Unit 6: Gases & The Kinetic Molecular Theory

1. CHM 1045: General Chemistry and Qualitative Analysis Unit 6:Gases & The Kinetic Molecular Theory Dr. Jorge L. Alonso Miami-Dade College – Kendall Campus Miami, FL • Textbook Reference: • Module # 8

2. Characteristics of Gases • Unlike liquids and solids, gases . . . . • Are highly compressible. • Expand to fill their containers. • Have extremely low densities. Condensed phases

3. Characteristics of Gases • Variables affecting the behavior of gases • Amount = number of moles () • Pressure (P) • Volume (V) • Temperature (T in Kelvin) {PropGases*}

4. F A P = Pressure Force = mass x acceleration Newton = 1kg . m/sec2 105 Newtons = (104kg)(10 m/sec2) • Pressure is the amount of force applied to an area. Approx. 12 miles 105 Newtons meter2 = 101.325 kPa = • Atmospheric pressure is the weight of air per unit of area.

5. Units of Pressure Torricelli’s • Atmosphere • 1.00 atm • = 760 mm Hg (torr) • = 101.325 kPa 760 mm Hg = weight of equal surface area of the atmosphere (Normal atmospheric pressure at sea level).

6. Barometer 33 ft H2O = weight of equal surface area of the atmosphere

7. Manometer instrument used to measure the difference in pressure between atmospheric pressure and that of a gas in a vessel. {Manometer}

8. Manometer Used to measure the difference in pressure between atmospheric pressure and that of a gas in a vessel. Pgas = 760 - 6 torrs Pgas = 760 torr Pgas = 760 + 6 torrs

9. Gas Laws Variables affecting gases: moles (η), pressure (P), volume (V) and Temperature (T) • Boyle’s Law • Compared: P versus V ( & T are held constant). • Charles’s Law • Compared: V versus T ( & P are held constant). • Avogadro’s Law • Compared: Vversusη(P & T are held constant). • Combined Gas Law • Compared: P vs Vvs. T ( is held constant). • Ideal Gas Law • Compared: P vs Vvs. ηvsT (no variable held constant). • Dalton’s Law of Partial Pressure • Compared: individual pressures of gases in a mixture

10. 1 P V Boyle’s Law: Pressure-Volume Relationship ( & T are held constant). 2 x The volume of a fixed quantity of gas at constant temperature is inversely proportional to the pressure. V ? V x ½ {Boyle’s Law}

11. 1 P k P V V= P ↑V ↓ = k • Also, P & V: inversely proportional OR • This means a plot of V versus 1/P will be a straight line. {PV.Graphs}

12. Boyle’s Law 

13. V T Charles’s Law: Temp. – Volume Relationship ( & P are held constant). • The volume of a fixed amount of gas at constant pressure is directly proportional to its absolute temperature. V ? V 2 x T 2 x {*Charles’s Law Liq N2}

14. V T = k V = kT VT or Charles’s Law • The volume of a gas is directly proportional to its absolute temperature. A plot of V versus T will be a straight line.

15. Charles’s Law

16. {AvogLaw} Avogadro’s Law: Moles-Volume Relationship (P & T are held constant). Vn V ? V 2 x  2 x • The volume is directly proportional to the number of moles of the gas. {Avogadro’s Law}

17. V = kn Vn or, • Mathematically, this means Avogadro’s Law {*Avogadro’s Law in Reactions}

18. Standard Temperature & Pressure (STP) and Molar Volume • Standard Temperature: 00C or 273K • Standard Pressure: 760 torr (1 atm) At STP the Molar Volume of any gas is 22.4 L (11.1 in)3 or (28.2cm)3 1 mole = 6.022 x 1023 part. = gMM = 22.4 L

19. Standard Temperature & Pressure (STP) and Molar Volume At STP the Molar Volume of any gas is 22.4 L 1 mole = 6.023 x 1023 part. = gMM = 22.4 L H2 = 2.0g O2 = 32.0g CO2 = 44.0 g Problem: At STP, what volume in mL would 75g of CO2 occupy?

