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Chem 300 - Ch 27/#2 Today’s To Do List l KMT (Continued) Bimolecular Collision Frequencies Mean Free Path

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Collision Frequency with Wall l Find the # molecules that collide with an area from a specific direction with speed u per unit time. l Integrate over all speeds & angles: Z wall = /4 = (8RT/ M) 1/2 = N A n/V = N a P/RT Z wall = (N a P/RT)(RT/2 M) 1/2

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Examples Z wall = (N a P/RT)(RT/2 M) 1/2 l 1. At 25 o C & 1 bar, Z wall = 2.88x10 23 s -1 -cm -2 l 2. Outside surface of Space Shuttle facing away from the Sun: T~1K, P=10 -10 bar Z = (6.02x10 23 )(10 -10 bar)/(0.0831L-bar/mol)(1K) x[(8.31J/mK)(1K)/(2 )(.028Kg)] 1/2 = 4.8x10 14 /s-cm 2 l 3. Avg speed: = (8RT/ M) 1/2 = 27.5 m/s l 4. u rms = (3RT/M) 1/2 = 29.8 m/s l 5. u pp = (2RT/M) 1/2 = 24.4 m/s

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Collision with Molecules l With an identical Stationery Molecule l With identical Moving Molecules l With different Moving Molecules l Total Collision Frequency of All Molecules with a Minimum Energy (E a )

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Stationery Molecule l To collide: Must come within a target area occupied by another molecule. Collision Cross Section ( ) = d 2 l Z A = = (8kT/ m) 1/2 =N A n/V

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Moving Target l Consider: A Relative Speed: = (8kT/ ) 1/2 A Relative Mass ( ) = m 1 m 2 /(m 1 + m 2 ) If Identical Molecules: m 1 = m 2 m ===> = m/2 = 2 1/2 l Z A = = 2 1/2

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Total Collision Freq. (Z AA ) l All Identical Molecules: Z AA =(½) Z A = (1/2 1/2 ) 2 l A & B Molecules: Z AB = A B AB AB = average of A & B cross sections

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Minimum KE Collisions l Collision freq./unit volume for molec. A & B with relative speed between u r and u r + du r – l dZ AB = l A B AB ( /kT) 3/2 (2/ ) 1/2 exp(- u r 2 /2kT)u r 3 du r l u r 3 favors more collisions among high-speed molecules.

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Mean Free Path l Average distance travelled between collisions (l): l l = /Z a = RT/(2 1/2 N a P) l Example: N 2 @ T=1K, P=10 -10 bar l l=(0.082)(1K)/[(2 1/2 x6.02x10 23 x 0.45x10 -18 m 2 x10 -10 bar)] = 2.1 m

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Example: He in upper atm T=1000K, P=10 -12 bar l = 0.218x10 -18 m 2 l l = 7x10 5 m = 467 mi l = 2x10 3 m/s l Avg time between collisions: t avg = l/ = 400 sec = 0.1 hr

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Perspective l Typical gas (N 2 ) @ 1 atm & 25 o C: l Collection of molec. Traveling with 350 m/s l Each molec. Collides with another every 1 ns l Between collisions it travels 100-10 3 molecular diameters. l Since d<

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Next Time l Debye-Huckel Theory l Odds and Ends

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