Kinetic Theory and the Behavior of Ideal & Real Gases Why study gases? n understanding of real world phenomena. n understanding of how science works. Gas Uniformly fills any container. Mixes completely with any other gas. Exerts pressure on its surroundings.
760 mm Hg (measured at 0 760 torr 0,35 Pa 0.35 kpa.03 bar 03 mb 4.7 lb in o C) manometer is used to measure the pressure inside closed containers Open-end manometer. (a) The pressure of the trapped gas, P gas equals the atmospheric pressure, P atm. Trapped gas pressure (b) higher and (c) lower than atmospheric pressure.
Compressing a gas increases its pressure. 3
Boyle s J-Tube Experiment V P 4
Jacques lexander Charles Law Boyle's Law : P Charles' Law : (when T V PV T V / T Gay - Lussac's Law : P / T V / T P / T (when V ) (when P P ) V ) 5
Example: What will be the the final pressure of a sample of oxygen with a volume of 850 m 3 at 655 torr and 5.0 o C if it is heated to 80.0 o C and given a final volume of 066 m 3? NLYSIS: Use the combined gas law with temperature in kelvins. SOLUTION: P V P V T T 3 850 m (80.0 73.)K 655 torr 3 066 m (5.0 73.)K 69 torr The combined gas law can be generalized to include changes in the number of moles of sample The ideal gas law is PV nrt R universal gas constant atm L 0.08 mol K 6
One mole of each gas occupies.4 at STP. Carbon dioxide is more dense that oxygen due to molar mass differences. Molar Mass of a Gas Molar Mass = drt/p d = density of gas T = temperature in Kelvin P = pressure of gas 5.4 7
The space above any liquid contains some of the liquid s vapor Example: sample of oxygen is collected over water at 0 o C and a pressure of 738 torr. What is the partial pressure of oxygen? NLYSIS: The partial pressure of oxygen is less than the total pressure. Get the vapor pressure e of water from table. (page 478). SOLUTION: P water vapor P gas 7.54 torr (738 7.54) torr 70. torr 8
Dalton s Law of Partial Pressures This is possible because the number of moles of each gas is directly proportional to its partial pressure Using the ideal gas equation for each gas n P V RT For a given mixture of gases, the volume and temperature is the same for all gases Using C=V/RT gives X PC PC PB C PZ C P P P P P P B total The partial pressure of a gas can be calculated using the total pressure and mole fraction P X P Z total 9
Gas volumes can be used in stoichiometry problems H ( g) O ( g) H O(g) volumes volume volumes (same temperature and pressure) volumes H ( g) volume O ( g) volumes H ( g ) volumes H O( g ) volumes O moles H moles H mole O ( g) just as ( g) mole O ( g) ( g) moles H O( g) ( g) volumes H O( g) moles H O( g) (a) Diffusion (b) Effusion The behavior of ideals gases can be explained Kinetic Molecular Theory So far we have considered what happens, but not why. In science, what always comes before why. 0
Kinetic Molecular Theory Postulates:. Gas particles are in rapid motion, colliding with container walls. Kinetic Molecular Theory Postulates:. Gas particles have negligible size compared to the distances between them.
Kinetic Molecular Theory Postulates: 3. Gas particles have no attraction for one another. Kinetic Molecular Theory Postulates: 4. bsolute temperature of the gas is a measure of the average kinetic energy of the gas particles. 5.6
Diffusion is the spontaneous intermingling of the molecules of one gas with another Effusion is the movement of gas molecules through a tiny hole into a vacuum The rates of both diffusion and effusion depend on the speed of the gas molecules The faster the molecules, the faster diffusion and effusion occur Thomas Graham studied effusion He found that the effusion rate of a gas was inversely proportional to the square root of the density (d) This is known as Graham s law effusion rate (constant P and T ) d effusion rate ( ) effusion rate ( B) d d Where M i is the molar mass of species i B M M B Diffusion The movement of one gas through another by thermal random motion. Diffusion is a very slow process in air because the mean free path is very short (for N 6x0-8 at STP it is 6.6x0 m). Given the nitrogen molecule s high velocity, the collision frequency is very high also (7.7x0 9 collisions/s). Diffusion also follows Graham's law: Rate of diffusion M 3
Diffusion of a gas particle through a space filled with other particles NH 3 (g) + HCl(g) = NH 4 Cl(s) HCl = 36.46 g/mol NH 3 = 7.03 g/mol Rate NH3 = The inverse relation between diffusion rate and molar mass. Due to it s light mass, ammonia travels.46 times as fast as hydrogen chloride NH 3 (g) + HCl(g) NH 4 Cl(s) 4
Relative Diffusion Rates of NH 3 and HCl Practical Example of Using Gas Density, Diffusion, Separation and Purification for Enriched Uranium Gaseous Diffusion Separation of Uranium 35 / 38 Gaseous Diffusion Separation of Uranium 35 / 38 Purified solid mixed U 3 O 8,UO 3,and, UO containing all uranium isotopes are converted to all isotopic forms of UF 6 (g) 5
Gaseous Diffusion Separation of Uranium 35 / 38 Purified solid mixed U 3 O 8,UO 3,and, UO containing all uranium isotopes are converted to all isotopic forms of UF 6 (g) 35 UF 6 vs 38 UF 6 0.7 % 99.8 % after approximately 000 runs 35 UF 6 is > 99% Purity Explain KMT Explain KMT on basis of the frequency of particle collisions with container walls. Explain KMT on basis of the velocity of particle collisions with container walls. When the gas volume is made smaller going from (a) to (b), the frequency of collisions per unit area of the containers wall increases. Thus the pressure increases Boyle s Law). 6
The kinetic theory and the pressure-temperature law (Gay-Lussac s law). The pressure increases from (a) to (b) as measured by the amount of mercury that must be added to maintain a constant volume. The kinetic theory and the temperature-volume law (Charles law). The pressure is the same in both (a) and (b). t higher temperatures the volume increases because the gas molecules have higher velocities. Kinetic Molecular Theory Particles are point masses in constant, random, straight line motion. Particles are separated by great distances. Collisions are rapid and elastic. No force between particles. Total energy remains constant. 7
Pressure ssessing Collision Forces Translational kinetic energy, Frequency of collisions, Impulse or momentum transfer, Pressure proportional to impulse times frequency k mu N v u V e I mu N P mu V Pressure and Molecular Speed Three dimensional systems lead to: N P m u 3 V u m is the modal speed u av is the simple average u rms u ssume one mole: PV=RT so: Pressure PV N 3 3RT N m u m u N m = M: 3RT M u Rearrange: u rms 3RT M 8
Distribution of Molecular Speeds u rms 3RT M Determining Molecular Speed Modify: PV=RT so: Solve for e k : Temperature PV N 3 m u RT e k N 3 3 N e k 3 R (T) N ( m u ) verage kinetic energy is directly proportional to temperature! 9
Gas Properties Relating to the Kinetic-Molecular Theory Diffusion Net rate is proportional to molecular l speed. Effusion related phenomenon. J. D. van der Waals corrected the ideal gas equation in a simple, but useful, way Plots of PV/nRT Versus P for Several Gases (00 K) 0
P measured n a V measured n a V measured : brings V measured measured P up nb nrt to ideal gas value nb : reduces measured V to ideal gas value a and b are known as the Van der Waals constants Substance Helium, He Neon, Ne Hydrogen, mmonia, Water, H O H NH 3 a b L atm mol L mol 0.034 0.0370 0.07 0.0709 0.0444 0.066 4.70 0.03707 5.464 0.03049