20. knT P nT P V nT P V V= k = Ideal-Gas Equation The Gas Laws: V 1/P (Boyle’s law) VT (Charles’s law) Vn (Avogadro’s law) Combining these, we get or or

21. nT P V P V nT 1 R R = k = Ideal-Gas Equation = The relationship then becomes PV = nRT

22. Ideal-Gas Equation: Useful for pure gas under one set of conditions. PV= nRT (torr) (L) = (mol) R (K) Units: {PV= nRT RapVideo} RapVideoLinkYouTube

23. Ideal Gas Law Problems What volume (in mL) would a 2.20 g sample of hydrogen gas (H2) at 50.00C occupying at 443 torr? PV= nRT V= nRT P R V = = 50.0 L

24. Ideal-Gas Equation: Densities of Gases PV = nRT For Ideal Gas Equation: Since Then and Dividing both sides of the equation on the left by V we get ( ) Where d = Density of Gas If we solve the equation for density, we get……..

25. Ideal-Gas Equation: Densities and Molecular Weigh of Gases Problem: What is the density of the oxygen in a tank in an AC room (25°C) and whose pressure gauge reads 25.0 atm Problem: A gas whose density is 0.0131 g/mL and is in a container at room temperature and whose pressure gauge reads 1.9 x 104 mmHg. What is its MW?

26. Ideal-Gas Equation: Densities & Molecular Weigh of Gases Problems What is the density (in g/mL) of SO2 at STP? PV = nRT ( ) = 2.62 g/L =

27. Ideal-Gas Equation: Densities & Molecular Weigh of Gases Problems What is the molecular weight of a gas whose density @ STP is 7.78 g/L? PV = nRT ( ) =

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30. Combining, we can get P2V2 T2 P1V1 T1 Combined Gas Law Equation The Gas Laws: V 1/P (Boyle’s law) VT (Charles’s law) The Combined Gas Law k = Useful for a constant amount of a pure gas under two different conditions.

31. P1V1 T1 P2V2 T2 = Combined Gas Law Equation Constant

32. P1V1 T1 P2V2 T2 = Combined Gas Law Problem A scuba diver takes a gas filled 1.0 L balloon from the surface where the temperature is 34 0C down to a depth of 66 ft (33 ft H2O = 1 atm). What volume will the gas balloon have at that depth if the temperature is 15 0C? V2 V2

33. Dalton’s Law ofPartial Pressures • The total pressure of a mixture of gases equals the sum of the pressures that each would exert if it were present alone. • In other words, • Ptotal = P1 + P2 + P3 + … • Pair = P N2 + PO2 + PH2O + …

34. Partial Pressures P of gas P of atm • When one collects a gas over water, there is water vapor mixed in with the gas. • To find only the pressure of the desired gas, one must subtract the vapor pressure of water from the total pressure.

35. Vapor Pressure of Water • Daltons Law: Ptotal = Pgas + PH2O • To find only the pressure of the desired gas, one must subtract the vapor pressure of water from the total pressure. Pgas = Ptotal - PH2O {Press on can}

36. Evaporation vs Boiling in terms of Vapor Pressure Patm Patm + + Pvap Pvap Vapor Pressure (v.p. or Pvap) Patm • Caused by the tendency of solids & liquids to evaporate to gaseous form. It is temperature (K.E.) dependent. = Pvap

37. Stoichiometry with Gases Mg (s) + 2HCl (aq) MgCl2 (aq) + H2 (g) Problem: If 2.0 g of Mg are reacted with excess HCl, what volume of H2 will be produced at 250C and 775 torr? At STP? PV = nRT

38. Kinetic-Molecular Theory A model that aids in our understanding of what happens to gas particles as environmental variables change. Main Tenets: • Gases consist of large numbers of molecules that are in continuous, random motion. 2.Collisionsbetween gas molecules and between gas molecules and the walls of the container mustbe completely elastic(energy may be transferred between molecules, but none is lost).

