Quantum Chemistry/Example 27

Calculate the frequency of a photon needed to excite CO from the ground vibrational state to the first vibrational state (k=1860 N/m).

$v_{0}={\frac {1}{2\pi }}\cdot {\biggl (}{\frac {k}{\mu }}{\biggr )}^{\frac {1}{2}}\qquad where\ \mu ={\frac {m_{1}\cdot m_{2}}{m_{1}+m_{2}}}$

C= 12.011u, O =15.999u

Plug the values into the reduced mass equation:

$\mu ={\frac {12.011u\cdot 15.999u}{12.011u+15.999u}}$

$\mu =6.861u$

Convert the reduced mass from units of u to kg

$1u=1.66054\times 10^{-27}kg$

$\mu =1.139\times 10^{-26}kg$

$v_{0}={\frac {1}{2\pi }}\cdot {\biggl (}{\frac {1860N/m}{1.139\times 10^{-26}kg}}{\biggr )}^{\frac {1}{2}}\qquad$

$\therefore v_{0}=6.43\times 10^{13}s^{-1}$