Lecture 2

1. Lecture 2 Ideal Gas Law Stull: pg 4-9, Wallace & Hobbs: 63-67

2. Atmospheric Pressure Atmospheric pressure change is approximately exponential with height, z. If temperature were uniform with height (it’s not) then: a= 0.0342 K/m, Po = scaled pressure = 101.325 kPa Hp = 7.29 Km = scale height for pressure, it is the e-folding distance for pressure curve.

3. Atmospheric Pressure Because pressure decreases with height monotonically (one direction), P can be used as a surrogate for a measure of altitude (Fig. 1.4, Stull). (Example in Stull, pg. 6)

5. Density • Density is defined as mass per unit volume. • Density increases as the number and molecular weight of molecules in a volume increase. • Average density at sea level = 1.255 kg m-3 (Table 1.2, Stull)

6. Density Because gases such as air are compressible, air density can vary over a wide range. Density decreases roughly exponentially with height in an atmosphere of uniform temperature. ρo = 1.225 kg/m3, scale height, Hp = 8.55 km.

7. Temperature When a group of molecules (microscopic) move in the same direction, motion is called wind (macroscopic). When they move in random directions, motion is associated with temperature. Higher temperatures, T are associated with greater average molecular speeds, v. a = 4 ·10-5 k m-2s2, scale height, mw= molecular weight of most gases.

8. Temperature Absolute units such as K (Kelvin) must be used for temperature in all thermodynamic and radiative laws! Standard average sea-level temperature is: T= 15°C = 288 K = 59 °F

9. Virtual Temperature For moist air, the gas constant changes because water vapor is less dense than dry air. To simplify things, a virtual temperature, Tv can be defined to include the effects of water vapor on density: where w is the water vapor mixing ratio (g water vapor / kg dry air) and all temperatures are in K. Moist air of temperature T behaves as dry air of temperature Tv.

10. Equation of State Because pressure is caused by the movement of molecules, we might expect the pressure P to be greater where there are more molecules (i.e., greater density ρ), and where they are moving faster (i.e., greater temperature T). The relationship between pressure, density, and temperature is called the Equation of State. For dry air the gases in the atmosphere have a simple equation of state known as the ideal gas law.

11. Ideal Gases • Molecules have zero volume • No intermolecular forces • Elastic collisions with walls of container • Works well for the Earth’s atmosphere

12. Molar Form

13. Variables • Pressure (p) • Volume (V) • Temperature (T) • Number of moles (n)

14. Ideal Gas Law Exercise: Determine units of R*

15. Universal Gas Constant R* = 8.314 Jmol-1K-1

16. Ideal Gas Law: Molar Form Memorize this!

17. Sealed Container cylinder Gas n fixed

18. Massless, Frictionless Piston piston Gas

19. Weights can be added or removed to change pressure Equilibrium  internal pressure = external pressure Gas

20. Keep T fixed, Vary p i.e., pressure and volume are inversely proportional (Boyle’s Law)

21. Increase Pressure

22. Increase Pressure

23. Increase Pressure Ideal gas law  V decreases

24. Increase Pressure Ideal gas law  V decreases

25. Increase Pressure Ideal gas law  V decreases

26. Decrease Pressure

27. Decrease Pressure

28. Decrease Pressure Ideal gas law  V increases

29. Decrease Pressure Ideal gas law  V increases

30. Decrease Pressure Ideal gas law  V increases

31. Keep p fixed, Vary T i.e., volume is directly proportional to temperature

32. Increase T Ideal gas law  V increases Apply Heat

33. Increase T Ideal gas law  V increases Apply Heat

34. Increase T Ideal gas law  V increases Apply Heat

35. Decrease T Ideal gas law  V decreases Apply Cooling

36. Decrease T Ideal gas law  V decreases Apply Cooling

37. Decrease T Ideal gas law  V decreases Apply Cooling

38. Fix V i.e., p is directly proportional to T

39. Increase T Ideal gas law  p increases Piston doesn’t move (internal pressure  external pressure) Pressure builds up Apply Heat

40. Exercise • n = 1.00 mol • V = 1.00 m3 • T = 300 K • Calculate p (in Pa)

41. Solution

42. Partial Pressure • Consider a mixture of k gases • Volume, V; temperature, T; pressure, p; # moles = n • ni = # moles of gas i • n1 + n2 +  + nk = n • Partial pressure, pi:

43. Dalton’s Law

44. Proof

45. Ratios Ratio of partial pressure to total pressure = mole fraction

46. Partial Volume

47. Volume Ratios Explains why mole fractions are also called volume fractions.

48. Ideal Gas Law – Mass Version • Start with molar version (1) Let m = mass Let M = molecular weight

49. Mass & Moles  (2)

50. Substitute (2) into (1)