In this chapter, we claimed that the NMR signals for \({ }^{2} \mathrm{H}\) (deuterium) and \({ }^{13} \mathrm{C}\) will not overlap \({ }^{1} \mathrm{H}\) spectra. Verify that this statement is true. Show that the frequency \((v)\) will be different for \({ }^{1} \mathrm{H},{ }^{2} \mathrm{H}\), and \({ }^{13} \mathrm{C}\) at a field strength \(\left(B_{0}\right)\) of \(4.7 \mathrm{~T}\). The gyromagnetic ratios for \({ }^{1} \mathrm{H}\), \({ }^{2} \mathrm{H}\), and \({ }^{13} \mathrm{C}\) are \(2.7 \times 10^{8}, 0.41 \times 10^{8}\), and \(0.67 \times 10^{8} \mathrm{rad}\) \(\mathrm{T}^{-1} \mathrm{~s}^{-1}\), respectively.

The Larmor frequency is crucial in NMR Spectroscopy because it tells us the rate at which the nucleus of an atom precesses around the direction of the applied magnetic field. This frequency is unique for each type of nucleus and depends on the magnetic field strength and the gyromagnetic ratio specific to that nucleus.The formula for calculating the Larmor frequency is: \[ v = \frac{\gamma B_0}{2π} \]Here,

• \(v\) represents the Larmor frequency
• \(\gamma\) stands for the gyromagnetic ratio
• \(B_0\) is the magnetic field strength

Understanding how to calculate the Larmor frequency helps in distinguishing between different types of nuclei like

• \(^{1}H\)
• \(^{2}H\)
• \(^{13}C\)

used in NMR experiments.

The gyromagnetic ratio is a fundamental property of nuclei that influences their behavior in a magnetic field. It represents the ratio between the magnetic moment and the angular momentum of the nucleus.Different nuclei have different gyromagnetic ratios influencing their Larmor frequencies. In the given exercise, we used the following gyromagnetic ratios:

• \(^{1}H\): 2.7 x 10^8 rad T^{-1} s^{-1}
• \(^{2}H\): 0.41 x 10^8 rad T^{-1} s^{-1}
• \(^{13}C\): 0.67 x 10^8 rad T^{-1} s^{-1}

A higher gyromagnetic ratio means a higher Larmor frequency for the same magnetic field strength. Therefore, \(^{1}H\) with a gyromagnetic ratio of 2.7 x 10^8 has a significantly higher Larmor frequency compared to \(^{2}H\) and \(^{13}C\).

In NMR spectroscopy, the strength of the external magnetic field (\(B_0\)) plays a crucial role in determining the Larmor frequency of a nucleus. Fragments in the exercise, we used a magnetic field strength of 4.7 \(T\) to calculate the Larmor frequencies.The relationship between magnetic field strength and Larmor frequency is direct. As you increase the magnetic field strength, the Larmor frequency also increases. This is why high-field NMR spectrometers provide greater resolution and sensitivity.For example, in the exercise, we saw that for a magnetic field strength of 4.7 \(T\):

• \(^{1}H\) has a Larmor frequency of approximately 202.5 MHz
• \(^{2}H\) is about 30.7 MHz
• \(^{13}C\) is around 50.1 MHz

This variation in frequency based on magnetic field strength and type of nucleus is what allows us to distinguish between different substances in NMR spectroscopy.