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Chapter 12: Q21P (page 508)

In Chapter 4 you determined the stiffness of the interatomic 鈥渟pring鈥� (chemical bond) between atoms in a block of lead to be 5 N/m, based on the value of Young鈥檚 modulus for lead. Since in our model each atom is connected to two springs, each half the length of the interatomic bond, the effective 鈥渋nteratomic spring stiffness鈥� for an oscillator is

4 脳 5 N/m = 20 N/m. The mass of one mole of lead is 207 g (0.207 kg). What is the energy, in joules, of one quantum of energy for an atomic oscillator in a block of lead?

Short Answer

Expert verified

The energy of one quantum of energy for an atomic oscillator in a block of lead is

8.04脳10-22J.

Step by step solution

01

Given data

k = 20 N/m

M = 0.207 kg

02

Solution

Mass of the one atom is,

m=MNA

The energy of one quantum of energy for an atomic oscillator in a block of

lead is,

E=鈩�蝇鈥�鈥�鈥�=h2rkm=h2rkM/NA鈥�=6.626脳10-342(3.14)200.207/6.022脳1023=8.04脳10-22J

Hence,

E=8.04脳10-22J

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Most popular questions from this chapter

At room temperature (293 K), calculate kBT in joules and eV.

Buckminsterfullerene, C60, is a large molecule consisting of 60 carbon atoms connected to form a hollow sphere. The diameter of a C60 molecule is about 7脳10-10 m. It has been hypothesized that C60 molecules might be found in clouds of interstellar dust, which often contain interesting chemical compounds. The temperature of an interstellar dust cloud may be very low; around 3 K. Suppose you are planning to try to detect the presence of C60 in such a cold dust cloud by detecting photons emitted when molecules undergo transitions from one rotational energy state to another. Approximately, what is the highest-numbered rotational level from which you would expect to observe emissions? Rotational levels are l= 0, 1, 2, 3, 鈥�

How many different ways are there to arrange 4 quanta among 3 atoms in a solid?

Young鈥檚 modulus for copper is measured by stretching a copper wire to be about 1.2脳1011N/m2. The density of copper is about 9g/cm3, and the mass of a mole is .Starting from a very low temperature, use these data to estimate roughly the temperature T at which we expect the specific heat for copper to approach 3 kB . Compare your estimate with the data shown on a graph in this chapter.

The interatomic spring stiffness for tungsten is determined from Young鈥檚 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 鈥渙scillators鈥� (masses on springs), note that since each oscillator is attached to two 鈥渟prings,鈥� and each 鈥渟pring鈥� 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: h潭=1.05脳10-34鈥�J.s(Planck鈥檚 constant divided by 2蟺), Avogadro鈥檚 number = 6.0221脳1023鈥�molecule/mole, kB=1.3807脳10-23鈥�J/K(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脳10鈭�23 J/K/atom.)
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