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Chapter 9: Q1CQ (page 402)

Consider a system of two identical objects heading straight toward each other. What would qualify and whit would disqualify the system as a thermodynamic systemin, and how, if at all, would this relate to the elasticity of the collision?

Short Answer

Expert verified

If the collision is inelastic, some energy will be lost to the surroundings. Also, it affects the evolution of the system and its properties that will lead to uncertainty about its expected behaviour.

Step by step solution

01

elastic collision

A collision defined as elastic occurs when the combined kinetic energy of the two bodies collides.

02

show how would relate to the elasticity of the collision.

The task is asking us to consider a system of two identical objects heading straight toward each other. We need to find the system that qualifies as a thermodynamic system and see how would relate to the elasticity of the collision.

Let's imagine the two identical objects heading directly towards each other. These two objects are not a thermodynamic system because they need a large group of particles from which precise measurements can be made. We would need to measure physical quantities such as pressure $P$ or temperature which is not easily and clearly defined or measured for small systems.

03

thermodynamic system given the simplicity of the two identical objects

But consider them to be a thermodynamic system given the simplicity of the two identical objects. Then precise measurements and extrapolations can be made of the properties of the individual particles. It is impossible for a large group of particles due to the several ways they could interact. In those cases, use density to describe it instead.

The system of two identical objects relates to the elasticity of the collision in that if the collision is perfectly elastic. Then, there would be losing of energy to the surroundings. Thus, it can model how the system will change with great precision.

Nevertheless, if the collision is inelastic, some energy will be lost to the surroundings. Also, it affects the evolution of the system and its properties that will lead to uncertainty about its expected behaviour.

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Most popular questions from this chapter

When a star has nearly bumped up its intimal fuel, it may become a white dwarf. It is crushed under its own enormous gravitational forces to the point at which the exclusion principle for the electrons becomes a factor. A smaller size would decrease the gravitational potential energy, but assuming the electrons to be packed into the lowest energy states consistent with the exclusion principle, "squeezing" the potential well necessarily increases the energies of all the electrons (by shortening their wavelengths). If gravitation and the electron exclusion principle are the only factors, there is minimum total energy and corresponding equilibrium radius.

(a) Treat the electrons in a white dwarf as a quantum gas. The minimum energy allowed by the exclusion principle (see Exercise 67) is
Uclocimns=310(3蟺2h3me32V)23N53

Note that as the volume Vis decreased, the energy does increase. For a neutral star. the number of electrons, N, equals the number of protons. If protons account for half of the white dwarf's mass M (neutrons accounting for the other half). Show that the minimum electron energy may be written

Uelectrons=9h280me(3蟺2M5mp5)131R2

Where, R is the star's radius?

(b) The gravitational potential energy of a sphere of mass Mand radius Ris given by

Ugray=-35GM2R

Taking both factors into account, show that the minimum total energy occurs when

R=3h28G(3蟺2me3mp5M)13

(c) Evaluate this radius for a star whose mass is equal to that of our Sun 2x1030kg.

(d) White dwarfs are comparable to the size of Earth. Does the value in part (c) agree?

Exercise 54 calculates the three oscillator distributions'E=0values in the special case wherekBTis15尉. Using a very common approximation technique. show that in the more general low-temperature limit,kBT鈮�蔚,theoccupation numbers become未/kBT,e5/kBT, and 1, for the distinguishable. boson. and fermion cases, respectively. Comment on these results. (Note: Although we assume thatkBT鈮�N蝇0/(2s+1). we also still assume that levels are closely spaced-that is kB鈫�鈮�h蝇0.),

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).

Consider a room divided by imaginary lines into three equal parts. Sketch a two-axis plot of the number of ways of arranging particles versus NleftandNrightfor the caseN=1023, Note that Nmiddleis not independent, being of courseN鈭�Nnght鈭�NleftYour axes should berole="math" localid="1658331658925" NleftandNright, and the number of ways should be represented by density of shading. (A form for numbers of ways applicable to a three-sided room is given in Appendix I. but the question can be answered without it.)

A. block has a cavity inside, occupied by a photon gas. Briefly explain what the characteristic of this gas should have to do with the temperature of the block.

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