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Chapter 6: Q. 6.36 (page 246)
Fill in the steps between equations 6.51 and 6.52, to determine the average speed of the molecules in an ideal gas.
The average speed of the molecules of an ideal gas is given by.
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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 .
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.
At room temperature, what fraction of the nitrogen molecules in the air are moving at less than?
Show explicitly from the results of this section thatfor an ideal gas.
Imagine a particle that can be in only three states, with energies -0.05 eV, 0, and 0.05 eV. This particle is in equilibrium with a reservoir at 300 K.
(a) Calculate the partition function for this particle.
(b) Calculate the probability for this particle to be in each of the three states.
(c) Because the zero point for measuring energies is arbitrary, we could just as well say that the energies of the three states are 0, +0.05 eV, and +0.10 eV, respectively. Repeat parts (a) and (b) using these numbers. Explain what changes and what doesn't.
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