Gases and the Kinetic Molecular Theory PDF

Gases and the Kinetic Molecular Theory 1 Application of the Gas Laws: Breathing For air to move in and out of the lungs, the pressure inside the lungs must change, forcing the lungs to change...

Gases and the Kinetic Molecular Theory 1 Application of the Gas Laws: Breathing For air to move in and out of the lungs, the pressure inside the lungs must change, forcing the lungs to change volume 2 Copyright ©McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 11: Gases and the Kinetic Molecular Theory 11.1 An Overview of the Physical States of Matter 11.2 Gas Pressure and Its Measurement 11.3 The Gas Laws and Their Experimental Foundations 11.4 Rearrangements of the Ideal Gas Law 11.5 The Kinetic-Molecular Theory: A Model for Gas Behavior 11.6 Real Gases: Deviations from Ideal Behavior 3 The Three States of Matter 4 Copyright ©McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. An Overview of the Physical States of Matter Gases can be distinguished from liquids and solids by several aspects of their behaviors Gas volume changes significantly with pressure Solid and liquid volumes are not greatly affected by pressure Gas volume changes significantly with temperature Gases expand when heated and shrink when cooled The volume change is 50 to 100 times greater for gases than for liquids and solids Gases flow very freely Gases have relatively low densities When a gas cools, its density increases because its volume decreases Gases form a solution in any proportions Gases are freely miscible with each other 5 Gas Pressure force Pressure = area Atmospheric pressure arises from the force exerted by atmospheric gases on the earth’s surface Atmospheric pressure decreases with altitude 6 A Mercury Barometer A barometer is used to measure atmospheric pressure At sea level, the column of Hg exerts the same pressure (weight/area) on the Hg surface in the dish as the atmosphere does: PHg = Patm 7 Copyright ©McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Common Units of Pressure Normal Atmospheric Pressure at Sea Unit Level and 0°C pascal (Pa); kilopascal (kPa) 1.01325×105 Pa; 101.325 kPa atmosphere (atm) 1 atm* millimeters of mercury (mmHg) 760 mmHg* torr 760 torr* pounds per square inch (lb/in2 or psi) 14.7 lb/in2 bar 1.01325 bar 1 Pa = 1 N/m2 1 atm = 101325 Pa = 760 mm Hg 1 torr = 1 mm Hg 1 bar = 1 x 105 Pa 8 The Gas Laws The gas laws describe the physical behavior of gases in terms of 4 variables: pressure (P) temperature (T) volume (V) amount (number of moles, n) Three gas laws express the effect of one variable on another, with the remaining two variables held constant An ideal gas is a gas that exhibits linear relationships among these variables No ideal gas actually exists, but most simple gases (e.g., N2, O2, H2) behave nearly ideally at ordinary temperatures and pressures 9 Boyle’s Law 10 Copyright ©Pearson Education, Inc. All rights reserved. Boyle’s Law At constant temperature (isothermal conditions) the volume occupied by a fixed amount of gas is inversely proportional to the external pressure 1 V∝ or PV = constant P At fixed T and n, P decreases as V increases P increases as V decreases 11 Charles’s Law 12 Copyright ©Pearson Education, Inc. All rights reserved. Charles’s Law At constant pressure (isobaric conditions), the volume occupied by a fixed amount of gas is directly proportional to its absolute (Kelvin) temperature V V∝T or = constant T At fixed P and n, V decreases as T decreases V increases as T increases 13 Avogadro’s Law At fixed temperature and pressure, the volume occupied by a gas is directly proportional to the amount (mol) of gas 14 Copyright ©Pearson Education, Inc. All rights reserved. Avogadro’s Law At fixed temperature and pressure, equal volumes of any ideal gas contain equal numbers of particles (or moles) V V∝n or = constant n At fixed P and T, V decreases as n decreases V increases as n increases 15 Gas Behavior at Standard Conditions STP or standard temperature and pressure specifies a pressure of 1 atm (760 torr) and a temperature of 0°C (273.15 K) The standard molar volume is the volume of 1 mol of an ideal gas at STP Standard molar volume = 22.4141 L or 22.4 L 16 Standard Molar Volume 17 Copyright ©McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Gay-Lussac’s Law At fixed volume, the pressure of a given mass of gas varies directly with the absolute temperature of the gas P P∝T or = constant T At fixed V and n, P decreases as T decreases P increases as T increases 18 The Ideal Gas Law PV = nRT R is the universal gas constant; the numerical value of R depends on the units used For 1 mol of gas at STP: PV 1 atm x 22.414L atm∙L R= = = 0.0821 nT 1 mol x 273.15K mol∙K J R = 8.314 mol∙K The ideal gas law can also be expressed by the combined equation: P1V1 P2V2 = n1T1 n2T2 19 Individual Gas Laws as Special Cases 20 Application of the Gas Laws: Breathing Breathing can be explained by Boyle’s law: When the diaphragm moves down, and the rib cage moves out, the volume of the lungs is expanded, which decreases the air pressure inside them The inside pressure is lower than atmospheric pressure: this causes new air to rush in, and we inhale When we breathe out, the reverse happens Exhaling air requires that we relax the diaphragm, which pushes against the lungs and slightly decreases their volume This slightly increases the pressure of the air in the lungs, and air is forced out 21 Copyright ©McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Sample Problem 1 Applying the Volume-Amount Relationship PROBLEM: A scale model of a blimp rises when it is filled with helium to a volume of 55.0 dm3. When 1.10 mol of He is added to the blimp, the volume is 26.2 dm3. How many more grams of He must be added to