H133: 1094 Session 8

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. T5 and T6 Group Problems [15 minutes]

1. Answer T5S.1 in a couple of sentences.
   
   
   
   
2. Do T6T.2. Use equation (T6.6) and the fact that "normal" objects have positive temperatures to justify your answer.
   
   
   
3. Do T6B.2. A simple application of (T6.6).
   
   
   
   
4. Do T6T.4 and T6T.5. These apply the Boltzmann factor formula as in equation (T6.20).
   
   
   

2. The Equilib Simulation [10 minutes]

Download Equilib.exe from the H133 page to your desktop. You have two Einstein solids, labeled A and B. When you first start, there are 400 oscillators in each (so 400/3 = 133 or so atoms each). The total energy is U = 2000 and U_A = 0 while U_B = 2000. With every "step", each oscillator has a chance to exchange one energy unit with one of its neighbors, selected at random. The two solids can exchange energy through the oscillators along their boundary.

1. Press "reset" and then "evolve" to get started. The graph shows you the energy in solid A. What is the equilibrium energy for A? Why does it tend toward equilibrium? How many steps (roughly) does it take to get to equilibrium? Determine roughly the size of fluctuations about equilibrium. That is, about how far away (in energy) does it get away from equilibrium as time goes on?
   
   
   
   
   
   
   
2. Now repeat the last part after switching the number of oscillators for each solid to 100. What is the size of the fluctuations now?
   
   
   
3. Let's test your observations against the theoretical result of problem T5A.1, which estimates how much the energy of a system (in this case, solid A) should fluctuate. The key result is equation (T5.11), which says that the magnitude of the fluctuations should decrease like one over the square-root of the number of oscillators. For the Einstein solid, what should the value of "a" be? Is equation (T5.11) consistent with your results from the last two parts? BONUS: Use (T5.11) and (T5.10) to make an absolute prediction of the fluctuations.
   
   
   
   
   

3. Nuclear Magnetic Resonance (NMR) PhET Simulation [10 minutes]

Start up the PhET applet "Simplified MRI" (under "Quantum Phenomena"). In the "Simplified NMR" section, there is an array of magnetic moments (from the protons in hydrogen initially) which can point up or down (spin-up or spin-down) with respect to a constant magnetic field (controlled at the right).

1. The energy display on the right shows how many are in the lower and upper energy levels (which are split by the magnetic field; vary the field to verify this). Set the magnetic field to 2 tesla. Record the ratio of up spins to down spins (since the spins are always flipping, watch for a while and take the most frequent ratio). Write an equation for this ratio as a Boltzmann factor that involves the energy difference between the level and the temperature.
   
   
   
   
   
2. Give it some power and adjust the frequency until the spins are in resonance. Describe what happens. Convert the resonant frequency to a photon energy in Joules and use your formula for the last part to calculate the approximate temperature of the system. [Hint: it is pretty cold!] BONUS: Verify your temperature by changing the field and getting a new photon energy.
   
   
   
   
   

4. T5S.3 with the StatMech Program [10 Minutes]

Download the StatMech program from the H133 webpage. N is the number of atoms and U the total internal energy. Set it up for N_A = N_B = 100 and U = 200e.

1. How many times more likely is the system to be found in the center macropartition than in the extreme macropartition where U_A = 0 and U_B = 200e?
   
   
   
2. What is roughly the range of values that U_A is likely to have more than 99% of the time? [Use graph.]
   
   
   
3. If U_A were initially to have the extreme value 0, how many times more likely is it to move to the next macropartition nearer the center than to remain in the extreme one?
   
   
   
4. Run again and answer these same three questions for N_A = N_B = 1000 and U = 200e. What is the effect of increasing just the system size by a factor of 10?
   
   
   
   
5. Now try N_A = N_B = 100 and U = 2000e. What is the effect of increasing just the energy available to the system by a factor of 10?
   
   
   
   
   

5. Additional Problems [if time permits]

1. Do T6S.4.
   
   
   
2. Do T6S.9.