Re: [Phys-l] App. for Was: Re: T dS versus dQ

[John Mallinckrodt <ajm@csupomona.edu>]

Bob LaMontagne wrote:

> it would be nice for the smiling professor to be able to give a  
> ballpark estimate of how large the temperature difference is  
> between slow and rapid adiabatic compression for specific instances  
> (say mach 0.0001 versus mach 0.9).

Okay.  I'll give it a very rough, back-of-the-envelope go, but no  
guarantees!

I figure that if you compress a monatomic gas to half its volume in a  
rigid container using a rigid piston moving at the sound speed,  
you'll hit something like half of the gas molecules one time and give  
them something like two to three times their original speed on  
average.  (I have in mind an average molecule that is moving  
generally toward the piston but at, say, a 45 degree angle in the lab  
frame and having an initial velocity equal to the rms speed which is  
34% faster than the piston is moving.  If that molecule makes an  
elastic collision with the piston, I calculate that it will leave  
with about 2.3 times its initial speed.)

If half the molecules have their speeds increased by a factor of 2.3  
then the gas would increase its total energy--and, therefore, its  
temperature by a factor of about 3.  This is to be compared with an  
isentropic compression that increases the temperature by 30%.

While I don't have a lot of confidence in this calculation, it seems  
to me that it might not be a bad estimate for a monatomic gas having  
a mean free path on the order of the cylinder length.  I'm not sure  
how it might need to be modified for a shorter mean free path (which  
would lead to a partial thermalization of the gas during the motion  
of the piston) or for a diatomic gas (which might have its  
vibrational and rotational modes excited by the collisions as well.)   
In any event, I'm certain that you would never actually see such a  
large temperature rise simply as a result of the substantial heat  
flow to the (actually non-rigid) walls that would take place with the  
first round of energetic gas collisions.

John Mallinckrodt
Cal Poly Pomona