H133: 1094 Session 9

Write your name and answers on this sheet and hand it in at the end.
After the indicated time, move on to the next activity, even if you are not finished!

1. T7: MBoltz and EBoltz Problems [12 minutes]

Start up MBoltz and EBoltz from the H133 webpage.

1. Do T7B.4. What should you choose for x1? For x2, does it matter if you choose 100 or 200? Why not? Argue that your result for the probability is reasonable.
   
   
   
   
   
2. As a warm-up problem for using EBoltz, do T7B.6.
   
   
   
   
   
3. Now for a more interesting exploration. The default energies for EBoltz are E = epsilon*n, as for the Einstein solid. What is the approximate average energy in the high-temperature limit? (Give it as a multiple of kT.)
   
   
   Now change the energy to be E = epsilon*n² [you get n² by typing n^2]. What is the approximate average energy in the high-temperature limit?
   
   
   Now try E = epsilon * sqrt(n) [this is how you write a square root]. Once again, what is the approximate average energy in the high-temperature limit? [NOTE: Don't try kT/epsilon much greater than 10, or it will take forever.]
   
   
   Have you discovered a pattern? What is it? Predict first and then test with E = epsilon*n³.
   
   
   

2. T8: Entropy of Monatomic Gas and Entropy Changes [15 minutes]

1. Start with some basic applications of equations (T8.7) and (T8.8). Do T8T.1 and T8T.2, explaining your answers.
   
   
   
   
2. Do T8T.5. What kind of process is this? Show your work.
   
   
   
   
   
3. Do T8B.8. [Hint: Phase changes are discussed in Section T8.4.]
   
   
   
   
4. The concept of a "suitable replacement process" (section T8.6) is important but sometimes tricky. You need to have the same initial and final macrostates and the process must be quasistatic. Use T8T.8 and T8T.9 to test your understanding. Only one answer each in T8T.8 and T8T.9 works; why do the others fail? In T8T.9, note that the ideal gas law applies to both the initial and final gases; your replacement process must be consistent.
   
   
   
   
   

3. Introduction to T9: Heat Engines [15 minutes]

A perfect engine, which converts thermal energy (heat) completely into work, would violate the 2nd Law of Thermodynamics. Flowing heat carries entropy from one object to another, but flowing work does not. (E.g., raising a weight or adiabatically compressing a gas doesn't change the entropy.) A real heat engine needs to expel the entropy the engine gets from the heat source (a hot reservoir at T_H), so there needs to be a cold reservoir at T_C.

According to the first law, |W| = |Q_H| - |Q_C|, where Q_H is the heat energy from the hot reservoir and Q_C is the heat energy sent to the cold reservoir. Then the efficiency e is given by
    e = benefit/cost = |W|/|Q_H| = 1 - |Q_C|/|Q_H| < (T_H-T_C)/T_H     (the < is really "less than or equal to").

1. To get warmed up at applying the idea of efficiency, do T9T.3, being careful to identify |W|, |Q_H|, and |Q_C| and then do T9T.5. Do you think the personal fan is practical?
   
   
   
   
   
2. Is there a symmetry between heat energy and work? Answer T9B.1.
   
   
   
   
3. Try an engineering application: the power plant proposed in T9S.8, making use of the basic efficiency relation above. Be careful with the temperatures given in centrigrade!
   
   
   
   
   
4. Let's finish with some T9S questions that we'll discuss on Thursday. Give a one or two sentence answer to each of the following:
   • T9S.2.
     
     
     
     
   • T9S.3.
     
     
     
     
   • T9S.4.