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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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The dissociation of molecular hydrogen into atomic hydrogen, can be treated as an ideal gas reaction using the techniques of Section 5.6. The equilibrium constant K for this reaction is defined as
whereis a reference pressure conventionally taken to beand the other P's are the partial pressures of the two species at equilibrium. Now, using the methods of Boltzmann statistics developed in this chapter, you are ready to calculate K from first principles. Do so. That is, derive a formula for K in terms of more basic quantities such as the energy needed to dissociate one molecule (see Problem 1.53) and the internal partition function for molecular hydrogen. This internal partition function is a product of rotational and vibrational contributions, which you can estimate using the methods and data in Section 6.2. (An molecule doesn't have any electronic spin degeneracy, but an H atom does-the electron can be in two different spin states. Neglect electronic excited states, which are important only at very high temperatures. The degeneracy due to nuclear spin alignments cancels, but include it if you wish.) Calculate K numerically at Discuss the implications, working out a couple of numerical examples to show when hydrogen is mostly dissociated and when it is not.
At room temperature, what fraction of the nitrogen molecules in the air are moving at less than?
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
Carefully plot the Maxwell speed distribution for nitrogen molecules at and at Plot both graphs on the same axes, and label the axes with numbers.
This problem concerns a collection of N identical harmonic oscillators (perhaps an Einstein solid or the internal vibrations of gas molecules) at temperature T. As in Section 2.2, the allowed energies of each oscillator are 0, hf, 2hf, and so on. (
a) Prove by long division that
For what values of x does this series have a finite sum?
(b) Evaluate the partition function for a single harmonic oscillator. Use the result of part (a) to simplify your answer as much as possible.
(c) Use formula 6.25 to find an expression for the average energy of a single oscillator at temperature T. Simplify your answer as much as possible.
(d) What is the total energy of the system of N oscillators at temperature T? Your result should agree with what you found in Problem 3.25.
(e) If you haven't already done so in Problem 3.25, compute the heat capacity of this system and check t hat it has the expected limits as and .
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