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Chapter 9: Q5CQ (page 402)

Defend or refuel the following claim: An energy distribution, such as the Boltzmann distribution. specifies the microstate of a thermodynamic system.

Short Answer

Expert verified

The statement above is not completely accurate. In principle, energy distributions such as the Boltzmann distribution determines the likelihood of several particles in a thermodynamic system to occupy a specific microstate. Simply put, energy distributions do not give an individual-particle energy, but a range of possible energies that can be occupied for a given temperature

Step by step solution

01

Boltzmann distribution

A Boltzmann distribution (also called Gibbs distribution) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of that state's energy and the temperature of the system.

02

reason for the enormous number of possible microstates

The reason for this is the enormous number of possible microstates to be inhabited resulting in the chaotic behaviour of the individual motion of the molecules. As a result, probability distributions are formulated to describe a statistical ensemble of microstates to describe the macroscopic properties of the system such as temperature and internal energy.

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Most popular questions from this chapter

Heat capacity (at constant volume) is defined as∂U/∂T. (a) Using a result derived in Example 9.6. obtain an expression for the heat capacity per unit volume, inJ/K⋅mi3, of a photon gas. (b) What is its value at300K?

Consider a system of one-dimensional spinless particles in a box (see Section 5.5) somehow exchanging energy. Through steps similar to those giving equation (9-27). show that

D(E)=m1/2Lℏπ21E1/2

Four distinguishable Hamonic oscillators a,b,c,andd may exchange energy. The energies allowed particleareEa=naℏω0: those allowed particlebareEb=nbℏω0, and so on. Consider an overall state (macro-state) in which the total energy is3ℏω0. One possible microstate would have particles a,b,andcin theirn=0states and particle d in itsn=3states that is,na,nb,ncnd=(0,0,0,3).

(a) List all possible microstates, (b) What is the probability that a given particle will be in itsn=0 state? (c) Answer part (b) for all other possible values of n. (d) Plot the probability versus n.

The fact that a laser's resonant cavity so effectively sharpens the wavelength can lead to the output of several closely spaced laser wavelengths, called longitudinal modes. Here we see how. Suppose the spontaneous emission serving as the seed for stimulated emission is of wavelength , but somewhat fuzzy, with a line width of roughly0.001nm either side of the central value. The resonant cavity is exactly60cm long. (a) How many wavelengths fit the standing-wave condition'? (b) If only u single wavelength were desired, would change the length of the cavity help? Explain.

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?
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