Re: [Phys-l] partial derivative at constant WHAT?

[John Denker <jsd@av8n.com>]

On 11/15/2008 12:58 PM, David Bowman wrote:

> >          - ∂ 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.  

In my CRC, the main table speaks of the fully-qualified 
"isothermal" compressibility.  This provides little 
guidances as to what unqualified terms such as "the" 
compressibility might mean.

There is another table that speaks of "the" compressibility
under conditions where I can't tell whether it's isothermal 
or adiabatic, and it probably doesn't matter.  So that's 
not much help either.

> If you want the adiabatic ones (maybe because you want to
> calculate the speed of sound) 

MINOR POINT:  That nicely illustrates one side of the
problem.  It is commonplace to find the speed of sound 
explained in terms of the density and "the" compressibility.  
Under ordinary conditions, and a wide range of other 
conditions besides, you'd better use the adiabatic 
compressibility.  Blithely assuming "the" compressibility 
means the isothermal compressibility will give you the 
wrong answer.  (Amusing historical note:  Issac Newton 
made this mistake.  But even if you're in good company, 
it's still a mistake.)

MORE IMPORTANTLY:  I don't want to argue about which
type of sloppy terminology is _more_ common.  I think
we can agree that *both* are common enough to pose a
major problem.  (If only one were common, we could
call it a "convention" and remove the ambiguity.)

IMHO this is only the tip of a very nasty iceberg.  I
see lots of textbooks where people write partial derivatives
without specifying what is to be held constant.  Such
expressions are just begging to be misunderstood.  

Long ago I promised myself that I would always write the 
"at constant ..." qualifiers on every partial derivative.  
The benefits far outweigh the costs.