
|__| (A) ... when the temperature goes up, the volume goes up,
and when the temperature goes down, the volume goes down.
__
|__| (B) ... when the temperature goes up, the volume goes down,
and vice versa.

For an analysis, see
http://www.av8n.com/physics/gas-laws.htm#sec-compare

I'd like to point out that the effect John discusses above is in no
way confined to ideal gases. The result is a *general* result that
*must* be obeyed for all ordinary fluidesque materials (isotropic &
homogeneous with changes in the volume and overall number of
particles being the only 2 relevant macro-parameters that can do
quasi-static macro-work). The result may even be more general than
this, but these (relatively loose) restrictions are simple enough to
be able to derive it in a straightforward manner.

The result in question is that the ratio of the adiabatic coefficient
of expansion to the isobaric coefficient of expansion is *negative*.

It can be shown by playing with equilibrium thermodynamic partial
derivatives that for *any* ordinary fluidesque material the ratio
of the adiabatic coefficient of expansion to the usual isobaric
coefficient of expansion is *always* -1/([gamma] - 1) no matter
*what* the equation of state happens to be. Here [gamma] is defined
as the ratio of the constant pressure specific heat to the constant
volume specific heat. It is also straightforward to show that the
quantity [gamma] - 1 *must* be non-negative by general stability
considerations. This makes -1/([gamma] - 1) negative (or at least
not ever positive). The reason for this is that more playing with
thermodynamic derivatives for such a system shows that [gamma] - 1
must always be equal to the expression:

V*T*([alpha])^2/(C_v*[kappa])

where [alpha] is defined as the usual isobaric coefficient of volume
expansion, C_v is the constant volume heat capacity, and [kappa] is
the ordinary isothermal compressibility. Now we know that the volume
V must always be positive; the absolute thermodynamic temperature T
must also always be positive (especially for a fluidesque system
whose Hamiltonian has no built-in intrinsic upper bound in energy
value); the value of ([alpha])^2, being a square of a real number,
must be non-negative; the heat capacity C_v must also be positive by
stability considerations so that the system is stable against thermal
fluctuations with the environment; and [kappa] must also be positive
by stability considerations so that the system is stable against
volume fluctuations with the environment. All these things together
mean that [gamma] - 1 is non-negative and thus -1/([gamma] - 1) is
also necessarily non-positive. Thus, when [alpha] is positive (as is
the case for an ideal gas) so that the system (sealed against
particle fluxes) expands when heated under constant pressure
conditions it *necessarily* must shrink when it temperature
increases adiabatically (by insulated compression). OTOH, for a
system like cold liquid water at a temperature below 3.98 deg C which
has a negative value for [alpha] and shrinks when heated under
isobaric conditions will *expand* when its temperature adiabatically
increases. In this situation the temperature actually *falls* when
cold water is adiabatically compressed.

But in any event, whether [alpha] is either positive *or* negative
the sign of the system's behavior under adiabatic conditions is
*opposite* to its behavior under isobaric conditions. And therefore
the ambiguity John discusses on his web site for answers (A) & (B)
above remains *regardless* of the sign of [alpha] and regardless of
the particular equation of state for the system.

On a similar note there is another related general thermodynamic
identity for ordinary fluidesque systems that I find quite remarkable
and deep. The identity in question is that [gamma] defined as the
ratio of the constant pressure specific heat to the constant volume
specific heat is also always necessarily equal to the ratio of the
usual isothermal compressibility to the adiabatic compressibility.
For some deep reason the ratio of a *thermal* quantity (i.e. specific
heat) taken to itself under different thermal conditions is always
equal to the ratio of a *mechanical* quantity (i.e. compressibility)
taken to itself under different thermal conditions.

David Bowman