Larmor Frequency Calculation Using Magnetogyric Ratio

The Larmor frequency ($\nu_L$) of a nucleus in Nuclear Magnetic Resonance (NMR) is directly proportional to the applied magnetic field strength ($B$) and the nucleus's magnetogyric ratio ($\gamma$). The relationship can be expressed as:

$ \nu_L = \frac{\gamma B}{2\pi} $

Since the magnetic field strength ($B = 1$ Tesla) is constant for both $^1H$ and $^{13}C$ in this problem, the Larmor frequency is directly proportional to the magnetogyric ratio:

$ \nu_L \propto \gamma $

Determining $^{13}C$ Larmor Frequency

We can use the ratio of Larmor frequencies for $^{13}C$ and $^1H$ to find the unknown frequency. The ratio of their frequencies will be equal to the ratio of their magnetogyric ratios:

$ \frac{\nu_{13C}}{\nu_{1H}} = \frac{\gamma_{13C}}{\gamma_{1H}} $

To find the Larmor frequency of $^{13}C$ ($\nu_{13C}$), we rearrange the formula:

$ \nu_{13C} = \nu_{1H} \times \frac{\gamma_{13C}}{\gamma_{1H}} $

Step-by-Step Calculation

1. Identify Given Values:
   • $^1H$ Larmor frequency ($\nu_{1H}$) = 42.57 MHz
   • $^1H$ magnetogyric ratio ($\gamma_{1H}$) = $26.75 \times 10^7 \text{ rad } T^{-1} s^{-1}$
   • $^{13}C$ magnetogyric ratio ($\gamma_{13C}$) = $6.72 \times 10^7 \text{ rad } T^{-1} s^{-1}$
   • Magnetic field strength ($B$) = 1 T (constant for both)
2. Substitute values into the formula:

   $ \nu_{13C} = 42.57 \text{ MHz} \times \frac{6.72 \times 10^7 \text{ rad } T^{-1} s^{-1}}{26.75 \times 10^7 \text{ rad } T^{-1} s^{-1}} $

3. Simplify and Calculate:

   The term $10^7 \text{ rad } T^{-1} s^{-1}$ cancels out.

   $ \nu_{13C} = 42.57 \text{ MHz} \times \frac{6.72}{26.75} $

   $ \nu_{13C} \approx 42.57 \text{ MHz} \times 0.251215 $

   $ \nu_{13C} \approx 10.6915 \text{ MHz} $

The calculated Larmor frequency for $^{13}C$ at 1 Tesla is approximately 10.69 MHz, which falls within the specified range.