Question

The interatomic spring stiffness for tungsten is determined from Young’s modulus measurements to be 90 N/m. The mass of one mole of tungsten is 0.185 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: ? = 1.0546 × 10−34 J · s (Planck’s constant divided by 2π), Avogadro’s number = 6.0221 × 1023 molecules/mole, kB = 1.3807 × 10−23 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 × 10−23 J/K/atom.)

204.91 E26 214.44 E27 22 3.85 E28 Figure 12.56

Answer 1

(a) 1 Quantum of energy = h squareroot x/M times NA for 4 oscillation x = 4 y = 1.0546 times 10^-34 squareroot 4 times 90/0.185 times 6.023 times 10^23 = 1.0546 times 10^-34 squareroot 1.172 times 10^27 (b) 1 Quantum = 3.6104 times 10^-21 J For q = 20, E = 3.6404 times 10^-21 times 20 = 7.2208 times 10^-20 J S_(20) = k_B In (No. of ways) = 1.38 times 10^-23 In (4.91 times 10^26) = 8.46 times 10^-22 J/K Delta E = E_q = 21 - 20 = 3.6104 times 10^-21 J S (21) = k_B In (No. of ways) = 1.38 times 10^-23 In (4.44 times 10^27) = 8.758 times 10^-22 Delta S = S (21) - S (20) = 0.325 times 10^-22 J/k T = Delta E/Delta S = 3.6104 times 10^-21 J/0.325 times 10^-22 J/k (C) Total internal energy of having n = 6 C atoms U = 3/2 n k_B T