# Quantum Chemistry/Example 17

The force constant of HCl is determined computationally to be 480 N/m. Given this information find the frequency of EM radiation required to excite the HCl molecule from its ground state to its first excited state.

## Solution

Given the bond constant (k) of HCl , we use the relationship between fundamental frequency and bond constant to find the bond constant.

${\displaystyle \nu _{0}={\frac {1}{2\pi }}{\left({\frac {k}{\mu }}\right)}^{1/2}}$

where ${\displaystyle K}$ is the bond constant.

${\displaystyle \mu }$ is the reduced mass of HCl.

To find the reduced mass of HCl the masses of H and Cl are multiplied and divided by the sum of the masses.

${\displaystyle \mu ={\frac {m_{1}\cdot m_{2}}{m_{1}+m_{2}}}}$

For HCl the reduced mass is calculated as

${\displaystyle \mu ={\frac {1.007842u\cdot 35.453u}{1.007842u+35.453u}}=0.979982u}$

convert to the SI unit of Kg

${\displaystyle 1u=1.66054\cdot 10^{-27}Kg}$

${\displaystyle 0.979983u=0.979983u\cdot 1.66054\cdot 10^{-27}Kg}$

${\displaystyle \mu =1.6273\cdot 10^{-27}Kg}$

To find the fundamental frequency

${\displaystyle \nu _{0}={\frac {1}{2\pi }}\left({\frac {K}{\mu }}\right)^{1/2}}$

${\displaystyle ={\frac {1}{2\pi }}\left({\frac {480N/m}{1.6273\cdot 10^{-27}Kg}}\right)^{1/2}}$
${\displaystyle =8.6438\cdot 10^{13}hz}$

After finding the fundamental frequency, the Energy at different quantum levels can be found by

${\displaystyle E_{v}=h\nu _{0}\left(v+{\frac {1}{2}}\right)}$

For the ground state i.e. ${\displaystyle v=0}$

${\displaystyle E_{0}=h\nu _{0}\left(0+{\frac {1}{2}}\right)}$
${\displaystyle E_{0}=h\nu _{0}\left({\frac {1}{2}}\right)}$

For the first excited state i.e. ${\displaystyle v=1}$

${\displaystyle E_{1}=h\nu _{0}\left(1+{\frac {1}{2}}\right)}$
${\displaystyle E_{1}=h\nu _{0}\left({\frac {3}{2}}\right)}$

The difference in energy between the two states is

${\displaystyle \Delta E=E_{1}-E_{0}}$
${\displaystyle \Delta E=h\nu _{0}\left({\frac {3}{2}}\right)-h\nu _{0}\left({\frac {1}{2}}\right)}$
${\displaystyle \Delta E=h\nu _{0}}$

and Energy is defined as Planck's constant multiplied by frequency

${\displaystyle \Delta E=hv}$
${\displaystyle hv=h\nu _{0}}$
${\displaystyle v=\nu _{0}=8.6438\cdot 10^{13}hz}$