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Chapter 28
Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
Thomas Engel, Philip Reid
Objectives
• Applications for nuclear magnetic resonance (NMR)
spectrum
• Applications for splitting of NMR peaks
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Outline
1. Intrinsic Nuclear Angular Momentum and Magnetic Moment
2. The Energy of Nuclei of Nonzero Nuclear Spin in a Magnetic
Field
3. The Chemical Shift for an Isolated Atom
4. The Chemical Shift for an Atom Embedded in a Molecule
5. Electronegativity of Neighboring Groups and Chemical Shifts
6. Magnetic Fields of Neighboring Groups and Chemical Shifts
7. Multiplet Splitting of NMR Peaks Arises through Spin–Spin
Coupling
8. Multiplet Splitting When More Than Two Spins Interact
9. Peak Widths in NMR Spectroscopy
10. Solid-State NMR
11. NMR Imaging
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.1 Intrinsic Nuclear Angular Momentum and Magnetic
Moment
•
•
As nuclear magnetic moment of the proton
weaker than electron magnetic moment, it has no
effect on the one-electron energy levels in the
hydrogen atom.
Nuclear magnetic moment, µ, is defined as
e
1
1
  gN
I  gN N  
2m proton
h
h
where I = nuclear angular momentum
e = unit of elementary nuclear charge (1.6×10-19 C)
mproton = mass of a proton
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.1 Intrinsic Nuclear Angular Momentum and Magnetic
Moment
•
βN is the nuclear magneton and nuclear
factor gN is characteristic of a particular
nucleus.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.2 The Energy of Nuclei of Nonzero Nuclear Spin in a
Magnetic Field
•
Larmor frequency,v, states that
1
v
B0 or   2v  B0
2
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.2 The Energy of Nuclei of Nonzero Nuclear Spin in a
Magnetic Field
•
In NMR spectroscopy a transition must be
induced between two different energy levels so
that the absorption/emission of the
electromagnetic energy can be detected.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 28.1
Calculate the two possible energies of the 1H nuclear
spin in a uniform magnetic field of 5.50 T.
b. Calculate the energy absorbed in making a transition
from the to the state. If a transition is made between
these levels by the absorption of electromagnetic
radiation, what region of the spectrum is used?
c. Calculate the relative populations of these two states
in equilibrium at 300 K.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
a. The two energies are given by


1
1
E   g N  N B0   5.5854 5.05110  27 5.50  7.76 10  26 J
2
2
b. The energy difference is given by


E  2 7.76 10 26  1.55 10 25 J
E 10.55  10  25
8 1
v


2
.
34

10
s
34
h
6.626 10
This is in the range of frequencies called radio
frequencies.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
c. The relative populations of the two states are
given by

    
 exp  

kT

   
1

    
2
  2  7.76 10 26 

  0.999962
  exp 
 23

 1.38110  300 
1  0.999962  3.8 105


From this result, we see that the populations of the
two states are the same to within a few parts per
million.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.3 The Chemical Shift for an Isolated Atom
•
•
When an atom is placed in a magnetic field, a
circulation current around the nucleus
generates a secondary magnetic field.
The z component of the induced magnetic field
is given by
0 
2

Bz 
3
cos
  1
3
4r
where µ0 = vacuum permeability
µ0 = induced magnetic moment
θ and r = spherical coordinates
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.3 The Chemical Shift for an Isolated Atom
•
The total field at the nucleus is given by the
sum of the external and induced fields,
Btotal  1   B0
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.4 The Chemical Shift for an Atom Embedded in a
Molecule
•
The frequency shift for an atom depends
linearly on the shielding constant, σ.
• This makes NMR a sensitive probe around a
nucleus with nonzero nuclear spin.
• Two factors responsible for chemical shift are:
1. Electronegativity of the neighboring group
2. Induced magnetic field of the neighboring
group
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.5 Electronegativity of Neighboring Groups and
Chemical Shifts
•
The chemical shifts for different classes of
molecules are strongly correlated with their
electron-withdrawing ability.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.6 Magnetic Fields of Neighboring Groups and
Chemical Shifts
•
The magnetic field at a 1H nucleus is a
superposition of the external field.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.6 Magnetic Fields of Neighboring Groups and
Chemical Shifts
•
NMR signal of a solution sample is generated by
the large number of molecules contained in the
sampling volume.
 average   individual
•
•
For 1H, the range of observed values for  is
about 10 ppm.
For nuclei in atoms that can exhibit both
paramagnetic and diamagnetic behavior,  can
vary over a much wider range.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.7 Multiplet Splitting of NMR Peaks Arises through
Spin–Spin Coupling
•
In a simulated NMR spectrum for ethanol,
individual peaks are split into multiplets.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.7 Multiplet Splitting of NMR Peaks Arises through
Spin–Spin Coupling
•
•
Multiplets arise as a result of spin–spin
interactions among different nuclei.
The spin energy operator for the noninteracting
spins is
Hˆ  B0 1   1 Iˆz1  B0 1   2 Iˆz2
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.7 Multiplet Splitting of NMR Peaks Arises through
Spin–Spin Coupling
•
Solving Schrödinger equation for the
corresponding eigenvalues gives
assume σ1>σ2
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 28.2
Show that the total nuclear energy eigenvalue for
the wave function    1 2 is
2
E2  
Solution:
hB0
 1   2 
2


