Chapter 13, Problem 13.4P

Interpretation Introduction

(a)

Interpretation:

The chemical shift of the ${\text{CH}}_{\text{3}}$ protons of dimethoxymethane in a spectrum at the magnetic field of $21,100$ gauss is to be stated.

Concept introduction:

Chemical shift of a proton is its resonance frequency in an applied magnetic field. Chemical shift can be expressed in $\text{Hz}$ as a frequency difference and in $\text{ppm}$ as their $\delta$ value. Chemical shift value in $\text{ppm}$ remain constant and chemical shift in $\text{Hz}$ changes at different magnetic field value.

Answer to Problem 13.4P

The chemical shift of the ${\text{CH}}_{\text{3}}$ protons of dimethoxymethane in a spectrum at the magnetic field of $\text{21, 100 gauss}$ is $296.6\text{Hz}$.

Explanation of Solution

Chemical shift value in $\text{ppm}$ remain constant in any field strength. Therefore the chemical shift value of ${\text{CH}}_{\text{3}}$ protons of dimethoxymethane in $\text{ppm}$ is $\text{3}\text{.30}\text{ppm}$.

The chemical shift in $\text{Hz}$ is calculated by the formula shown below.

$\Delta \nu =\delta \times {\nu }_0$…(1)

But,

${\nu }_0=\frac{{\gamma }_{\text{H}}}{2\pi }\times B_0$

Substitute the value of ${\nu }_0=\frac{{\gamma }_{\text{H}}}{2\pi }\times B_0$ in equation (1).

$\Delta \nu =\delta \times \frac{{\gamma }_{\text{H}}}{2\pi }\times B_0$ … (2)

Where,

• $\Delta \nu$ is the frequency difference.

• ${\gamma }_{\text{H}}$ is the gyromagnetic ratio, $26753\text{radians}{\text{gauss}}^{-1}{\text{s}}^{-1}$.

• $B_0$ is the magnetic field.

• $\delta$ is the chemical shift value in $\text{ppm}$.

• ${\nu }_0$ is the absorption frequency.

Substitute the value of $\delta$ as $\text{3}\text{.30}\text{ppm}$; $B_0$ as $21,100$ gauss; ${\gamma }_{\text{H}}$ as $26753\text{radians}{\text{gauss}}^{-1}{\text{s}}^{-1}$ in equation (1).

$\begin{array}{rll}\Delta \nu & =& \delta \times \frac{{\gamma }_{\text{H}}}{2\pi }\times B_0\\ & =& 3.30\text{ppm}\times \frac{26753\text{radians}{\text{gauss}}^{-1}{\text{s}}^{-1}}{2\left(3.14\right)\text{radians}}\times 21100\text{gauss}\\ & =& 296626018\text{ppm}{\text{s}}^{-1}\end{array}$

The unit of $\Delta \nu$ is converted from $\text{ppm}{\text{s}}^{-1}$ to ${\text{s}}^{-1}$ by using the formula $\left(\text{ppm}{\text{s}}^{-1}\times {10}^{-6}={\text{s}}^{-1}\right)$.

Therefore,

$\begin{array}{rll}\Delta \nu & =& 296626018\text{ppm}{\text{s}}^{-1}\times {10}^{-6}\\ & =& 296.6{\text{s}}^{-1}\end{array}$

The unit $\text{1}{\text{s}}^{-1}$ is equal to $\text{1}\text{Hz}$.

So,

$296.6{\text{s}}^{-1}=296.6\text{Hz}$

The value of $\Delta \nu$ in $\text{Hz}$ is shown below.

$\Delta \nu =296.6\text{Hz}$

Therefore, the chemical shift of the ${\text{CH}}_{\text{3}}$ protons of dimethoxymethane in a spectrum at the magnetic field of $\text{21, 100 gauss}$ is $296.6\text{Hz}$.

Conclusion

The chemical shift of the ${\text{CH}}_{\text{3}}$ protons of dimethoxymethane in a spectrum at the magnetic field of $\text{21, 100 gauss}$ is $296.6\text{Hz}$.

