Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid Chapter 28 Nuclear Magnetic Resonance...

Physical Chemistry 2Physical Chemistry 2ndnd Edition EditionThomas Engel, Philip Reid

Chapter 28 Chapter 28 Nuclear Magnetic Resonance SpectroscopyNuclear Magnetic Resonance Spectroscopy

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Physical Chemistry 2nd EditionChapter 28: Nuclear Magnetic Resonance Spectroscopy

ObjectivesObjectives

• Applications for nuclear magnetic resonance (NMR) spectrum

• Applications for splitting of NMR peaks



OutlineOutline

1.Intrinsic Nuclear Angular Momentum and Magnetic Moment2.The Energy of Nuclei of Nonzero Nuclear Spin in a Magnetic

Field3.The Chemical Shift for an Isolated Atom4.The Chemical Shift for an Atom Embedded in a Molecule5.Electronegativity of Neighboring Groups and Chemical Shifts6.Magnetic Fields of Neighboring Groups and Chemical Shifts7.Multiplet Splitting of NMR Peaks Arises through Spin–Spin

Coupling8.Multiplet Splitting When More Than Two Spins Interact9.Peak Widths in NMR Spectroscopy10.Solid-State NMR11.NMR Imaging



28.1 28.1 Intrinsic Nuclear Angular Momentum and Intrinsic Nuclear Angular Momentum and Magnetic Magnetic Moment 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

where I = nuclear angular momentum

e = unit of elementary nuclear charge (1.6×10-

19 C) mproton = mass of a proton

hhgI

m

eg NN

protonN

11

2



28.1 Intrinsic Nuclear Angular Momentum and 28.1 Intrinsic Nuclear Angular Momentum and Magnetic Magnetic Moment Moment

• βN is the nuclear magneton and nuclear factor gN is characteristic of a particular nucleus.



28.2 The Energy of Nuclei of Nonzero Nuclear 28.2 The Energy of Nuclei of Nonzero Nuclear Spin in a Spin in a Magnetic Field Magnetic Field

• Larmor frequency,v, states that

00 2or 2

1BvBv



28.2 The Energy of Nuclei of Nonzero Nuclear 28.2 The Energy of Nuclei of Nonzero Nuclear Spin in a Spin in a Magnetic Field 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.



Example 28.1Example 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.



SolutionSolution

a. The two energies are given by

b. The energy difference is given by

This is in the range of frequencies called radio frequencies.

18

34

25

2526

1034.210626.6

1055.10

1055.11076.72

sh

Ev

JE

JBgE NN2627

0 1076.750.510051.55854.52

1

2

1



SolutionSolution

c. The relative populations of the two states are given by

From this result, we see that the populations of the two states are the same to within a few parts per million.

5

23

26

108.3999962.01

21

999962.030010381.1

1076.72expexp

kT



28.3 28.3 The Chemical Shift for an Isolated AtomThe 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

where µ0 = vacuum permeability

µ0 = induced magnetic moment

θ and r = spherical coordinates

1cos34

23

0

r

Bz



28.3 28.3 The Chemical Shift for an Isolated AtomThe Chemical Shift for an Isolated Atom

• The total field at the nucleus is given by the sum of the external and induced fields,

01 BBtotal



28.4 The Chemical Shift for an Atom Embedded 28.4 The Chemical Shift for an Atom Embedded in a in a Molecule 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:

• Electronegativity of the neighboring group • Induced magnetic field of the neighboring

group



28.5 Electronegativity of Neighboring Groups 28.5 Electronegativity of Neighboring Groups and and Chemical Shifts Chemical Shifts

• The chemical shifts for different classes of molecules are strongly correlated with their electron-withdrawing ability.



28.6 Magnetic Fields of Neighboring Groups and 28.6 Magnetic Fields of Neighboring Groups and Chemical Shifts Chemical Shifts

• The magnetic field at a 1H nucleus is a superposition of the external field.



28.6 Magnetic Fields of Neighboring Groups and 28.6 Magnetic Fields of Neighboring Groups and Chemical Shifts Chemical Shifts

• NMR signal of a solution sample is generated by the large number of molecules contained in the sampling volume.

• 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.

individualaverage



28.7 Multiplet Splitting of NMR Peaks Arises 28.7 Multiplet Splitting of NMR Peaks Arises through through Spin–Spin Coupling Spin–Spin Coupling

• In a simulated NMR spectrum for ethanol, individual peaks are split into multiplets.




• Multiplets arise as a result of spin–spin interactions among different nuclei.

• The spin energy operator for the noninteracting spins is

21

ˆ1ˆ1ˆ2010 zz IBIBH




• Solving Schrödinger equation for the corresponding eigenvalues gives

assume σ1>σ2




Show that the total nuclear energy eigenvalue for the wave function is

Solution:

212

210

2 2

Bh

E

22

210

2010

20102

212

2112

2112

21ˆ1ˆ1ˆ21

E

Bh

Bh

Bh

IBIBH zz




• The splitting between levels 2 and 3 and the energy shifts of all four levels for interacting spins emphasize the spin–spin interactions.




• For the noninteracting spin case, E2-E1=E4-E3 and E3-E1=E4-E2.

• NMR spectrum contains only two peaks corresponding to the frequencies:




• In general the energy correction is

• To solve for eigenfunctions, we have

2121**

12

2

21ˆˆ114

ddIIJh

E xxj




Show that the energy correction to

4/ is 21 1222 hJE



SolutionSolution

We evaluate

21

2

21

22

21

2

12

2

2121

2121

2121

12

2

2121212112

2

2

214

2*1*

214

2*1*

214

2*1*

4

21ˆˆ2*1*

21ˆˆ2*1*

21ˆˆ2*1*4

21ˆˆˆˆˆˆ2*1*4

ddh

ddhi

ddh

Jh

ddII

ddII

ddII

Jh

ddIIIIIIJh

E

zz

yy

xx

zzyyxx



SolutionSolution

Because of the orthogonality of the spin functions, the first two integrals are zero and

Note that because J12 has the units of s-1, hJ has the unit joule.

44

4212*1*

4

4 122

12

22

2112

22

2

JhJ

h

hddJ

h

hE




• 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

AX

AXAXAA

A mmJB

v

2

1




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.



SolutionSolution

Turning on each of the interactions in sequence results in the following diagram:



SolutionSolution

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.



28.9 Multiplet Splitting When More Than Two Spins 28.9 Multiplet Splitting When More Than Two Spins InteractInteract

• 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 radio-frequency field is dissipated to the surrounding.



28.9 Multiplet Splitting When More Than Two 28.9 Multiplet Splitting When More Than Two Spins InteractSpins Interact

• Heisenberg uncertainty principle states that

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.

t

1vor 1

h

tE



28.10 Solid-State NMR28.10 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

where rij = distance between the dipoles

θij = angle between the magnetic field direction and

the vector connecting the dipoles

1cos33 2

3 ij

ij

jidd hr

v



28.10 Solid-State NMR28.10 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.



28.11 NMR Imaging28.11 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.



28.11 NMR Imaging28.11 NMR Imaging

• Below are the properties in NMR which provide image contrast without adding foreign substances:

1. Relaxation times T1 and T2

2. Chemical shifts3. Flow rates

• Chemical shift imaging is used to localize metabolic processes and to follow signal transmission in the brain.

Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid Chapter 28 Nuclear Magnetic Resonance...

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