1. Show that for an ideal gas Cheistry 432 Proble Set 11 Spring 2018 Solutions P V 2 3 < KE > where <KE> is the average kinetic energy of the gas olecules. P 1 3 ρ v2 KE 1 2 v2 ρ N V P V 1 3 N v2 2 3 N 1 2 v2 2 3 KE 2. Calculate the average, root ean square and ost probable speeds for argon gas atos at a teperature of 25 C. 3 v rs [ 31.381 10 23 JK 1 ] 1/2 298K 431.4 s 1 39.95/6.022 10 26 kg 8 v π 8 1/2 v rs 397.4 s 1 3π v 2 2 1/2 v rs 352.2 s 1 3 1
3. Deterine an expression for < vx 2 > for a gas, where v x is the x-coponent of the velocity. Calculate < vx 2 > for argon gas at 25 C and copare the result with the average speed found in proble 2. 1/2 vx 2 vx 2 exp v2 x 2π 2 π 2π 2 k BT v 2 x v 2 1 3 [ 2 ] 3/2 vx 2 1/2 v rs 3 249.1 s 1 4. For one ole of argon gas at 25 C and 1 atosphere pressure, calculate a the collision frequency for a single argon ato; 2σ v P z 2.36 10 18 2 397.4 s 1 1.381 10 23 JK 1 1.01 10 5 N 2 4.96 10 9 s 1 298K b the total nuber of collisions per unit tie per unit volue; Z AA 1 2 zρ 1 2 z P 1 2 4.96 109 s 1 1.01 10 5 N 2 1.381 10 23 JK 1 298K 6.09 1034 s 1 3 c the ean free path for argon atos; λ v 397.4 s 1 z 4.96 10 9 s 8.01 1 10 8 and 2
d the collision frequency of the argon atos with the walls of the container. P Z w 2π 1.01 10 5 N 2 [2π39.95/6.022 10 26 kg1.381 10 23 JK 1 298K] 1/2 2.44 1027 2 s 1 The collision cross section for argon atos can be found in Table 16.1 on page 406 of the text book. 5. Show that the velocity necessary for an object to escape fro the gravitational attraction of the earth is given by v 2 2GM R e where G 6.670 10 11 3 kg 1 s 2, M is the ass of the earth 5.98 10 24 kg and R e is the radius of the earth 6.38 10 6. Calculate the teperature at which the average speed of argon atos equals the escape velocity. kinetic energypotential energy at surface of earth v 2 2GM R 1 2 v2 GM R 26.67 10 11 N 2 kg 1 5.98 10 24 kg 11182 s 1 6.38 10 6 11182 s 1 8 π T 11182 s 1 2 π39.95/6.022 10 26 kg 81.381 10 23 JK 1 235875K 6. The Maxwell-Boltzann speed distribution for an ideal two-diensional gas is given by F v v exp v2 /2. Deterine an expression for the ost probable speed of an ideal two-diensional gas, and calculate the fraction of olecules in an ideal two-diensional gas having speeds less than the ost probable speed. F v [ ] /2kT e v2 1 v2 0 at v 3
Let Then f fraction y v2 2 v 0 dy v 2 /2 ve v2 /2 dv vdv f e y dy 0 1 e v 2 /2 1 exp 2 1 e 1/2.393 7. For a two-diensional gas, the Maxwell-Boltzann speed distribution is given by F v v exp v2 /2. Derive an expression for the ratio of the average speed to the ost probable speed for a two-diensional gas. v v 2 e v2 /2 dv 0 3/2 π 2 π 4 2 F v 1 v v e v2 /2 0 at Then v v v π 2 8. The Maxwell-Boltzann speed distribution for a two-diensional gas is given by F v v e v2 /2. For a two-diensional gas derive an expression for the fraction of olecules having speeds greater than the average speed. v vf vdv v 2 e v2 /2 dv 0 0 4
3/2 π 2 π 4 2 f ve v2 /2 dv v 1 2 e v2 /2 v e v 2 /2 2 { exp } π e π/4 0.46... 2 2 9. A saple of xenon gas at a pressure of 1.00 bar is known to undergo 1.13 10 9 collisions per second with a ean-free path of 1.9 10 7. Assuing xenon is well represented by a hard sphere, calculate the cross-sectional area, σ, and radius of the Xe atos. σ λz v 1.9 10 7 1.13 10 9 s 1 214.7 s 1 T π 8k B v 2 8 π 131.3 π 6.022 10 kg 26 81.381 10 23 J K 1 214.7 s 1 2 286 K z 2 v σ P k BT z P 2 v 1.381 10 23 J K 1 286 K1.13 10 9 s 1 21.00 105 N 2 214.7 s 1 d 2.16 10 10 r d/2 1.08 10 10 1.47 10 19 2 πd 2 10. A sall hard cube having a length of 1.0 10 5 eters on a side is placed into a saple of argon gas at a teperature of 298K and a pressure of 2.00 bar. Calculate the nuber of collisions per second the argon atos ake with the cube. You should assue all 6 faces of the cube are exposed to the argon gas. Z w P 2π 2.00 10 5 N 2 39.9 kg 2π 1.381 10 6.022 10 23 J K 1 298 K 26 4.83 10 27 2 s 1 nuber of collisions per second with cube Z w A 4.83 10 27 2 s 1 6.0 10 10 2 2.9 10 18 s 1 5
11. The average speed of argon atos in a saple of argon gas is 728.5 s 1, and the radius of an argon ato is 1.9 10 10. Calculate the ean free path of argon atos in the saple when the total pressure is 3.00 bar. 8 v π 39.9 kg T π v 2 8k B π 728.5 s 1 2 6.022 10 26 81.381 10 23 J K 1 1000 K σ πd 2 π3.8 10 10 2 4.5 10 19 2 λ v z k BT 2σP 1.381 10 23 J K 1 1000. K 24.5 10 19 2 3.00 10 5 N 2 7.2 10 8 12. In a country called Flatland, a two-diensional cheist places 7.0 10 15 two-diensional, ideal-gas atos having the ass of real argon inside a square box each side of which is 1.0 eters in length. Derive an expression for v x, the average x coponent of the velocity in the positive direction for the atos. Use the result to calculate the nuber of collisions per second per eter the atos ake with any of the walls of the square box at a teperature of 298K. Recall the noralized probability density for the x coponent of the velocity is given by v x fv x 1/2 e vx 2/2k BT 2π 1/2 v x e v2 x/2k B T dv x 2π 0 2π 1/2 1 2 2 2π Letting A be the area swept by a single olecule in tie t and L be the length perpendicular to the direction of otion A v x tl The total nuber of collisions in tie t A the area density ρ of the atos in the square box. Then the collisions frequency per unit tie per unit length is given by Z w ρ v x ρ 6 2π
7.0 1015 1.0 2 1/2 1.381 10 23 J K 1 298 K 39.9 kg 2π 6.022 10 26 7.0 10 17 1 s 1 13. The N 2 olecule has a natural vibrational frequency of 2360 c 1 and can be assued to be spherical with a diaeter of 3.85 Å. Calculate the nuber of vibrations an average N 2 olecule akes between collisions at 25 C and a pressure of 1 atosphere. 2σ v P z 2[π3.85 10 10 2 1.01 10 5 N 2 ] 1.38 10 23 JK 1 298K 81.381 10 23 JK 1 298K π 28/6.022 10 26 kg 7.69 10 9 collisions s 1 2360c 1 2.998 10 10 c s 1 7.08 10 13 vibrations s 1 7.08 10 13 vibrations 7.69 10 9 collison 9227 vibrations collision 7