1. Kinetic Theory of Gases Lecturer: Professor Stephen T. Thornton

2. Reading Quiz
A gas cylinder is used to fill dozens of balloons with helium gas over a weekend. Which of the following statements is most true on Monday?
The remaining helium atoms in the cylinder have less kinetic energy.
The pressure in the cylinder and the rms speed of the remaining helium atoms increases.
The pressure in the cylinder and the rms speed of the remaining helium atoms decreases.
The pressure in the cylinder decreases, but the rms speed of the remaining helium atoms remains constant.

3. Reading Quiz
Answer: D
The pressure in the cylinder goes down, but the rms speed of the helium atoms only depends on the ambient temperature, which remains constant.

4. Last Time Thermal expansion - more Brownian motion Ideal gas law
Moles, Avogadro's number, etc.

5. Today Kinetic energy of molecules Maxwell distribution Phase changes
Vapor pressure and humidity
Van de Waals Equation
Mean free path
Diffusion

6. Kinetic Theory of Gases
James Clerk Maxwell, a great Scottish mathematical physicist, did much of the early work in statistical theory, but died at age 48.
Ludwig Boltzmann despaired so much about his work in statistical theory that he committed suicide in 1906.
Paul Ehrenfest, who picked up Boltzmann’s work, committed suicide in 1933.
Now it is our turn to study the subject.
Perhaps we should proceed cautiously!

7. Kinetic theory of gases
Assumptions:
Container has large number N of identical molecules, each of mass m.
Molecules bounce around and are far apart.
Molecules collide elastically with wall and each other.

8. Force Exerted by a Molecule on the Wall of a Container

10. In a real gas, the molecules are moving around with different speeds
In a real gas, the molecules are moving around with different speeds. Maxwell did a more complete calculation (see later) using the average velocity and included molecules moving in all 3 directions. He found

11. We have found a clear connection between microscopic behavior and macroscopic variables.

12. Conceptual Quiz: Imagine that that the temperature of a fixed-volume container of ideal gas changes from 1000C to 2000C. What happens to the pressure and the average kinetic energy of the molecules? A) Pressure goes up; average kinetic energy goes up by factor of 2. B) Pressure goes up; average kinetic energy goes up, but less than a factor of 2. C) Pressure goes down; average kinetic energy goes up by some unknown amount. D) Pressure goes down; average kinetic energy goes up by a factor of 2.

13. Answer: B Pressure goes up according to ideal gas law
Answer: B Pressure goes up according to ideal gas law. Kinetic energy is proportional to Kelvin temperature, not Celsius temperature, so K increases, but not by a factor of K to 473 K

15. The Maxwell Speed Distribution
The molecules in a gas will not all have the same speed; their distribution of speeds is called the Maxwell distribution:

16. Note that these speeds go as the square root of the temperature and inversely as the square root of the mass.

17. Hydrogen Gas Molecules. A 1
Hydrogen Gas Molecules. A 1.0-mol sample of hydrogen gas has a temperature of 27°C. (a) What is the total kinetic energy of all the gas molecules in the sample? (b) How fast would a 65-kg person have to run to have the same kinetic energy?
Giancoli, 4th ed, Problem 18-6

18. Uranium Isotopes. Two isotopes of uranium,
(the superscripts refer to their atomic masses), can be separated by a gas diffusion process by combining them with fluorine to make the gaseous compound UF6 . Calculate the ratio of the rms speeds of these molecules for the two isotopes, at constant T. Find masses in Appendix or on Internet.
Giancoli, 4th ed, Problem 18-15

19. Real Gases and Changes of Phase
The curves here represent the behavior of the gas at different temperatures. The cooler it gets, the further the gas is from ideal.
In curve D, the gas becomes liquid; it begins condensing at (b) and is entirely liquid at (a). The point (c) is called the critical point.
Figure PV diagram for a real substance. Curves A, B, C, and D represent the same substance at different temperatures (TA > TB > TC > TD).

20. A PT diagram is called a phase diagram; it shows all three phases of matter. The solid-liquid transition is melting or freezing; the liquid-vapor one is boiling or condensing; and the solid-vapor one is sublimation.
Phase diagram of water.
Figure Phase diagram for water (note that the scales are not linear).

21. Dry ice at atmospheric pressure, -78.5 0C
The triple point is the only point where all three phases can coexist in equilibrium.
Phase diagram of carbon dioxide.
The solid form is called “dry ice”.
Figure Phase diagram for carbon dioxide.
Dry ice at atmospheric pressure, C

22. The Solid-Liquid Phase Boundary Look at point A
The Solid-Liquid Phase Boundary Look at point A. If we increase the pressure, the ice melts. This is why ice skate blades glide so easily.

23. A Liquid in Equilibrium with its Vapor
A Liquid in Equilibrium with its Vapor. In equilibrium the number of molecules evaporating into gas is equal to number from gas condensing into liquid.

24. The Vapor-Pressure Curve for Water
The Vapor-Pressure Curve for Water. A liquid boils at the temperature at which its vapor pressure equals the external pressure.

