1. 2013 General Chemistry I 1 Chapter 4. THE PROPERTIES OF GASES 2013 General Chemistry I THE NATURE OF GASES THE GAS LAWS 4.1 Observing Gases 4.2 Pressure 4.3 Alternative Units of Pressure 4.4 The Experimental Observations 4.5 Applications of the Ideal Gas Law 4.6 Gas Density 4.7 The Stoichiometry of Reacting Gases 4.8 Mixtures of Gases

2. 2013 General Chemistry I 2 Prelude to Chapters 4 and 5. The Three States of Matter

3. 2013 General Chemistry I 3 An Overview of the Physical States of Matter The Distinction of Gases from Liquids and Solids 1. Gas volume changes greatly with pressure. 2. Gas volume changes greatly with temperature. 3. Gases have relatively low viscosity. 4. Most gases have relatively low densities under normal conditions. 5. Gases are miscible.

4. 2013 General Chemistry I 4 THE NATURE OF GASES (Sections 4.1-4.3) 4.1 Observing Gases Many of physical properties of gases are very similar, regardless of the identity of the gas. Therefore, they can all be described simultaneously. Samples of gases large enough to study are examples of bulk matter – forms of matter that consist of large numbers of molecules Compressibility – the act of reducing the volume of a sample of a gas Expansivity - the ability of a gas to fill the space available to it rapidly Two major properties of gases:

5. 2013 General Chemistry I 5 4.2 Pressure - – SI unit of pressure is the pascal (Pa) – Pressure arises from the collisions of gas molecules on the walls of the container.

6. 2013 General Chemistry I 6  Barometer – A glass tube, sealed at one end, filled with liquid mercury, and inverted into a beaker also containing liquid mercury (Torricelli) where h = the height of a column, d = density of liquid, and g = acceleration of gravity (9.80665 ms -2 ) Measurement of Pressure

7. 2013 General Chemistry I 7  Manometer - Two types of Hg manometer: (a) open-tube and (b) closed tube system This is a U-shaped tube filled with liquid and connected to an experimental system, whose pressure is being monitored.

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10. 10 Example 4.3 135s A student attaches a glass bulb containing neon gas to an open-tube manometer and calculates the pressure of the gas to be 0.890 atm. (a)If the atmospheric pressure is 762 Torr, what height difference between the two sides of the mercury in the manometer did the student find? (b) Which side is higher, the side of the manometer attached to the bulb or the side open to the atmosphere? (c) If the student mistakenly switches the numbers for the sides of the manometer when recording the data in the laboratory notebook, what would be the reported pressure in the gas bulb? 135

11. 2013 General Chemistry I 11 Solution to Exercise 4.3

12. 2013 General Chemistry I 12 4.3 Alternative Units of Pressure - 1 bar = 10 5 Pa = 100 kPa - 1 atm = 760 Torr = 1.01325×10 5 Pa (101.325 kPa) - 1 Torr ~ 1 mmHg mbar Weather map

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14. 2013 General Chemistry I 14 THE GAS LAWS (Sections 4.4-4.6) 4.4 The Experimental Observations  Boyle’s law: For a fixed amount of gas at constant temperature, volume is inversely proportional to pressure. This applies to an isothermal system (constant T) with a fixed amount of gas (constant n).

15. 2013 General Chemistry I 15 - For isothermal changes between two states (1 and 2),

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17. 2013 General Chemistry I 17  Charles’s law: For a fixed amount of gas under constant pressure, the volume varies linearly with the temperature. This applies to an isobaric system (constant P) with a fixed amount of gas (constant n).

18. 2013 General Chemistry I 18 - Kelvin temperature scale T = 0 K = -273.15 o C, when V → 0. - Celsius temperature scale t ( o C) = T (K) - 273.15 0 o C = 273.15 K The Kelvin Scale of Temperature If a Charles’ plot of V versus T (at constant P and n) is extrapolated to V = 0, the intercept on the T axis is ~-273 o C.

19. 2013 General Chemistry I 19 Another aspect of gas behavior (Gay-Lussac’s Law) This applies to an isochoric system (constant V) with a fixed amount of gas (constant n).

