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Chapter 6: Q 6.4 (page 225)

Estimate the partition function for the hypothetical system represented in Figure 6.3. Then estimate the probability of this system being in its ground state.

Short Answer

Expert verified

Hence the partition function isz≈3and the probability isP=13

Step by step solution

01

Given information

The partition function for the hypothetical system represented in Figure 6.3

02

Explanation

Consider figure 6.3, where the x axis represents energy and the y axis represents probability. Because the system's ground state has an energy of 0, the height of the bar at this point must be 1, hence the real height of the bar indicates the unit of length we'll use to solve the problem. Because the sum of the heights of the nine bars is about 13 cm, the sum of the heights must be this number divided by 44 cm in relation to the first bar; this is known as the partition function (the summation of the Boltzmann factors), so:

Z=13cm4.4cm=2.95≈3Z≈3

The probability is:

P=1Ze-E(s)/kT

At ground state E(s)=0, hence

P=1Z=13P=13

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Most popular questions from this chapter

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, Ei-E¯,fori=1to5. Call these deviations ΔEi.

(b) Compute the average of the squares of the five deviations, that is, ΔEi2¯. Then compute the square root of this quantity, which is the root-mean- square (rms) deviation, or standard deviation. Call this number σE. Does σEgive a reasonable measure of how far the individual values tend to stray from the average?

(c) Prove in general that

σE2=E2¯-(E¯)2

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.

Fill in the steps between equations 6.51 and 6.52, to determine the average speed of the molecules in an ideal gas.

Consider a large system of Nindistinguishable, noninteracting molecules (perhaps an ideal gas or a dilute solution). Find an expression for the Helmholtz free energy of this system, in terms of Z1, the partition function for a single molecule. (Use Stirling's approximation to eliminate the N!.) Then use your result to find the chemical potential, again in terms ofZ1.

Apply the result of Problem 6.18 to obtain a formula for the standard deviation of the energy of a system of N identical harmonic oscillators (such as in an Einstein solid), in the high-temperature limit. Divide by the average energy to obtain a measure of the fractional fluctuation in energy. Evaluate this fraction numerically for N = 1, 104, and 1020. Discuss the results briefly.

Each of the hydrogen atom states shown in Figure 6.2 is actually twofold degenerate, because the electron can be in two independent spin states, both with essentially the same energy. Repeat the calculation given in the text for the relative probability of being in a first excited state, taking spin degeneracy into account. Show that the results are unaffected.
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