2­18 NMR and energy levels
2.7 Exercises
E 2­1
In a proton spectrum the peak from TMS is found to be at 400.135705 MHz.
What is the shift, in ppm, of a peak which has a frequency of
400.136305 MHz? Recalculate the shift using the spectrometer frequency,
spec quoted by the manufacturer as 400.13 MHz rather than TMS in the de-
nominator of Eq. 2.2:
ppm = 106
×
 - TMS
spec
.
Does this make a significant difference to the value of the shift?
E 2­2
Two peaks in a proton spectrum are found at 1.54 and 5.34 ppm. The spec-
trometer frequency is quoted as 400.13 MHz. What is the separation of these
two lines in Hz and in rad s-1?
E 2­3
Calculate the Larmor frequency (in Hz and in rad s-1) of a carbon-13 res-
onance with chemical shift 48 ppm when recorded in a spectrometer with
a magnetic field strength of 9.4 T. The gyromagnetic ratio of carbon-13 is
+6.7283 × 107 rad s-1 T-1.
E 2­4
Consider a system of two weakly coupled spins. Let the Larmor frequencyOf course in reality the Larmor
frequencies out to be tens or
hundreds of MHz, not 100 Hz!
However, it makes the numbers
easier to handle if we use these
unrealistic small values; the
principles remain the same,
however.
of the first spin be -100 Hz and that of the second spin be -200 Hz, and let
the coupling between the two spins be -5 Hz. Compute the frequencies of
the lines in the normal (single quantum) spectrum.
Make a sketch of the spectrum, roughly to scale, and label each line with
the energy levels involved (i.e. 1­2 etc.). Also indicate for each line which
spin flips and the spin state of the passive spin. Compare your sketch with
Fig. 2.7 and comment on any differences.
E 2­5
For a three spin system, draw up a table similar to that on page 2­10 showing
the frequencies of the four lines of the multiplet from spin 2. Then, taking
0,2 = -200 Hz, J23 = 4 Hz and the rest of the parameters as in Fig. 2.11,
compute the frequencies of the lines which comprise the spin 2 multiplet.
Make a sketch of the multiplet (roughly to scale) and label the lines in the
same way as is done in Fig. 2.11. How would these labels change if J23 =
-4 Hz?
On an energy level diagram, indicate the four transitions which comprise
the spin 2 multiplet, and which four comprise the spin 3 multiplet.
E 2­6
For a three spin system, compute the frequencies of the six zero-quantum
2.7 Exercises 2­19
transitions and also mark these on an energy level diagram. Do these six
transitions fall into natural groups? How would you describe the spectrum?
E 2­7
Calculate the line frequencies and intensities of the spectrum for a system of
two spins with the following parameters: 0,1 = -10 Hz, 0,2 = -20 Hz,
J12 = 5 Hz. Make a sketch of the spectrum (roughly to scale) indicating
which transition is which and the position of the Larmor frequencies.
E 2­8
The spectrum from a strongly-coupled two spin system showed lines at the Make sure that you have your
calculator set to "radians" when
you compute sin 2.
following frequencies, in Hz, (intensities are given in brackets): 32.0 (1.3),
39.0 (0.7), 6.0 (0.7), 13.0 (1.3). Determine the values of the coupling constant
and the two Larmor frequencies. Show that the values you find are consistent
with the observed intensities.
frequency (Hz)
1510 20 25 30 35 40
Fig. 2.18 The AB part of an ABX spectrum
E 2­9
Figure 2.18 shows the AB part of an ABX spectrum. Disentangle the two
subspectra, mark in the rough positions of the effective Larmor frequencies
and hence estimate the size of the AX and BX couplings. Also, give the value
of the AB coupling.