Question

An antihistamine shows sharp peaks at Raman shifts of $\overline{v}=488,725,875,925,$and $1350{\mathrm{cm}}^{-1}$. At what wavelengths in nanometers would the Stokes and anti-Stokes lines for the antihistamine appear if the source were

(a) a helium-neon laser (632.8 nm)?

(b) an argon-ion laser (488.0 nm)?

Short answer

(a) The wavelengths of Stokes lines of Raman shifts of$488,725,875,925,350{\mathrm{cm}}^{-1}$for the antihistamine appear if the source is a helium-neon laser is $653,663,670,672,692\mathrm{nm}$.

The wavelengths of anti-Stokes lines of Raman shifts of localid="1650527198234" $488,725,875,925,1350{\mathrm{cm}}^{-1}$for the antihistamine appear if the source is a helium-neon laser is localid="1651401344244" $614,605,600,598,583\mathrm{nm}$.

(b)

The wavelengths of Stokes lines of Raman shifts of $488,725,875,925,1350{\mathrm{cm}}^{-1}$for the antihistamine appear if the source is an argon-ion laser are $500,506,509,511,522\mathrm{nm}$.

The wavelengths of Stokes lines of Raman shifts of$488,725,875,925,1350{\mathrm{cm}}^{-1}$for the antihistamine appear if the source is an argon-ion laser are localid="1651401334063" $476,471,468,467,458\mathrm{nm}$.

Step-by-step solution

Part (a) Step 1: Given information

The values of Raman shift are $488,725,875,925,350{\mathrm{cm}}^{-1}$.

Part (a) Step 2: Expression for the Raman shift for stoke line 

The expression for the Raman shift for stoke line is as follows:

$∆\overline{\nu }=\overline{{\nu }_{ex}}-\overline{{\nu }_v}...\left(1\right)$

Part (a) Step 3: Expression for the Raman shift for the anti-stoke line

The expression for the Raman shift for the anti-stoke line is as follows:

$∆\overline{\nu }=\overline{{\nu }_v}-\overline{{\nu }_{ex}}...\left(2\right)$

Part (a) Step 4: Expression for the wavelength

The expression for the wavelength is as follows:

$\overline{{\nu }_{ex}}=\frac{1}{{\lambda }_{ex}}...\left(3\right)$

Part (a) Step 5: Expression for the excitation wavelength  

The expression for excitation wavelength is as follows:

${\lambda }_{ex}=\frac{1}{\overline{{\nu }_{ex}}}...\left(4\right)$

Part (a) Step 6: Wavelength of the stokes line at 488 cm-1

Substitute the value in equation (3) as follows:

$\begin{array}{rcl}{\overline{v}}_{ex}& =& \frac{1}{632.8\mathrm{nm}}\\ & =& \frac{1}{632.8\mathrm{nm}\left({\displaystyle \frac{{10}^{-7}\mathrm{cm}}{1\mathrm{nm}}}\right)}\\ & =& 158\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (1) as follows:

$\begin{array}{rcl}488{\mathrm{cm}}^{-1}& =& 158\times {10}^2{\mathrm{cm}}^{-1}-\overline{v_v}\\ \overline{v_v}& =& 158\times {10}^2{\mathrm{cm}}^{-1}-488{\mathrm{cm}}^{-1}\\ & =& 153.14\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

$\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{153.14\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 653\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 653\mathrm{nm}\end{array}$

Part (a) Step 7: Wavelength of the stokes line at 725 cm-1

Substitute the value in equation (1) as follows:

$\begin{array}{rcl}725{\mathrm{cm}}^{-1}& =& 158\times {10}^2{\mathrm{cm}}^{-1}-\overline{v_v}\\ \overline{v_v}& =& 158\times {10}^2{\mathrm{cm}}^{-1}-725{\mathrm{cm}}^{-1}\\ & =& 150.75\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

$\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{150.75\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 663\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 663\mathrm{nm}\end{array}$

Part (a) Step 8: Wavelength of the stokes line at  875 cm-1

Substitute the value in equation (1) as follows:

$\begin{array}{rcl}875{\mathrm{cm}}^{-1}& =& 158\times {10}^2{\mathrm{cm}}^{-1}-\overline{v_v}\\ \overline{v_v}& =& 158\times {10}^2{\mathrm{cm}}^{-1}-875{\mathrm{cm}}^{-1}\\ & =& 149.25\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

localid="1651401435203" $\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{149.25\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 670\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 670\mathrm{nm}\end{array}$

Part (a) Step 9: Wavelength of the stokes line at  925 cm-1

Substitute the value in equation (1) as follows:

$\begin{array}{rcl}925{\mathrm{cm}}^{-1}& =& 158\times {10}^2{\mathrm{cm}}^{-1}-\overline{v_v}\\ \overline{v_v}& =& 158\times {10}^2{\mathrm{cm}}^{-1}-925{\mathrm{cm}}^{-1}\\ & =& 148.75\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

