Problem 29 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

The magnetogyric ratio for a \(^{1} \mathrm{H}\) nucleus is \(26.7519 \times 10^{7} \mathrm{T}^{-1} \mathrm{s}^{-1}\) Calculate the magnetic field strength (in tesla) required to give a Larmor frequency of \(400 \mathrm{MHz}\). (Section 10.7 ) Further questions on using IR and NMR spectroscopies in the context of structure elucidation are given at the end of Chapter 12 \((p .606)\)

Step-by-Step Solution

Answer

The magnetic field strength required is approximately 9.398 T.

Step 1: Understanding the Relationship

To solve this problem, we recognize that the Larmor frequency () is related to the magnetic field strength (B) and the magnetogyric ratio () through the equation:\[ \omega = \gamma B \]where \( \omega \) is the angular Larmor frequency in radians per second, \( \gamma \) is the magnetogyric ratio, and \( B \) is the magnetic field strength in tesla.

Step 2: Convert Larmor Frequency to Angular Frequency

Given the Larmor frequency in megahertz (MHz), we first convert it to hertz (Hz) and then use the relationship \( \omega = 2 \pi f \) to find the angular frequency in radians per second.\[ f = 400 \text{ MHz} = 400 \times 10^6 \text{ Hz} \]\[ \omega = 2 \pi \times 400 \times 10^6 \text{ Hz} \]

Step 3: Solve for Magnetic Field Strength

Plug the values for \( \omega \) and the given \( \gamma \) into the formula \( \omega = \gamma B \) and solve for \( B \):\[ B = \frac{\omega}{\gamma} \]Substitute \( \gamma = 26.7519 \times 10^7 \text{ T}^{-1}\text{s}^{-1} \) into the equation:\[ B = \frac{2\pi \times 400 \times 10^6}{26.7519 \times 10^7} \]

Step 4: Calculate Result

Perform the arithmetic to find the magnetic field strength:Calculate \( \omega = 2 \pi \times 400 \times 10^6 \approx 2.513 imes 10^9 \) rad/s.Then, compute the magnetic field:\[ B = \frac{2.513 \times 10^9}{26.7519 \times 10^7} \approx 9.398 \text{ T} \]

Key Concepts

Magnetogyric Ratio

The magnetogyric ratio, often denoted as \( \gamma \), is a crucial parameter in magnetic resonance phenomena, particularly in nuclear magnetic resonance (NMR) and electron spin resonance (ESR). It represents the ratio of the magnetic moment to the angular momentum of a particle. In simpler terms, it's a factor that tells us how much the magnetic moment of a particle, like a nucleus, will precess in a magnetic field.
The magnetogyric ratio is unique for each type of nucleus or particle. For a proton \((^{1} \mathrm{H})\), \( \gamma \) is approximately \(26.7519 \times 10^{7} \mathrm{T}^{-1} \mathrm{s}^{-1}\).

• This value signifies how strongly a nucleus will interact with a magnetic field.
• It's critical for calculating the Larmor frequency in NMR.
• Nuclei with different \( \gamma \) values will resonate at different frequencies in the same magnetic field.

Understanding \( \gamma \) helps us predict how different nuclei will behave in a magnetic field, which is vital for applications like MRI (Magnetic Resonance Imaging).

Magnetic Field Strength

Magnetic field strength, denoted as \(B\), is a measure of the intensity of a magnetic field. In the context of NMR, it's expressed in tesla (T). The strength of the magnetic field determines the Larmor frequency, or the rate at which a nucleus precesses around the field.

• Higher magnetic field strengths result in higher Larmor frequencies.
• The field strength is crucial in differentiating between various nuclei based on their resonant frequencies.
• In chemical analysis, a stronger field gives better resolution and sensitivity, allowing for detailed structure elucidation.

Modern NMR spectrometers can operate at very high magnetic field strengths, enhancing their ability to resolve complex molecular structures. This capability is essential for chemists and biochemists studying the arrangement of atoms within a molecule.

NMR Spectroscopy

NMR spectroscopy is a powerful analytical technique used to determine the structure of molecules. It exploits the magnetic properties of certain atomic nuclei and is based on the principle of resonant absorption of radiofrequency radiation by nuclei in a magnetic field.
Nuclei that have an odd number of protons or neutrons have a magnetic moment and can be analyzed using NMR. Here's how the process generally works:

• A sample is placed in a strong magnetic field.
• Radiofrequency pulses are applied, causing nuclei to absorb energy and precess around the magnetic field vector.
• The absorbed energy is detected and analyzed to yield spectra that offer insights into molecular structure.

NMR is particularly effective because it allows researchers to observe the local environment around a nucleus. This makes it invaluable for chemical analysis, providing detailed information about the structure, dynamics, reaction state, and chemical environment of molecules.

Angular Frequency

Angular frequency, often represented by \( \omega \), is a measure of how fast something rotates or oscillates and is usually expressed in radians per second. In the context of NMR, it refers to the frequency at which the nuclear spins precess around the magnetic field lines.

• Angular frequency is directly proportional to the magnetic field strength and the magnetogyric ratio, given by \( \omega = \gamma B \).
• This formula highlights the relationship between how quickly nuclear spins precess (\( \omega \)) and both the magnetogyric ratio (\( \gamma \)) and the strength of the applied magnetic field (\( B \)).

In practical terms, understanding angular frequency is essential for calculating the Larmor frequency, which is the specific frequency where resonance occurs in NMR spectroscopy. This knowledge allows scientists to fine-tune the NMR parameters to identify specific atomic nuclei accurately.