Rotational Vibrational Spectra of Diatomic molecules.

The radiation energy absorbed in IR region brings about the simultaneous change in the rotational and vibrational energies of the molecule. This is observed in the fine structure of rotation bands. This combined spectrum is called Vibrational-rotational spectrum. The rotational and vibrational changes are considered as independent events and there is no interaction between them. Thus, the total energy change of the vibrating rotator is taken as the sum of vibrational and rotational energy changes. i.e. $$\Delta E = \Delta { E }_{ J } + \Delta { E }_{ v }\qquad …(1)$$ where , $$\Delta { E }_{ J }$$ is rotational energy change = $$\frac { { h }^{ 2 } }{ 8{ \pi }^{ 2 }I } 2J$$ and $$\Delta { E }_{ v }$$ is vibrational energy change = $$\left( v’ – v \right) h\omega$$ Substituting these values in equation (1) we get, $$\Delta E = \left( v’ – v \right) h\omega + \frac { { h }^{ 2 } }{ 8{ \pi }^{ 2 }I } 2J\qquad …(2)$$ As the total transitions are quantised, $$\therefore \Delta E = h\upsilon = hc\overline { \upsilon }$$ On comparing this with equation (2),$$\ hc\overline { \upsilon } = \left( v’ – v \right) h\omega + \frac { { h }^{ 2 } }{ 8{ \pi }^{ 2 }I } 2J$$ $$c\overline { \upsilon } = \left( v’ – v \right) \omega + \frac { h }{ 8{ \pi }^{ 2 }I } 2J$$ $$\overline { \upsilon } = \left( v’ – v \right) \frac { \omega }{ c } + \frac { h }{ 8{ \pi }^{ 2 }Ic } 2J$$ $$\overline { \upsilon } = \left( v’ – v \right) \overline { \omega } + \frac { h }{ 8{ \pi }^{ 2 }Ic } 2J$$ $$\overline { \upsilon } = \left( v’ – v \right) \overline { \omega } + 2BJ\qquad …(3)$$ where $$B$$ ir rotational constant = $$\frac { h }{ 8{ \pi }^{ 2 }Ic }$$ For harmonic oscillations, $$\left( v’ – v \right) = \Delta v = \pm 1$$ as per selection rule. Therefore eq (3) becomes, $$\therefore \overline { \upsilon } = \overline { \omega } + 2BJ\qquad …(4)\$$ Therefore, the frequency separation of successive rotational lines in vibration-rotation band will be given by, $$\Delta \overline { \upsilon } = \left[ \overline { \omega } + 2BJ’ \right] – \left[ \overline { \omega } + 2BJ \right]$$ $$\Delta \overline { \upsilon } = \overline { \omega } + 2BJ’ – \overline { \omega } – 2BJ$$ $$\Delta \overline { \upsilon } = 2BJ’ – 2BJ$$ $$\Delta \overline { \upsilon } = 2B\left( J’ – J \right) \qquad …(5)$$ As per the selection rule $$\left( J’ – J \right) = \Delta J = \pm 1$$ $$\boxed { \Delta \overline { \upsilon } = 2B }$$ This gives the frequency seperation of successive rotation lines in Rotational-Vibrational band.

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