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Unformatted text preview: Answers to Physics 176 One-Minute Questionnaires Lecture date: January 27, 2011 When does the kinetic model spectacularly fail (if ever)? The simple kinetic models we discussed in class fail when quantum mechanics becomes important, e.g., when the details of how a gas molecule collides with another gas molecule or with molecules in the wall of the container become important. One case this happens is the one you already thought about in the first homework assignment, when the de Broglie wavelength of the gas molecules becomes comparable to their spacing (some combination of high density and low temperature). A second case arises when the magnitude of the thermal energy kT becomes comparable to or smaller than the separation Δ E be- tween energy levels of a molecule or energy levels of a surface. (Surfaces often act like isolated quantum systems, with properties different than molecules or solids.) This usually happens at lower temperatures. Kinetic models can be generalized to include quantum mechanics in which case they can be arbitrarily accurate. But in that case they lose their convenience for providing quick useful insights; one would have to write a complicated computer program and use a powerful computer to study a quantum mechanical kinetic model. How can the Karo syrup unwind in exactly the same way that it wound up? A related question was: “How do we mathematically describe the demo about the dye in corn syrup?” A key reason for the apparent reversibility is that the diffusion con- stant D for the dye within the Karo syrup is unusually small: a dye molecule is unable to travel far by a random walk because the syrup has long chains of sugar molecules that tangle with one another and so these long molecules can not move easily, trapping the dye molecules to close to their original positions. As to why one can return to the original ink pattern after “unwinding” the syrup, this is a fairly simple mechanical effect of the fluid: as you turn the handle, you apply a shear stress that smears the thin cylinder of injected ink into an extremely fine concentric spiral sheet of ink that is difficult to 1 see by eye. As long as the diffusion constant is sufficiently small, this thin spiral of ink does not diffuse away so you can recover the ink by unwinding. You might enjoy borrowing the mixing apparatus and trying some ex- periments of your own. For example, how many turns can one make before losing the ink pattern? What if you raise the temperature of the syrup, does ink diffusion increase and the effect goes away? What happens with water or other liquids instead of carp syrup? What happens if you start diluting the cargo syrup partially with water? (I would be glad to give you extra credit if you would like to explore this or pursue other thermal-physics related projects.) A mathematical description of the syrup as you turn the crank would involve the so-called Navier-Stokes equations of fluid dynamics. These equa- tions lie beyond the range of this course, but I can discuss them with you...
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