Physics 103, September 26, 2007
The most important things in § 5 of the text.
• The idea of a gas being composed of many small particles whose interactions we can understand using concepts like pressure and energy. This is called the “kinetic theory” of gases. • The derivation of the relation between force on the walls of a container and the speed, mass, density, etc., of the gas within in.
• The ideal gas law, in a slightly modified form: P V = NkBT . 3 • Average kinetic energy of a gas particle: hKi = 2 kB T . From this you can derive the 1 2 rms velocity using K = 2 mv . • A new unit of energy: 1 eV = 1.602 × 10−19 J. • The idea of mean free path. You should understand how it was derived. There will be a problem related to this onto next week’s homework set. • The modeling of viscosity using the kinetic theory of gases has great historical importance, and it is a useful exercise to work through, but it is not important for you to remember the details.
Some problems we might work through in class.
1. HLA Problem 5.13. For STP, use 273 K and 1 atm.
13. A box ofH2 gas is at STP a. What is the pressure of the gas in i. Torr ii. pascals iii. atmospheres b. What is the root-mean-square velocity of the H2 molecules? c. The temperature of the gas of H2 is tripled and the number c~· molecules is halved. All else remains the same. A. What is the new pressure in the box? B. How does each molecule's average kinetic energy change froc its value at STP?
3 2. In the above example, assume that the H2 gas is in a box with volume 1 m . What is the total kinetic energy of the gas particles in the box at the original temperature? What would it be if the temperature were increased by 20 K? If energy costs $0.10 per kilowatt-hour, how much does it cost to heat the box by this much? How much would it cost to heat the lecture hall from freezing to a comfortable temperature?
Continued on the other side.... 3. HLA Exercise 5.11.
11. An object moving with a velocity of 11kms-1
ill leave the Earth and newer r.eturn. At what atoms have Vrms to this escaRe
4. The temperature calculated in the previous problem is substantially higher than the temperature of the Earth’s atmosphere. Hence one would not expect hydrogen to escape the Earth’s atmosphere. Yet, essentially all the hydrogen originally in the Earth’s atmosphere has escaped. Why? (Hydrogen is by far the most common element in the solar system, and in the Universe as a while. The Sun, Jupiter, and Saturn are all made up primarily of hydrogen, and Uranus and Neptune have atmospheres dominated by hydrogen.)
5. A bubble forms at the bottom of a glass of liquid of mass density ˆ and depth d. The bubble floats up to the top of the liquid. Assume that no molecules are added to or removed from the bubble on its upward trip. If the bubble is spherical, What is the relationship between the radius of the bubble at the top to the radius of the bubble on the bottom?
6. A concentrated sample of gas placed within a larger volume of gas will di use outward. For example, gaseous particles from a bottle of perfume will slowly di use through a room. This happens via collisions between gas molecules; each collision essentially randomly re-orients the motion of the particles. It turns out thatp over time t, the distance over which the gas di uses outward is proportional to t. (This is usually quantified as an rms distance from the source of the gas.) Calculate how long it would take gas particles to di use around our lecture hall:
(a) Estimate the mean free path, l, and typical speed of the gas molecules in the lecture hall. From this calculate the amount of time for a molecule to travel the distance l. p (b) Assuming that the distance traveled is proportional to t, estimate how long it takes for particles to di use across the lecture hall.