Calculate the nuclear g_I factor for a free ¹H nucleus.  The magnetogyric ratio for a free ¹H nucleus is 26.752 × 10⁷ T^–1 s^–1.

Calculate the difference in the populations of the two nuclear spin states of ¹H nuclei in a magnetic field of 10.0 T at a temperature of 298 K.

Protons in a high-field nuclear magnetic resonance spectrometer resonate at frequencies close to 800 MHz.  Calculate the strength of the magnetic field of the spectrometer.

A nuclear magnetic resonance transition is shifted from the reference in a 400 MHz NMR spectrometer by 529 Hz.  Calculate the chemical shift.

The two fine-structure components of a nuclear magnetic resonance transition are observed at chemical shifts of 2.142 and 2.208 in a 300 MHz NMR spectrometer.  Calculate the coupling constant.

The coupling constant, ³J_HH, observed for coupling between two protons in the backbone of a protein has been measured as 5.2 Hz.  The Karplus equation,

with coefficients A = +7 Hz, B = –1 Hz, and C = +5 Hz may be used to estimate the H–N–C_a–H dihedral angle.  Which of the following angles is consistent with the observed coupling constant?

At low temperatures, peaks in the NMR spectrum of p-nitrophenol, C₆H₄NO₂OH, are observed as doublets split by ~100 Hz, because the molecule exists in equilibrium, as both protonated and deprotonated forms.  At higher temperatures, the peaks coalesce to produce a single broadened peak with a line width equivalent to the separation between the individual peaks.  Estimate the rate constant for proton exchange.

Two peaks, corresponding to different conformations of a molecule are observed at chemical shifts of 2.9 and 3.8 when the NMR spectrum is recorded at low temperatures using a 500 MHz spectrometer.  At what rate of interconversion will the two signals collapse to a single line?

Calculate the strength of the magnetic field necessary to produce a difference in the energy of the two spin states of an electron of 10^–26 J.

The ESR spectrum of the quinoxaline anion features fine structure resulting from the coupling of the electron spin to the hydrogen nuclei.  The observed coupling constants are shown below.  Calculate the spin densities on each of the carbon atoms at the positions labelled.