Question

The ${\mathbf{\text{Li}}}_{\mathbf{\text{2}}}$molecule (${}^{\mathbf{7}}\mathbf{\text{Li}}$isotope) shows a very weak infrared line in its vibrational spectrum at a wavelength of$\mathbf{2}\mathbf{.}\mathbf{85}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{m}\mathbf{.}$Calculate the force constant for the ${\mathbf{\text{Li}}}_{\mathbf{\text{2}}}$molecule.

Short answer

The force constant for ${\text{Li}}_{\text{2}}$molecule is$\text{k}=25.42\text{kg}/{\text{s}}^2$

Step-by-step solution

Definition of vibration spectrum

The part of a molecular spectrum in which the bands arise from quantized changes in the energy of mutual atomic vibrations within the molecule.

Given information

Calculate the force constant for${\text{Li}}_{\text{2}}$. The given data is as follows,

Wavelength $\lambda =2.85\times {10}^{-5}m$.

Calculation of force constant 

The formula of harmonic oscillation is as follows.

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

where,$k$ is the force constant and$\mu$ is the mass,$\nu$ is frequency

Now, use the Table for reduced mass of the molecule and mass of lithium atom.

$\text{m}\left[{}_3{}^7\text{Li}\right]=7.0160040\text{u}$

Calculate the reduced mass

$\begin{array}{rcl}\mu & =& \frac{m{\left[{}_3{}^{7}Li\right]}^2}{2·m\left[{}_3{}^{7}Li\right]}\\ & =& \frac{{(7.0160040u)}^2}{2·7.0160040u}\\ & =& 3.508002u\\ & =& 3.508002u·1.6605387·{10}^{-27}{\text{kgu}}^{-1}\\ & =& 5.83·{10}^{-27}\text{kg}\end{array}$

The reduced mass is $\mu =3.508002u$

Adding all value in formula

Lastly, add all value in formula of frequency.

$$\begin{gathered} \nu =\frac{1}{2\pi }\sqrt{\frac{k}{\mu }} \\ 1.05·{10}^{13}{\text{s}}^{-1}=0.1591·\sqrt{\frac{k}{5.83·{10}^{-27}\text{kg}}} \\ \frac{1.05·{10}^{13}{\text{s}}^{-1}}{0.159}=\sqrt{\frac{k}{5.83·{10}^{-27}\text{kg}}} \\ {\left(\frac{1.05·{10}^{13}{\text{s}}^{-1}}{0.159}\right)}^2=\frac{k}{5.83·{10}^{-27}\text{kg}} \\ {\left(\frac{1.05·{10}^{13}{\text{s}}^{-1}}{0.159}\right)}^2·5.83·{10}^{-27}\text{kg}=k \\ \text{k}=25.42\text{kg}/{\text{s}}^2 \end{gathered}$$

The force constant for ${\text{Li}}_{\text{2}}$molecule is$\text{k}=25.42\text{kg}/{\text{s}}^2$