Question

Calculate the fundamental stretching frequency and wavenumber for a C–H bond, considering that the atoms vibrate independently of other groups on a carbon atom and that the force constant is 4800/nm

Answer 1

The fundamental stretching wavenumber $\bar\nu$ can be calculated from the force constant $k$ (4800 N/m, or kg/s²) using the following equation:

$\bar\nu = \frac{1}{2\pi c}\sqrt{\frac{k}{\mu}}$ ,

where $c$ is the speed of light 3·10¹⁰ cm/s, $\mu$ is the reduced mass and $k$ is the force constant.

The reduced mass of C and H is (taken from the molar masses in kg/mol and Avogadro's number 6.022·10²³ mol^-1):

$\mu= \frac{m_Cm_H}{m_C + m_H} = \frac{12·1·10^{-3}}{(12 +1)·6.022·10^{23}} = 1.53·10^{-27}$ kg.

Finally, the fundamental stretching wavenumber is:

$\bar\nu = \frac{1}{2\pi·3·10^{10}}\sqrt\frac{4800}{1.53·10^{-27}} = 9397$ cm^-1.

And the frequency:

$\nu = \bar{\nu}c = 2.819·10^{14}$ s^-1.