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Calculate the relative numbers of Br2 molecules ( ᷉v = 321 cm−1) in the second and first excited vibrational states at (i) 298 K, (ii) 800 K.
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- Calculate the relative numbers of Cl2 molecules ( ᷉v = 559.7 cm−1) in the ground and first excited vibrational states at (i) 298 K, (ii) 500 K.The vibrational temperature of a molecule prepared in a supersonic jet can be estimated from the observed popula- tions of its vibrational levels, assuming a Boltzmann distri- bution. The vibrational frequency of HgBr is 5.58 × 1012 s-1, and the ratio of the number of molecules in the n = 1 state to the number in the n = 0 state is 0.127. Estimate the vibra- tional temperature under these conditions.b. The energy difference between consecutive vibrational states is 1.0 x 1020 J for a molecule. (i) Calculate the population ratio, n4/n¡, for this system at 298 K and discuss the significance of this ratio in terms of the distribution of molecules in the higher vibrational energy states. (ii) Estimate the vibrational partition function at 298 K. (iii) Estimate the fundamental vibration wave number for this molecule. h = 6.626 x 10-3ª J s k= 1.38 x 1023 J K' c = 2.998 x 10® m s''
- Estimate the ratio of the number of molecules in the first excited vibrational state of the molecule N2 to the number in the ground state, at a temperature of 450 K. The vibrational frequency of N2 is 7.07 × 1013 s-1.Estimate the ratio of the number of molecules in the firstexcited vibrational state of the molecule N2 to the numberin the ground state, at a temperature of 450 K. The vibrational frequency of N2 is 7.07 × 1013 s-1.Derive an expression for the mean energy of a collection of molecules that have three energy levels at 0, ε, and 3ε with degeneracies 1, 5, and 3, respectively.
- 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.]The rotational constant for CO is 1.9314 cm−1 and 1.6116 cm−1 in the ground and first excited vibrational states, respectively. By how much does the internuclear distance change as a result of this transition?A molecule in a liquid undergoes about 1.0 × 1013 collisions in each second. Suppose that (i) every collision is effective in deactivating the molecule vibrationally and (ii) that one collision in 100 is effective. Calculate the width (in cm−1) of vibrational transitions in the molecule.
- What is the most highly populated rotational level of Cl2 at (i) 25 °C, (ii) 100 °C? Take ᷉ B = 0.244 cm−1.Calculate the CO and CS bond lengths in OCS from the rotational constants B(16O12C32S) = 6081.5MHz, B(16O12C34S) = 5932.8MHz.The three normal modes of water are the symmetric stretch (3652 cm¹), the antisymmetric stretch (3756 cm¹), and the bend (1595 cm¹). (a) Calculate the molecular vibrational partition function of water at 500 K. (b) At 500 K, what fraction of water molecules have the bend excited to v₂=1. What fraction of water molecules have the symmetric stretch excited to v₁=1? Why do more molecules have the bend excited? (c) At 500 K, what fraction of water molecules have both v2-1 and v₁=1 excited?