PHYS 213 :: Physics Illinois :: University of Illinois at Urbana-Champaign

James Scholar Assignment #3

This assignment has several parts:

To gain some background, read at least 4 of the following articles on this topic:
Required Reading
MunichNegTempFAQ.pdf
NegativeTempWikipedia.pdf
Pick 2 More From the List Below
CarrNegTemp.pdf
MPG-NegTemp.pdf
PurcellNegTemp.pdf
RamseyNegTemp.pdf
SciAmDemons-Bennett.pdf

  1. Write a ~3/4-page essay, roughly explaining what it means for systems to have negative temperature, and when this can be expected to arise.
  2. Next, solve the following problems, based on the sort of calculations we’ve already done in class:
    • Consider a two-level atom; call the ground state ‘g’, excited state 'e', energy gap Δ, and temperature T. Calculate P(atom is in state e, T ͢ ∞). Now consider a collection of 100 such atoms ("System 1") and calculate the entropy S of all 100 atoms, in this high-T limit.
    • Instead imagine starting with all of the atoms in the excited state (obviously not achievable using thermal processes alone). What is the entropy S?
    • What happens if we put this totally excited system in thermal contact with another 100 two-level atoms, all initially in their ground states ("System 2")? In particular, calculate the initial (before bringing the two systems together) and final (after the systems have reached equilibrium) entropies.
    • Next, calculate the entropy when only the first quantum of energy, △, has transferred from System 1 to System 2.
    • Finally, calculate the approximate initial temperature of System 1, using our previous definition: 1/T = dS/dU; your answer should be in terms of △ and fundamental constants.
    • Now repeat #c, but assuming that System 2 is initially at near-infinite temperature. Explain what happens, and what this must mean for how 'hot' System 1 (initially with all atoms excited) must be.
  3. Which of the following *could* be made to have a negative temperature? Explain your reasoning.
    • Electron magnetic moments in a magnetic field
    • A three-level atom
    • A simple harmonic oscillator
    • A gas of helium atoms in the room
  4. Finally, in this article, MPG-NegTemp.pdf, it is suggested that systems with negative temperatures allow one to extract more work, i.e., to have a higher efficiency, in principle above 100%. Do you think this is a reasonable claim, and why/why not?

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