Chapter 3 The Kinetic Theory of Gases 3.1. Ideal Gases Experimental Laws and the Equation of State

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1 Chater 3 The Knetc Theory of Gases 3.1. Ideal Gases Exermental Laws and the Equaton of State Molecular Model of an Ideal Gas 3.3. Mean Free Path 3.4. The Boltzmann Dstrbuton Law and The Dstrbuton of Molecular Seeds 3.5. The Molar Secfc Heats of an Ideal Gas 3.6. The Equartton of Energy Theorem 3.7. The dabatc Exanson of an Ideal Gas

2 Overvew In ths chater, we consder the hyscs of gases at the mcroscoc level: gas conssts of atoms that fll ther contaner s volume and the volume s a result of the freedom of the atoms sread throughout the contaner. The temerature s a measure of the knetc energy of the atoms. The ressure exerted by a gas s roduced by the collsons of the atoms wth the contaner s wall. The knetc theory of gases relates the moton of the atoms to the volume, ressure, and temerature of the gas.

3 Ideal gases truly deal gas does not exst n nature. But all real gases aroach the deal state at low enough denstes: + Molecules are far enough aart, so they do not nteract wth one another. + n deal gas obeys the deal gas law (see next sldes).

4 vogadro s Number 3.1. Ideal Gases vogadro s law Equal volumes of gases, at the same temerature and ressure, contan the same number of molecules. Useful unt to measure the szes of our samles s mole. One mole s the number of atoms n a 12 g samle of carbon-12 In chater 2, secton 2.3, one mole contans 6.02 x elementary unts N mol N : vogadro s number

5 Number of moles contaned n a samle of any substance: n N N where N s the number of molecules n the samle n M samle M M samle mn where M samle s the mass of the samle M s the molar mass (the mass of 1 mol) m s the mass of one molecule

6 Problem 1 (. 530) Fnd the mass n klograms of 7.5x10 24 atoms of arsenc (s), whch has molar mass of 74.9 g/mol. The number of moles n: n M M arsenc 24 N Marsenc nm M 23 N (g)

7 Exermental Laws and the Equaton of State Boyle s Law (Boyle-Marotte): For a gven mass, at constant temerature (sothermal), the ressure tmes the volume s a constant for an deal gas. constant Robert Boyle ( ) Glenn Research Center/NS

8 Charles s Law: For a gven mass, at constant ressure (sobarc), the volume s drectly roortonal to the temerature. constant T Jacques Charles ( ) Glenn Research Center/NS

9 Gay-Lussac s Law: For a gven mass, at constant volume (sochorc), the ressure s drectly roortonal to the temerature. constant T J. L. Gay-Lussac ( ) Glenn Research Center/NS

10 Equaton of State: Boyle s Law: constant Charles s Law: constant T Gay-Lussac s Law: constant T The gas laws of Boyle, Charles and Gay-Lussac can be combned nto a sngle equaton of state: nrt (deal gas law) where s the absolute ressure n s the number of moles of gas T s the temerature (n K) R s a constant, called the gas constant R 8.31J mol 1 1 K

11 The Boltzmann constant k s: nr k N N R N kn 8.31J mol kn 23 K mol J K (N s the number of molecules) -1 NkT (deal gas law)

12 Samle Problem (. 510) cylnder contans 12 L of oxygen at 20 0 C and 15 atm. The temerature s rased to 35 0 C, and the volume s reduced to 8.5 L. What s the fnal ressure of the gas n atmosheres? ssume that the gas s deal. Key equaton: t state : t the fnal state f: nrt f nrt f nrt f f f f f T We must convert temeratures n C 0 to that n K: T (273 T T T 20) K 293 K; T f (273 35) K 308 (15 atm)(308 K)(12 L) f 22.3 atm (293 K)(8.5 L) f f K

13 Problem 4 (. 530) quantty of deal gas at 10 0 C and 100 kpa occues a volume of 3.0 m 3. (a) How many moles of the gas are resent? (b) If the ressure s now rased to 300 kpa and the temerature s rased to 30 0 C, how much volume does the gas occuy? ssume no leaks. T 2 (a) nrt ; n RT 100 n kpa R Pa; (b) t any state (,, T): T K : 1 2 T J 3.0 m nr 3 mol ;T (moles) nrt 11 T 2 T 1 1 K K T (m T )

14 Problem 5 (. 530) The best laboratory vacuum has a ressure of about atm, or 1.01x10-13 Pa. How many gas molecules are there er cubc centmeter n such a vacuum at 293 K? nrt The number of moles: n RT ; R 8.31J mol 1 1 K Pa; 1cm m ;T 293 K The number of molecules: N n N N (molecules)

15 Chater 1: densty: Revew ρ m ressure: F fluds at rest: 0 ρgh absolute ressure atmoshere ressure gauge ressure Pascal s law: F F 0 0 rchmede s rncal: F b ρ flud g : buoyant force

16 Equaton of contnuty: v1 2v2 1 constant Bernoull s equaton: Chater 2: ρv ρgy1 2 ρv ρgy T(K) T(C 0 ) constant thermal exanson: Lnear exanson: (solds) rea exanson: (solds) olume exanson: (solds and lquds) Δ ΔL LT Δ T; 2 T; 3 Heat caacty: Q C ΔT C (T f T )

17 secfc caacty: Q cm ΔT cm (T f T ) Latent heat: Q Lm Heat Transfer Mechansms: Conducton: P cond Q t k T H T L C (Unt: W = J/s) steady-state: P cond k 2 (T L H 2 - T X ) k 1 (T L X 1 - T C ) If the slab conssts of n materals: P cond (T n 1 H - T (L /k C ) )

18 Chater 3: Three secal cases: 1. dabatc rocesses: Q 0 ΔEnt W 2. Constant-volume (sochorc) rocesses: W 0 ΔE Q 3. Cyclcal rocesses: ΔE nt 0 Q W nt

19 Work done by the gas: -Exanson: W 0 -Comresson: W 0 Energy transferred as heat Q: -Heat transferred to the gas (recevng energy as heat): Q 0 -Heat transferred from the gas (releasng energy as heat): Q 0 B W area BC W 0 C W area BC W 0 B C