advertisement

Old exam questions Most probable speed 3 2004 HE 3.1, 3.2 HE 4.1, 4.2 HE 5.1 F3 2005 HE 3.1, 3.2 HE 4.1, 4.2 HE 5.1, 5.2 F 3, 5a-c 2007 E1 4, 5 E2 1, 2, 3, 4 ... - ⎛ m ⎞ 2 2 − mu 2 2 k BT ⎟⎟ u e F (u ) = 4π ⎜⎜ ⎝ 2πk BT ⎠ 2006 M 3, 4 2008 E2 1, 2, 3, 4, 5 2 ⌧ Mean speed ⎛ m F (u ) = 4π ⎜⎜ ⎝ 2πk BT ... - ⌧ ... - 3 ⌧ Root mean square speed 3 ⎞ 2 2 − mu 2 2 k BT ⎟⎟ u e ⎠ 4 ⎛ m F (u ) = 4π ⎜⎜ ⎝ 2πk BT ... - ⌧ 3 ⎞ 2 2 − mu 2 2 k BT ⎟⎟ u e ⎠ 5 Example: T of a single molecule? Maxwell-Boltzmann distribution u mp = 2 k BT m u av = 8k BT πm ump = 2 k BT m 8k B T uav = πm urms ump = Suppose you have a single molecule of N2 in a box. Its speed is found to be 575 m/sec. What is its temperature? 3k BT = m u mp = 2 k BT m 2 k BT m u av = 8 k BT πm urms = 3k BT m Thermodynamic temperature is the property that determines if two systems are in thermal equilibrium with each other. ... - ⌧ 6 7 ⌧ ... - continued… Example: Average KE of a gas molecule? Revised: At what temperature would a single molecule of N2 moving at 575 m/sec have kinetic energy equal to kBT? Suppose you have a mole of N2 in a box. The average speed for gas molecules is found to be 575 m/sec. What is the average kinetic energy of a molecule of the gas? u mp = ... - ⌧ 8 2 k BT m u av = ... - 8 k BT πm urms = ⌧ 3k BT m 9 Molecular collisions in a gas Collisions per sec for one molecule z=4 π • How many collisions does a gas molecule make per sec? • How far does a gas molecule travel between collisions? • How manyy total collisions per sec are there in a sample of gas? ⌧ ... - 10 Total # of collisions in a sample Z tot = ... - N 2 ... - N 2 RT σ V M ⌧ 11 Mean free path ⎛ N RT ⎞ ⎜4 π σ 2 ⎟ ⎜ ⎟ V M ⎝ ⎠ ⌧ 12 ... - ⌧ 13 Example: Nitrogen transport kinetics Diffusion coefficient and molecular shape Suppose you have a mole of N2 in a box at STP. What are 1) the number of collisions per second for a N2 molecule, 2) the total number of molecular collisions in the box in a millisecond, millisecond 3) the characteristic time between collisions for a typical N2 molecule and 4) the typical distance a molecule flies between collisions? σ = 3.74 × 10-10 m ... - ux M = 28 g/mol ⌧ 14 Microscopic diffusion constant ... - ⌧ 15 Einstein relationship d 2x dx m 2 = Fc , x (t ) − f dt dt < x 2 >= 2 Dt ux ... - ⌧ 16 ... - ⌧ 17 Diffusion of a sphere Diffusion of nonspherical particles f 0 = 6πηa ux What is f for a sphere? h ? 2 2⎛a⎞ ⎜ ⎟ 3⎝b⎠ f = f0 ⎛ a⎞ ln⎜ 2 ⎟ − 0.30 ⎝ b⎠ 3 2 aeff = 3 3ab ... - ⌧ 18 Example: diffusion coefficient of a nanotube ... - η = 1.0 × 10-3 Pa·sec a/b = 1 × 104 A carbon nanotube is 5 nm in diameter and has a lengthto-diameter ratio of 10,000. What is the diffusion coefficient of this nanotube in water at T = 25 °C? C? 2 19 ⌧ continues 2 aeff = 3 3ab 2 f 0 = 6πη a a = 2.5 × 10-5 m b = 2.5 × 10-9 m η = 1.0 × 10-3 Pa·sec 2 2⎛a⎞ ⎜ ⎟ 3⎝b⎠ f = f0 ⎛ a⎞ ln⎜ 2 ⎟ − 0.30 ⎝ b⎠ 3 ... - ⌧ 20 ... - ⌧ 21