Interpretation Introduction

(b)

Interpretation:

The chemical shift of the ${\text{CH}}_{\text{3}}$ protons of dimethoxymethane in a spectrum at the magnetic field of $141,000$ gauss is to be stated.

Concept introduction:

Chemical shift of a proton is its resonance frequency in an applied magnetic field. Chemical shift can be expressed in $\text{Hz}$ as a frequency difference and in $\text{ppm}$ as their $\delta$ value. Chemical shift value in $\text{ppm}$ remain constant and chemical shift in $\text{Hz}$ changes at different magnetic field value.

Answer to Problem 13.4P

The chemical shift of the ${\text{CH}}_{\text{3}}$ protons of dimethoxymethane in a spectrum at the magnetic field of $141,000\text{gauss}$ gauss is $1982.20\text{Hz}$.

Explanation of Solution

Chemical shift value in $\text{ppm}$ remain constant in any field strength. Therefore the chemical shift value of ${\text{CH}}_{\text{3}}$ protons of dimethoxymethane in $\text{ppm}$ is $\text{3}\text{.30}\text{ppm}$.

The chemical shift in $\text{Hz}$ is calculated by the formula shown below.

$\Delta \nu =\delta \times {\nu }_0$ …(1)

But,

${\nu }_0=\frac{{\gamma }_{\text{H}}}{2\pi }\times B_0$

Substitute the value of ${\nu }_0=\frac{{\gamma }_{\text{H}}}{2\pi }\times B_0$ in equation (1).

$\Delta \nu =\delta \times \frac{{\gamma }_{\text{H}}}{2\pi }\times B_0$ …(2)

Where,

• $\Delta \nu$ is the frequency difference.

• ${\gamma }_{\text{H}}$ is the gyromagnetic ratio, $26753\text{radians}{\text{gauss}}^{-1}{\text{s}}^{-1}$.

• $B_0$ is the magnetic field.

• $\delta$ is the chemical shift value in $\text{ppm}$.

• ${\nu }_0$ is the absorption frequency.

Substitute the value of $\delta$ as $\text{3}\text{.30}\text{ppm}$; $B_0$ as $141,000$ gauss;

${\gamma }_{\text{H}}$ as $26753\text{radians}{\text{gauss}}^{-1}{\text{s}}^{-1}$ in equation (1).

$\begin{array}{rll}\Delta \nu & =& \delta \times \frac{{\gamma }_{\text{H}}}{2\pi }\times B_0\\ & =& 3.30\text{ppm}\times \frac{26753\text{radians}{\text{gauss}}^{-1}{\text{s}}^{-1}}{2\left(3.14\right)\text{radians}}\times 141000\text{gauss}\\ & =& 1982192820\text{ppm}{\text{s}}^{-1}\end{array}$

The unit of $\Delta \nu$ is converted from $\text{ppm}{\text{s}}^{-1}$ to ${\text{s}}^{-1}$ by using the formula $\left(\text{ppm}{\text{s}}^{-1}\times {10}^{-6}={\text{s}}^{-1}\right)$.

Therefore,

$\begin{array}{rll}\Delta \nu & =& 1982192820\text{ppm}{\text{s}}^{-1}\times {10}^{-6}\\ & =& 1982.19{\text{s}}^{-1}\\ & \approx & 1982.20{\text{s}}^{-1}\end{array}$

The unit $\text{1}{\text{s}}^{-1}$ is equal to $\text{1}\text{Hz}$.

So,

$1982.20{\text{s}}^{-1}=1982.20\text{Hz}$

The value of $\Delta \nu$ in $\text{Hz}$ is shown below.

$\Delta \nu =1982.20\text{Hz}$

Therefore, the chemical shift of the ${\text{CH}}_{\text{3}}$ protons of dimethoxymethane in a spectrum at the magnetic field of $141,000\text{gauss}$ is $1982.20\text{Hz}$.

Conclusion

The chemical shift of the ${\text{CH}}_{\text{3}}$ protons of dimethoxymethane in a spectrum at the magnetic field of $141,000\text{gauss}$ is $1982.20\text{Hz}$.