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Chapter 9: Q6CQ (page 402)

Not surprisingly. in a collection of oscillators, as in other thermodynamic systems, raising the temperature causes particles' energies to increase. Why shouldn鈥檛 point be reached where there are more panicles in some high energy state than in a lower energy. state? (The fundamental idea, not a formula that might arise from it. is the object.)

Short Answer

Expert verified

Whereby increasing the temperature causes an increase in the particles' energies, doing so warrants a drastic change in the number of ways a particular energy state can be occupied. As a result, particles occupying the least energy state will still tend to have the most freedom to re-allocate the energy in response to an increase in temperature.

Step by step solution

01

thermodynamic system

The field of science known as thermodynamics studies temperature, heat, and the interactions between heat and other types of energy.

02

Step 2:higher energy states are more probable than lower energy states.

Therefore still, the most likely energy state that can be occupied by particle is of that corresponding to the lowest energy state. For this reason, it not plausible for the energy to be re-distributed in way that higher energy states are more probable than lower energy states.

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Most popular questions from this chapter

The Fermi velocityVfis defined byEF=12msF2, whereEFis the Fermi energy. The Fermi energy for conduction electron in sodium is 3 IV. (a) Calculate the Fermis velocity (b) What would be the wavelength of an electron with this velocity? (c) If each sodium atom contributes one conduction electron to the electron gas and sodium atom are spaced roughly0.37nmapart. If it is necessary, by the criteria of equation (9-43), to treat the conduction electron gas as quantum gas?

The exact probabilities of equation (9-9) rest on the claim that the number of ways of addingNdistinct non-negative integer to give a total ofM is (M+N-1)![M!(N-1)!] . One way to prove it involves the following trick. It represents two ways that Ndistinct integers can add toM-9 and5, respectively. In this special case.

The X's represent the total of the integers, M-each row has 5. The 1'srepresent "dividers" between the distinct integers of which there will of course be N-Ieach row has8 . The first row says thatn1 is3 (three X'sbefore the divider between it andn2 ), n2is0 (noX's between its left divider withn1 and its right divider withn3 ),n3 ) is1 . n4throughn6 are0 , n7is1 , and n8and n9are0 . The second row says that n2is 2. n6is 1, n9is2 , and all othern are0 . Further rows could account for all possible ways that the integers can add toM . Argue that properly applied, the binomial coefficient (discussed in AppendixJ ) can be invoked to give the correct total number of ways for anyN andM .

The electrons鈥� contribution to the heat capacity of a metal is small and goes to 0as T鈫�0. We might try to calculate it via the total internal energy, localid="1660131882505" U=鈭�EN(E)D(E)dE, but it is one of those integrals impossible to do in closed form, and localid="1660131274621" N(E)FDis the culprit. Still, we can explain why the heat capacity should go to zero and obtain a rough value.

(a) Starting withN(E)FDexpressed as in equation (34), show that the slope N(E)FDdEatE=EFis-1(4kBT).

(b) Based on part (a), the accompanying figure is a good approximation to N(E)FDwhen Tis small. In a normal gas, such as air, whenTis raised a little, all molecules, on average, gain a little energy, proportional to kBT. Thus, the internal energy Uincreases linearly with T, and the heat capacity, 鈭�U鈭�T, is roughly constant. Argue on the basis of the figure that in this fermion gas, as the temperature increases from 0to a small value T, while some particles gain energy of roughly kBT, not all do, and the number doing so is also roughly proportional to localid="1660131824460" T. What effect does this have on the heat capacity?

(c)Viewing the total energy increase as simply 鈭�U= (number of particles whose energy increases) (energy change per particle) and assuming the density of states is simply a constant Dover the entire range of particle energies, show that the heat capacity under these lowest-temperature conditions should be proportional to kBREFT. (Trying to be more precise is not really worthwhile, for the proportionality constant is subject to several corrections from effects we ignore).

We based the exact probabilities of equation (9-9) on the claim that the number of ways of addingN distinct nonnegative integer quantum numbers to give a total ofM is{M+N-1)!/M!(N-1)!. Verify this claim (a) for the caseN=2,M=5and(b)for the case.

N=5,M=2

Using the result of part (a) in Exercise 74 , determine the number of photons per unit volume in outer space. whose temperature - the so-called cosmic background temperature-is2.7K .
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