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Chapter 13: Problem 11

When placed in a powerful magnetic field, there is a small population bias for the \({ }^{1} \mathrm{H}\) and \({ }^{13} \mathrm{C}\) nuclei to be aligned with the magnetic field, and they precess.

Short Answer

Expert verified
Answer: When magnetic nuclei like hydrogen-1 and carbon-13 are placed in a strong magnetic field, their energy levels split into distinct, quantized states related to the orientation of their magnetic moment relative to the magnetic field. This results in a small population bias, preferentially occupying one of the states, usually the lower-energy state. Additionally, these nuclei precess around the magnetic field lines, with their orientation changing so that their axis of rotation traces out a cone around the magnetic field direction. This phenomenon of alignment, population bias, and precession is fundamental to understanding nuclear magnetic resonance (NMR).

Step by step solution

01

Understanding the phenomenon

When nuclei of \({ }^{1} \mathrm{H}\) (hydrogen) and \({ }^{13} \mathrm{C}\) (carbon) are placed in a strong magnetic field, they tend to align with it. This means that their magnetic dipoles will preferentially point in a direction parallel to the magnetic field lines. Due to quantum mechanics, there will be a small preference (population bias) for one specific alignment, either parallel or anti-parallel to the field (justifying the use of the words "small population bias"). Additionally, these nuclei will precess around the magnetic field lines, which means that they will rotate about their axis parallel to the field direction.
02

Alignment and Population Bias

When placed in a powerful magnetic field, nuclei like \({ }^{1} \mathrm{H}\) and \({ }^{13} \mathrm{C}\) will have their energy levels split into distinct, quantized states, each corresponding to a different orientation of the nuclei's magnetic moment relative to the magnetic field. Typically, there will be a lower-energy state (aligned with the magnetic field) and a higher-energy state (anti-aligned). Because of this quantization, there will be a small population bias for the nucleus to preferentially be in one of these states (usually the lower-energy state), as there are more ways for a system to be in that state than in the other states.
03

Precession Around the Magnetic Field

The magnetic moment/minor of a nucleus like \({ }^{1} \mathrm{H}\) or \({ }^{13} \mathrm{C}\) will also precess around the direction of the magnetic field. Precession is a change in the orientation of the axis of rotation of the nucleus so that it traces out a cone around the direction of the magnetic field. The precession frequency, called the Larmor frequency, is determined by the strength of the magnetic field and the gyromagnetic ratio \((\gamma)\) of the nucleus. It can be calculated using the following equation: \(\omega_\mathrm{L} = -\gamma B\) where \(\omega_\mathrm{L}\) is the Larmor frequency, \(B\) is the magnetic field strength, and \(\gamma\) is the gyromagnetic ratio for the specific nucleus.
04

Connection to Nuclear Magnetic Resonance (NMR)

The phenomena of alignment, population bias, and precession are fundamental to our understanding of nuclear magnetic resonance (NMR). NMR is a technique used to investigate the local chemical environment and structures of molecules. In an NMR experiment, the sample is placed in a strong magnetic field, causing the nuclei to align and precess around the magnetic field lines. Then, an electromagnetic pulse is applied, causing the nuclei to absorb energy and transition between energy states. By analyzing the emitted signals when the nuclei return to their original energy states, scientists can gather valuable information about the sample's molecular structure.

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Most popular questions from this chapter

Following is a \({ }^{1} \mathrm{H}\)-NMR spectrum of 2-butanol. Explain why the \(\mathrm{CH}_{2}\) protons appear as a complex multiplet rather than as a simple quintet.

The percent scharacter of carbon participating in a \(\mathrm{C}-\mathrm{H}\) bond can be established by measuring the \({ }^{13} \mathrm{C}-{ }^{1} \mathrm{H}\) coupling constant and using the relationship $$ \text { Percent scharacter }=0.2 \mathrm{~J}\left({ }^{13} \mathrm{C}-{ }^{1} \mathrm{H}\right) $$ The \({ }^{15} \mathrm{C}-{ }^{1} \mathrm{H}\) coupling constant observed for methane, for example, is \(125 \mathrm{~Hz}\), which gives \(25 \%\) scharacter, the value expected for an \(s p^{3}\) hybridized carbon atom. (a) Calculate the expected \({ }^{13} \mathrm{C}-{ }^{1} \mathrm{H}\) coupling constant in ethylene and acetylene. (b) In cyclopropane, the \({ }^{19} \mathrm{C}-{ }^{1} \mathrm{H}\) coupling constant is \(160 \mathrm{~Hz}\). What is the hybridization of carbon in cyclopropane?

Calculate the index of hydrogen deficiency of these compounds. (a) Aspirin, \(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\) (b) Ascorbic acid (vitamin \(\mathrm{C}\) ), \(\mathrm{C}_{6} \mathrm{H}_{g} \mathrm{O}_{6}\) (c) Pyridine, \(\mathrm{C}_{5} \mathrm{H}_{5} \mathrm{~N}\) (d) Urea, \(\mathrm{CH}_{4} \mathrm{~N}_{2} \mathrm{O}\) (e) Cholesterol, \(\mathrm{C}_{2} \mathrm{H}_{45} \mathrm{O}\) (f) Dopamine, \(\mathrm{C}_{8} \mathrm{H}_{11} \mathrm{NO}_{2}\)

Chemical shift, \(\delta\), is defined as the frequency shift from tetramethyl silane (TMS) divided by the operating frequency of the spectrometer. \- The resonance signals in \({ }^{\mathrm{l}}\) H-NMR spectra are reported according to how far they are shifted from the resonance signal of the 12 equivalent hydrogens in TMS. \- The resonance signals in \({ }^{13}\) CNMR spectra are reported according to how far they are shifted from the resonance signal of the four equivalent carbons in TMS.

The natural abundance of \({ }^{19} \mathrm{C}\) is only \(1.1 \%\). Furthermore, its sensitivity in NMR specroscopy (a measure of the energy difference between a spin aligned with or against an applied magnetic field) is only \(1.6 \%\) that of \({ }^{1} \mathrm{H}\). What are the relative signal intensiies expected for the \({ }^{1} \mathrm{H}-\mathrm{NMR}\) and \({ }^{13} \mathrm{C}-\mathrm{NMR}\) spectra of the same sample of \(\mathrm{Si}\left(\mathrm{CH}_{5}\right)_{4}\) ?
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