Chapter 16, Problem 16.46P

Interpretation Introduction

Interpretation:

It is to be shown how the equation that relates the operating frequency of an NMR spectrophotometer to the gyromagnetic ratio of the nucleus of interest and the magnitude of the external magnetic field is derived from Equation 16-1, ${\text{E}}_{\text{photon}}{\text{= ΔE}}_{\text{spin}}$.

Concept introduction:

A spectrometer’s operational frequency (${\text{ν}}_{\text{op}}$) is the frequency at which a proton absorbs radiation when subjected to the spectrometer’s magnetic field given by the equation:

${\text{ν}}_{op}\text{ = }\frac{\gamma B_{ext}}{2\pi }$

where ${\text{ν}}_{\text{op}}$ is the operational frequency, $\gamma$ is the gyromagnetic ratio which is constant, and ${\text{B}}_{\text{ext}}$ is the external magnetic field. Operating frequency is constant for a given magnetic field strength. When the energy difference between two spin states equal to the energy of the photon, the photon will be absorbed.

Equations 16-1 and 16-3 are as follows:

$\frac{{\text{γB}}_{\text{ext}}}{\text{2π}}{\text{= ΔE}}_{\text{spin}}$ -------------------------------------------- equation 1

${\text{ΔE}}_{\text{spin}}{\text{=ΔE}}_{\text{photon}}\text{= hν}$--------------------------------------- equation 2