25. Pressure Cooker. A pressure cooker is a sealed pot designed to cook food with the steam produced by boiling water somewhat above 100°C. The pressure cooker in the figure uses a weight of mass m to allow steam to escape at a certain pressure through a small hole (diameter d) in the cooker’s lid. If d = 3.0 mm, what should m be in order to cook food at 120°C? Assume that atmospheric pressure outside the cooker is x 105 Pa.
Giancoli, 4th ed, Problem 18-35

26. Vapor Pressure and Humidity
An open container of water can evaporate, rather than boil, away. The fastest molecules are escaping from the water’s surface, so evaporation is a cooling process as well.
The inverse process is called condensation.
When the evaporation and condensation processes are in equilibrium, the vapor just above the liquid is said to be saturated, and its pressure is the saturated vapor pressure.
Figure Vapor appears above a liquid in a closed container.

27. Partial pressure is the pressure each component of a mixture of gases would exert if it were the only gas present. The partial pressure of water in the air can be as low as zero, and as high as the saturated vapor pressure at that temperature.
Relative humidity is a measure of the saturation of the air.

28. When the humidity is high, it feels muggy; it is hard for any more water to evaporate.
The dew point is the temperature at which the air would be saturated with water.
If the temperature goes below the dew point, dew, fog, or even rain may occur.
Figure Fog or mist settling around a castle where the temperature has dropped below the dew point.

29. Vapor Pressure and Humidity
The saturated vapor pressure increases with temperature.
Fog tends to occur during winter mornings when it is cold, and the air is saturated with water vapor.

30. The molecules having higher speed are able to leave a sweat drop on the skin, resulting in a lower temperature and drawing heat from the skin. This in turn heats the drop and the process occurs again.

31. Speed Distribution for O2 and H2 at 20 ºC

32. Do demos Boiling by cooling Boiling by reducing pressure

33. Skipped as quiz on 4/16/12

37. Van der Waals Equation of State
To get a more realistic model of a gas, we include the finite size of the molecules and the range of the intermolecular force beyond the size of the molecule.
Figure Molecules, of radius r, colliding.

38. We assume that some fraction b of the volume is unavailable due to the finite size of the molecules. We also expect that the pressure will be reduced by a factor proportional to the square of the density, due to interactions near the walls. This gives the Van der Waals equation of state; the constants a and b are found experimentally for each gas:
a is an indication of the attraction between molecules. b is a measure of the volume of the individual molecules.

39. The PV diagram for a Van der Waals gas fits most experimental data quite well.
Figure PV diagram for a van der Waals gas, shown for four different temperatures. For TA, TB, and TC (TC is chosen equal to the critical temperature), the curves fit experimental data very well for most gases. The curve labeled TD, a temperature below the critical point, passes through the liquid–vapor region. The maximum (point b) and minimum (point d) would seem to be artifacts, since we usually see constant pressure, as indicated by the horizontal dashed line (and Fig. 18–4). However, for very pure supersaturated vapors or supercooled liquids, the sections ab and ed, respectively, have been observed. (The section bd would be unstable and has not been observed.)

40. Mean Free Path Molecules in ideal gases have many collisions per
second – billions in the
case of air at STP – but between collisions the molecules interact very little.
The average distance the molecule travels between collisions, called the mean free path, can be calculated.
Figure Zigzag path of a molecule colliding with other molecules.

41. The mean free path can be calculated, given the average speed, the density of the gas, the size of the molecules, and the relative speed of the colliding molecules. The result:
Figure Molecule at left moves to the right with speed vav. It collides with any molecule whose center is within the cylinder of radius 2r.

42. Diffusion
Even without stirring, a few drops of dye in water will gradually spread throughout. This process is called diffusion.
Figure A few drops of food coloring (a) dropped into water, (b) spreads slowly throughout the water, eventually (c) becoming uniform.

43. Diffusion occurs from a region of high concentration to a region of lower concentration.
Figure Diffusion occurs from a region of high concentration to one of lower concentration (only one type of molecule is shown).

44. The rate of diffusion is given by:
If concentration C is mol/m3, J is mol/s passing a given point.
In this equation, D is the diffusion constant.
If concentration C
is kg/m3, J is kg/s
passing a given point.
Paper chromotograhpy

45. Best one: http://serendip.brynmawr.edu/exchange/diffusion/applet
Diffusion simulations
Best one:

46. Relative Humidity. Air that is at its dew point of 5°C is drawn into a building where it is heated to 20°C. What will be the relative humidity at this temperature? Assume constant pressure of 1.0 atm. Take into account the expansion of the air.
Giancoli, 4th ed, Problem 18-38

47. Mean Free Path. The mean free path of molecules at STP is measured to be about 5.6 x 10-8 m. Estimate the diameter of a CO2 molecule. (b) Do the same for He gas for which the mean free path is ~25 x 10-8 m at STP.
Giancoli, 4th ed, Problem 18-48