You have a vibrational degree of freedom that can be treated like a harmonic oscillator with a spacing between energy levels of 538.03 cm1, What is the probability that this degree of freedom is in state v= 1 at a temperature of 1,138.2 K. Give your answer with at least 3 significant figures.
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- Determine the number of total degrees of freedom and the number of vibrational degrees of freedom for the following species. a Hydrogen sulfide, H2S b Carbonyl sulfide, OCS c The sulfate ion, SO42 d Phosgene, COCl2 e Elemental chlorine, Cl2 f A linear molecule having 20 atoms g A nonlinear molecule having 20 atomsAs noted in lecture, the rigid rotor model can be improved by recognizing that in a realistic anharmonic potential, the bond length increases with the vibrational quantum number v. Thus, the rotational constant depends on v, and it can be shown that By = Be – ae(v +). For 'H®Br, B = 8.473 cm1 and a = 0.226 cm². Use this information to calculate the bond length for HBr a) as a rigid rotor, and b) as a nonrigid rotor in the ground vibrational state. Find a literature value for this bond length (cite your source) and compare your answers. Under what conditions would you expect the nonrigid rotor to be a significantly better model?The rotational constant of 12C16O is 57.65 GHz. Calculate the value of J for the most populated level at (a) 300 K and (b) 1000 K.
- A diatomic molecule has a rotational constant of 8.0 cm-1, and vibrational frequency of 1200 cm 1. What is the energy of the state with v = 1 and j = 6 relative to the lowest energy state, E = E(v=0,j=0)? 0Calculate the ratio of the populations in the first two rotational energy levels of carbon monoxide, the lowest J=0 energy level and the higher J = 1 energy level, at 300 K if the energy difference between the levels is 3.8 cm-1and the degeneracies gJ of the two levels are g0 = 1 and g1 = 3, respectively. (You will see in Section 20.3 that there are 2J 1 1 rotational quantum states at each energy level EJ.)J.G. Dojahn et al. (J. Phys. Chem. 100, 9649 (1996)) characterized the potential energy curves of the ground and electronic states of homonuclear diatomic halogen anions. These anions have a 2Σu+ ground state and 2Πg, 2Πu, and 2Σg+ excited states. To which of the excited states are electric-dipole transitions allowed from the ground state? Explain your conclusion.
- What can we predict about fluorescence intensity as a function of temperature from Boltzmann’s distribution?Consider the diatomic molecule AB modeled as a rigid rotor (two masses separated by a fixed distance equal to the bond length of the molecule). The rotational constant of the diatomic AB is 25.5263 cm-1. (a) What is the difference in energy, expressed in wavenumbers, between the energy levels of AB with J = 10 and J = 6? (b) Consider now a diatomic A'B', for which the atomic masses are ma 0.85 mA and mB' 0.85 mB and for its bond length ra'B' = 0.913 rAB. What is the difference in energy, expressed in wavenumbers, between the energy levels of the A'B' molecule with J = 9 and J = 7?Assuming the vibrations of a ¹4N2 molecule are equivalent to those of a harmonic oscillator with a force constant kr = 2293.8 N/m, what is the zero-point energy of vibration of this molecule? Use m(14N) = 14.0031 mu.
- Develop an expression for the value of J corresponding to the most highly populated rotational energy level of a diatomic rotor at a temperature T remembering that the degeneracy of each level is 2J + 1. Evaluate the expression for ICl (for which ᷉ B = 0.1142 cm−1) at 25 °C. Repeat the problem for the most highly populated level of a spherical rotor, taking note of the fact that each level is (2J + 1)2-fold degenerate. Evaluate the expression for CH4 (for which ᷉ B = 5.24 cm−1) at 25 °C. Hint: To develop the expression, recall that the first derivative of a function is zero when the function reaches either a maximum or minimum value.The first five vibrational energy levels of ¹H¹27 I are at 1144.83, 3374.90, 5525.51, 7596.66, and 9588.35 cm¹. Treating the molecule as an anharmonic oscillator, estimate the dissociation energy of the molecule in units of reciprocal centimetres (cm-¹). [Note: m(¹H) = 1.0078 u, m(¹271) = 126.9045 u; assume the second order anharmonicity constant, Ye, to be zero.] [Note: Use graph paper in your answer.]7 The fundamental vibrational wavenumber ( ṽ) for 1H 127I molecule is 23096 cm-1 A. Determine the force constant (k) of 1H 127I B. Calculate the value of for 2H 127I. Show all calculations and the units. Explain the reasoning