2. The Gas Laws

3. Kinetic Theory  All molecules are in constant motion.  Evidence: Perfume molecules moving across a room..

4. 1. A gas is composed of particles molecules or atoms Considered to be hard spheres far enough apart that we can ignore their volume. Between the molecules is empty space. The Kinetic Theory of Gases Makes three descriptions of gas particles

5. 2. The particles are in constant random motion. Move in straight lines until they bounce off each other or the walls. 3. All collisions are perfectly elastic

6. For example: Average speed of an oxygen molecule is 1656 km/hr at 20ºC The molecules don’t travel very far without hitting each other so they move in random directions.

7. Kinetic Energy and Temperature Temperature is a measure of the Average kinetic energy of the molecules of a substance. Higher temperature faster molecules. At absolute zero (0 K) all molecular motion would stop.

8. Kinetic Molecular Theory Postulates Evidence 1. Gases are tiny molecules in mostly empty space. The compressibility of gases. 2. There are no attractive forces between molecules. Gases do not clump. 3. The molecules move in constant, rapid, random, straight-line motion. Gases mix rapidly. 4. The molecules collide classically with container walls and one another. Gases exert pressure that does not diminish over time. 5. The average kinetic energy of the molecules is proportional to the Kelvin temperature of the sample. Charles’ Law

9. 8 Elastic vs. Inelastic Collisions 8 v1v1 elastic collision inelastic collision POW v2v2 v3v3 v4v4

10. Characteristics of Gases Gases expand to fill any container. – random motion, no attraction Gases are fluids (like liquids). – no attraction Gases have very low densities. – no volume = lots of empty space Courtesy Christy Johannesson www.nisd.net/communicationsarts/pages/chem

11. Characteristics of Gases Gases can be compressed. –n–no volume = lots of empty space Gases undergo diffusion & effusion. –r–random motion Courtesy Christy Johannesson www.nisd.net/communicationsarts/pages/chem

12. Kinetic Molecular Theory Ideal gas law… – have no volume. – have elastic collisions. – are in constant, random, straight-line motion. – don’t attract or repel each other. – have an avg. KE directly related to Kelvin temperature Courtesy Christy Johannesson www.nisd.net/communicationsarts/pages/chem

13. Real Gases Particles in a REAL gas… – have their own volume – attract each other Gas behavior is most ideal… – at low pressures – at high temperatures – in nonpolar atoms/molecules Courtesy Christy Johannesson www.nisd.net/communicationsarts/pages/chem

14. Properties of Gases V = volume of the gas (liters, L) T = temperature (Kelvin, K) P = pressure (atmospheres, atm) n = amount (moles, mol) R = gas constant (R = 0.082057 L atm / (mol K Ideal gases PV = nRT Gas properties can be modeled using math. Model depends on:

15. Kinetic theory of gases and … Compressibility of Gases Boyle’s Law P  collision rate with wall Collision rate  number density Number density  1/V P  1/V Charles’ Law P  collision rate with wall Collision rate  average kinetic energy of gas molecules Average kinetic energy  T P  TP  T

16. Kinetic theory of gases and … Avogadro’s Law P  collision rate with wall Collision rate  number density Number density  n P  nP  n Dalton’s Law of Partial Pressures Molecules do not attract or repel one another P exerted by one type of molecule is unaffected by the presence of another gas P total =  P i

17. Ideal Gas Law Equation

18. Direct vs. Indirect Charles’ LawPV = nRT Gay-Lussac’s LawPV = nRT Boyles’ LawPV = nRT Direct Indirect

19. Kinetic Theory and the Gas Laws Dorin, Demmin, Gabel, Chemistry The Study of Matter, 3 rd Edition, 1990, page 323 (newer book) original temperature original pressure original volume increased temperature increased pressure original volume increased temperature original pressure increased volume (a)(b)(c) 10

20. Molar Volume Timberlake, Chemistry 7 th Edition, page 268 1 mol of a gas @ STP has a volume of 22.4 L 273 K n He = 1 mole (4.0 g) V He = 22.4 L P = 1 atm 2 2 273 K n N = 1 mole (28.0 g) V N = 22.4 L P = 1 atm 2 2 273 K n O = 1 mole (32.0 g) V O = 22.4 L P = 1 atm 2 2

21. V 2 V Volume and Number of Moles Zumdahl, Zumdahl, DeCoste, World of Chemistry  2002, page 413 3 V n = 1 n = 2 n = 3

22. A Gas Sample is Compressed Zumdahl, Zumdahl, DeCoste, World of Chemistry  2002, page 429

23. Avogadro's Hypothesis Equal volumes of gases at the same T and P have the same number of molecules. V = n(RT/P) = kn This means, for example, that number of moles goes up as volume goes up. *Amedeo Avogadro (1776 - 1856) 1 mole = 6.022 x 10 23 Vn V and n are directly related. twice as many molecules

24. Avogadro’s Hypothesis N2N2 H2H2 Ar CH 4 At the same temperature and pressure, equal volumes of different gases contain the same number of molecules. Each balloon holds 1.0 L of gas at 20 o C and 1 atm pressure. Each contains 0.045 mol or 2.69 x 10 22 molecules of gas.

