**What wavelength of light is required to excite this vibration? The vibration frequency of HCI is**$v= 8.85 \times 10^{13} \space s^{-1}$.

This problem aims to familiarize us with **vibrating molecules** and the **energy** they dissipate or absorb from their surroundings. This problem requires the core knowledge of **chemistry** along with **molecules** and their **movements.**

Let’s first look at **molecular vibration.** Molecules that have only **two atoms** vibrate by merely forcing closer and then repelling. For instance, the **nitrogen** $(N_2)$ molecule and **oxygen** $(O_2)$ molecules vibrate simply. Whereas molecules that contain $3$ or more atoms **oscillate** in more **complicated** patterns. For instance, **Carbon dioxide** $(CO_2)$ molecules have $3$ **distinct** vibration manners.

## Expert Answer

We can define the **energy** of a **vibrating molecule** as a **quantized** mechanism that is much similar to the **vibrancy** of an electron in the **hydrogen** $(H_2)$ atom. The mathematical equation to calculate the different energy levels of a **vibrating** molecule is given as:

\[ E_n = \left( n + \dfrac{1}{2} \right) \space hv\]

Where,

The $n$ is the **quantum number** with the positive values of $1, 2, 3, \space …$.

The variable $h$ is **Planck’s constant** and is given as $h = 6.262 \times 10^{-34} \space Js$.

And, $v$ is the vibrating **frequency** of **HCI** and is given as $v= 8.85 \times 10^{13} \space s^{-1}$.

The **minimum energy** required to vibrate the HCI can be calculated by finding the **difference** between the **energies** of the two lowest **quantum** numbers.

So finding the **energies** at **quantum** number $n =1, 2$ and subtracting to find the **minimum energy** required to vibrate the HCI:

\[E_1 = \left(1 + \dfrac{1}{2} \right) hv = \left(1 + \dfrac{1}{2} \right) (6.262 \times 10^{-34}).(8.85 \times 10^{13})\]

\[E_1 = 8.796015 \times 10^{-20}\]

\[E_2 = \left(2 + \dfrac{1}{2} \right) hv = \left(1 + \dfrac{1}{2} \right) (6.262 \times 10^{-34}).(8.85 \times 10^{13})\]

\[E_1 = 1.466 \times 10^{-19}\]

Now finding the **difference** using this equation:

\[\Delta E = E_2 – E_1\]

\[=1.466 \times 10^{-19} \space – \space 8.796015 \times 10^{-20}\]

$\Delta E$ comes out to be:

\[\Delta E = 5.864 \times 10^{-20} \space J\]

Now find the **wavelength** of the light that can **excite** this **vibration.**

The generic **formula** for calculating $\Delta E$ is given as:

\[\Delta E = \dfrac{hc}{ \lambda }\]

Rearranging it for the **wavelength** $\lambda$:

\[\lambda = \dfrac{hc}{\Delta E}\]

**Inserting** the values and **solving** to find the $\lambda$:

\[\lambda = \dfrac{ (6.262 \times 10^{-34}).(3.00 \times 10^{8}) }{ 5.864 \times 10^{-20} }\]

$\lambda$ comes out to be:

\[\lambda = 3390 \space nm\]

## Numerical Answer

The **Minimum energy** required to vibrate the HCI is $\Delta E = 5.864 \times 10^{-20} \space J$.

The **wavelength** of the light that can excite this **vibration** is $3390 \space nm$.

## Example

What **wavelength** of light is required to excite the **vibration** of $3.867 \times 10^{-20} \space J$?

**Formula** is given as:

\[\lambda = \dfrac{hc}{\Delta E}\]

**Inserting** the values and **solving** to find the $\lambda$:

\[\lambda=\dfrac{ (6.262 \times 10^{-34}).(3.00 \times 10^{8}) }{ 3.867 \times 10^{-20} }\]

$\lambda$ comes out to be:

\[\lambda=4.8 \space \mu m\]