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Chapter 6: Q 6.2 (page 224)
Prove that the probability of finding an atom in any particular energy level is , where and the "'entropy" of a level is k times the logarithm of the number of degenerate states for that level.
Hence proved that the probability of finding an atom in any particular energy level is:
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Imagine a world in which space is two-dimensional, but the laws of physics are otherwise the same. Derive the speed distribution formula for an ideal gas of nonrelativistic particles in this fictitious world, and sketch this distribution. Carefully explain the similarities and differences between the two-dimensional and three-dimensional cases. What is the most likely velocity vector? What is the most likely speed?
Equations 6.92 and 6.93 for the entropy and chemical potential involve the logarithm of the quantity . Is this logarithm normally positive or negative? Plug in some numbers for an ordinary gas and discuss.
Consider an ideal gas of highly relativistic particles ( such as photons or fast-moving electrons) whose energy-momentum relation is instead of . Assume that these particles live in a one-dimensional universe. By following the same logic as above, derive a formula for the single particle partition function,, for one particle in the gas.
A particle near earth's surface traveling faster than about has enough kinetic energy to completely escape from the earth, despite earth's gravitational pull. Molecules in the upper atmosphere that are moving faster than this will therefore escape if they do not suffer any collisions on the way out.
(a) The temperature of earth's upper atmosphere is actually quite high, around Calculate the probability of a nitrogen molecule at this temperature moving faster than , and comment on the result.
(b) Repeat the calculation for a hydrogen molecule and for a helium atom, and discuss the implications.
(c) Escape speed from the moon's surface is only about . Explain why the moon has no atmosphere.
In the real world, most oscillators are not perfectly harmonic. For a quantum oscillator, this means that the spacing between energy levels is not exactly uniform. The vibrational levels of an H2 molecule, for example, are more accurately described by the approximate formula
where is the spacing between the two lowest levels. Thus, the levels get closer together with increasing energy. (This formula is reasonably accurate only up to about n = 15; for slightly higher n it would say that En decreases with increasing n. In fact, the molecule dissociates and there are no more discrete levels beyond n 15.) Use a computer to calculate the partition function, average energy, and heat capacity of a system with this set of energy levels. Include all levels through n = 15, but check to see how the results change when you include fewer levels Plot the heat capacity as a function of . Compare to the case of a perfectly harmonic oscillator with evenly spaced levels, and also to the vibrational portion of the graph in Figure 1.13.
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