Re: [Phys-l] T dS versus dQ

[John Mallinckrodt <ajm@csupomona.edu>]

Carl's note along with a hint from bc's recent message prompt me to  
suggest a model for very rapid compression in which work is done  
against the initial pressure, P_o, plus a dynamic pressure given by  
(1/2) rho_o (2v)^2 where v is the piston velocity and 2v is the  
velocity to which the impacted gas is accelerated.

If we express v in terms of a dimensionless fraction of the rms speed  
of the gas molecules, i.e., v = alpha*v_rms = alpha * sqrt(3*P_o/ 
rho_o) then the first law

E_f  =  E_o  +  W

becomes

(f/2)nRT_f = (f/2)nRT_o + (P_o + 6 alpha^2 P_o)* | delta_V |

where f is the number of degrees of freedom for the gas.  The ideal  
gas law then gives us

T_f/T_o = 1 + (2/f)*(1 + 6 alpha^2)*( | delta V |/ V_o)

For a diatomic gas (f = 5) compressed to half its initial volume, the  
formula gives us (begin monospace font)

v/v_rms     T_f/T_o
  0.1         1.2
  0.3         1.3
  0.6         1.6
  1.0         2.4
  2.0         6

Notice that a) not until the piston speed is nearly half the rms  
speed of the gas molecules does this formula give final temperatures  
that are even as large as those predicted for an infinitely slow,  
isentropic compression and that b) for piston speeds much greater  
than the rms speed, the formula gives enormous final temperatures  
that would never begin to be realized due to rapid conduction of heat  
to the walls.  I conclude that the formula has very limited  
applicability and, thus, as Carl does, that the Adiabatic Gas Law  
apparatus suffers no noticeable ill-effects due to the non- 
quasistatic nature of the compression.

John Mallinckrodt
Cal Poly Pomona

Carl Mungan wrote:

> I think there's no practical problem with the Adiabatic Gas Law
> apparatus. To be adiabatic (no heat transfer) the process cannot be
> too slow because in the real world no thermal insulation is perfect
> (especially since there's not just conduction but also radiation to
> worry about). On the other hand, to be reversible (isentropic) the
> process cannot be too rapid - compared to typical molecular speeds of
> the gas. Well these two limits leave a big working range - any quick
> motion of the piston by a human arm is bound to be just fine. Carl