<v 1 > 1/ 2. = 444 ms -1 PN RT. ρ = (πd 2 ) C rel ρ = m -3 s -1. = = mol L -1 s -1

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1 TOPIC - THE COLLISIO THEORY Example of solved problems. For molecular oxygen at 5 o C, (a) calculate the collision frequency z and (b) the collision density Z at a pressure of atm. Given: d oxy nm. nswer: The collision diameter of O is 0.36 nm or m (a) Collision frequency z (πd ) C rel ρ (πd ) <v > ρ Collision cross-section σ πd π( m) m Relative mean speed, <v > C rel <v > < v > 8 8.3kg m s 98K 3 0 π kg mol ms - <v > < v > ( ms- ) 67.9 ms - Molecule density ρ V P RT 3 3 (atm)(6.0 0 mol )(0 L m ) ( atm L K mol )(98 K).60 5 m -3 So z (πd ) <v > ρ( m )( 67.9 ms - )(.60 5 m -3 ) s - (b) Collision density Z z ρ (πd ) C rel ρ m -3 s - 3 ( m s )(0 m L ).90 8 mol L - s mol. Large vacuum chambers have been built for testing space vehicles at 0-7 Pa. Calculate the number of molecular impacts per square meter of wall per second for molecular CO (d 0.0 nm) at 35 o C? nswer: Use formula of collision with wall, J Calculate one item after another: i. Molecule density: 7 3 P 0 Pa mol 3 ρ.3 0 m 3 V RT 8.35 Pa m K mol K 6 3

2 mmmmiiiinnnn hhhhii. Mean speed : 8RT kg m s K mol 308 K v ms π M π 0 kg mol iii. Put together: Zw J (.3 0 m )(385.07ms 3 ).50 5 m - s ml container with a holeµm in diameter is filled with hydrogen. This container is placed in an evacuated chamber at 0 o C. How long will it take for 80% of the hydrogen to effuse out? nswer: d J dt d < v > dt V t 0 ln d o t < v > V < v > t V t 7355s 89 t 0 dt 00 ln 0 V < v > 6 0 π ( m) m m ms t 3 V volume < v > mean speed area, π 0 no. of molecules t. n effusion cell has a circular hole of diameter.50 mm. If the molar mass of the solid in the cell is 60 g mol - and its vapor pressure is Pa at 00 K, by how much will the mass of the solid decrease in a period of h? nswer: P 8RT P J Zw o m t RT πm (πrtm ) P o m t PMo t (πrtm ) ( πrtm Pa 60 0 kg mol π (.50 0 m) 60 60s (π 8.35kg m s kg mol ).0 0 kg 0 )gggg kg m s Pa 7

3 Formula for collisions Same molecule Different molecules. Collision frequency, or z (πd ) C rel ρ z collision per second, or ( π d ) C rel () ρ collision of a single molecule.. Collision density or total number of coll. or Z z ρ Z ( π d ) C rel () ρ ρ -coll. per volume per second. πd C rel ρ 3. Collision flux or collision with wall, or with surface, or coll. through a hole, or coll. per area s - or rate of effusion. Z w J d w dt M t o Exersice a. What is the mean speed of nitrogen molecule relative to another nitrogen molecule at 90 K?(66 m s - ). What is the average relative speed of hydrogen molecules with respect to oxygen molecules at 300 K? (837 m s - ) 3. Calculate the average number of molecules per cubic centimeter for nitrogen gas at bar pressure and 5 o C? (.30 9 ). The pressure in interplanetary space is estimated to be of the order of 0 - Pa. Calculate the collision frequency for an atom present in the space. ssume that only hydrogen atoms (d 0. nm) are present and that the temperature is 000 K and.( s - ) 5. How many collisions does a single r atom (d 0.3 nm) make in.0 s when the temperature is 5 o C and the pressure is.0 µatm? Given: r: M R ( s - ) 6. How many collisions per second does an (d 0.3 nm) molecule make at an altitude of 5 km where the temperature is 7 K and the pressure kpa. (.0 6 s - ) 7. Calculate the collision frequency for hydrogen molecule in hydrogen gas at.0 bar pressure and 0 o C? Given: d H.7 Ǻ. (.0 7 s - ) 8

