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Re: [Phys-l] T dS versus dQ



I subscribe to Astronomy Picture of the Day

http://antwrp.gsfc.nasa.gov/apod/astropix.html

Today (Feb 23) they have a great picture of the waves preceding a rocket as it ascends through the air. It seemed relevant to our discussion of waves ahead of a moving piston. The waves reoriented ice crystals in the air making the waves visible. (APOD has an archive - just in case you are not receiving this on the 23rd.)

Bob at PC



________________________________________
From: phys-l-bounces@carnot.physics.buffalo.edu [phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of John Mallinckrodt [ajm@csupomona.edu]
Sent: Thursday, February 11, 2010 7:30 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] T dS versus dQ

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

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