Quantitative Exercise 9.4. Tip 9/14/2015. Quantitative analysis of an ideal gas

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1 Chapter 9 - GASES 9. Quantitative analysis of gas 9.4 emperature 9.5 esting the ideal gas Quantitative analysis of an ideal gas We need more simplifying assumptions. Assume that the particles do not collide with each other they collide only with the walls of the container, exerting pressure on the walls. his is a reasonable assumption for a gas of low density. Assume that the collisions of particles with the walls are elastic. his makes sense, because the pressure of the gas in a closed container remains constant, which would not happen if the particles' kinetic energy decreased. Quantitative analysis of an ideal gas Imagine a gas moving inside a cubic container with sides of length L. he wall exerts a force on the particle, and the particle in turn exerts an equal-magnitude and oppositely directed force on the wall. Quantitative analysis of an ideal gas When the particles have high speed, they (1) hit the walls of the container more frequently and () exert a greater force during the collisions. Both factors lead to a greater pressure. hus it is the speed squared not just the speed of the particles that affects the pressure. ip Quantitative Exercise 9.4 Estimate the average speed of air particles at normal conditions, when the air is at atmospheric pressure (1.0 x 10 5 N/m ) and 1 mole of the air particles (6.0 x 10 molecules) occupies.4 L or.4 x 10 m. Although air is composed of many types of particles, we will assume that the air particles have an average mass m air = 4.8 x 10 6 kg/particle. 1

2 ime interval between collisions Why does it take 5 to 10 minutes for the smell of perfume to travel across a room if the average speed of air particles is v = 480 m/s? he average distance between particles in a gas is about D =. x 10 7 cm. erhaps we cannot assume that the gas particle collisions can be ignored in fact, they collide about 109 times per second. Even though the gas particles are moving fast, their migration from one place to another is slow. emperature emperature is often measured with a liquid thermometer. he bulb and part of the tube are filled with a liquid that expands predictably when heated and that shrinks when cooled. In the experiment, we can hypothesize that the ball expands because the impulses of the particles against the inside walls are larger when the gas is warm. his would happen if the particles were moving faster. Based on this reasoning, we can hypothesize that the of a gas is related to the speed of the random motion of its particles. Data show that /N is the same for all gases in all containers when they are at the same. his leads to the following equation: N = k N = KE Conclusions: /N only depends on the of the gas. emperature is related to the average Kinetic energy per particle of the gas. Difficulty: Kinetic energy is always a positive number.

3 [a] [a] CONDIIONS IN HE BAH /N for one mole of gas in a.4 L container at two different s. RESSURE OLUME /N = k Boltzmann constant "k Named after German hysicist Ludwig Boltzmann Ice water () 1.01x10 5 N/m.4x10 - m.77x10-1 J Boiling water (+100) 1.84x10 5 N/m.4x10 - m 5.154x10-1 J Find the constant "k Find the s of the gas k = 1.8x10 J degree. Absolute (Kelvin) scale and ideal gas law Absolute (Kelvin) scale and ideal gas law = = = = Gas 1 Gas Gas 1 Gas 0000 Linear (Gas 1) Linear (Gas ) [ o C] Linear (Gas 1) Linear (Gas ) [ o C] Absolute (Kelvin) scale and ideal gas law We need a scale on which the zero point is the lowest possible. hat way, all s will be positive. he lowest possible on the new scale is 0; on the Celsius scale, it would be 7.15 C. his scale is called the absolute scale or the Kelvin scale. It was invented by William homson, Lord Kelvin of Largs, in Ideal gas law Because one mole has Avogadro's number of particles, N = nn A. Substituting this, we get:

4 Universal Gas Constant R 6.0 x 10 particles N A = (constant number) mole J 1.8 x 10 k = degree. (constant number) Find the combination of these two constant numbers. R = 8. J K mole Ideal gas law Universal Gas Law N = k N = KE Find an expression for KE emperature and particle motion We can now connect the average kinetic energy of the gas particles to the absolute of the gas: KE = k he of a gas is an indication of the average random translational kinetic energy of the particles in the ideal gas. emperature and particle motion You have two containers holding identical gases that have been sitting in the same room for a long time. One container is large and the other one is small. Which one has a higher? Because the average kinetic energy per particle is the same in each container, the s of the two gases are the same. he total kinetic energy of the particles in the large container is larger because it contains more particles. 4

5 emperature and particle motion What will happen if you mix a container of hot gas with a container of cold gas? he faster-moving particles of the hot gas will collide with the slower-moving particles of the cold gas. Following a collision, on average the faster-moving particle will be moving more slowly than before, and the slower particle will be moving more rapidly. Eventually, the particles of the two gases will have the same average kinetic energy and, therefore, the same. his is called thermal equilibrium. Quantitative Exercise 9.5 Estimate the average speed of air particles in a typical room. Air consists of particles whose average molar mass is 9 x 10 kg/mole. KE = k m = M N A KE = m v Room = 0 o C Microscopic vs Macroscopic approach = Nmv Microscopic quantities that are NO measured directly = nr Macroscopic quantities that can be measured directly ESING HE IDEAL GAS LAW ISOHERMAL ROCESS Boyle s Law = nr = constant ISOHERMAL ROCESS COMLEE HE SHAE OF HE GRAHS 1 1 = 5

6 ISOCHORIC ROCESS Gay-Lussac s Law = nr = nr = constant 1 = 1 ISOCHORIC ROCESS COMLEE HE SHAE OF HE GRAHS ISOBARIC ROCESS Charles Law = nr = nr = constant 1 = 1 ISOBARIC ROCESS COMLEE HE SHAE OF HE GRAHS ACIE LEARNING GUIDE CLASSWORK ALG ALG 9.4. ALG 9.4. ALG ALG HOMEWORK ALG 9..1 ALG 9.. ALG 9.. ALG 9..4 ALG 9..5 ALG 9..6 ALG 9..7 ALG