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Chapter 6: Q. 6.46 (page 255)
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.
Yes, the logarithm is normally positive for ordinary gas as is positive.
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Use the Maxwell distribution to calculate the average value of for the molecules of an ideal gas. Check that your answer agrees with equation 6.41.
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 .
Consider a large system of indistinguishable, noninteracting molecules (perhaps an ideal gas or a dilute solution). Find an expression for the Helmholtz free energy of this system, in terms of , the partition function for a single molecule. (Use Stirling's approximation to eliminate the .) Then use your result to find the chemical potential, again in terms of.
The most common measure of the fluctuations of a set of numbers away from the average is the standard deviation, defined as follows.
(a) For each atom in the five-atom toy model of Figure 6.5, compute the deviation of the energy from the average energy, that is, . Call these deviations .
(b) Compute the average of the squares of the five deviations, that is, . Then compute the square root of this quantity, which is the root-mean- square (rms) deviation, or standard deviation. Call this number . Does give a reasonable measure of how far the individual values tend to stray from the average?
(c) Prove in general that
that is, the standard deviation squared is the average of the squares minus the square of the average. This formula usually gives the easier way of computing a standard deviation.
(d) Check the preceding formula for the five-atom toy model of Figure 6.5.
Derive equation 6.92 and 6.93 for the entropy and chemical potential of an ideal gas.
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