Quantum Chemistry/Example 21

Write a question and its solution that shows the calculation of the vibrational force constant of H19F. Show what the force constant would have to be for HF to absorb visible red light instead of IR light.

If H19F has a vibrational fundamental wavenumber of 4141.3 cm, What is the bond force constant? What would the force constant need to be to absorb visible red light instead of IR light? H has a n exact mass of 1.00782 u, and 19F has an exact mass of 18.998403 u.

Solution:

First, calculate the reduced mass for H19F.

$$\mu=\frac{m_1m_2}{m_1+m_2}$$

$$\mu=\frac{(1.00782)(18.998403)}{1.00782+18.998403}$$

$$\mu=0.95705 \ \text{u} \times 1.66\times10^{-27} \text{kg} \ \text{u}^{-1}$$

$$\displaystyle \mu = 1.589 \times{10^{-27}} \ \text{kg}$$

Now, solve for the force constant:

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

$$2\pi\cdot c\cdot \tilde{\nu}={\left({\frac {k}{\mu }}\right)^{\frac{1}2}} $$

$$\frac{k}{\mu}=4\pi^2c^2\tilde{\nu}^2 $$

$$k=4\pi^2c^2\tilde{\nu}^2\mu $$

$$k=4\pi^2(2.998\times10^{10} \text{cm} \cdot \text{s}^{-1})^2(4141.3 \ \text{cm})^2(1.589\times10^{-27} \ \text{kg}) $$

$$k= 966.99 \text{N}\cdot \text{m}^{-1} $$

Therefore, the bond force constant is 967 N m-1.

Red light is around 700 nm, or 700×10-9 m. Converted to wavenumber, that is 14285.71 cm-1. Substituting this in,

$$k=4\pi^2(2.998\times10^{10} \text{cm} \cdot \text{s}^{-1})^2(14285.71 \ \text{cm})^2(1.589\times10^{-27} \ \text{kg}) $$

$$k= 11506.7 \ \text{N}\cdot \text{m}^{-1} $$

Therefore, the bond force constant would have to be 11506 N m-1 to absorb visible light.