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Chapter 9: Q3CQ (page 402)

Given an arbitrary thermodynamic system, which is larger. the number of possible macro-states. or the numberof possible microstates, or is it impossible to say? Explain your answer. (For most systems, both are infinite, but il is still possible to answer the question,)

Short Answer

Expert verified

None can be measured; we know that there are more possible microstates

Step by step solution

01

microstates

Microstate is a term that describes the microscopic properties of a thermodynamic system.

02

possible number of microstates

Greater number is of the possible microstates, this is only theoretical simply because no one can keep track of each molecule and its position.

Knowing the number of molecules on each side would be knowing the macro-state of the system, but knowing each molecule where it is, specifying the actual molecule, is giving more options - microstates.

Although none can be measured, we know that there are more possible microstates.

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Most popular questions from this chapter

Verify that the probabilities shown in Table 9.1 for four distinguishable oscillators sharing energy 2δEagree with the exact probabilities given by equation (9-9).

We claim that the famous exponential decrease of probability with energy is natural, the vastly most probable and disordered state given the constraints on total energy and number of particles. It should be a state of maximum entropy ! The proof involves mathematical techniques beyond the scope of the text, but finding support is good exercise and not difficult. Consider a system of11oscillators sharing a total energy of just5hω0 . In the symbols of Section 9.3. N=11andM=5 .

  1. Using equation(9-9) , calculate the probabilities ofn , being0,1,2, and3 .
  2. How many particlesNn , would be expected in each level? Round each to the nearest integer. (Happily. the number is still 11. and the energy still5hω0 .) What you have is a distribution of the energy that is as close to expectations is possible. given that numbers at each level in a real case are integers.
  3. Entropy is related to the number of microscopic ways the macro state can be obtained. and the number of ways of permuting particle labels withN0 ,N1,N2 , and N3fixed and totaling11 is11!(N0!N1!N2!N3!) . (See Appendix J for the proof.) Calculate the number of ways for your distribution.
  4. Calculate the number of ways if there were6 particles inn=0.5 inn=1 and none higher. Note that this also has the same total energy.
  5. Find at least one other distribution in which the 11 oscillators share the same energy, and calculate the number of ways.

  6. What do your finding suggests?

Suppose we have a system of identical particles moving in just one dimension and for which the energy quantization relationship isE=bn2/3, wherebis a constant andan integer quantum number. Discuss whether the density of states should be independent ofE, an increasing function ofE, or a decreasing function ofE.

Classically, what would be the average energy of a particle in a system of particles fine to move in the xy-plane while rotating about the i-axis?

Using the relationship between temperature and MandN given in (9-16) and that betweenE andn in (9-6), obtain equation (9-17) from (9- 12). The first sum given In Exercise 30 will be useful.
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