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Thermal Physics 224 Final exam 8.25 am, Wednesday December 12, 2007 Autumn 2007 Instructor: David Cobden
You have 2 hours. Begin at 8.25 and hand your exam to me by the time I leave the room at 10.25.
Attempt all the questions.
Please write your name on every page and your SID on the first page.
Write all your working on these question sheets. Use this front page for extra working. It is important to show your calculation or derivation. Some of the marks are given for showing clear and accurate working and reasoning.
Watch the blackboard for corrections or clarifications during the exam.
This is a closed book exam. No books, notes or calculators allowed.
ܷ݀ܰ݀ߤ ܸ݀ െ ܵ݀ܶൌ.
ܰ ൎ 6 ൈ 10 ଶଷ^ ݇ ൌ 1.4 ൈ 10ି ଶଷ^ JK-1^ ܴܰൌ (^) ݇ ൌ 8.3 JK-1^ e = 1.6 × 10-19^ C
1 mole of gas occupies 24 liters (= 0.024 m^3 ) at T = 300 K and p = 1 bar (= 10^5 Pa)
- [5] Two possible energies of a molecule in surroundings at temperature T are E 1 and E 2. What is the ratio of the respective probabilities P 1 and P 2 of finding it with these energies?
- [12] Assume that the earth’s atmosphere is an ideal gas with a uniform temperature T. Write down the form of the probability distribution ܲ ሻݖሺ that a given molecule of mass m will be found at a height z. (Think of comparing the probabilities of being at two different heights. Use the gravitational constant g ). Deduce an expression for the variation of density, and therefore pressure with height, if the pressure at ground level ( z = 0) is p 0.
- [15] A dipole has two energy levels, ܧଵ ߂ ൌ and ܧଶ ߂െ ൌ. Find its partition function at temperature T and show that the mean energy is ܧത ൌ െ߂ tanh ߂ߚ, where ߚൌ 1/݇ (^) ܶ.
- [10] Assume that the pressure inside the balloon is 1% greater than that outside and assume that CO 2 is an ideal gas. How much heat, if any, is generated if the balloon bursts? (Consider enthalpy. Ignore the energetics of the rubber.)
- [5] Does the total entropy increase or remain the same during this process, and why?
- [8] State the fundamental statistical mechanical definitions of temperature and pressure.
- [6] Is temperature ever negative in a real system? What would happen if something at negative temperature were brought in contact with something at positive temperature?
- [8] Define the Gibbs free energy G in terms of U and other macroscopic variables, and give or deduce the thermodynamic identity relating small changes in G to those in T , p and N.
- [8] From this you can deduce that ቀ డீ డ ቁ்^ ,ே ܸൌ^.^ Write down the other two relationships for
derivatives of G that can be deduced immediately from the thermodynamic identity.
- [8] What thermodynamic condition determines the positions of phase boundaries on a phase diagram, and why?
[8] Here is part of the phase diagram of CO 2 (from Wikipedia.) Identify the triple point and the critical point and say what happens at each of them.
- [10] Sketch the variation with T of the Gibbs free energy of an amount of CO 2 at a pressure p = 20 bar, taking into account the answer to Question 14, and add comments.
- [10] Sketch the variation with p of the Gibbs free energy at T = 250 K, and add comments.
- [8] Describe what would happen if a cup of liquid CO 2 kept at 250 K and 20 bar were suddenly brought into a room at normal pressure (1 bar).
