Hmwk_14_Solutions

Hmwk_14_Solutions - Physics 221 Fall 2008 Homework #14...

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Physics 221 Fall 2008 Homework #14 Solutions Ch. 18:4; Ch. 19 Due Tues, Dec 9, 2008 14.1 According to the rule of Dulong and Petit, the heat capacity of solids should be the same per mole of atoms . (a) What is the predicted molar heat capacity of table salt (NaCl)? How does this compare with the measured value at room temperature in Table 17.3? The molar heat capacity at constant volume is predicted by the rule of Dulong and Petit to be C V = 2(3R) = 6(8.31 J/mol K) = 49.9 J/mol K. (The factor of 2 comes from 2 atoms per formula unit) The value in Table 17.3 is 51.4 J/mol K, which is within about 3% of the calculated value. (b) The rule of Dulong and Petit is obtained from classical physics. Quantum physics begins to be important as the temperature decreases, and results in a decrease in the heat capacity with decreasing temperature as seen in Fig. 18.21 on p. 629 of the text. According to quantum mechanics, for very low temperatures the heat capacity at constant volume per mole of atoms arising from lattice vibrations obeys the “Debye T 3 law”, which is C V = R 12 ± 4 5 ² ³ ´ µ · T ¸ D ² ³ ´ µ · 3 where R is the molar gas constant, ± D is a characteristic “Debye temperature” associated with a particular material, and both T and ± D are expressed in units of K. How much heat does it take to heat up a 1.00 g chunk of silicon (Si, M = 28.1 g/mol, ± D = 645 K) from 1.0 K to 10.0 K? (Hint: Since dQ = n C V ( T ) dT , you need to do an integration) Compare this value with the amount of heat it takes to heat up 1.00 g of Si from 1.0 ˚C to 10.0 ˚C according to the rule of Dulong and Petit. The number of moles of Si is n = (1.00 g)/(28.1 g/mol) = 0.0356 mol . Q = nC V ( T ) dT 1 . 0 K 10.0 K ± = nR 12 ² 4 5 ³ D 3 ´ µ · ¸ ¹ T 4 4 º » ¼ ½ ¾ ¿ T = 1 . 0 K T = 10 . 0K = nR 3 4 5 D 3 ´ µ · ¸ ¹ T 4 º » ½ ¾ T = 1 . 0 K T = 10 . 0K = ( 0 . 0356 mol) ( 8 . 31 J/mol K) 3 4 5 ( 645 K ) 3 ² ³ ´ µ · ( 10 . 0 K) 4 ¸ ( 1 . 0 K ) 4 ² ³ µ = 0.644 mJ. At high temperatures, heating the same amount of Si up by the same temperature change requires a heat of Q = n C V ² T = n (3 R ) ² T = (0.0356 mol)3(8.31 J/mol K)(9.0 K) = 8.0 J . This is about a factor of 12400 times larger than the low temperature value!
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2 14.2 A triatomic molecule consists of 3 atoms arranged along a straight line. The molecules can translate and rotate but not vibrate. (a)
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This note was uploaded on 01/28/2010 for the course PHYSICS 221 taught by Professor Johnson during the Fall '06 term at Iowa State.

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Hmwk_14_Solutions - Physics 221 Fall 2008 Homework #14...

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