Chem390_solution_2 - 2 Chem390_solution_2.nb Chem3900 Solution to Problem Set 2 Problem 1 1 The eigenvalue for the energy of a harmonic oscillaotor

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Chem39 0 0 : Solution to Problem Set 2 Problem 1 : ± 1 ² ± The eigenvaluefor the energy of a harmonic oscillaotor is E v ² ± v ³ 1 ³ 2 ² h ´ . The probability of finding an oscillator in the v µ th state is given by P v ± 1 ³ Q ² Exp ± E v ³± k B T ²² . Q ± T ± ² ² Simplify ³´ v ² 0 ³ Exp ³ ´ µ v µ 1 2 · h · k B T ¸¸ ; P ± v ± ² ² Exp ³ ´ µ v µ 1 2 · h · k B T ¸¹ Q ± T ² ; AVGv ² Simplify ³´ v ² 0 ³ µ vP ± v ²·¸ 1 ± 1 ²³ h ´ TkB ± 2 ² We need to solve v ·² 1 for the temperature T. Solve ³ 1 ²² 1 ´ 1 µ¸ h · TkB . constJSI . ·¹ º 2.75 º 10 13 s ´ 1 » ,T ¸ ±± T µ 1904.05 K ²² ± 3 ² Let' s first plot v · for the given T range. 1 ´ 1 µ¸ h · TkB . constJSI . ·¹ º 2.75 º 10 13 s ´ 1 »¶ . T ¹ TK; Plot ± » , ¼ T, 0.001, 2000 ½ , PlotRange ¹ ¼ 0, 1.2 ½ , AxesLabel ¹ ¼ "T µ K · ", " ¼ v ½ " ½² 500 1000 1500 2000 T ³ K ´ 0.2 0.4 0.6 0.8 1 v · The above plot shows that the average quantum number v stays near zero below about 500 K and then it increases almost linearly at higher temperatures. Since the average energy is proportional to v and C V is the rate of the energy increase with T, this plot is consistent with the Figure 17.4, where the C V slowly increases near 500 K and then stays constant at higher temperatures. Problem 2 : Let' s first define p j s and the partition function Q ´ T µ . g ² ¼ 5, 3, 1, 5 ½ ; E Oxy ² ¼ 0.00 eV, 0.02 eV, 0.03 eV, 1.97 eV ½ ; Q ± T ± ² ² ´ j ² 1 4 g ±± j ²² Exp ³ ´ E Oxy ±± j ²² k B T ¸ . constJSI; p ² g Exp ³ ´ E Oxy k B T ¸ ¹ Q ± T ²¶ . constJSI µ 5 ³ 0. K T 5 ³ ± 22 860.8 K T ²³ ± 348.133 K T ² 3 ³ ± 232.089 K T ² 5 ³ 0. K T , 3 ³ ± 232.089 K T 5 ³ ± 22 860.8 K T ²³ ± 348.133 K T ²
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This note was uploaded on 05/04/2009 for the course CHEM 3900 taught by Professor Park during the Spring '08 term at Cornell University (Engineering School).

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Chem390_solution_2 - 2 Chem390_solution_2.nb Chem3900 Solution to Problem Set 2 Problem 1 1 The eigenvalue for the energy of a harmonic oscillaotor

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