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Transcript
Chem 3500 Fall 2006 P. Hogan Magnetic Resonance NMR spectroscopy is used to study the magnetic properties of nuclei. Only nuclei that possess a spin have magnetic properties and have a quantum spin number, I. During this course we will study two different nuclei with a quantum spin number I = ½, the 1 H nucleus and the 13 C nucleus. A charged nucleus that has a spin will posses a magnetic moment µ. Equation 1 defines the relationship between the quantum spin number and the magnetic moment. µ = ?hI/2p (eq. 1) In the above relationship, h is Planck’s constant (h = 6.2608 X 10-34), and ? is the gyromagnetic ratio of the nucleus has units of 107 T-1 s -1 where T is units Tesla. The gyromagnetic ratio of 1H is 26.75, and the gyromagnetic ratio of 13C is 6.73. The quantum spin number I also determines the number of possible spin states that a nucleus can possess. There are 2I + 1 spin states . Thus the 1 H and 13C nuclei each have two spin states corresponding to I = ½ and I = - ½. In the absence of external magnetic field, these spin states are equal in energy and equally populated. If a sample of nuclei are subjected to a magnetic field B0 (units Tesla) along a direction designated as the z axis, the energies of the nuclei with two different spin states will no longer be equal in energy. The will be a slightly higher population of nuclei aligned with the direction of the magnetic field B0 (+ z) than against (-z) . The force of B0 on the magnetic moment causes the magnetic moment vector to move in a circular orbit about the z axis (coincidental with B0), a motion called precession. The orbits of the two populations are shown below in figure 1. The angle and magnitude of the vector quantity µ is a function of the atomic spin number. z B0 z µ Iz = +1/2 Iz = -1/2 µ Figure 1. The precession of the magnetic moment with a +z component is shown on the right while the precession of the magnetic moment with a – z component is shown on the left. Precession of the magnetic moment around B0 occurs with an angular frequency ? 0, also called the Larmor frequency. The Larmor frequency can be expressed in terms of linear frequency ?0 (eq. 2) The energy difference between the lower energy spin state (Iz = +1/2) and the higher energy spin state (Iz = -1/2) is given by Planck’s relationship (eq. 3) ? 0 = 2 p ?0 (eq. 2) ? E = h?0 (eq. 3) The magnitude of the angular frequency ? 0, is proportional to B0. The constant of proportionality between ? 0 and B0 is ?, the gyromagnetic ration of the nucleus. This relationship is given in equation 4. ? 0= ?B0 (eq. 4) This quantity can also be expressed in terms of linear frequency (eq. 5) ?0 = ?B0/2 p (eq. 5) Thus the energy difference between the two spin states can be written in terms of the gyromagnetic ratio and a magnetic field of value B0 (eq. 6). ? E = ? h B0/2 p (eq. 6) From equation 5, we can see that the frequency of precession is directly proportional to the magnitude of the magnetic field B0, and consequently (see eq. 6) the energy difference between the two spin states is also directly proportional to the magnitude of the magnetic field B0. An NMR experiment is conducted by application of a circularly polarized radiofrequency field B1 into the xy plane. The radiofrequency field B1 will consist of both an electric component and magnetic component. The magnetic component of B1 will interact with bulk magnetization of the sample. Bulk magnetization is illustrated in figure 2. z z M y x y x Figure 2. The picture on the right is a representation of the magnetic mo ments of a collection of nuclei in the absence of a magnetic field with magnitude B0 . There is no net magnetization within the samp le. In the picture on the right, the collection of nuclei have been exposed to a field B0 along a direction designated along the z axis. More nuclei have magnetic moments possessing a +z component than nuclei possessing magnetic moments with a –z component. Thus, there is a net bulk magnetization M of the sample in the +z direction. Figure 3 illustrates a vector representation of the circularly polarized radiofrequency field B1. The vector representing the direction and magnitude of B1 makes a circular orbit around the z axis within the xy plane. If the angular frequency of B1 is equal to the Larmor frequency of the nuclei, the precessing nuclei will be under the influence of a constant magnetic field and energy will be absorbed by the nuclei. This condition is called resonance. The bulk magnetization of the sample will be tipped off the z axis and begin to move toward the xy plane. As long as the field B1 is applied the magnetization M will rotate around B1. If we choose the laboratory frame of reference to be rotating with B1 at the Larmor frequency, the vector representing B1 will appear to remain stationary along the x axis. The magnetization vector M will now precess around B1 within the yz plane with the frequency ?B1/2 p (figure 3) . z z ?