Problem 5. 2D World Consider a mono-atomic ideal gas in a two dimensional world, so the velocities are labeled by v = ()

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Problem 5 2d World Consider A Mono Atomic Ideal Gas In A Two Dimensional World So The Velocities Are Labeled By V 1 (68.05 KiB) Viewed 43 times

Problem 5 2d World Consider A Mono Atomic Ideal Gas In A Two Dimensional World So The Velocities Are Labeled By V 2 (35.63 KiB) Viewed 43 times

Problem 5 2d World Consider A Mono Atomic Ideal Gas In A Two Dimensional World So The Velocities Are Labeled By V 3 (57.82 KiB) Viewed 43 times

Problem 5. 2D World Consider a mono-atomic ideal gas in a two dimensional world, so the velocities are labeled by v = () (a) Use Jacobians to show that the "volume element is (15) dude vdu do a(,0) where we = vcos 8 and uy = v sin 0. with ue (0.00) and 0 € (0.26. It is under- stood that these expressions are meant to be integrated over. The double bars mean determinant and then absolute value of the Jacobian matrix (...) 100) (16) (b) Consider the change of coordinates from arcos O and y-rsin 0. Write down the y Jacobian in analogy with (a). The columns of the Jacobian form vectors e es (17) . av 30 (18) 30 Compute the norms of these vectors ledanded and show that the vectors are orthogonal Qualitatively interpret these vectors and their lengths by referring to Fig. 3. Note that the volume element is le drllende sirke the vectors are orthogonal (c) (Optional. No work) in three dimensions show that the Jacobian of the map - sincoso, yrsino.cosroos, is () sin cos reos cose - siol drdodo sin sin cos costo sino sin Odrdedo CON - sin 0 (19) P | 00:02:39 | ara
dr rde ē, de erdr Figure 3: Cylindrical coordinates in two dimensions, (r...) e Figure 4: Spherical coordinates
Look at the columns and find the lengths of e, dr, e do and e do Interpret these vectors and lengths by looking at Figl 4 where the spherical coordinatos explained (d) Write down the normalized Maxwell velocity distribution, P(1) - Puddy and, using the Jacobian of part (@) and an integral over 0. determine the normalized speed distribution 07 (1) - Pudu (20) Describe in plain speak and a simple picture (like Fig. 3) what we are doing with the whole Jacobian + Integral over 8" steps You should find that all factors of have canceled in your final expression for P(u) in two dimensions. You can check your result by doing the next item. (e) Compute (mu) using the speed distribution. You should find T. Is your result consistent with the equipartition theorem? Explain. Please be explicit about how to do the integral. If you get stuck try changing variables to a dimensionless energy (1) Consider a box of gas with N particles and density n. The box has a small hole. In three dimensions we found that the total flux (number per area per time) escaping the holes (21) and the pressure is determined by (?) vin P (2) (22) where n is the density. Show that the two dimensional versions of these results is - (23) $ = n) ) In (0) Pame) (24) where is the flux (number per length per second). The first step is to find the differ- ential flux do in two dimensions, generalizing Eq. (13) which is for three dimensions Discussion: In two dimensions we have, (u) = (sk. T/2m) and (P) = 21,7/m.so the results of part (e) establish that: p=nk, (25) क P V2ππι (26) It is noteworthy that these last relations are the same as their 3D counterparts. Indeed, other derivations (see our Book problem 6.9) make it clear that these last relations hold in all dimensions Sometimes people were 780,0) to mean the determinant of the Jacobian matrix, rather than Just the matrix itself. Our books this notation, described in appendix C