lew@ihuxr.UUCP (08/30/83)
This is a followup to my previous posting "Trapezoidal holes & the 2nd Law", which was a response to Alan Wendt's net.general posting, "Second Law of thermodynamics repealed?" To recap, Alan noted that holes with a trapezoidal cross-section (I take this to mean they are frustrums of cones) will allow up to three times as many particles to pass through from the small hole side, as from the large hole side. (I can believe this after drawing a few trajectories.) He went on to remark that this would seem to provide a work-free "pump" which would cause the pressure on one side of a wall of such holes to build up spontaneously, in violation of the 2nd Law. He noted that the imperfect reflection of the gas molecules would probably prevent this from happening and then stated, "It seems strange to me that the second law of thermodynamics should depend so critically on the characteristics of molecular impact with surfaces." In my previous submission I agreed with Alan that this was the reason the "hole pump" wouldn't work, but I stated that this was a fundamental rather than an accidental escape clause. I would like to make some further remarks aimed at reinforcing this seemingly facile claim. The key to reconciling this with your intuition is to recognize the difference in scale between the mechanical model and the thermal model. We have a "strange loop" here which should satisfy Doug Hofstadter. The kinetic theory of gases builds a thermal model by statistically analyzing a mechanical model. However, if we attempt to construct such a model as Alan suggested when he said, "Can somebody make a such a wall which is thin enough and with lots of sufficiently small holes (45% angles are fine)? Perhaps by dropping very fine sand on a very thin sheet of glass in a vacuum?" ... the components of our model are each a thermal system! But the power and scope of kinetic theory is such that we can readily reconcile this loop. Consider a "gas" of this fine sand. How fine? Let's make it truly fine with a diameter of .01 millimeter. This gives a mass of about e-12 kg. Now we put the sand in a box in a vacuum (think space shuttle) at a temperature of 300K. There are 3 translational degrees of freedom for each grain of sand. So we expect to find: 1/2 * m * v^2 = 3/2 * k * T k is Boltzmann's constant, equal to 1.4 e-23 J/K. This gives v = e-4 m/sec which means that if the box is ten centimeters on a side the thermalized sand particles will take twenty minutes to traverse it. More to the point, the particles will be displacing their own diameters in 1/10 second. It shouldn't be hard to imagine that in the 1/100 second or so of contact time with the walls, the thermal vibration of the wall and of the particle itself will perturb the trajectory by a magnitude of order 1. In fact, this is built in to the calculation since we required that the translational degrees of freedom carry 1/2kT just like all the other (internal) degrees of freedom. Note that we should remove the quotes from the term "gas" describing our model. It is a legitimate thermal system in real thermal equilibrium with the more familiar internal modes. Its heat capacity is very,very small compared to the heat capacity of the bulk of the sand and the box, but that's OK. This is why we are not surpised that sand settles down to the bottom of a box instead of flying all over the place like we usually imagine a gas. It's all just a question of magnitude. Lew Mammel, Jr. ihuxr!lew