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Chapter 12: Q18P (page 508)
How many different ways are there to arrange 4 quanta among 3 atoms in a solid?
The numbers of ways to arrange 4 quanta among 12 oscillations is 1,365.
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A box contains a uniform disk of mass M and radius R that is pivoted on a low-friction axle through its centre (Figure 12.58). A block of mass m is pressed against the disk by a spring, so that the block acts like a brake, making the disk hard to turn. The box and the spring have negligible mass. A string is wrapped around the disk (out of the way of the brake) and passes through a hole in the box. A force of constant magnitude F acts on the end of the string. The motion takes place in outer space. At time the speed of the box is , and the rotationalspeed of the disk is . At time the box has moved a distance x, and the end of the string has moved a longer distance d, as shown.
(a) At time , what is the speed of the box? (b) During this process, the brake exerts a tangential friction force of magnitude f. At time , what is the angular speed of the disk? (c) At time , assume that you know (from part b) the rotational speed of the disk. From time to time , what is the increase in thermal energy of the apparatus? (d) Suppose that the increase in thermal energy in part (c) is . The disk and brake are made of iron, and their total mass is . At time their temperature was . At time , what is their approximate temperature?
How does the speed of sound in a gas change when you raise the temperature from ? Explain briefly.
The interatomic spring stiffness for tungsten is determined from Young’s modulus measurements to be 90 N. The mass of one mole of tungsten is 0185 kg . If we model a block of tungsten as a collection of atomic “oscillators” (masses on springs), note that since each oscillator is attached to two “springs,” and each “spring” is half the length of the interatomic bond, the effective interatomic spring stiffness for one of these oscillators is 4 times the calculated value given above.Use these precise values for the constants: (Planck’s constant divided by 2π), Avogadro’s number = , (the Boltzmann constant). (a) What is one quantum of energy for one of these atomic oscillators? (b) Figure 12.56 contains the number of ways to arrange a given number of quanta of energy in a particular block of tungsten. Fill in the blanks to complete the table, including calculating the temperature of the block. The energy E is measured from the ground state. Nothing goes in the shaded boxes. Be sure to give the temperature to the nearest 0.1 kelvin. (c) There are about 60 atoms in this object. What is the heat capacity on a per-atom basis? (Note that at high temperatures the heat capacity on a per-atom basis approaches the classical limit of 3kB = 4.2×10−23 J/K/atom.)
Suppose that the entropy of a certain substance (not anEinstein solid) is given by, where ais a constant. Whatis the energy Eas a function of the temperature T?
Consider the exponential function . Evaluate this function for x = 1, 10,000, and 0.01.
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