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21. 2013 General Chemistry I 21 Avogadro’s Principle  Under the same conditions of temperature and pressure, a given number of gas molecules occupy the same volume regardless of their chemical identity. - This defines molar volume

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23. 2013 General Chemistry I 23 This is formed by combining the laws of Boyle, Charles, Gay-Lussac and Avogadro.  The ideal gas law: Gas constant, R = PV/nT. It is sometimes called a “universal constant” and has the value 8.314 J K -1 mol -1 in SI units, although other units are often used (Table 4.2). The Ideal Gas Law

24. 2013 General Chemistry I 24 -The ideal gas law, PV = nRT, is an equation of state that summarizes the relations describing the response of an ideal gas to changes in pressure, volume, temperature, and amount of molecules; it is an example of a limiting law. (it is strictly valid only in some limit: here, as P  0.) Table 4.2. The Gas Constant, R

25. 2013 General Chemistry I 25 - Standard ambient temperature and pressure (SATP) 298.15 K and 1 bar, molar volume at SATP = 24.79 L·mol -1 - Standard temperature and pressure (STP) 0 o C and 1 atm (273.15 K and 1.01325 bar) - Molar volume at STP - For conditions 1 and 2, - Molar volume 4.5 Applications of the Ideal Gas Law

26. 2013 General Chemistry I 26 EXAMPLE 4.4 In an investigation of the properties of the coolant gas used in an air-conditioning system, a sample of volume 500 mL at 28.0 o C was found to exert a pressure of 92.0 kPa. What pressure will the sample exert when it is compressed to 30 mL and cooled to -5.0 o C?

27. 2013 General Chemistry I 27 Calculating the pressure of a given sample

28. 2013 General Chemistry I 28 Using the combined gas law when one variable is changed

29. 2013 General Chemistry I 29 Using the combined gas law when two variables are changed

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31. 2013 General Chemistry I 31 Example 4.25 143s A sample of methane gas, CH 4, was slowly heated at a constant pressure of 0.90 bar. The volume of the gas was measured at a series of different Temperatures and a plot of volume vs. temperature was constructed. The slope of the line was 2.88×10 -4 L K -1. What was the mass of the sample of methane? Solution

32. 2013 General Chemistry I 32 4.6 Gas Density Molar concentration of a gas at STP (where molar volume is 22.4141 L): Molar concentration of a gas is the number moles divided by the volume occupied by the gas. Density, however, does depend on the identity of the gas. This value is the same for all gases, assuming ideal behavior.

33. 2013 General Chemistry I 33 Density at STP For a given P and T, the greater the molar mass, the greater its density. At constant T, the density increases with P. In this case, P is increased either by adding more material or by compression (reduction of V). Raising T allows a gas to expand at constant P, increases V and therefore reduces its density. Gas Density Relationships

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35. 2013 General Chemistry I 35 -Molar volumes of gases are generally > 1000 times those of liquids and solids. e.g. V m (gases) = ~ 25 L mol -1 ; V m (liquid water) = 18 mL mol -1 -Reactions that produce gases from condensed phases can be explosive. e.g. sodium azide (NaN 3 ) for air bags 4.7 The Stoichiometry of Reacting Gases

36. 2013 General Chemistry I 36 EXAMPLE 4.6 The carbon dioxide generated by the personnel in the artificial atmosphere of submarines and spacecraft must be removed form the air and the oxygen recovered. Submarine design teams have investigated the use of potassium superoxide, KO 2, as an air purifier because this compound reacts with carbon dioxide and releases oxygen: 4 KO 2 (s) + 2 CO 2 (g) → 2 K 2 CO 3 (s) + 3 O 2 (g) Calculate the mass of KO 2 needed to react with 50 L of CO 2 at 25 o C and 1.0 atm. V m = 24.47 Lmol -1 ; 1 mol CO 2 -> 2 mol KO 2 ; M KO2 = 71.10 gmol -1

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38. 2013 General Chemistry I 38 Example 4.59 148s A 15.0-mL sample of ammonia gas at 1.00×10 2 Torr and 30 o C is mixed with 25.0 mL of hydrogen chloride gas at 1.50×10 2 Torr and 25 o C, and the following reaction takes place: NH 3 (g) + HCl(g) NH 4 Cl(s) (a) Calculate the mass of NH 4 Cl that forms.

