Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re[2]: magnetic lines like these?



Somebody asked: > Suppose a mixture of two gases, such as He and Ar in
thermal equilibrium, is inside a long cylindrical tube. A thin
alluminum window, at one end of the tube, breakes [removed very
rapidly] and the mixture is ejected into the surrounding vacuum. Each
atom gains a non-random velocity component along the axis. Is the
average non-random velocity component for the He atoms the same as for
the Ar atoms?<

I (Philip Zell) responded I think the thermal velocities
of the atoms before the window breaks will determine their velocity
components along the axis of the tube after the window breaks. ....<<

Ludwig replied: The argument was quite convincing for a short tube (in
comparison with the mean free paths of molecules). You assume that
"thermal equilibrium is established only by collisions between the atoms
and the walls of the tube".<<<

>>>Is the situation different at high pressures, when gas particles
collide very frequent?<<<

Seems to me that when collision rates are high, a couple of scenarios are
possible. We could assume that the gases are always in thermal
equilibrium with each other, even though their temperatures are changing
(I assume the gases are cooling as would be expected during a Joule
expansion), because collisions are so frequent that the gases are
thermalized over small increments of expansion. In this case, could we
not argue that the thermal velocities still determine their velocity
components along the axis of the tube? If so, then helium would still be
the faster expander, on average. If we can't assume thermal equilibrium
during the expansion, the problem is messier.

One idea I had was to forget the thermodynamics approach and treat this
like a rocket problem. Because the gas atoms transfer momentum to the
closed end of the tube, the tube, in turn transfers momentum to the
atoms, conserving total momentum. So the question becomes, how is
momentum distributed among the atoms? Assuming equal numbers of helium
and argon atoms, I can't think of any reason to believe that momentum
would be preferentially transferred to one species. So, if we assume
equal amounts of momentum get transferred to the helium and argon atoms,
then the less massive atoms would once again have to move faster than the
more massive atoms.

A third approach would be to treat this like a fluid flow problem.
Assuming that the tube is horizontal, Bernoulli's equation tells us that
the pressure drop along a stream line equals the gain in the kinetic
energy density of the gas. Since the pressure drop would be the same for
both gases, the gain in KE density would be the same for both gases.
Because the gases are contained in the same tube, the volumes of the
gases are also equal. So then the kinetic energy of a given volume of
either gas would have to be the same. But the 2 equal volumes would not
weigh the same...helium would weigh less, so the bulk helium flow would
have to be faster to have the same KE.


>>>Ludwig asked further: Do small drops ejected from the valve of a
pressure cooker travel faster than those whose masses are considerably
larger? Why should we assume that - and + ions ejected from the Sun
(mostly e and p) have very differnt velocities? Ludwik Kowalski<<<

Taking the second question first, is it not assumed that energy is
partitioned equally between electrons and protons, so electrons move
faster because of their smaller mass?

RE: the pressure cooker: now we're dealing with the liquid form of one
particular material (water). At a given location, two droplets of
unequal diameters will experience the same pressure but unequal forces.
E.g., if one droplet has twice the diameter of the other, it will
experience 4 times the force. But the larger droplet has 8 times the
volume of the smaller droplet, and, therefore, it has 8 times the mass of
the smaller droplet. Therefore, the larger droplet will experience an
acceleration that is half that experienced by the smaller droplet. Looks
to me like the smaller one is faster again.

Philip Zell
zell@act.org