4 8. Calculate the collision density for molecular chlorine at 35 o C and 770 mmhg. The collision diameter is cm. (.0 35 s - m 3 ) 9. What is the average time between collisions of an oxygen molecule in oxygen gas at 0 - bar and 50 o C (d 3.60 Ǻ)? 0. Calculate the mean free path for chlorine gas (d 0.5 nm) at 0. Pa and 5 o C? (0.03 m). How many molecules of O strike the wall per unit area per unit time at 600 mmhg at 5 o C? (d O 3.60Å )(.0 7 m - s - ). Large vacuum chambers have been built for testing space vehicles at 0-6 Pa. Calculate the number of molecules of nitrogen (d nm) impact per square meter of wall per second at 8 o C. (.60 m - s - ) 3. solid surface with dimensions.5 mm3.0 mm is exposed to He gas at 90 Pa and 500 K. How many collision do the He atoms make with this surface in 5s?(He: M R.00) ( m - s - ) Exercise b. box contains H (molecule b) and He (molecule c) at a total pressure of. atm. If the mixture contains 8% H by weight, calculate the ratio of z b (b)/z c (c)? Given d b 0.7 nm and d c.0 nm.( ). gas mixture contains H at /3 atm and O at /3 atm at 7 o C. Calculate the number of collision per second of H with other H, O with other O, and H with O (d H 0.7 nm, d O 0.36 nm) ( s -,.0 9 s -,.70 9 ). 3. t 30 km above the Earth s surface (roughly in the middle of the stratosphere), the temperature is roughly 000 K and the gas density is molecules/m 3. ssuming (d 0.38 nm) is representative of the stratosphere, determine z ; undergoes in this region of the stratosphere in s. ( ).. n equal number of moles of H (d 0.7 nm) and Cl (d 0.5 nm) are mixed and held at 98 K and the total pressure of kpa. (a) Calculate the collision frequencies z and z, where hydrogen is component and chlorine is component. (b) Calculate the collision density, Z. (.0 8 s -,.0 8 s -,.300 mol L - s - ) 5. For O (d m) at 0-3 bar at 5 o C, (a) What is the collision density Z? (b) What is the average time between collisions of a single molecule?(.60 mol L - s -,.60-7 s). 9

5 6. The average surface temperature of Mars is 0 K, and the surface pressure is.7 torr. The Martian atmosphere is mainly CO and with smaller amounts of r, O, CO, H O, and e. Considering only the two main components, we approximate the Martian atmospheric composition as xco 0.97 and x The collision diameters are dco.6 Ǻ and d 3.7 Ǻ. For gas at 0 K at the Martian surface, calculate (a) the collision rate for one particular CO molecule with other CO molecules; (b) the collision rate for one particular molecule with CO molecules;(c) the number of collisions per second made by one particular molecule;(d) the number of CO - collisions per second in.0 cm container of volume 0-5 cm 3 holds three molecules of gas b, which we label b, b, and b 3. In s, there are two b -b collisions, two b -b 3 collisions, and two b -b 3 collsions. Find z b (b) and Z bb. The vapor pressure of water at 5 o C is 360 Pa. If every water molecule that strikes the surface of liquid water sticks, what is the rate of evaporation of molecules from a square meter of surface? (.0 6 m - s - ) ml container with a hole0µm in diameter is filled with hydrogen. This container is placed in an evacuated chamber at 0 o C. How long will it take for 90% of the hydrogen to effuse out? (37 min) 9. Exactly dm 3 of nitrogen, under a pressure of bar, will it takes 5.80 minutes to effuse through an orifice. How long will it take for Helium to effuse under the same conditions? (.9 min) 0. The best laboratory vacuum pump can generate a vacuum of about ntorr. t 5 o C and assuming that air consists of molecules with a collision diameter of 395 pm, calculate the mean free path of the gas. (.60 m). Dry air had a CO mole fraction of Calculate the total mass of CO that strikes 0.5 cm of one side of a green leaf in 0 s in dry air at 39 o C and atm? (0.33 g ). certain sample of a pure oxygen ( d 3.6 Ǻ) has mean speed, <v> 50 m s - and the average time between two successive collisions of a given molecule with other molecules is.00-0 s. Find the mean free path and molecule density in this gas. (.80-7 m, 9.60 m -3 ) 0

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