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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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Estimate the probability that a hydrogen atom at room temperature is in one of its first excited states (relative to the probability of being in the ground state). Don't forget to take degeneracy into account. Then repeat the calculation for a hydrogen atom in the atmosphere of the star UMa, whose surface temperature is approximately 9500 K.
In this section we computed the single-particle translational partition function,, by summing over all definite-energy wave functions. An alternative approach, however, is to sum over all position and momentum vectors, as we did in Section 2.5. Because position and momentum are continuous variables, the sums are really integrals, and we need to slip a factor of to get a unitless number that actually counts the independent wavefunctions. Thus we might guess the formula
role="math" localid="1647147005946"
where the single integral sign actually represents six integrals, three over the position components and three over the momentum components. The region of integration includes all momentum vectors, but only those position vectors that lie within a box of volume . By evaluating the integrals explicitly, show that this expression yields the same result for the translational partition function as that obtained in the text.
In the numerical example in the text, I calculated only the ratio of the probabilities of a hydrogen atom being in two different states. At such a low temperature the absolute probability of being in a first excited state is essentially the same as the relative probability compared to the ground state. Proving this rigorously, however, is a bit problematic, because a hydrogen atom has infinitely many states.
(a) Estimate the partition function for a hydrogen atom at 5800 K, by adding the Boltzmann factors for all the states shown explicitly in Figure 6.2. (For simplicity you may wish to take the ground state energy to be zero, and shift the other energies according!y.)
(b) Show that if all bound states are included in the sum, then the partition function of a hydrogen atom is infinite, at any nonzero temperature. (See Appendix A for the full energy level structure of a hydrogen atom.)
(c) When a hydrogen atom is in energy level n, the approximate radius of the electron wavefunction is , where ao is the Bohr radius, about 5 x 10-11 m. Going back to equation 6.3, argue that the PdV term is Tot negligible for the very high-n states, and therefore that the result of part (a), not that of part (b), gives the physically relevant partition function for this problem. Discuss.
Use Boltzmann factors to derive the exponential formula for the density of an isothermal atmosphere, already derived in Problems 1.16 and 3.37. (Hint: Let the system be a single air molecule, let s1 be a state with the molecule at sea level, and let s2 be a state with the molecule at height z.)
Consider a system of two Einstein solids, where the first "solid" contains just a single oscillator, while the second solid contains 100 oscillators. The total number of energy units in the combined system is fixed at 500. Use a computer to make a table of the multiplicity of the combined system, for each possible value of the energy of the first solid from 0 units to 20. Make a graph of the total multiplicity vs. the energy of the first solid, and discuss, in some detail, whether the shape of the graph is what you would expect. Also plot the logarithm of the total multiplicity, and discuss the shape of this graph.
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