Physical Chemistry 2 nd Edition

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

Outline • Intrinsic Nuclear Angular Momentum and Magnetic Moment • The Energy of Nuclei of Nonzero Nuclear Spin in a Magnetic Field • The Chemical Shift for an Isolated Atom • The Chemical Shift for an Atom Embedded in a Molecule • Electronegativity of Neighboring Groups and Chemical Shifts • Magnetic Fields of Neighboring Groups and Chemical Shifts • Multiplet Splitting of NMR Peaks Arises through Spin–Spin Coupling • Multiplet Splitting When More Than Two Spins Interact • Peak Widths in NMR Spectroscopy • Solid-State NMR • NMR Imaging

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 where I = nuclear angular momentum e = unit of elementary nuclear charge (1.6×10-19 C) mproton = mass of a proton

28.1 Intrinsic Nuclear Angular Momentum and Magnetic Moment • βNis the nuclear magneton and nuclear factor gN is characteristic of a particular nucleus.

28.2 The Energy of Nuclei of Nonzero Nuclear Spin in a Magnetic Field • Larmor frequency,v, states that

28.2 The Energy of Nuclei of Nonzero Nuclear Spin in a Magnetic Field • In NMR spectroscopya transition must be induced betweentwo different energy levels so that the absorption/emission of the electromagnetic energycan be detected.

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.

Solution a. The two energies are given by b. The energy difference is given by This is in the range of frequencies called radio frequencies.

Solution 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.

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 where µ0= vacuum permeability µ0 = induced magnetic moment θ and r = spherical coordinates

28.3 The Chemical Shift for an Isolated Atom • The total field at thenucleus is given by the sum of the external and induced fields,

28.4 The Chemical Shift for an Atom Embedded in a Molecule • The frequency shift for an atom depends linearly on theshielding constant, σ. • This makes NMR a sensitive probe around a nucleus withnonzero nuclear spin. • Two factors responsible for chemical shift are: • Electronegativity of the neighboring group • Induced magnetic field of the neighboringgroup

28.5 Electronegativity of Neighboring Groups and Chemical Shifts • The chemical shifts for different classes of molecules arestrongly correlated with their electron-withdrawing ability.

28.6 Magnetic Fields of Neighboring Groups and Chemical Shifts • The magnetic field at a 1H nucleus is a superposition of the external field.

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. • 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.

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.

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

28.7 Multiplet Splitting of NMR Peaks Arises through Spin–Spin Coupling • Solving Schrödinger equation for the corresponding eigenvalues gives assume σ1>σ2

Example 28.2 Show that the total nuclear energy eigenvalue for the wave function is Solution:

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.

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:

28.7 Multiplet Splitting of NMR Peaks Arises through Spin–Spin Coupling • In general the energy correction is • To solve for eigenfunctions, we have

Example 28.3 Show that the energy correction to

Solution We evaluate

Solution 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.

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

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.

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

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.

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 timeT1 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 Spins 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.

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 where rij = distance between the dipoles θij = angle between the magnetic field direction and the vector connecting the dipoles

28.10 Solid-State NMR • Carry out NMR experiments on solids because: • Many materials are solids so option of obtaining solution spectra is not available • Molecular anisotropy of the chemical shift can be obtained from solid-state NMR spectra.

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.

28.11 NMR Imaging • Below are the properties in NMR which provide image contrast without adding foreign substances: • Relaxation times T1 and T2 • Chemical shifts • Flow rates • Chemical shift imaging is used to localize metabolic processes and to follow signal transmission in the brain.

Physical Chemistry 2 nd Edition