14 The IDEAL GAS LAW. and KINETIC THEORY Molecular Mass, The Mole, and Avogadro s Number. Atomic Masses

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1 14 The IDEAL GAS LAW and KINETIC THEORY 14.1 Molecular Mass, The Mole, and Avogadro s Number Atomic Masses The SI Unit of mass: An atomic mass unit is de ned as a unit of mass equal to 1/12 of the mass of an atom of carbon-12 ( 12 6 C). It is equal to 1: kg (see below). Symbol: u (in some older texts: amu)

2 A Z X nomenclature: A is the mass number; Z is the atomic number; N is the neutron number: A=Z+N. For a given chemical element Z is xed, e.g. Z=6 for carbon, while N may vary, giving several isotopes. [WEBSITE: UPSCALE at UofT: - look for Nuclear Constants ] Let m be the mass of atom / molecule: m = () molar mass N A number of moles: n = N N A = m mass per mole Periodic Table on the inside back cover of text: carbon in nature: m (C) = 12:011 u N.B. m 12 6 C = 12 u (exactly) Molecular Mass: the sum of the masses of its component atoms, e.g. H 2 O:

3 2(1:00794 u) + 15:9994 u = 18:0153 u N.B. He, Ne, Ar, Xe are monatomic; while H 2, N 2, O 2 are diatomic, as gases in nature. The Mole and Avogadro s Number The SI Unit of amount of substance: A mole is de ned as the amount of substance of a system that contains as many elementary entities as there are atoms in 0:012 kilograms of carbon-12. The elementary entities must be speci ed and may be atoms, molecules, ions, electrons, other particles, or speci ed groups of particles. Symbol: mol Avogadro s number N A = 6: mol 1 [CODATA inside front cover] CAUTION: this SI Unit refers to a GRAM-MOLE of substance! One mole of carbon-12 has a mass of 12

4 grams, or kg. One mole of natural carbon has a mass of grams, or kg. ADDED CALCULATIONS (see Ex. 6 - later) m 12 6 C = 0:012 kg 6: = 1: kg m (1 u) = : kg = 1: kg If we use a kilogram-mole we must count 1000 times as many molecules: 6: = 6: m (H) = 1:00794 kg 6: = 1: kg which is slightly more than the mass of 1 1H, because of deuterium, 2 1 H. m (proton) = 1: kg Ex. 1: Hope Diamond - monatomic carbon

5 (44:5 carats) 0:2 g 1 carat = 8:9 g 8:9 g 1 mol 12:011 g = 0: 741 mol 0: 741 mol 6: atoms 1 mol = 4: atoms Rosser Reeves Ruby - Al 2 O 3 m (Al 2 O 3 ) = 2 (26:9815 u) + 3 (15:9994 u) = 101:96 u (138 carats) 0:2 g 1 carat = 138=5 g = 27: 6 g 27: 6 g 1 mol 101:96 g = 0: 271 mol 0: 271 mol 6: molecules 1 mol = 1: molecules 14.2 The IDEAL GAS LAW

6 Recollect Chapter 12 and Figures 12.3, 12.4, but now use the Kelvin scale instead of Celsius DRAW FIG PRESSURE versus TEMPERATURE at CONSTANT VOLUME The graph is a straight line which extrapolates to absolute zero. Also the pressure is proportional to the number of molecules, and inversely proportional to the volume (Boyle s Law). We assume that the density is low enough so that the molecules do not interact except during elastic collisions. GOOD SIMULATION ON WEB:

7 IDEAL GAS LAW: The absolute pressure P of an ideal gas is directly proportional to the Kelvin temperature T and the number of moles n of the gas and is inversely proportional to the volume of the gas: P V = nrt (14:1) where R is the universal gas constant and has the value 8.31 J= mol K 1 in SI units. If N is the total number of particles: N = nn A, so P V = nn A RNA T = NkT where Boltzmann s constant k = R NA = 1: J K 1 Then the ideal gas law is P V = NkT (14:2)

