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Chapter 6: Q 6.16 (page 231)
Prove that, for any system in equilibrium with a reservoir at temperature T, the average value of the energy is
where . These formulas can be extremely useful when you have an explicit formula for the partition function.
Hence proved,
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The energy required to ionise a hydrogen atom is 13.6 eV, so you might expect that the number of ionised hydrogen atoms in the sun's atmosphere would be even less than the number in the first excited state. Yet at the end of Chapter 5 I showed that the fraction of ionised hydrogen is much larger, nearly one atom in 10,000. Explain why this result is not a contradiction, and why it would be incorrect to try to calculate the fraction of ionised hydrogen using the methods of this section.
Suppose you have 10 atoms of weberium: 4 with energy 0 eV, 3 with energy 1 eV, 2 with energy 4 eV, and 1 with energy 6 eV.
(a) Compute the average energy of all your atoms, by adding up all their energies and dividing by 10.
(b) Compute the probability that one of your atoms chosen at random would have energy E, for each of the four values of E that occur.
(c) Compute the average energy again, using the formula
Cold interstellar molecular clouds often contain the molecule cyanogen (CN), whose first rotational excited states have an energy of 4.7x 10-4 eV (above the ground state). There are actually three such excited states, all with the same energy. In 1941, studies of the absorption spectrum of starlight that passes | through these molecular clouds showed that for every ten CN molecules that are in the ground state, approximately three others are in the three first excited states (that is, an average of one in each of these states). To account for this data, astronomers suggested that the molecules might be in thermal equilibrium with some "reservoir" with a well-defined temperature. What is that temperature?
Although an ordinary H2 molecule consists of two identical atoms, this is not the case for the molecule HD, with one atom of deuterium (i.e., heavy hydrogen, 2H). Because of its small moment of inertia, the HD molecule has a relatively large value of At approximately what temperature would you expect the rotational heat capacity of a gas of HD molecules to "freeze out," that is, to fall significantly below the constant value predicted by the equipartition theorem?
In problem 6.20 you computed the partition function for a quantum harmonic oscillator :, where is the spacing between energy levels.
(a) Find an expression for the Helmholtz free energy of a system of harmonic oscillators.
(b) Find an expression for the entropy of this system as a function of temperature. (Don't worry, the result is fairly complicated.)
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