Question

The resonance frequency of \( ^1H \) nuclei is 300 MHz in an NMR spectrometer. If the spectrometer is operated at 12 T magnetic field, the resonance frequency (in MHz) of the same \( ^1H \) nuclei is ..................

Hint:

The resonance frequency in NMR can be calculated using the Larmor equation and knowing the gyromagnetic ratio and magnetic field strength. Be sure to convert units where necessary.

Answer 1

The resonance frequency \( \nu \) of a nucleus in a magnetic field is given by the Larmor equation: \[ \nu = \gamma B \] where \( \gamma \) is the gyromagnetic ratio, and \( B \) is the magnetic field strength.
The gyromagnetic ratio \( \gamma \) can be related to the nuclear magneton \( \beta_N \) and the nuclear g-factor \( g_N \) by: \[ \gamma = \frac{g_N \beta_N}{h} \] where:
- \( \beta_N = 5.05 \times 10^{-27} \, \text{J T}^{-1} \),
- \( g_N = 5.586 \) for \( ^1H \),
- \( h = 6.63 \times 10^{-34} \, \text{J s} \).
First, calculate the gyromagnetic ratio for \( ^1H \): \[ \gamma = \frac{5.586 \times 5.05 \times 10^{-27}}{6.63 \times 10^{-34}} = 4.22 \times 10^7 \, \text{Hz/T} \] Step 1: Calculate the resonance frequency.
The magnetic field is given as \( B = 12 \, \text{T} \), so the resonance frequency is: \[ \nu = \gamma B = (4.22 \times 10^7 \, \text{Hz/T})(12 \, \text{T}) = 5.06 \times 10^8 \, \text{Hz} \] Converting to MHz: \[ \nu = 5.06 \times 10^8 \, \text{Hz} \times \frac{1 \, \text{MHz}}{10^6 \, \text{Hz}} = 506 \, \text{MHz} \] Final Answer: \[ \boxed{506} \]