Re: [Phys-l] d ln V / d ln P = ??

["David Bowman" <David_Bowman@georgetowncollege.edu>]

Regarding the topic brought up by John D's question:
 
> Hi Folks --
>
> This is mostly a terminology question.
>
> By way of background:  the quantity
>
>          - ? P  |
>     Ks = ------ |                       [1]
>          ? ln V |S
>
> is called the adiabatic bulk modulus, and its inverse is call
> the adiabatic compressibility.  It is common but sloppy to call
> them "the" bulk modulus and "the" compressibility.
 
I was under the impression that these 'sloppy' names that are
unmodified by appropriate descriptive adjectives were more
commonly used of the *isothermal* quantities rather than for
the adiabatic ones.  For instance if you look up the
compressibility/bulk modulus for various substances in the CRC
tables I believe you will find the isothermal ones tabulated
there.  If you want the adiabatic ones (maybe because you want to
calculate the speed of sound) you need to work them out for
yourself using the isothermal ones & the specific heat ratio, ?.
If you can't find the ratio ? listed it, too, can be worked out
from one of the specific heats (commonly c_p), the density, the
isobaric coefficient of expansion, the isothermal compressibility,
& the temperature.
 
BTW, this brings up a general thermodynamic result that I think has
some real profundity.  The result I have in mind is
 
c_p/c_v = K_s/K_T
 
IOW, the ratio of the isobaric specific heat to the isochoric
specific heat is always necessarily equal to the ratio of the
adiabatic bulk modulus to the isothermal bulk modulus.  This
identity eerily equates a ratio of *thermal* properties to a ratio
of *mechanical* properties under all equilibrium conditions for all
isotropic substances.  The result has even greater generality than
this.  A generalization of it even applies to systems for which
macro-work isn't even done on them via volume changes against
pressure.  For instance for magnetic a spin-system a generalized
version of the result holds for the magnetic susceptibility ratio
rather than the compressibility ratio, and for an electrically
polarizable system a version of it holds for the ratio of
electrical polarizabilities.  Of course the corresponding specific
heat ratio in these other cases is not that of the isobaric to
isochoric specific heats but, but the constant magnetic field
specific heat to the constant magnetization specific heat, or the
constant electric field specific heat to the constant electrical
polarization specific heat.
 
In general, if we have a system where macro-work can be done by the
system on the surroundings according to the differential
relationship dw = X dx where X is the generalized thermodynamic
force congugate to the macro-parameter x, then the following
identity of ratios holds: c_X/c_x = ?_T/?_S .  Here c_X is the
constant-X specific heat, c_x is the constant-x specific heat,
?_T is the isothermal susceptibility and ?_S is the adiabatic
susceptibility where each susceptibility is given by a derivative
of the form
 
? = -?ln(x)/?X
 
with the appropriate quantities held fixed.
 
> Reference: http://en.wikipedia.org/wiki/Bulk_modulus
>
> So the question is, what do we call the quantity
>
>          - ? ln P |
>     ?  = -------- |                     [2]
>           ? ln V  |S
>
> It is obviously dimensionless. 
>
> For ideal gases, it is equal to the adiabatic index i.e. the
> ratio of specific heats i.e. ? ....  We don't need a new
> name for that;  there are more than enough names already.
>
> On the other hand, for water at 1 atm, the ratio of specific
> heats is close to unity, whereas the dimensionless modulus
> in eq. [2] is 22000 times greater than that.
>
> If it helps, note that I am actually more interested in the
> compressibility-like quantity
>
>          - ? ln V |
>     ?? = -------- |                     [3]
>           ? ln P  |S
>
> Is there a conventional name for either of [2] or [3], and/or
> can anybody suggest a nice name?
 
I don't know of any commonly used single-word name for either of
these dimensionless quantities.  But I'm not all that intimately
acquainted with the extant literature either.  It's not a very
flashy name, but it has the advantage of accuracy if quantity [3]
is called the 'adiabatic compressibility pressure product'.  It does
have the disadvantage of being four words.
 
David Bowman