PHYS 224, Winter 2003
Thermal Physics

Homework and quizzes

Homework will be set, on this web page, each Wednesday and handed in at class on the following Wednesday. The homework will be marked out of 10. The questions marked with asterisks will be carefully graded, and worth 8 marks together. The other 2 marks are for trying all the set questions. These questions are chosen to sparsely cover the required material. You will greatly improve your understanding by attempting other questions (eg from Giancoli) that are not listed below. After the homework deadline the solutions will be posted here. No late answers will be accepted except in special circumstances which you must explain to me. The homework and quizzes together count for up to 20% of the final grade (see exams and grading).

There will be a homework session with the TA every week (Tuesday, 12pm, C231). You are also encouraged to discuss the homework or other matters with me during my office hours (Monday 1.30-2.30 and Wednesday 1.30-2.30, B432)

Past homework solutions are available by request.

Quizzes

Quizzes are 10-minute tests which will be given during the class, usually on Mondays. Their point is mainly to monitor your understanding of concepts recently introduced in class, and to stimulate your thought. You will be encouraged to interact with the other students while doing them. Full marks (ten points) will be given for handing in a short test with your name on it and an effort to solve the problem. We will use your answers to try to help us find out what concepts students have most difficulty with.

Quiz 1 - pressure in U-tubes
Quiz 2 (with solutions) - continuity, Bernoulli and streamlines
Quiz 3 (with solutions) - pressure, capillary, thermal expansion
Quiz 4 (with solutions) - distribution functions
Quiz 5 (with solutions) - First law
Quiz 6 (with solutions) - Equipartition of energy
Quiz 7 (with solutions) - Isochoric vs isobaric heat capacity
Quiz 8 (with solutions) - Reversibility and the Carnot Cycle
Quiz 9 (with solutions) - West Coast Weather

Homework set 1 - due Wednesday 15 January

The following are problems (not 'questions'!) from Giancoli chapter 13. (Remember, * = graded carefully).

15,18,23* - static pressure
20 - Pascal's principle
29,33,35,38 - buoyancy
43 - continuity eqn
45,47,51 - Bernoulli eqn
58,92*,93,96 - combined continuity and Bernoulli eqns

Homework set 2 - due Wednesday 22 January

More problems from Giancoli Chapter 13:
61 - viscosity
66 - application of Poiseuille flow
67 - turbulence
72, 74 - surface tension

A1 - Draw a sketch, and show that the rotation of a ball suspended over an air blower tilted away from the vertical is appropriate to produce a magnus force which tends to push the ball back into the jet.

A2* - Poiseuille flow:
Derive Eqn. (13-12) for Poiseuille (steady) flow in a pipe. [Use the following method if you like: Assume that the liquid speed v(r) is only a function of distance r from the axis of the tube. This means the flow is laminar, with the same symmmetry as the boundary conditions. Now, require that the balance of all forces (pressure and viscous) acting on a cylindrical portion of the fluid of radius r coaxial with the center of the tube must be zero in the steady state. This gives you v(r). Then integrate to get the total flow.]

A3. - surface tension:
Estimate the height to which water will rise in a glass capillary tube of internal diameter d. [Assume the contact angle phi in Fig. 13-35 is small, and balance the surface tension force on the column against the weight of the water].

and some problems from Giancoli chapter 17:
7,28 - degrees F to C to K
75*,33,76 - ideal gas law
52 - ideal gas temperature scale

Homework set 3 - due Wednesday 29 January

From Giancoli Chapter 17:
12,17,18 - thermal expansion
48* - density of molecules in air

From Chapter 18:
2 - speed of molecules in gas
14,15 - distributions
30,31,32* - van der Waals equation

Homework set 4 - due Wednesday 5 February

Chapter 18:
33,39,42,61 - mean free path
44 - diffusion

B.* Assume the earth's atmosphere is entirely made of a stationary ideal gas, nitrogen (m = 28 a.u.), at a uniform temperature T = 300 K, and assume the surface is all ocean with no land masses.
B1. Using dp/dy = - (density)g, show that the variation of molecular density with height obeys a Boltzmann distribution: f(E) = f0 exp(-E/kT), where E is the gravitational energy of a molecule at a given height y. (Take E=0 at sea level.)
B2. What is the probability that a particular molecule will be found less than 1 km above sea level?
B3. What is the mean gravitational energy of the molecule as it diffuses throughout the atmosphere?
B4. What is its mean kinetic energy?
B5. What is its rms velocity?
B6. What is its average velocity?
B7. Estimate the average molecular spacing at sea level.
B8. Estimate the mean free path at sea level.
B9. Estimate the typical time taken for a molecule to diffuse 1 m.
In each part above where possible give both an algebraic and a numerical answer.

Homework set 5 - due Wednesday 12 February

The following Giancoli problems from Chapter 19:
11,16,20,25*,28 - specific and latent heats
36,38,40,43,46* - first law
48 - ideal diatomic gas
Also, one quick question with notes on diffusion.

Homework set 6 - due Wednesday 19 February

Giancoli problems from Chapter 19:
54,58 - ideal gases
60,62,65* - adiabatic expansion
69,73,75,93,102* - thermal conduction
85,88 - heat capacity

Homework set 7 - due Friday 28 February

C1 - What is the temperature of an object which emits black body radiation with a peak intensity at a wavelength of 0.5 microns? Note: Wiens' law and the Planck distribution are given in section 38-1 of Giancoli.
C2 - From the Planck distribution as a function of frequency (nu), show that the total intensity of black body radiation, integrated over all frequencies, is proportional to T^4. This gives the T^4 in the Stefan-Boltzmann law. (You don't actually need to do an integral.)

Giancoli Chapter 20 problems:
4,8* - engines and efficiency
13,16,21 - Carnot engine and maximum efficiency
23,27,30* - refrigerators
34,41,63 - entropy

Homework set 8 - due Friday 7 March

All numbered problems are from Giancoli.

20-67 - Stirling engine. Can you relate the cycle shown in Fig 20-17 to the operation of the engine shown on this web page?

20-45 and 20-48* - entropy change calculations
20-54 - Boltzmann's law

During the time of the missing class on Wednesday March 5, please read sections 18-3 to 18-5 and answer these problems:
18-19*, 18-20, 18-22, 18-24 - phases of water.
18-31 - (try it again now if you didn't do it before) critical point of a van-der-Waals gas. What is going on in the brown shaded region in Fig. 18-11?

Homework set 9 - due Friday 14 March

There are no starred questions this week! Have a go at the following questions. You will get 8 marks for handing in your attempts and 10 for getting the answers. You can find the derivations in any book on thermodynamics or work them out for yourself.

D0. Read the hyperphysics sections on the thermodynamic potentials G, H and F.

D1. In a 'throttling' process, used to cool and liquify gases, the gas is passed through a nozzle from a high to a low pressure region. The process happens fast so there is no heat flow. Show that the enthalpy H=U+pV of the gas is unchanged during the process.

D2. The Gibbs free energy is defined by G = U + pV - TS. Show that dG = -SdT + Vdp. Derive a Maxwell relation between S, T, V and p from this equation.