The model of a diatomic molecule presented in this chapter is called the rigid rotor- harmonic...

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The model of a diatomic molecule presented in this chapter is called the rigid rotor- harmonic oscillator model. The rotational-vibrational energy in this approximation is E, = (n+ )hv + BJ(J+ 1) This expression can be improved in a number of ways. The harmonic oscillator approximation can be modified to include terms that reflect the deviations from harmonic behavior (an- harmonicity) as the vibrational energy of the molecule increases. This is done by quantum mechanical perturbation theory, which gives E, = (n+ )hv- xdn+ thv+. where x, is a small constant called the anharmonicity constant. In addition to this, there is a correction due to the fact that the molecule is not a rigid rotor and, in fact, stretches some as the molecule rotates with greater energy. This is also handled by perturbation theory and gives E, = BJ(J+ 1) - DJJ+ 1)* + . where D is a small constant called the centrifugal distortion constant. Lastly, there exists a coupling between the rotational and vibrational modes of the molecule, since its moment of inertia changes as the molecule vibrates. Putting all this together gives Er = (n+ )hv + BJ(J+ 1) – x.(n + })*hv – DJ(J+ 1)? - aln+ J(J+ 1) where a is the rotation-vibration coupling constant. These terms, which correct the rigid rotor-harmonic oscillator approximation, are usually quite small. Using this more rigorous expression for ɛ, show that the molecular partition function can be written in the form q(V, T)=9r-hoqcorr where 2kT (D B B 9cerr = 1+ 2Bhv + x. + higher-order terms in D, a, and x. (elhy - 1) Calculate the effect of qeorr on E and Cv for O2 at 300°K, given the following values of the spectroscopic parameters: x. =0.0076, D=4.8 x 10- cm-, and a = 0.016 cm-. %3D The model of a diatomic molecule presented in this chapter is called the rigid rotor- harmonic oscillator model. The rotational-vibrational energy in this approximation is E, = (n+ )hv + BJ(J+ 1) This expression can be improved in a number of ways. The harmonic oscillator approximation can be modified to include terms that reflect the deviations from harmonic behavior (an- harmonicity) as the vibrational energy of the molecule increases. This is done by quantum mechanical perturbation theory, which gives E, = (n+ )hv- xdn+ thv+. where x, is a small constant called the anharmonicity constant. In addition to this, there is a correction due to the fact that the molecule is not a rigid rotor and, in fact, stretches some as the molecule rotates with greater energy. This is also handled by perturbation theory and gives E, = BJ(J+ 1) - DJJ+ 1)* + . where D is a small constant called the centrifugal distortion constant. Lastly, there exists a coupling between the rotational and vibrational modes of the molecule, since its moment of inertia changes as the molecule vibrates. Putting all this together gives Er = (n+ )hv + BJ(J+ 1) – x.(n + })*hv – DJ(J+ 1)? - aln+ J(J+ 1) where a is the rotation-vibration coupling constant. These terms, which correct the rigid rotor-harmonic oscillator approximation, are usually quite small. Using this more rigorous expression for ɛ, show that the molecular partition function can be written in the form q(V, T)=9r-hoqcorr where 2kT (D B B 9cerr = 1+ 2Bhv + x. + higher-order terms in D, a, and x. (elhy - 1) Calculate the effect of qeorr on E and Cv for O2 at 300°K, given the following values of the spectroscopic parameters: x. =0.0076, D=4.8 x 10- cm-, and a = 0.016 cm-. %3D

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Lom It 2KT 3D 2Bhu Och e At T 300 k Xe 00076 cmt 48X106 ... View the full answer