Hˆ  2   B0 1   1 Iˆz1  B0 1   2 Iˆz2  1 2
h
h
 B0 1   1  1 2  B0 1   2  1 2
2
2
hB0
 1   2  1 2

2
 E2 2
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.7 Multiplet Splitting of NMR Peaks Arises through
Spin–Spin Coupling
•
The splitting between levels 2 and 3 and the
energy shifts of all four levels for interacting
spins emphasize the spin–spin interactions.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.7 Multiplet Splitting of NMR Peaks Arises through
Spin–Spin Coupling
•
•
For the noninteracting spin case, E2-E1=E4-E3
and E3-E1=E4-E2.
NMR spectrum contains only two peaks
corresponding to the frequencies:
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.7 Multiplet Splitting of NMR Peaks Arises through
Spin–Spin Coupling
•
In general the energy correction is
4 2
E j 
J12   * 1 * 1Iˆ1x Iˆ2 x 1 2d 1d 2
h
•
To solve for eigenfunctions, we have
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 28.3
Show that the energy correction to
 2   1 2 is E2  hJ12 / 4
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
We evaluate


4 2
E2 
J12   * 1 * 2  Iˆ1x Iˆ2 x  Iˆ1 y Iˆ2 y  Iˆ1z Iˆ2 z  1 2 d 1d 2
h
  * 1 * 2  Iˆ1x Iˆ2 x  1 2 d 1d 2 
 

2
4

J12    * 1 * 2  Iˆ1 y Iˆ2 y  1 2 d 1d 2 


h
  * 1 * 2  Iˆ Iˆ  1 2 d d 
1z 2 z
1
2 
 








 h2 
   * 1 * 2   1 2 d 1d 2

4
 




 i 2h2 
4 2

J12    * 1 * 2 
 1 2 d 1d 2 
h


 4 


2
  * 1 * 2  h  1 2 d d 
1
2 


 
4




Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
Because of the orthogonality of the spin functions, the
first two integrals are zero and
4 2  h 2 
4 2  h 2 
h 2 J12
   J12   *1 * 2 1 2d 1d 2 
   J12  
E2 
h  4
h  4
4
Note that because J12 has the units of s-1, hJ has the
unit joule.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.8 Multiplet Splitting of NMR Peaks Arises through
Spin–Spin Coupling
•
•
Many organic molecules have more than two
inequivalent protons that are close to generate
multiplet splittings.
The frequencies for transitions in a system
involving the nuclear spin A can be written as
 A B1   A 
vA 
  J AX mX mA
2
X A
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 28.4
Using the same reasoning as that applied to the AX2
case, predict the NMR spectrum for an AX3 spin
system. Such a spectrum is observed for the
methylene protons in the molecule CH3-CH2-CCl3
where the coupling is to the methyl group hydrogens.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
Turning on each of the interactions in sequence
results in the following diagram:
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
The end result is a quartet with the intensity ratios
1:3:3:1. These results can be generalized to the rule
that if a 1H nucleus has n equivalent 1H neighbors, its
NMR spectral line will be split into n+1 peaks. The
relative intensity of these peaks is given by the
coefficients in the expansion of (1+x)n, the binomial
expression.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.9 Multiplet Splitting When More Than Two Spins Interact
•
•
•
•
The ability of any spectroscopic technique is
limited by the width of the peaks in frequency.
When 2 different NMR active nuclei have
characteristic frequencies closer than the width
of the peaks, it is difficult to distinguish them.
Thus the change in the magnetization vector
M with time must be considered.
Relaxation time T1 determines the rate at
which the energy absorbed from the radiofrequency field is dissipated to the surrounding.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.9 Multiplet Splitting When More Than Two Spins
Interact
•
Heisenberg uncertainty principle states that
Et
1
 1 or v 
h
t
where Δt = lifetime of the excited state,
Δv = width in frequency of the spectral line
•
In the NMR experiment, T2 is equivalent to ∆t
and determines the width of the spectral line,
∆n.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.10 Solid-State NMR
•
•
Solids do not have well-separated narrow peaks
as direct dipole–dipole coupling between spins
is not averaged to zero.
The frequency shift resulting from direct
coupling between two dipoles i and j is
vd d 
3i  j
3
ij
hr
3 cos
2
 ij  1
where rij = distance between the dipoles
θij = angle between the magnetic field direction and
the vector connecting the dipoles
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.10 Solid-State NMR
• Carry out NMR experiments on solids because:
1. Many materials are solids so option of obtaining
solution spectra is not available
2. Molecular anisotropy of the chemical shift can
be obtained from solid-state NMR spectra.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.11 NMR Imaging
•
•
•
NMR spectroscopy is important in imaging the
interior of solids.
In NMR imaging, a magnetic field gradient is
superimposed onto the constant magnetic field.
Resonance frequency of a given spin depends
not only on the identity of the spin and local
magnetic field.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
28.11 NMR Imaging
•
Below are the properties in NMR which provide
image contrast without adding foreign
substances:
1. Relaxation times T1 and T2
2. Chemical shifts
3. Flow rates
•
Chemical shift imaging is used to localize
metabolic processes and to follow signal
transmission in the brain.
Chapter 28: Nuclear Magnetic Resonance Spectroscopy
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd