Sol_Problem_of_the_Week_–_14

Sol_Problem_of_the_Week_–_14 - among 9...

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Solution - Problem of the Week – 14 Consider a nanoparticle consisting of 3 atoms. The interatomic spring stiffness for this substance is 20 N/m, and the mass of one mole of the substance is 0.070 kg. (a) Using our model (the Einstein model of a solid), how many oscillators are in this nanoparticle? Each atom can oscillate in three mutually perpendicular directions. There are therefore 9 oscillators. (b) What is the energy required to raise one of these oscillators from its ground state to its first excited state? Show your work. We must first calculate the mass of a single atom. m = 0.07 kg / mole 6 ! 10 23 atoms / mole = 1.17 ! 10 " 25 kg / atom ! E = ! k m = 1.05 " 10 # 34 J $ sec 20 N / m 1.17 " 10 # 25 kg = 1.37 " 10 # 21 J = 8.6 " 10 # 3 eV (c) Calculate the entropy of the particle when 3 quanta of energy have been added to the particle. Show your work. We must calculate the number of ways three quanta of energy can be distributed
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Unformatted text preview: among 9 oscillators. ! = ( N " 1 + q )! q ! N " 1 ( ) ! = (9 " 1 + 3)! 3! 9 " 1 ( ) ! = 11! 3! 8 ( ) ! = 11 # 10 # 9 3 # 2 = 165 S = k B ln ! = 1.38 " 10 # 23 J / K $ ln(165) = 7.05 " 10 # 23 J / K (d) Calculate the approximate temperature of this particle when 3 quanta of energy have been added to the particle. Show your work. The addition of 3 quanta of energy has increase the entropy of the system from zero to the value calculated in part (c). 1 T = dS dE int , T ! " E int " S = 4.11 # 10 $ 21 J 7.05 # 10 $ 23 J / K = 58.3 K (e) Calculate the approximate heat capacity of the particle when 3 quanta of energy have been added to the particle. Show your work. C = ! E system / N atoms ! T = 4.11 " 10 # 21 J / 3 58.3 K = 2.35 " 10 # 23 J / K...
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This note was uploaded on 12/07/2011 for the course PHYS 172 taught by Professor ? during the Fall '08 term at Purdue.

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Sol_Problem_of_the_Week_–_14 - among 9...

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