39. 2013 General Chemistry I 39 148s

40. 2013 General Chemistry I 40 (b) Identify the gas in excess and determine the pressure of the excess gas at 27 o C after the reaction is complete (in the combined volume of the original two flasks). 148s

41. 2013 General Chemistry I 41 4.8 Mixtures of Gases – A mixture of gases that do not react with one another behaves like a single pure gas.  Partial pressure: The total pressure of a mixture of gases is the sum of the partial pressures of its components (John Dalton). P = P A + P B + … for the mixture containing A, B, … - Humid gas: P = P dry air + P water vapor (P water vapor = 47 Torr at 37 o C)  mole fraction: the number of moles of molecules of the gas expressed as a fraction of the total number of moles of molecules in the sample.

42. 2013 General Chemistry I 42 EXAMPLE 4.7 Air is a source of reactants for many chemical processes. To determine how much air is needed for these reactions, it is useful to know the partial pressures of the components. A certain sample of dry air of total mass 1.00 g consists almost entirely of 0.76 g of nitrogen and 0.24 g of oxygen. Calculate the partial pressures of these gases when the total pressure is 0.87 atm.

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44. 2013 General Chemistry I 44 Chapter 4. THE PROPERTIES OF GASES 2012 General Chemistry I MOLECULAR MOTION REAL GASES 4.9 Diffusion and Effusion 4.10 The Kinetic Model of Gases 4.11 The Maxwell Distribution of Speeds 4.12 Deviations from Ideality 4.13 The Liquefaction of Gases 4.14 Equations of State of Real Gases

45. 2013 General Chemistry I 45 MOLECULAR MOTION (Sections 4.9-4.11) 4.9 Diffusion and Effusion  Diffusion: gradual dispersal of one substance through another substance  Effusion: escape of a gas through a small hole into a vacuum

46. 2013 General Chemistry I 46  Graham’s law: At constant T, the rate of effusion of a gas is inversely proportional to the square root of its molar mass: Strictly,Graham’s law relates to effusion, but it can also be used for diffusion. For two gases A and B with molar masses M A and M B,

47. 2013 General Chemistry I 47 Rate of effusion and average speed increase as the square root of the temperature:  Combined relationship: The average speed of molecules in a gas is directly proportional to the square root of the temperature and inversely proportional to the square root of the molar mass.

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49. 2013 General Chemistry I 49 Example 4.73 152s A sample of argon gas effuses through a porous plug in 147 s. Calculate the time required for the same amount of (a) CO 2, (b) C 2 H 4, (c) H 2, and (d) SO 2 to effuse under the same conditions of pressure and temperature.

50. 2013 General Chemistry I 50 4.10 The Kinetic Model of Gases  Kinetic molecular theory (KMT) of gases makes four assumptions: 1. A gas consists of a collection of molecules in continuous random motion. 2. Gas molecules are infinitesimally small points. 3. The molecules move in straight lines until they collide. 4. The molecules do not influence one another except during collisions. - Collision with walls: consider molecules traveling only in one dimensional x with a velocity of v x.

51. 2013 General Chemistry I 51 The change in momentum (final – initial) of one molecule: -2mv x = 2mv x momentum change for the wall All the molecules within a distance v x  t of the wall and traveling toward it will strike the wall during the Interval  t. If the wall has area A, all the particles in a volume Av x  t will reach the wall if they are traveling toward it.

52. 2013 General Chemistry I 52 The number of molecules in the volume Av x  t is that fraction of the total volume V, multiplied by the total number of molecules: The average number of collisions with the wall during the interval  t is half the number in the volume Av x  t: The total momentum change = number of collisions × individual molecule momentum change

53. 2013 General Chemistry I 53 Force = rate of change of momentum = (total momentum change)/  t for the average value of Mean square speed: Pressure on wall:

54. 2013 General Chemistry I 54 where v rms is the root mean square speed, or - The temperature is proportional to the mean square speed of the molecules in a gas. - This was the first acceptable physical interpretation of temperature: a measure of molecular motion.

