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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鈮�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鈮�3Z鈮�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

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

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where 系 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 鈮�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 kT/系. 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.

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You should find蟽E=kTC/k

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