make it rise? Assume constant T and P. 22 The Ideal Gas Law and Gas Density m m density d = and n (moles) = V ℳ m PV = RT ℳ m ℳP = V RT ℳP d= RT The density of a gas is directly proportional to its molar mass inversely proportional to its temperature directly proportional to the pressure 23 The Ideal Gas Law and Molar Mass m n= ℳ m PV = RT ℳ mRT 𝓜= PV 24 Mixtures of Gases Gases mix homogeneously in any proportions Each gas in a mixture behaves as if it were the only gas present The pressure exerted by each gas in a mixture is called its partial pressure Dalton’s Law of partial pressures states that the total pressure in a mixture is the sum of the partial pressures of the component gases Ptotal = P1 + P2 + … + PN The partial pressure of a gas A is proportional to its mole fraction XA: PA = ΧA x Ptotal nA XA= ntotal 25 Air Composition Percentage composition by volume in dry air: 21% O2; 78% N2; 1% Ar Average molar mass of air: (0.21· 32 + 0.78 · 28 + 0.1 · 40) / (0.21 + 0.78 + 0.1) = 29 g/mol 26 The Ideal Gas Law and Stoichiometry 27 Sample Problem 2 Using Gas Variables to Find Amounts of Reactants and Products PROBLEM: Solid lithium hydroxide is used to "scrub" CO2 from the air in spacecraft and submarines; it reacts with the CO2 to produce lithium carbonate and water. What volume of CO2 at 23°C and 716 torr can be removed by reaction with 395 g of lithium hydroxide? 28 Kinetic-Molecular Theory Kinetic-molecular theory: model that accounts for macroscopic gas behavior at the level of individual particles (atoms or molecules) Postulate 1: Gas particles are tiny with large spaces between them → the volume of each particle is so small compared to the total volume of the gas that it is assumed to be zero Postulate 2: Gas particles are in constant, random, straight-line motion except when they collide with each other or with the container walls Postulate 3: Collisions are elastic, meaning that colliding particles exchange energy but do not lose any energy due to friction → their total kinetic energy (Ek) is constant. Between collisions the particles do not influence each other by attractive or repulsive forces 29 Molecular Speed Each particle has a molecular speed (u) The most probable speed (peak of the curve) increases as the temperature increases 𝐸𝑘 = 𝑐 × 𝑇 Molecular speeds for N2 gas at three temperatures: 30 Copyright ©McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Molar Mass and Molecular Speed At the same T, the most probable molecular speed increases as the molar mass decreases → lighter molecules move faster, on average, than heavier molecules 31 Copyright ©McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Molecular View of the Gas Laws 32 Copyright ©Pearson Education, Inc. All rights reserved. Kinetic Energy and Gas Behavior At a given T, all gases have the same average kinetic energy: 1 Ek = mass × speed2 2 Kinetic energy depends on both the mass and the speed of a particle At the same T, a heavier gas particle moves more slowly than a lighter one At a given T, equimolar samples of any gases exert the same pressure (force per unit area) and, thus, occupy the same volume Temperature is a measure of the average kinetic energy of the particle: 3 𝑅 3 𝐸𝑘 = 𝑇 = 𝑘𝐵 𝑇 2 𝑁𝐴 2 kB = R/NA is the Boltzmann constant where R is the universal gas constant, and NA is the Avogadro’s number kB = 1.38∙10-23 J/K 33 Graham’s Law of Effusion Effusion is the process by which a gas escapes through a small hole in its container into an evacuated space. Diffusion is the movement of one gas (or fluid) through another 34 Copyright ©Pearson Education, Inc. All rights reserved. Real Gases: Deviations from Ideal Behavior The kinetic-molecular model describes the behavior of ideal gases Real gases deviate from this behavior: Real gases have real volume → gas particles are not points of mass, but have volumes determined by the sizes of their atoms and the bonds between them Real gases do experience attractive and repulsive forces between their particles Real gases deviate most from ideal behavior at low temperature and high pressure 35 Molar Volume of Some Common Gases Gas Molar Volume (L/mol) Boiling Point (°C) He 22.435 −268.9 H2 22.432 −252.8 Ne 22.422 −246.1 Ideal gas 22.414 — Ar 22.397 −185.9 N2 22.396 −195.8 O2 22.390 −183.0 CO 22.388 −191.5 Cl2 22.184 −34.0 NH3 22.079 −33.4 At STP (0 °C and 1 atm) 36 Deviations With Increasing External Pressure At moderately high Pext, PV/RT vales are lower than ideal values (1) because of particle volume 37 Copyright ©McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Effect of Interparticle Attractions As Pext rises, V decreases and the particles get closer → interparticle attraction lowers the force of impact with the container wall → the gas pressure is decreased, and PV/RT < 1 38 Copyright ©McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Effect of Particle Volume As Pext rises, the free volume decreases and becomes less than the container volume → V is given by the container volume → PV/RT > 1 39 Copyright ©McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Van der Waals Equation The van der Waals equation adjusts the ideal gas law to take into account the real volume of the gas particles (→ adjust V down) the effect of interparticle attractions (→ adjust P up) Van der Waals equation for n moles of a real gas 𝑛2 𝑎 𝑃 + 2 𝑉 − 𝑛𝑏 = 𝑛𝑅𝑇 𝑉 The constant a relates to factors that influence the attraction between particles The constant b relates to particle volume 40 Van der Waals Constants Gas a​​ (atm  L2/mol2) ​b (L/mol) He 0.034 0.0237 Ne 0.211 0.0171 Ar 1.35 0.0322 Kr 2.32 0.0398 Xe 4.19 0.0511 H2 0.244 0.0266 N2 1.39 0.0391 O2 1.36 0.0318 Cl2 6.49 0.0562 CH4 2.25 0.0428 CO 1.45 0.0395 CO2 3.59 0.0427 NH3 4.17 0.0371 H2O 5.46 0.0305 41