55. 2013 General Chemistry I 55 EXAMPLE 4.7 What is the root mean square speed of nitrogen Molecules in air at 20 o C?

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57. 2013 General Chemistry I 57 A molecular description of Boyle’s Law 153s

58. 2013 General Chemistry I 58 A molecular description of Charles’s Law 153s

59. 2013 General Chemistry I 59 A molecular description of Avogadro’s Law 153s

60. 2013 General Chemistry I 60 A molecular description of Dalton’s law of partial pressures 153s

61. 2013 General Chemistry I 61 4.11 The Maxwell Distribution of Speeds v = a particle’s speed  N = the number of molecules with speeds in the range between v +  v N = total number of molecules; M = molar mass f(v) = Maxwell distribution of speeds For an infinitesimal range, average speed Maxwell derived equation 22, for calculating the fraction of gas molecules having the speed v at any instant, from the kinetic model.

62. 2013 General Chemistry I 62 - Molar mass (M) dependence: as M increases, the fraction of molecules with speeds greater than a specific speed decreases. - Temperature dependence: as T increases, the fraction of molecules with speeds greater than a specific speed increases.

63. 2013 General Chemistry I 63 REAL GASES (Sections 4.12-4.14) 4.12 Deviations from Ideality - Gases condense to liquids when cooled or compressed (attraction). - Liquids are difficult to compress (repulsion). Deviation from ideal gases - Deviations from the ideal gas law are significant at high pressures and low temperatures (where significant intermolecular interactions exist).

64. 2013 General Chemistry I 64  Compression factor (Z): the ratio of the actual molar volume of the gas to the molar volume of an ideal gas under the same conditions. For an ideal gas, Z = 1 Long range attractions; smaller Z, condensation of gases Short range repulsions; larger Z, low compressibility of liquids and solids, finite molecular volume

65. 2013 General Chemistry I 65 - For many gases, attractions dominate at low pressure (Z 1).

66. 2013 General Chemistry I 66 4.13 The Liquefaction of Gases  Joule-Thomson effect: when attractive forces dominate, a real gas cools as it expands. – In this case expansion requires energy, which comes from the kinetic energy of the gas, lowering the temperature. – The effect is used in some refrigerators and to effect the condensation of gases such as oxygen, nitrogen, and argon.

67. 2013 General Chemistry I 67 The Linde refrigerator for the liquefaction of gases i.e. Adiabatic cooling; temperature decrease under isentropic expansion of any gas (w 0)

68. 2013 General Chemistry I 68 4.14 Equations of State of Real Gases  Virial equation:  van der Waals equation: or pressure reduced due to attractions between pairs of molecules –nb volume excluded since molecules cannot overlap (repulsions) b volume excluded by 1 mol ~ molar volume in the liquid state

69. 2013 General Chemistry I 69 The effect of intermolecular attractions on measured gas pressure P ideal = P + a(n/V) 2 (P: actual pressure) actual pressure is smaller than the ideal pressure 161s

70. 2013 General Chemistry I 70 The effect of molecular volume on measured gas volume V ideal = V - nb actual volume is greater than the ideal volume 161s

71. 2013 General Chemistry I 71 Virial expansion of the van der Waals equation At low particle densities

72. 2013 General Chemistry I 72 Table 4.5Van der Waals Parameters for some Common Gases

73. 2013 General Chemistry I 73  Model of gas 1. A large number of gas molecules in ceaseless, random, and straight motion. 2. The average speed and the spread of speeds increase with T and decrease with m. 3. Molecules travel in straight lines until they collide with other molecules or the container wall. 4. Widely separated. Intermolecular forces have only a weak effect on the properties. 5. Repulsions increase the molar volume, whereas attractions decrease the molar volume.

74. 2013 General Chemistry I 74 EXAMPLE 4.9 Refrigerant gas (a = 16.2 L 2 atm mol –2, b = 0.084 L/mol), 1.50 mol in 5.00 L at 0 o C; Estimate the pressure. Use expanded form of the van der Waals equation: P = nRT/(V – nb) - an 2 /V 2

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