Chapter 6: 6.24 (page 236)
For an molecule, the constant is approximately . Estimate the rotational partition function for an molecule at room temperature.
The rotational partition function is .
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Fill in the steps between equations 6.51 and 6.52, to determine the average speed of the molecules in an ideal gas.
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.
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.
You might wonder why all the molecules in a gas in thermal equilibrium don't have exactly the same speed. After all, when two molecules collide, doesn't the faster one always lose energy and the slower one gain energy? And if so, wouldn't repeated collisions eventually bring all the molecules to some common speed? Describe an example of a billiard-ball collision in which this is not the case: The faster ball gains energy and the slower ball loses energy. Include numbers, and be sure that your collision conserves both energy and momentum.
Some advances textbooks define entropy by the formula
where the sum runs over all microstates accessible to the system and is the probability of the system being in microstate .
(a) For an isolated system, role="math" localid="1647056883940" for all accessible states . Show that in this case the preceding formula reduces to our familiar definition of entropy.
(b) For a system in thermal equilibrium with a reservoir at temperature,role="math" localid="1647057328146" . Show that in this case as well, the preceding formula agrees with what we already know about entropy.
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