Complete the following table. Which nucleus requires the least energy to flip its spin at this applied field? Which nucleus requires the most energy? \begin{tabular}{lccc} \hline Nucleus & Applied Field (tesla, T) & Radio Frequency \((\mathrm{MHz})\) & Energy \((\mathrm{J} / \mathbf{m o l})\) \\ \hline\({ }^{1} \mathrm{H}\) & \(7.05\) & 300 & \\ \({ }^{13} \mathrm{C}\) & \(7.05\) & \(75.5\) & \(-\) \\ \({ }^{19} \mathrm{~F}\) & \(7.05\) & 282 & \(-\) \\ \hline \end{tabular}

To find the energy required to flip the spin for each nucleus, we need to first convert the radio frequency from MHz to Hz by multiplying with 10^6. After that, we can apply the formula E = h * f to find the energy. Since the given unit for energy is in J/mol, we also need to account for that by multiplying by Avogadro's number (6.022 x 10^23). For \({ }^{1} \mathrm{H}\): Frequency (Hz) = 300 MHz * 10^6 = 3.00 x 10^8 Hz Energy = (6.626 x 10^(-34) J*s) * (3.00 x 10^8 Hz) * (6.022 x 10^23) Energy = 119.7 J/mol For \({ }^{13} \mathrm{C}\): Frequency (Hz) = 75.5 MHz * 10^6 = 7.55 x 10^7 Hz Energy = (6.626 x 10^(-34) J*s) * (7.55 x 10^7 Hz) * (6.022 x 10^23) Energy = 29.95 J/mol For \({ }^{19} \mathrm{~F}\): Frequency (Hz) = 282 MHz * 10^6 = 2.82 x 10^8 Hz, Energy = (6.626 x 10^(-34) J*s) * (2.82 x 10^8 Hz) * (6.022 x 10^23) Energy = 112.71 J/mol

Now we can compare the energy for each nucleus to determine which one requires the least energy and which one requires the most energy. Least energy required: \({ }^{13} \mathrm{C}\) (29.95 J/mol) Most energy required: \({ }^{1} \mathrm{H}\) (119.7 J/mol) Updated table: \begin{tabular}{lccc} \hline Nucleus & Applied Field (tesla, T) & Radio Frequency \((\mathrm{MHz})\) & Energy \((\mathrm{J} / \mathbf{m o l})\) \\\ \hline\({ }^{1} \mathrm{H}\) & \(7.05\) & 300 & 119.7\\\ \({ }^{13} \mathrm{C}\) & \(7.05\) & \(75.5\) & 29.95 \\\ \({ }^{19} \mathrm{~F}\) & \(7.05\) & 282 & 112.71 \\\ \hline \end{tabular} The nucleus that requires the least energy to flip its spin is \({ }^{13} \mathrm{C}\), and the nucleus that requires the most energy is \({ }^{1} \mathrm{H}\).