25. Volume vs. Quantity of Gas 0 0.2 0.4 0.6 0.8 1.0 Volume (L) 2 4 6 8 10 14 Number of moles 12 16 18 20 22 24 26 The graph shows there is a direct relationship between the volume and quantity of gas. Whenever the quantity of gas is increased, the volume will increase. 1 mole = 22.4 L @ STP

26. Same Gas, Volume, and Temperature, but… Dorin, Demmin, Gabel, Chemistry The Study of Matter, 3 rd Edition, 1990, page 316

27. Same Gas, Volume, and Temperature, but… different numbers of moles Dorin, Demmin, Gabel, Chemistry The Study of Matter, 3 rd Edition, 1990, page 316

28. Adding and Removing Gases 100 kPa 200 kPa 100 kPaDecreasing Pressure

29. In a smaller container - molecules have less room to move. They hit the sides of the container more often. This causes an increase in pressure. As volume decreases: pressure increases. Changing the Size of the Container

30. Pressure Is caused by the collisions of molecules with the walls of a container force/unit area SI units = Newton/meter 2 = 1 Pascal (Pa)

31. Pressure KEY UNITS AT SEA LEVEL 101.325 kPa (kilopascal) or 101,325 Pa 1 atm 760 mm Hg 760 torr 14.7 psi Courtesy Christy Johannesson www.nisd.net/communicationsarts/pages/chem Sea level

32. Evangelista Torricelli Developed first method of measuring pressure… The device was called a “barometer” Baro = weight Meter = measure Measuring Pressure

33. Barometer Vacuum Height of column 29.92 in. (76 cm) Air pressure Mercury

34. Barometer- measures atmospheric pressure Zumdahl, Zumdahl, DeCoste, World of Chemistry  2002, page 401 Empty space (a vacuum) Hg Weight of the mercury in the column Weight of the atmosphere (atmospheric pressure)

35. Barometers Mount Everest Sea level On top of Mount Everest Sea level fraction of 1 atm average altitude (m)(ft) 100 1/25,48618,000 1/38,37627,480 1/1016,13252,926 1/10030,901101,381 1/100048,467159,013 1/1000069,464227,899 1/10000096,282283,076

36. Manometer –measures contained gas pressure U-tube Manometer Courtesy Christy Johannesson www.nisd.net/communicationsarts/pages/chem

37. lower pressure higher pressure Manometer P1P1 PaPa height 750 mm Hg 130 mm higher pressure 880 mm Hg P a = h = +- lower pressure 620 mm Hg P 1 = P a P 1 < P a

38. Manometer PbPb PaPa 750 mm Hg P a =

39. lower pressure Manometer PaPa height 70 mm Hg 130 mm lower pressure 620 mm Hg P a = h = -

40. 880 mm Hg higher pressure higher pressure Manometer PaPa height 750 mm Hg 130 mm P a = h = +

41. Reading a Vernier A Vernier allows a precise reading of some value. In the figure to the left, the Vernier moves up and down to measure a position on the scale. This could be part of a barometer which reads atmospheric pressure. The "pointer" is the line on the vernier labelled "0". Thus the measured position is almost exactly 756 in whatever units the scale is calibrated in. If you look closely you will see that the distance between the divisions on the vernier are not the same as the divisions on the scale. The 0 line on the vernier lines up at 756 on the scale, but the 10 line on the vernier lines up at 765 on the scale. Thus the distance between the divisions on the vernier are 90% of the distance between the divisions on the scale. 756 750 760 770 Scale 5 0 10 Vernier http://www.upscale.utoronto.ca/PVB/Harrison/Vernier/Vernier.html

42. If we do another reading with the vernier at a different position, the pointer, the line marked 0, may not line up exactly with one of the lines on the scale. Here the "pointer" lines up at approximately 746.5 on the scale. If you look you will see that only one line on the vernier lines up exactly with one of the lines on the scale, the 5 line. This means that our first guess was correct: the reading is 746.5. 5 0 10 750 740 760 What is the reading now? 741.9 http://www.upscale.utoronto.ca/PVB/Harrison/Vernier/Vernier.html