=?B1 /2p M M ?0 y y B1 B1 x ? = ?0 x Figure 3. The depiction on the right shows the direction and frequency of the circularly polarized radiofrequency field B1 . The vector representing the direction and magnitude of B1 orbits the z-axis. If the angular frequency of this orbit is equal to the Larmor frequency of precession, the magnetization vector will be tipped away from the z-axis. The depiction on the right shows what an observer in a frame rotating at the Larmor frequency would observe. The field B1 appears to be stationary, while M precesses around B1 . If B1 is applied for a duration p ?B1/2 (25% of the time needed for M to make a full rotation around B1, we say we have applied a 90° pulse. This is illustrated in figure 4. In a laboratory frame rotating at the Larmor frequency the magnetization vector M appears to tip and become aligned with the y axis. In the stationary laboratory frame of reference the bulk magnetization M, now appears to precess in the xy plane (figure 4). z z z M M B1 x y B1 x y y M x M ? = ?0 Figure 4. The first picture is in from the perspective of the rotating frame as the pulse of B1 begins. The middle picture is from the perspective of the rotating frame after the 90° pulse. The third picture is from the perspective of the stationary frame after the 90° pulse. A detector placed along the y axis sees an oscillating magnetic field at the Larmor frequency. Thus the magnitude of the signal varies as a cosine function with time. If the B1 field is turned off after the 90° pulse, and the magnetization in along the y-axis is measured over time at the resonance frequency an exponential decay of y magnetization (My ) over time would be observed (figure 5). Any process that returns the x and y magnetizations to their equilibrium values of zero is called spin-spin relaxation and is generally a first order process with time constant T2. Eventually the magnetization vector M returns to it’s equilibrium status in which M consists of only a +z component. Any process that returns the z magnetization to it’s equilibrium value of an excess of +1/2 spins is called spin-lattice relaxation and is generally a first order process with time constant T1. z z M y y x ? = ?0 M x Figure 5. The depiction on the left shows the path of the bulk magnetization vector M after B1 is turned off. Eventually M returns to it’s equilibrium status (right depiction) in which there are no x and y components to the magnetization only a small +z component. If the magnitude of the My is plotted vs. time, the result is called an FID. (free induction decay, figure 6). An FID can be manipulated mathematically by using a Fourier transformation which transforms the data into a plot of magnitude of My vs. frequency. My t Figure 6. An FID of a system containing a single type of nucleus. The Chemical Shift Nuclei in different chemical environments have different resonance frequencies. These resonance frequencies are referred to as chemical shifts. The resonance frequency depends greatly upon the electronic environment of the nucleus. The electrons surrounding a given create a local magnetic field that attenuates the applied magnetic field B0. The relative populations of the two spin states depend on the energy difference between them. The population ratio of the lower to higher energy spin states is given by eq. 7. [Iz =+1/2]/[Iz =-1/2] = e-(?E/kT) (eq. 7) In eq. 7, ? E is the energy difference between two spin states, k is Boltzmann’s constant (1.38065 X 10-23 J/K) and T is temperature in units Kelvin. Recall from eq. 6 that the enrgy difference between the two spin states is proportional to the applied magnetic field. ? E = ? h B0/2 p (eq. 6) This equation can be applied to a bare nucleus, however when a nucleus is surrounded by electrons the actual magnetic field that the nucleus feels is counteracted in part by the electrons around the nucleus. Thus the value of actual magnetic field felt by the nucleus can be written as the following: Beffective = B0 - Blocal (eq. 8) In the above equation Blocal is the magnetic field created by electrons surrounding the nucleus and Beffective is the actual magnetic field seen by a nucleus. Equation 6 can now be rewritten to reflect the actual energy difference between the spin states based upon the actual magnetic field Beffective seen by the nucleus. ? E = ? h Beffective/2 p (eq. 9) Equation 4 defined the relationship between the Larmor frequency of a nucleus and the applied magnetic field. This equation can now be rewritten as the following. ? effective= ?Beffective (eq. 10) The new Larmor frequency ? effective is the observed resonance frequency and is different for every nucleus in a chemically different environment. When a nucleus is surrounded by an area of high electron density, the Blocal term in eq. 8 