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Chapter 9: Q2CQ (page 402)

What information would you need to specify the macro-state of the air in a room? What information would you need to specify the microstate?

Short Answer

Expert verified

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

The word "macro-state" refers to the macroscopic characteristics of a thermodynamic system.

Step by step solution

01

microstates and macro-states

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

The word "macro-state" refers to the macroscopic characteristics of a thermodynamic system.

To specify what microstate the system is in, a detailed description is needed. We can define a macro-state by specifying the value of macroscopic variable.

02

physical properties of microstates

For typical systems, the number of microstates is very big, and they represent the system is far more particular than considered. For example, let's consider a box filled with gas or the air in a room. We have no way to measure the exact position and momentum of every gas molecule, and they are insignificant even if we could measure them. Instead, we are usually interested in a small number of macroscopic variables: the total energy of the system, the total number of gas molecules, the volume of space it takes up, etc. These physical quantities are measurable and have practical importance.

03

macro-states

A macro-state is defined by defining the value of every macroscopic variable. The density of states is the quantity of microstates that make up a microstate. It is written asΩ(E,V,…)Ω(E,V,…) , where the arguments are the macro-macroscopic

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Most popular questions from this chapter

Derivation of equation(9−40): Our model for calculatingE¯is equation (9-26), whose denominator is the total number of particlesNand whose numerator is the total energy of the system, which we here callUtotal. State with the denominator:

N=∫0∞N(E)∩(E)dE

Insert the quantum gas density of states and an expression for the distribution. using±to distinguish the Bose-Einstein from the Fermi-Dirac. Then change variables:E=y2, and factorBe+r2/kUTout of the denominator. In the integrand will be a factor

(1∓1Be−y2/kBT)−1

Using,(l∓ε)−1≅1±ε a sum of two integrals results, each of Gaussian form. The integral thus becomes two terms in powers of1/B. Repeat the process. but instead find an expression forUtotalin terms of1/B, using

U|ntal=∫0∞EN(E)D(E)dE

Divide your expression forUtotalby that forN. both in terms of1/B. Now1/Bcan safely be eliminated by using the lowest-order expression forNin terms of1/B.

The Fermi energy in a quantum gas depends inversely on the volume, Basing your answer on Simple Chapter 5 type quantum mechanics (not such quaint notions as squeezing classical particles of finite volume into a container too small). Explain why.

Determine the density of statesD(E)for a 2D infinite well (ignoring spin) in whichEnx,ny=(nx2+ny2)π2ℏ22mL2

The Stirling approximation.J!≡2πJJ+1/2e-J, is very handy when dealing with numbers larger than about100 . Consider the following ratio: the number of ways Nparticles can be evenly divided between two halves of a room to the number of ways they can be divided with60%on the right and40%on the left.

(a) Show, using the Stirling approximation, that the ratio is approximately4046065Nfor largeN.

(b) Explain how this fits with the claim that average behaviours become more predictable in large systems.

You have six shelves, one above the other and all above the floor, and six volumes of an encyclopedia, A, B, C, D, E and F.

(a) list all the ways you can arrange the volumes with five on the floor and one on the sixth/top shelf. One way might be(ABCDE_,_,_,_,_F).

(b) List all the ways you can arrange them with four on the floor and two on the third shelf.

(c) Show that there are many more ways, relative to pans (a) and (b), to arrange the six volumes with two on the floor and two each on the first and second shelves. (There are several ways to answer

this, but even listing them all won't take forever it's fewer than.)

(d) Suddenly, a fantastic change! All six volumes are volume X-it's impossible to tell them apart. For each of the three distributions described in parts (a), (b), and (c), how many different (distinguishable) ways are there now?

(e) If the energy you expend to lift a volume from the floor is proportional to a shelf's height, how do the total energies of distributions (a), (b), and (c) compare?

(I) Use these ideas to argue that the relative probabilities of occupying the lowest energy states should be higher for hosons than for classically distinguishable particles.

(g) Combine these ideas with a famous principle to argue that the relative probabilities of occupying the lowest states should he lower for fermions than for classically distinguishable particles.
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