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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 } ...(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) homega Substituting these values in equation (1) we get, Delta E = left( v' - v right) homega + frac { { h }^{ 2 } }{ 8{ pi }^{ 2 }I } 2J ...(2) As the total transitions are quantised, therefore Delta E = hupsilon = hcoverline { upsilon } On comparing this with equation (2), hcoverline { upsilon } = left( v' - v right) homega + frac { { h }^{ 2 } }{ 8{ pi }^{ 2 }I } 2J coverline { 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 (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 ...(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 } = 2Bleft( J' - J right) ...(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.