43. 750 740 760 If we do another reading with the vernier at a different position, the pointer, the line marked 0, may not line up exactly with one of the lines on the scale. Here the "pointer" lines up at approximately 746.5 on the scale. If you look you will see that only one line on the vernier lines up exactly with one of the lines on the scale, the 5 line. This means that our first guess was correct: the reading is 746.5. 5 0 10 What is the reading now? 756.0 http://www.upscale.utoronto.ca/PVB/Harrison/Vernier/Vernier.html

44. 750 740 760 Here is a final example, with the vernier at yet another position. The pointer points to a value that is obviously greater than 751.5 and also less than 752.0. Looking for divisions on the vernier that match a division on the scale, the 8 line matches fairly closely. So the reading is about 751.8. In fact, the 8 line on the vernier appears to be a little bit above the corresponding line on the scale. The 8 line on the vernier is clearly somewhat below the corresponding line of the scale. So with sharp eyes one might report this reading as 751.82 ± 0.02. This "reading error" of ± 0.02 is probably the correct error of precision to specify for all measurements done with this apparatus. 5 0 10 http://www.upscale.utoronto.ca/PVB/Harrison/Vernier/Vernier.html

45. Dalton’s Law of Partial Pressures V and T are constant P1P1 P2P2 P total = P 1 + P 2

46. Consider a case in which two gases, A and B, are in a container of volume V. P A = n A RT V P B = n B RT V n A is the number of moles of A n B is the number of moles of B P T = P A + P B X A = nAnA n A + n B X B = nBnB n A + n B P A = X A P T P B = X B P T P i = X i P T

47. A sample of natural gas contains 8.24 moles of CH 4, 0.421 moles of C 2 H 6, and 0.116 moles of C 3 H 8. If the total pressure of the gases is 1.37 atm, what is the partial pressure of propane (C 3 H 8 )? P i = X i P T X propane = 0.116 8.24 + 0.421 + 0.116 P T = 1.37 atm = 0.0132 P propane = 0.0132 x 1.37 atm= 0.0181 atm

48. A. Gas Stoichiometry Moles  Liters of a Gas Moles  Liters of a Gas – STP - use 22.4 L/mol – Non-STP - use ideal gas law Non-STP Problems Non-STP Problems – Given liters of gas? start with ideal gas law – Looking for liters of gas? start with stoichiometry conv.

49. Gas Law Calculations Ideal Gas Law PV = nRT Ideal Gas Law PV = nRT Dalton’s Law Partial Pressures P T = P A + P B Dalton’s Law Partial Pressures P T = P A + P B Charles’ Law Charles’ Law T 1 = T 2 V 1 = V 2 Boyle’s Law Boyle’s Law P 1 V 1 = P 2 V 2 Gay-Lussac T 1 = T 2 P 1 = P 2 Combined Combined T 1 = T 2 P 1 V 1 = P 2 V 2 Avogadro’s Law Avogadro’s Law Add or remove gas Manometer Manometer Big = small + height R = 0.0821 L atm / mol K 1 atm = 760 mm Hg = 101.3 kPa Bernoulli’s Principle Bernoulli’s Principle Fast moving fluids… create low pressure Density Density T 1 D 1 = T 2 D 2 P 1 = P 2 Graham’s Law Graham’s Law diffusion vs. effusion

50. B. Graham’s Law Diffusion Diffusion – Spreading of gas molecules throughout a container until evenly distributed. Effusion Effusion – Passing of gas molecules through a tiny opening in a container

51. B. Graham’s Law KE = ½mv 2 Speed of diffusion/effusion Speed of diffusion/effusion – Kinetic energy is determined by the temperature of the gas. – At the same temp & KE, heavier molecules move more slowly. – Larger m  smaller v because…

52. B. Graham’s Law Graham’s Law Graham’s Law – Rate of diffusion of a gas is inversely related to the square root of its molar mass. Ratio of gas A’s speed to gas B’s speed

53. Determine the relative rate of diffusion for krypton and bromine. Kr diffuses 1.381 times faster than Br 2. B. Graham’s Law The first gas is “Gas A” and the second gas is “Gas B”. Relative rate mean find the ratio “v A /v B ”.

54. A molecule of oxygen gas has an average speed of 12.3 m/s at a given temp and pressure. What is the average speed of hydrogen molecules at the same conditions? B. Graham’s Law Put the gas with the unknown speed as “Gas A”.

55. An unknown gas diffuses 4.0 times faster than O 2. Find its molar mass. B. Graham’s Law The first gas is “Gas A” and the second gas is “Gas B”. The ratio “v A /v B ” is 4.0. Square both sides to get rid of the square root sign.