Question

Problem 3: vibrational degrees of freedom of an ideal diatomic molecule FIG. 2: Vibrational degrees of freedom in a diatomic molecule. In addition to rotational degrees of freedom, the diatomic molecule has a vibrational energy associated with the stretching of the interatomic bond. Quantum mechanically, the vibrational energy is quantized into non-degenerate energy levels, Evib = (n + 1/2) hf, where n can take any non-negative integer value, i.e., n = 0, 1, 2, 3, . . . (i) The exponent of the Boltzmann factor has to be dimensionless, which allows us to define a characteristic temperature Ovib = hf/kB. The fundamental frequency of vibration of a HCl molecule is f = 8.66 x 10^13 Hz. Calculate the value of Ovib for the HCl molecule. (ii) By considering the partition function Zvib for the vibrational degrees of freedom, and making use of the fact that the average vibrational energy is given by -? ln Zvib/?? (where ? = 1/(kBT)), derive an exact analytic expression for the associated heat capacity Cv. Show that at high temperatures, Cv agrees with what one expects from the Equipartition Theorem, and at low temperatures, Cv behaves in accordance with the Third Law of Thermodynamics. (Classically, the stretching of the bond gives rise to (I) a stored potential energy which is quadratic in the displacement of the bond length from its equilibrium value, and (II) a kinetic energy associated with the relative displacement of the two atoms, which also goes as the square of the corresponding momentum.)

Problem 3: vibrational degrees of freedom of an ideal diatomic molecule

FIG. 2: Vibrational degrees of freedom in a diatomic molecule.

In addition to rotational degrees of freedom, the diatomic molecule has a vibrational energy associated with the stretching of the interatomic bond. Quantum mechanically, the vibrational energy is quantized into non-degenerate energy levels,

Evib = (n + 1/2) hf,

where n can take any non-negative integer value, i.e., n = 0, 1, 2, 3, . . .
(i) The exponent of the Boltzmann factor has to be dimensionless, which allows us to define a characteristic temperature Ovib = hf/kB. The fundamental frequency of vibration of a HCl molecule is f = 8.66 x 10^13 Hz. Calculate the value of Ovib for the HCl molecule.
(ii) By considering the partition function Zvib for the vibrational degrees of freedom, and making use of the fact that the average vibrational energy is given by -? ln Zvib/?? (where ? = 1/(kBT)), derive an exact analytic expression for the associated heat capacity Cv. Show that at high temperatures, Cv agrees with what one expects from the Equipartition Theorem, and at low temperatures, Cv behaves in accordance with the Third Law of Thermodynamics.
(Classically, the stretching of the bond gives rise to (I) a stored potential energy which is quadratic in the displacement of the bond length from its equilibrium value, and (II) a kinetic energy associated with the relative displacement of the two atoms, which also goes as the square of the corresponding momentum.)

Problem 3: vibrational degrees of freedom of an ideal diatomic molecule

FIG. 2: Vibrational degrees of freedom in a diatomic molecule.

In addition to rotational degrees of freedom, the diatomic molecule has a vibrational energy associated with the stretching of the interatomic bond. Quantum mechanically, the vibrational energy is quantized into non-degenerate energy levels,

Evib = (n + 1/2) hf,

where n can take any non-negative integer value, i.e., n = 0, 1, 2, 3, . . .
(i) The exponent of the Boltzmann factor has to be dimensionless, which allows us to define a characteristic temperature Ovib = hf/kB. The fundamental frequency of vibration of a HCl molecule is f = 8.66 x 10^13 Hz. Calculate the value of Ovib for the HCl molecule.
(ii) By considering the partition function Zvib for the vibrational degrees of freedom, and making use of the fact that the average vibrational energy is given by -? ln Zvib/?? (where ? = 1/(kBT)), derive an exact analytic expression for the associated heat capacity Cv. Show that at high temperatures, Cv agrees with what one expects from the Equipartition Theorem, and at low temperatures, Cv behaves in accordance with the Third Law of Thermodynamics.
(Classically, the stretching of the bond gives rise to (I) a stored potential energy which is quadratic in the displacement of the bond length from its equilibrium value, and (II) a kinetic energy associated with the relative displacement of the two atoms, which also goes as the square of the corresponding momentum.)

Added by Timothy F.

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