$\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{148.75\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 672\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 672\mathrm{nm}\end{array}$

Part (a) Step 10: Wavelength of the stokes line at  1350 cm-1

Substitute the value in equation (1) as follows:

$\begin{array}{rcl}1350{\mathrm{cm}}^{-1}& =& 158\times {10}^2{\mathrm{cm}}^{-1}-\overline{v_v}\\ \overline{v_v}& =& 158\times {10}^2{\mathrm{cm}}^{-1}-1350{\mathrm{cm}}^{-1}\\ & =& 144.50\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

$\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{144.50\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 692\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 692\mathrm{nm}\end{array}$

Part (a) Step 11: Wavelength of the anti-stokes line at 488 cm-1

Substitute the value in equation (2) as follows:

$\begin{array}{rcl}488{\mathrm{cm}}^{-1}& =& \overline{v_v}-158\times {10}^2{\mathrm{cm}}^{-1}\\ \overline{v_v}& =& 488{\mathrm{cm}}^{-1}+158\times {10}^2{\mathrm{cm}}^{-1}\\ & =& 162.88\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

$\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{162.88\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 614\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 614\mathrm{nm}\end{array}$

Part (a) Step 12: Wavelength of the anti-stokes line at 725 cm-1

Substitute the value in equation (2) as follows:

$\begin{array}{rcl}725{\mathrm{cm}}^{-1}& =& \overline{v_v}-158\times {10}^2{\mathrm{cm}}^{-1}\\ \overline{v_v}& =& 725{\mathrm{cm}}^{-1}+158\times {10}^2{\mathrm{cm}}^{-1}\\ & =& 165.25\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

$\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{165.25\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 605\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 605\mathrm{nm}\end{array}$

Part (a) Step 13: Wavelength of the anti-stokes line at 875 cm-1

Substitute the value in equation (2) as follows:

$\begin{array}{rcl}875{\mathrm{cm}}^{-1}& =& \overline{v_v}-158\times {10}^2{\mathrm{cm}}^{-1}\\ \overline{v_v}& =& 875{\mathrm{cm}}^{-1}+158\times {10}^2{\mathrm{cm}}^{-1}\\ & =& 166.75\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

localid="1651401522522" $\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{166.75\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 600\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 600\mathrm{nm}\end{array}$

Part (a) Step 14: Wavelength of the anti-stokes line at 925 cm-1

Substitute the value in equation (2) as follows:

$\begin{array}{rcl}925{\mathrm{cm}}^{-1}& =& \overline{v_v}-158\times {10}^2{\mathrm{cm}}^{-1}\\ \overline{v_v}& =& 925{\mathrm{cm}}^{-1}+158\times {10}^2{\mathrm{cm}}^{-1}\\ & =& 167.25\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

$\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{167.25\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 598\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 598\mathrm{nm}\end{array}$

Part (a) Step 15: Wavelength of the anti-stokes line at 1350 cm-1

Substitute the value in equation (2) as follows:

$\begin{array}{rcl}1350{\mathrm{cm}}^{-1}& =& \overline{v_v}-158\times {10}^2{\mathrm{cm}}^{-1}\\ \overline{v_v}& =& 1350{\mathrm{cm}}^{-1}+158\times {10}^2{\mathrm{cm}}^{-1}\\ & =& 171.50\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

localid="1651401549431" $\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{171.50\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 683\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 683\mathrm{nm}\end{array}$

Part (b) Step 1: Wavelength of the stokes line at 488 cm-1

Substitute the value in equation (3) as follows:

$\begin{array}{rcl}{\overline{v}}_{ex}& =& \frac{1}{488\mathrm{nm}}\\ & =& \frac{1}{488\mathrm{nm}\left({\displaystyle \frac{{10}^{-7}\mathrm{cm}}{1\mathrm{nm}}}\right)}\\ & =& 204.9\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (1) as follows:

$\begin{array}{rcl}488{\mathrm{cm}}^{-1}& =& 204.9\times {10}^2{\mathrm{cm}}^{-1}-\overline{v_v}\\ \overline{v_v}& =& 158\times {10}^2{\mathrm{cm}}^{-1}-488{\mathrm{cm}}^{-1}\\ & =& 200\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

localid="1651401571942" $\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{200\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 500\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 500\mathrm{nm}\end{array}$

Part (b) Step 2: Wavelength of the stokes line at 725 cm-1

Substitute the value in equation (1) as follows:

$\begin{array}{rcl}725{\mathrm{cm}}^{-1}& =& 204.9\times {10}^2{\mathrm{cm}}^{-1}-\overline{v_v}\\ \overline{v_v}& =& 204.9\times {10}^2{\mathrm{cm}}^{-1}-725{\mathrm{cm}}^{-1}\\ & =& 197.65\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

$\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{197.65\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 506\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 506\mathrm{nm}\end{array}$

Part (b) Step 3: Wavelength of the stokes line at 875 cm-1

Substitute the value in equation (1) as follows:

