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B.Sc. Chemistry - Rotational Spectroscopy.

09/06/2017 0 0

Rotational spectroscopy is concerned with the measurement of the energies of transitions between quantized rotational states of molecules in the gas phase.

The rotational energy levels, F(J), of the molecule can be expressed as,

$F\left(J\right)=BJ\left(J+1\right)\qquad J=0,1,2,...$

$B={h \over {8\pi ^{2}cI_{B}}}={h \over {8\pi ^{2}cI_{C}}}$

For a diatomic molecule,

$I=\frac{m_1m_2}{m_1 +m_2}d^2$

$\Delta J=J^{\prime} - J^{\prime\prime} = \pm 1$ . Thus, the locations of the lines in a rotational spectrum will be given by

${\tilde \nu }_{{J^{{\prime }}\leftrightarrow J^{{\prime \prime }}}}=F\left(J^{{\prime }}\right)-F\left(J^{{\prime \prime }}\right)=2B\left(J^{{\prime \prime }}+1\right)\qquad J^{{\prime \prime }}=0,1,2,...$

$J^{\prime\prime}$ denotes the lower level and $J^{\prime}$ denotes the upper level involved in the transition.

The number of molecules in an excited state with quantum number J, relative to the number of molecules in the ground state, N_J/N₀ is given by

$\frac{N_J}{N_0}=e^{-\frac{E_J}{kT}} =e^{-\frac {BhcJ(J+1)}{kT}}$

Lets solve an example:

Solution:

Note: The value of Boltzmann's constant is approximately 1.3807 x 10 ^-²³ joule s per kelvin (J. K ^-¹ ).

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