39. Kinetic-Molecular Theory Main Tenets: 3.Attractive and repulsive forces between gas molecules are negligible. 4. The combined volumeof all the molecules of the gasis negligible (excluded volume) relative to the total volume in which the gas is contained.

40. Kinetic-Molecular Theory Main Tenets: 5. Energy can be transferred between molecules during collisions, but the average kinetic energy of the molecules does not change with time, as long as the temperature of the gas remains constant. @ 100 0C 6. The average kinetic energy (KE=½mv2) of the molecules is proportional to the absolute temperature. {KE T(K)}

41. Diffusion Effusion Movement of molecules from an area of high concentration to an area of low concentration until equilibrium is reached (homogeneity). The escape (diffusion) of gas molecules through a tiny hole into an evacuated space.

42. Effect of Molecular Mass on Rate of Effusion and Diffusion Thomas Graham (1846): rate of diffusion is inversely proportional to the square root of its molar mass Kinetic Energy per individual molecule:

43. Dropper with Br (l) Rate of Diffusion & Effusion Thomas Graham (1846): rate of diffusion is inversely proportional to the square root of its molar mass {BrDiffusion} Comparing the rates of two gases: Graham’s Law of Diffusion and Effusion of Gases {GasDiff}

44. Effusion and Diffusion • This is the most widespread uranium enrichmentmethod. Uranium is reacted with fluorine to make uranium hexafluoride gas: 235UF6 & 238UF6 • The physical principle is that the diffusion speed of a gas molecule depends on the mass of the molecule: the lighter ones diffuse faster and get through a porous material easier. • In gas diffusion units, uranium-hexafluoride gas diffuses through an etched foil made of either an aluminum alloy or teflon, due to artificially maintained difference in pressure. The lighter molecules(i.e. those containing 235U) get through easier to the other side, therefore the gas accumulating there will be richer in 235U.

45. Gas Centrifugation The gas centrifuge is essentially a bowl, in which there is a rotor spinning round at a very high speed. The gas (UF6) directed to the centrifuge is forced to spin by the rotor. Due to the centrifugal force the heavier molecules (those which contain 238U) will accumulate near the wall of the bowl, while the lighter molecules containing 235U will stay closer to the center of the centrifuge.  

46. Boltzmann Distributions The Maxwell–Boltzmann distribution is the statistical distribution of molecular speeds in a gas. It corresponds to the most probable speed distribution in a collisionally-dominated system consisting of a large number of non-interacting particles. {Boltzman Plot}

47. Kinetic Energy of Gas Molecules Kinetic Energy per individual molecule: ☺ ☺ Kinetic Energy per mole: Combining above equations and solving for velocity we get: • The root-mean square velocity of gases is a very close approximation to the average gas velocity. ☺ • To calculate this correctly: • The value of R = 8.314 kg m2/s2 K mol • Mm= molar mass, and it must be in kg/mol.

48. The Kinetic-Molecular Theory • Example: What is the root mean square velocity of N2 molecules at room T, 25.0oC? • To calculate this correctly: • The value of R = 8.314 kg m2/s2 K mol • And M must be in kg/mol.

49. The Kinetic-Molecular Theory Problem: What is the root mean square velocity of He atoms at room T, 25.0oC? You do it! • To calculate this correctly: • The value of R = 8.314 kg m2/s2 K mol • And M must be in kg/mol. GasMW He 4 N2 28 SF6 146 • Can you think of a physical situation that proves He molecules have a velocity that is so much greater than N2 molecules? • What happens to your voice when you breathe He(g) or SF6 (g)?

50. Ideal vs.Real Gases In the real world, the behavior of gases only conforms to the ideal-gas equation at relatively high temperature and low pressure.