is large. This leads to a lower resonance frequency as seen by eq. 10. When a nucleus is an area of high electron density, the nucleus is said to be shielded from the applied magnetic field. When a nucleus is surrounded by an area of low electron density, the Blocal term in eq. 8 is small. This leads to a higher resonance frequency as seen by eq. 10. When a nucleus is an area of low electron density, the nucleus is said to be deshielded from the applied magnetic field. In order to standardize and simplify the reported chemical shifts of a given resonance frequency of a nucleus, an internal standard in an NMR experiment will typically be chosen. The standard of choice for 1H and 13C NMR experiments is tetramethylsilane (CH3 )4Si, often abbreviated as TMS. Silicon is much less electronegative than carbon, and therefore both the carbons and protons are shielded to a much larger extent that carbons and protons of most organic compounds. For this reason the resonance frequency of the protons and carbons of TMS have been arbitrarily assigned “0”. The chemical shift, most often abbreviated as the symbol d, is the difference between the magnitude of the resonance frequency of a measured nucleus and that of TMS. The resonance frequencies also called chemical shifts of nuclei are typically not reported in units Hz. Chemical shifts instead are reported in parts per million (ppm), and are obtained by measuring the difference in frequencies (in units Hz) between a given nuclei and the relevant nucleus of TMS and dividing the value of that number by the operating frequency of the spectrometer (in units MHz). d (ppm) = ? observed (Hz) – ?TMS(Hz)/operating frequency of the spectrometer (MHz) The chemical shift of a given measured nucleus is independent of the operating frequency of the spectrometer. The resonance frequencies recorded in NMR experiment will typically be plotted in ppm. The signal corresponding to TMS will be placed at “0” at the right hand of the spectrum. The ppm numbers will increase from right to left. Signals to the left of the TMS signal are said to be downfield of TMS. Almost all measured nuclei will have resonance frequencies downfield of The resonance frequencies of TMS. In general the right hand side of the spectrum is referred to as the upfield portion of the spectrum and the left hand side is referred to as the downfield part of the spectrum. Essential features of an 1H NMR spectrum The chemical shift of alkanes and functionalized alkanes (1H NMR) The extent to which a given proton nucleus is shielded from the applied magnetic field B0 is dictated by the nature of the carbon to which the proton is attached and nature of any other substituents bonded to the carbon atom. As we have seen, due to the electropositive nature of the silicon atom, the protons of tetramethylsilane (CH3 )4Si are shielded from the applied magnetic field more than most other protons of organic molecules. To begin our discussion of chemical shifts, we will focus first on alkanes and functionalized alkanes. The protons of the simplest alkane, methane (CH4), have a chemical shift d = 0.23. Carbon is slightly more electronegative than hydrogen, and thus there is somewhat less electron density around the hydrogen nuclei of methane that there is around the hydrogen nuclei of TMS. The hydrogen nuclei of methane are deshielded relative to those of TMS. Substitution of a carbon-hydrogen bond in methane with a carbon-carbon bond leads to further deshielding of resultant hydrogen nuclei. This can be viewed as replacina hydrogen with a relatively more electronegative group, further dishielding the protons by removing some electron density from the carbon to which they are attached. The chemical shift of ethane (CH3CH3) is 0.86. The methylene group of propane (CH3CH2CH3) is further deshielded relative to the methyl groups of ethane and has a chemical shift of 1.33. Finally the methane proton of 2-methylpropane ((CH3)3CH) are found at d = 1.56. In general the protons of simple alkanes will be found in the region 0.2-1.8 ppm. Substitution of a carbon-hydrogen bond with an electronegative atom or group will contribute to the deshielding of protons attached to the carbon under examination. The chemical shift of protons attached to the carbon of an alcohol functionality for some simple alcohols are shown below. Note that with increasing carbon substitution, further deshielding occurs. The chemical shift of the protons of ethereal carbons are similar to those of their alcohol counterparts, however note that the protons of of the ester portion of methyl acetate are deshielded relative to those of ethers or alcohols. The shifts of the methyl halides are shown below. As expected the relative electronegativity of the halide dictates the relative chemical shift of the methyl protons. Replacement of carbon hydrogen bonds with additional carbon-halogen bonds results