Question

The precession frequency of the protons in a laboratory NMR spectrometer is \(15.35850 \mathrm{MHz}\). The magnetic moment of the proton is \(1.410608 \cdot 10^{-26} \mathrm{~J} / \mathrm{T}\), while its spin angular momentum is \(0.5272863 \cdot 10^{-34} \mathrm{~J}\) s. Calculate the magnitude of the magnetic field in which the protons are immersed.

Short answer

Question: Calculate the magnitude of the magnetic field in which the protons are immersed, given the precession frequency of 15.35850 MHz, the magnetic moment of the proton as 1.410608 x 10^(-26) J/T, and the spin angular momentum of the proton as 0.5272863 x 10^(-34) J*s. Answer: The magnitude of the magnetic field in which the protons are immersed is approximately 0.578 T.

Step-by-step solution

Write down the given values

We are given the following values: - Precession frequency, \(\nu = 15.35850 \mathrm{MHz}\) - Magnetic moment of the proton, \(\mu = 1.410608 \times 10^{-26} \mathrm{J/T}\) - Spin angular momentum of the proton, \(s = 0.5272863 \times 10^{-34} \mathrm{J\cdot s}\)

Convert the precession frequency to Hz

We need to convert the precession frequency from MHz to Hz: \(\nu = 15.35850 \mathrm{MHz} = 15.35850 \times 10^6 \mathrm{Hz}\)

Rearrange the Larmor equation to solve for the magnetic field

The Larmor equation is given by: \(\nu = \frac{\mu B}{s}\) We need to solve for \(B\), the magnetic field magnitude. Rearranging the Larmor equation, we get: \(B = \frac{\nu \cdot s}{\mu}\)

Plug in the given values and calculate the magnetic field magnitude

Now, we can plug in the given values to find the magnetic field magnitude: \(B = \frac{(15.35850 \times 10^6 \mathrm{Hz}) \cdot (0.5272863 \times 10^{-34} \mathrm{J\cdot s})}{1.410608 \times 10^{-26} \mathrm{J/T}}\) Computing the values, we get: \(B \approx 0.578 \mathrm{T}\)

State the final answer

The magnitude of the magnetic field in which the protons are immersed is approximately \(0.578 \mathrm{T}\).

Key concepts

Precession Frequency

Precession frequency is a fundamental concept in NMR spectroscopy. It refers to the rate at which the magnetic moment of a nucleus rotates around an external magnetic field. This frequency is measured in Hertz (Hz), indicating how many complete rotations occur each second.

In the context of NMR, the common unit used for precession frequency is Megahertz (MHz). To convert from MHz to Hz, one must multiply by a factor of 10⁶. For instance, a given frequency of 15.35850 MHz is equivalent to 15,358,500 Hz.

Understanding precession frequency is crucial as it relates to the energy levels of nuclear spins and their ability to absorb radiofrequency. In an NMR experiment, the nucleus precesses at a unique frequency, which helps in identifying different chemical environments within a molecule.

Magnetic Moment

The magnetic moment is a vector quantity that represents the magnetic strength and orientation of a nucleus with respect to an external magnetic field. For protons, the magnetic moment is a defining factor in how they respond to applied magnetic fields during NMR analysis.

It is typically expressed in joules per tesla (J/T), a unit that depicts the torque a magnetic dipole experiences in a field. The higher the magnetic moment, the more responsive the nucleus is to an external magnetic field. The value of the magnetic moment for a proton also influences the energy transition states during NMR spectroscopy, thus impacting the spectral data obtained from experiments.

Spin Angular Momentum

Spin angular momentum is an intrinsic property of nuclei, arising from their spin. It is a quantum mechanical concept often denoted by the letter ‘s’. Similar to the magnetic moment, it influences how nuclei behave in magnetic fields.

Expressed in joules per second (J⋅s), spin angular momentum represents the „spinning“ motion of nuclei. Protons possess spin angular momentum due to their charge and rotation, which collectively contribute to their behavior in a magnetic field.

Understanding spin angular momentum is essential in NMR, as it helps explain why certain nuclei interact differently with magnetic fields, leading to the observable spectral patterns unique to NMR spectroscopy.

Larmor Equation

The Larmor Equation is a critical formula used in NMR to relate precession frequency to both the magnetic moment and spin angular momentum of a nucleus. The equation is written as: \[ u = \frac{\mu B}{s} \] where \( u \) is the precession frequency, \( \mu \) is the magnetic moment, \( B \) is the magnetic field, and \( s \) is the spin angular momentum.

This equation essentially describes how the frequency at which nuclei precess is directly proportional to the magnetic field applied and inversely proportional to their spin angular momentum.

• High magnetic moments or low spin angular momentum result in higher precession frequencies.
• Likewise, stronger magnetic fields will also increase the precession frequency.

Applying the Larmor Equation helps in determining unknown values, such as calculating the magnetic field when the other parameters are known, as shown in the example exercise.