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Chapter 9: Q31E (page 404)

Show that equation (9- 16) follows from (9-15) and (9- 10).

Short Answer

Expert verified

The value forkBTisln(1+NM)=h蝇0kBT

鈬�kBT=h蝇0ln(1+NM)

Step by step solution

01

Given data

The angle between the vertices of a tetrahedron is109.5掳, when the vertices of a tetrahedron are four vertices of a cube symmetrically chosen so that no two are adjacent.

02

Step 2:

Consider the cube in the given image. In order to find the angle between the vertices of a tetrahedron, let us form a tetrahedron using the following vertices.

03

To show the equation using natural logarithm.

MNh蝇0=h蝇0eh蝇0kBT-1

鈬�MN=1eh蝇0kBT-1

鈬�eh蝇0kBT-1=NM

鈬�eh蝇0kBT=1+NMn

ln(eh蝇0kBT)=ln(1+NM)

鈬�h蝇0kBT=ln(1+NM)

鈬�ln(1+NM)=h蝇0kBT

鈬�kBT=h蝇0ln(1+NM)

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Most popular questions from this chapter

Example 9.2 obtains a ratio of the number of particles expected in the n = 2state lo that in the ground state. Rather than the n = 2state, consider arbitrary n.

(a) Show that the ratio is numberofenergyEnnumberofenergyE1=n2e-13.6cV(1-n-2)/kBT

Note that hydrogen atom energies are En=-13.6eV/st2.

(b) What is the limit of this ratio as n becomes very large? Can it exceed 1? If so, under what condition(s)?

(c) In Example 9.2. we found that even at the temperature of the Sun's surface(~6000K), the ratio for n = 2 is only 10-8 . For what value of nwould the ratio be 0.01?

(d) Is it realistic that the number of atoms with high n could be greater than the number with low n ?

Determine the density of statesD(E)for a 2D infinite well (ignoring spin) in whichEnx,ny=(nx2+ny2)蟺2鈩�22mL2

Obtain an order-of-magnitude value for the temperature at which helium might begin to exhibit quantum/ superfluid behaviour. See equation (9.43). (Helium's specific gravity is about 0.12.)

By considering its constituents, determine the dimensions (e.g. length, distance over lime. etc.) of the denominator in equation(9鈭�26). Why is the result sensible?

The diagram shows two systems that may exchange both thermal and mechanical energy via a movable, heat-conducting partition. Because both Eand Vmay change. We consider the entropy of each system to be a function of both:S(E,V). Considering the exchange of thermal energy only, we argued in Section 9.2 that was reasonable to define1Tas未S未E. In the more general case, PTis also defined as something.

a) Why should pressure come into play, and to what mightPTbe equated.

b) Given this relationship, show thatdS=dQT(Remember the first law of thermodynamics.)
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