Chem 113A HW4 Solutions

Chem 113A HW4 Solutions - Chem 113A Homework Assignment...

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Chem 113A - Homework Assignment Week 4 Due March 2 nd in class at the beginning of lecture 1) According to the experimental data table from your book (Table 5.1), the force constant of H 127 I is 291 N m −1 . a. What is the reduced mass of H 127 I ? ? = ? 1 ? 2 ? 1 + ? 2 = 1(127) 1 + (127) ? = 0.992 ? = 1.65 × 10 −27 ?𝑔 b. Calculate the fundamental vibrational frequency of H 127 I in units of cm −1 . ? ̃ = 1 2𝜋? ? ? = 1 2𝜋? 291 𝑁? −1 1.65 × 10 −27 ?𝑔 = 2.2 × 10 3 ?? −1 c. From which region of the electromagnetic spectrum would H 127 I absorb photons? This wavenumber corresponds to the IR region of the spectrum d. What is the zero-point energy of H 127 I ? ? 0 = 1 2 ℎ? = 2.2 × 10 −20 𝐽 = 0.14 ?𝑉 2) The harmonic oscillator model assumes that the potential experienced by the oscillator is perfectly parabolic. Real molecules however do not experience a perfectly harmonic potential. The Morse potential is one of the simplest anharmonic potentials for which we can express the potential in a simple functional form: 𝑉(𝑥) = ? 𝑒 (1 − ? −?𝑥 ) 2 Where ? 𝑒 and ? can be determined experimentally. a. Use a Taylor-Maclaurin series expansion of the Morse potential to show that the harmonic oscillator potential is a good first-order approximation of the Morse potential. The Taylor expansion of 𝑉(𝑥) is: 𝑉(𝑥) = 𝑉(0) + ( ?𝑉 ?𝑥 ) 𝑥=0 𝑥 1 1! + ( ?𝑉 2 ?𝑥 2 ) 𝑥=0 𝑥 2 2! + ⋯ ( ?𝑉 ?𝑥 ) = 2? 𝑒 ?(? −?𝑥 − ? −2?𝑥 ) ( ?𝑉 ?𝑥 ) 𝑥=0 = 2? 𝑒 ?(? −?∗0 − ? −2?∗0 ) = 0 ( ? 2 𝑉 ?𝑥 2 ) = −2? 𝑒 ?(?? −?𝑥 − 2?? −2?𝑥 ) ( ? 2 𝑉 ?𝑥 2 ) 𝑥=0 = −2? 𝑒 ?(?? −?∗0 − 2?? −2?∗0 ) = 2? 𝑒 ? 2 We may set the zero of the potential to any arbitrary value, so 𝑉(0) = 0 . We then have: 𝑉(𝑥) = 0 + 0 ∗ 𝑥 + 2?

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