8 Ex. 2: Alveoli air sacs absolute pressure: P = 10 5 Pa radius: r = 0:125 mm, V = 4 3 r3 body temperature: T = 310 K h N = P kt V = (105 Pa) 43 (0: m) 3i (1: J K 1 )(310 K) = 1: Since air is 14% O 2, (85% N 2, 15% Argon), number of oxygen molecules =0:14N = 2: CEx. READ 3 - Beer Bubbles expand when they rise!-! Problem 22

9 CO 2 bubble at depth of 0.2 m: volume V 2 ; pressure P 2 = P 1 + gh, P 1 = P atm = 1: Pa Ideal gas law (or Boyle s law): P 1 V 1 = P 2 V 2 V 1 V = P 2 2 P = P atm+gh 1 1: P atm = 1+ gh P atm = 1+ (1000)(9:8)(0:2) 1: = i.e. a 2% increase in volume. Boyle s Law - TRSP FIG P _ 1=V P i V i = P f V f (14:3) Ex. 4: Scuba Diving - READ Charles Law (Gay-Lussac) V i T I = V f T f (14:4)

10 14.3 KINETIC THEORY of GASES The Distribution of Molecular Speeds TRSP FIG Maxwell-Boltzmann curves Kinetic Theory TRSP FIG The essence of this idea goes back to Newton:. Calculate the pressure due to molecules impacting the walls of a container of gas (assume square of side L). Consider motion along the x -direction: average force = final momentum initial momentum time between collisions = ( mv) (mv) 2L=v = mv2 L Total force: F = N mv 2 3 L

11 where v 2 is the square of the root-mean-square speed: v rms = v 21 2 Then P = F L 2 = N mv 2 rms 3 L 3 Since V = L 3, P V = 2 3 N 1 2 mv 2 rms (14:5) We see that the bracket is just the average translational kinetic energy KE = 1 2 mv2 rms : P V = 2 3 N KE Comparing with the ideal gas law, P V = NkT : KE = 1 2 mv2 rms = 3 2kT (14:6) THUS the absolute Kelvin temperature is expressed in terms of the mechanics of particles. They move faster when the temperature is hotter, and vice versa.

12 CEx.5 - A single particle does not have a temperature. Ex. 6: Speeds of Molecules (14:6)! v rms = 1 2KE 2 m Take T = 293 K: KE = 3 2 kt = 3 2 1: J K 1 (293 K) = 6: J () m = molar mass N A v rms (N 2 ) = 9 m= s v rms (O 2 ) = 9 m= s 2(6: J) 2 = 510: 28 g mol 1 =(6: mol 1 ) 2(6: J) 2 = 477: 32 g mol 1 =(6: mol 1 )

13 v rms (H 2 ) = 6 m= s 2(6: J) 2 =1911: 2 g mol 1 =(6: mol 1 ) CEx. 7 - Hydrogen escapes the atmosphere of Earth. The escape velocity is 2GM R 1 2 = 2(6: )(5: ) 2 6:3810 = 11182: m= s 6!Problem 32: At what temperature is v rms (H 2 ) of hydrogen equal to this escape velocity? 1911 m= s 1 T K = m= s! T = K = 10032: K ALT: (14:6)! T = mv2 rms 3k

14 ! 2 g mol 1 =(6: mol 1 )(11182: m= s) 2 3(1: J K 1 ) = K Brownian motion of pollen in water - Einstein in 1905 showed that this was due to impacts of molecules. The Internal Energy of a Monatomic Ideal Gas This is useful in Chapter 15. For a monatomic ideal gas: U = N A 32 kt, or since N A k = R: U = 2 3 nrt (14:7) *14.4 Di usion - READ IF INTERESTED- (MCAT?) [TRSP FIG ]

Chapter 14. The Ideal Gas Law and Kinetic Theory

Chapter 14. The Ideal Gas Law and Kinetic Theory Chapter 14 The Ideal Gas Law and Kinetic Theory 14.1 Molecular Mass, the Mole, and Avogadro s Number To facilitate comparison of the mass of one atom with another, a mass scale know as the atomic mass