$\begin{array}{rcl}875{\mathrm{cm}}^{-1}& =& 204.9\times {10}^2{\mathrm{cm}}^{-1}-\overline{v_v}\\ \overline{v_v}& =& 204.9\times {10}^2{\mathrm{cm}}^{-1}-875{\mathrm{cm}}^{-1}\\ & =& 196.15\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

localid="1651401602906" $\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{196.15\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 509\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 509\mathrm{nm}\end{array}$

Part (b) Step 4: Wavelength of the stokes line at 925 cm-1

Substitute the value in equation (1) as follows:

$\begin{array}{rcl}925{\mathrm{cm}}^{-1}& =& 204.9\times {10}^2{\mathrm{cm}}^{-1}-\overline{v_v}\\ \overline{v_v}& =& 204.9\times {10}^2{\mathrm{cm}}^{-1}-925{\mathrm{cm}}^{-1}\\ & =& 195.65\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

localid="1651401624739" $\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{195.65\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 511\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 511\mathrm{nm}\end{array}$

Part (b) Step 5: Wavelength of the stokes line at 1350 cm-1

Substitute the value in equation (1) as follows:

$\begin{array}{rcl}1350{\mathrm{cm}}^{-1}& =& 204.9\times {10}^2{\mathrm{cm}}^{-1}-\overline{v_v}\\ \overline{v_v}& =& 204.9\times {10}^2{\mathrm{cm}}^{-1}-1350{\mathrm{cm}}^{-1}\\ & =& 191.4\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

localid="1651401640160" $\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{191.4\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 522\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 522\mathrm{nm}\end{array}$

Part (b) Step 6: Wavelength of the anti-stokes line at 488 cm-1

Substitute the value in equation (4) as follows:

$\begin{array}{rcl}488{\mathrm{cm}}^{-1}& =& \overline{v_v}-204.9\times {10}^2{\mathrm{cm}}^{-1}\\ \overline{v_v}& =& 488{\mathrm{cm}}^{-1}+204.9\times {10}^2{\mathrm{cm}}^{-1}\\ & =& 209.78\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

localid="1651401650333" $\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{209.78\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 476\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 476\mathrm{nm}\end{array}$

Part (b) Step 7: Wavelength of the anti-stokes line at 725 cm-1

Substitute the value in equation (2) as follows:

$\begin{array}{rcl}725{\mathrm{cm}}^{-1}& =& \overline{v_v}-204.9\times {10}^2{\mathrm{cm}}^{-1}\\ \overline{v_v}& =& 725{\mathrm{cm}}^{-1}+204.9\times {10}^2{\mathrm{cm}}^{-1}\\ & =& 212.15\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

localid="1651401667809" $\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{212.15\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 471\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 471\mathrm{nm}\end{array}$

Part (b) Step 8: Wavelength of the anti-stokes line at 875 cm-1

Substitute the value in equation (2) as follows:

$\begin{array}{rcl}875{\mathrm{cm}}^{-1}& =& \overline{v_v}-204.9\times {10}^2{\mathrm{cm}}^{-1}\\ \overline{v_v}& =& 875{\mathrm{cm}}^{-1}+204.9\times {10}^2{\mathrm{cm}}^{-1}\\ & =& 213.65\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

localid="1651401687139" $\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{213.65\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 468\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 468\mathrm{nm}\end{array}$

Part (b) Step 9: Wavelength of the anti-stokes line at 925 cm-1

Substitute the value in equation (2) as follows:

$\begin{array}{rcl}925{\mathrm{cm}}^{-1}& =& \overline{v_v}-204.9\times {10}^2{\mathrm{cm}}^{-1}\\ \overline{v_v}& =& 925{\mathrm{cm}}^{-1}+204.9\times {10}^2{\mathrm{cm}}^{-1}\\ & =& 214.15\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

localid="1651401699023" $\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{214.15\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 467\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 467\mathrm{nm}\end{array}$

Part (b) Step 10: Wavelength of the anti-stokes line at 1350 cm-1

Substitute the value in equation (2) as follows:

$\begin{array}{rcl}1350{\mathrm{cm}}^{-1}& =& \overline{v_v}-204.9\times {10}^2{\mathrm{cm}}^{-1}\\ \overline{v_v}& =& 1350{\mathrm{cm}}^{-1}+204.9\times {10}^2{\mathrm{cm}}^{-1}\\ & =& 218.40\times {10}^2{\mathrm{cm}}^{-1}\end{array}$

Substitute the value in equation (4) as follows:

$\begin{array}{rcl}{\lambda }_{ex}& =& \frac{1}{218.40\times {10}^2{\mathrm{cm}}^{-1}}\\ & =& 458\times {10}^{-7}\mathrm{cm}\left(\frac{{10}^7\mathrm{nm}}{1\mathrm{cm}}\right)\\ & =& 458\mathrm{nm}\end{array}$