Question

Calculate the absorption frequency corresponding to the $-\mathrm{C}-\mathrm{H}$stretching vibration treating the group as a simple diatomic $\mathrm{C}-\mathrm{H}$molecule with a force constant of $k=5\times {10}^2\mathrm{N}/\mathrm{m}$. Compare the calculated value with the range found in correlation charts (such as the one shown in Figure 17-6). Repeat the calculation for the deuterated bond.

Short answer

The absorption frequency corresponding to the $-\mathrm{C}-\mathrm{H}$stretching vibration treating the group as a single diatomic $\mathrm{C}-\mathrm{H}$ molecule is $2984.65{\mathrm{cm}}^{-1}$. The calculated value lies near the range found in the correlation charts. The absorption frequency for $\mathrm{C}-\mathrm{D}$molecule is $2218.32{\mathrm{cm}}^{-1}$.

Step-by-step solution

Step 1. Given information  

The value of force constant of $\mathrm{C}-\mathrm{H}$molecule $k=5\times {10}^2\mathrm{N}/\mathrm{m}$Determine the absorption frequency for $-\mathrm{C}-\mathrm{H}$and $-\mathrm{C}-\mathrm{D}$stretching vibration should be determined and compare the $\mathrm{C}-\mathrm{H}$value with the range found in the charts.

Step 2. Absorption frequency expression 

The expression for the wavenumber is:

$\overline{v}=\frac{1}{2\pi c}\sqrt{\frac{k}{\mu }}$...... (I)

$\mu =\frac{m_1{m}_2}{m_1+m_2}$

Substitute data-custom-editor="chemistry" $\frac{m_1{m}_2}{m_1+m_2}$for data-custom-editor="chemistry" $\mu$in Equation (I).

$\overline{v}=\frac{1}{2\pi c}\sqrt{\frac{k}{\left(\frac{m_1{m}_2}{m_1+m_2}\right)}}$...... (II)

Step 3. The calculation for absorption frequency of -C-Hbond

Substitute values in Equation (II):

$$\begin{gathered} \overline{v}=\frac{1}{2\times 3.14\times 3\times {10}^{10}\mathrm{cm}/\mathrm{s}}\sqrt{\frac{5.0\times {10}^2\mathrm{N}/\mathrm{m}}{\left({\displaystyle \frac{12\mathrm{g}/\mathrm{mol}\times 1\mathrm{g}/\mathrm{mol}}{12\mathrm{g}/\mathrm{mol}+1\mathrm{g}/\mathrm{mol}}}\right)}} \\ =\left(5.307\times {10}^{-12}\mathrm{s}/\mathrm{cm}\right)\sqrt{\frac{5.0\times {10}^2\mathrm{N}/\mathrm{m}}{\left({\displaystyle \frac{12\mathrm{g}/\mathrm{mol}}{13\mathrm{g}/\mathrm{mol}}}\right)}} \\ =\left(5.307\times {10}^{-12}\mathrm{s}/\mathrm{cm}\right)\sqrt{\frac{5.0\times {10}^2\mathrm{N}/\mathrm{m}}{9.230\times {10}^{-4}\mathrm{kg}/\mathrm{mol}}} \\ =\left(5.307\times {10}^{-12}\mathrm{s}/\mathrm{cm}\right)\sqrt{\frac{5.0\times {10}^2\mathrm{N}/\mathrm{m}}{9.230\times {10}^{-4}\mathrm{kg}/\mathrm{mol}\times {\displaystyle \frac{1\mathrm{mol}}{6\times {10}^{-23}\mathrm{atom}}}\times 1\mathrm{atom}}} \\ =\left(5.307\times {10}^{-12}\mathrm{s}/\mathrm{cm}\right)\sqrt{\frac{5.0\times {10}^2\mathrm{kg}/{\mathrm{s}}^2}{1.538\times {10}^{-27}\mathrm{kg}}} \\ =\left(5.307\times {10}^{-12}\mathrm{s}/\mathrm{cm}\right)\sqrt{3.164\times {10}^{29}{\mathrm{s}}^{-2}} \\ =\left(5.307\times {10}^{-12}\mathrm{s}/\mathrm{cm}\right)\left(5.624\times {10}^{14}{\mathrm{s}}^{-1}\right) \\ =2984.65{\mathrm{cm}}^{-1} \end{gathered}$$

Step 4. Compare the calculated value with the range found in correlation charts

The absorption frequency of $-\mathrm{C}-\mathrm{H}$lies within the range of $2850{\mathrm{cm}}^{-1}$to $2970{\mathrm{cm}}^{-1}$in the correlation charts. Therefore, the calculated value for the absorption frequency for$-\mathrm{C}-\mathrm{H}$ molecule is near to the experimental value.

Step 5. The calculation for absorption frequency of -D-H bond

Substitute values in Equation (II):

$$\begin{gathered} \overline{v}=\frac{1}{2\times 3.14\times 3\times {10}^{10}\mathrm{cm}/\mathrm{s}}\sqrt{\frac{5.0\times {10}^2\mathrm{N}/\mathrm{m}}{\left({\displaystyle \frac{12\mathrm{g}/\mathrm{mol}\times 2\mathrm{g}/\mathrm{mol}}{12\mathrm{g}/\mathrm{mol}+2\mathrm{g}/\mathrm{mol}}}\right)}} \\ =\left(5.307\times {10}^{-12}\mathrm{s}/\mathrm{cm}\right)\sqrt{\frac{5.0\times {10}^2\mathrm{N}/\mathrm{m}}{\left({\displaystyle \frac{24\mathrm{g}/\mathrm{mol}}{14\mathrm{g}/\mathrm{mol}}}\right)}} \\ =\left(5.307\times {10}^{-12}\mathrm{s}/\mathrm{cm}\right)\sqrt{\frac{5.0\times {10}^2\mathrm{N}/\mathrm{m}}{1.714\times {10}^{-3}\mathrm{kg}/\mathrm{mol}}} \\ =\left(5.307\times {10}^{-12}\mathrm{s}/\mathrm{cm}\right)\sqrt{\frac{5.0\times {10}^2\mathrm{N}/\mathrm{m}}{1.714\times {10}^{-3}\mathrm{kg}/\mathrm{mol}\times {\displaystyle \frac{1\mathrm{mol}}{6\times {10}^{-23}\mathrm{atom}}}\times 1\mathrm{atom}}} \\ =\left(5.307\times {10}^{-12}\mathrm{s}/\mathrm{cm}\right)\sqrt{\frac{5.0\times {10}^2\mathrm{kg}/{\mathrm{s}}^2}{2.856\times {10}^{-27}\mathrm{kg}}} \\ =\left(5.307\times {10}^{-12}\mathrm{s}/\mathrm{cm}\right)\sqrt{1.750\times {10}^{29}{\mathrm{s}}^{-2}} \\ =\left(5.307\times {10}^{-12}\mathrm{s}/\mathrm{cm}\right)\left(4.18\times {10}^{14}{\mathrm{s}}^{-1}\right) \\ =2218.32{\mathrm{cm}}^{-1} \end{gathered}$$

Step 6. Conclusion 

The absorption frequency corresponding to the $-\mathrm{C}-\mathrm{H}$stretching vibration treating the group as a single diatomic $\mathrm{C}-\mathrm{H}$molecule is $2984.65{\mathrm{cm}}^{-1}$. The calculated value lies near the range found in the correlation charts.

The absorption frequency for $\mathrm{C}-\mathrm{D}$molecule is $2218.32{\mathrm{cm}}^{-1}$.