5.1. A crystal is composed of identical atoms which only interact with one another very weakly. Each has a set of three equally spaced singlet energy levels, E₁. E₂, E, such that E-E₁ = 1023J and E-E, 2x 103 J. Calculate the temperature at which (a) there are equal numbers (to within 1 per cent) of atoms in each of the three states. (b) 99 per cent of the atoms are in the ground state E₁. (c) If the energy level E, is doubly degenerate calculate the temperature for (b). (d) Why in (a) cannot truly equal populations be achieved? 5.2. In an assembly of 1023 simple harmonic oscillators each has a frequency 10¹³ Hz. Calculate (ignoring the zero-point energy) the mean thermal energy of the system at 2 K, 20 K, 200 K. and 2000 K. Discuss the significance of your answers. in what temperature range could the system be treated classically? 5.3. Using the Dulong-Petit relation calculate the (classical) thermal energy of 1 g mole of a material at 300 K. Aluminium has a Debye 6 of about 430 K. Estimate the thermal energy per gram mole at 300 K with the aid of Fig. 5.3. Explain why the two results are different. 54. What factors determine the Debye # of a particular element? Estimate the Debye e of gold from the following information. For gold, the atomic weight is 197, the density is 1.9 x 10 kg m and the velocity of sound is 2100 ms. For copper the atomic weight is 63-5, the density is 8900 kg m³, the velocity of sound is 3800 ms, and the Debye is 348 K. Copper has the same structure as gold. 25 20 3 mol K- o. 15 -1941(7/6J mol K-1 Fig. 5.3, the specific heat per mole calculated using Debye theory. Below T/e = 0.1, the curve follows the cubic relation 00 0-2 04 0-6 0-8 10 12 (TIA) 55 At low temperatures the Debye of NaCl and KCl. which have the same crystal structure, are 310 K and 230 K respectively The lattice specific heat of KCI at 5K is 3.8 x 10J mol K- Estimate the lattice specific heat of NaCl a: 5 K and of KCl at 2 K 5.6 Estimate the change in the thermal energy of 100 g of copper (atomic weight 63-5. Debye 9348 K) when it is cooled (a) from 300 to 4 K. (b) from 78 to 4 K. and (c) from 20 to 4 K. If the latent heat of liquid helium is 2700 J 1-1. estimate how much helium would be required to produce the cooling in each case. (This calculation will over-estimate the quantity because the cooling power of the cold gas is neglected.) 5.7. The Debye of diamond is 2000 K, its density is 3500 kg m³, its atomic weight is 12 and the interatomic spacing is 0-15 nm. Calculate the velocity of sound in diamond. Make an estimate of the dominant phonon wavelength at 300 K (sce section 6.4) and calculate the frequency of lattice vibration to which this cor- responds. Answers to above Problems. Chapter 5. pp. 93-4 5.1. (a) 144 K: (b) 0-16 K: (c) 0-136 K; (d) T = x required. 5.2. 6-3 x 1097 J; 6-3 x 108 J; 71 J: 2700 J; >500 K. 5.3. 7470 J mol: 4000 J mol"¹. 5.4. 169 K. - 5.5. 1.55 x 10J mol-1.2.4 x 10-³ J mol¯¹. 5.6. (a) 8000J: (b) 640 J. (c) 2-89 J: 31: 0-24 1: 10 31. - 5.7. 1-2x 10 ms: 1 nm: 1-2x 10¹³ Hz.