in further deshielding of the resultant protons. Alkanes attached to sp 2 hybridized carbons are also deshielded relative to their unfunctionalized counterparts. An sp2 hybridized carbon can be considered an electronegative substituent relative to an alkane, because of the lower relative energy (more s character) of the sp2 orbitals. Some characteristic examples are shown below. Chemical Equivalence and the Number of Signals in an NMR spectrum Any nuclei that can be exchanged by a symmetry operation are said to be chemically equivalent. Within organic molecules the there are two common types of chemically equivalent sets of nuclei: homotopic and enantiotopic. Nuclei that can not be exchanged by a symmetry operation are chemically inequivalent. The number of signals in a 1 N NMR spectrum tells us how many types of chemically inequivalent protons are contained within the compound being studied. Chemically equivalent nuclei contribute equally to their respective signal in an NMR spectrum. In the 13C spectrum the number of signals will tell us how many chemically inequivalent carbon nuclei are contained within the compound being studied. Nuclei that can be exchanged by a rotational symmetry are said to be homotopic. Homotopic nuclei are chemically equivalent. Some examples of molecules containing sets of homotopic protons and associated rotational axes are shown below. Protons of methylene groups that can not be exchanged by rotational operations, but can be exchanged by a mirror plane are said to be enantiotopic. Enantiotopic protons are chemically equivalent. These protons are called enantiotopic because replacement of one proton of the pair by another atom or group produces the enantiomerthat results when the other proton is replaced by the same group. The geminal methylene protons of butane are enantiotopic. If geminal protons of a methylene group can not be exchanged by a symmetry operation, they are diasterotopic. Like the above test for enantiotopicity, replacement of one proton of the pair by another atom or group produces a diastereomer of the molecule that results when the other proton is replaced by the same group. Any geminal methylene protons in a molecule containing a chiral center are diastereotopic. Diastereotopic protons are not chemically equivalent. The diasterotopicity test is shown below on 2-butanol. Less obvious is the fact that a chiral center is not a necessarily required for geminal methylene protons to be diastereotopic. Examine the diethyl acetal of acetaldehyde. Note that in conformationally mobile systems that chemically inequivalent protons can switch positions through carbon-carbon bond rotation and contribute to the same resonance. Spin-Spin Coupling and the Coupling Constant: Simple First Order Patterns Spin-spin coupling is a phenomenon observed when spin information is transferred from one nucleus to another. In the context of organic molecules, generally three types of coupling are observed; one bond couplings such as the coupling between a proton and the carbon nucleus to which is attached, two bond couplings such as coupling between protons on the same carbon, and three bond couplings like couplings between protons of vicinal carbons. Four bond couplings can be important in the context of aromatic systems and will be considered later. When spin information is transferred between chemically inequivalent nuclei, the resultant resonances of the respective nuclei are split into multiplets. Multiplets are described according to the number of signals associated with nucleus. In order to understand why coupling will cause multiplicities with regards proton resonances, examine a hypothetical molecule in which two protons in very different chemical environments are coupled to one another. We will label one proton HA and the other proton HX (the reasons for this nomenclature will be apparent later). Each proton has a spin of ½ and can exist two spin states: those whose magnetic moments are aligned with B0 which we will designate the +½ spin state, and those whose magnetic moments are aligned against B0 which we will designate the -½ spin state. The relative population of these two spin states only differ in the parts per million, and for the purposes of this discussion will be equal. Consider that half of the HA protons in the sample have HX neighbors that posses the +½ spin state and the other half of the HA protons in the sample have HX neighbors that posses the ½ spin state. The result is that there are two magnetically and energetically different populations of HA, and therefore the signal corresponding to HA is comprised of two resonances of equal intensity reflecting the two spin states corresponding to HX. A similar analysis reveals that HX will also appear as two resonances of equal intensity reflecting the two spin states corresponding to HA. The multiplicity of HA and HX are referred to as doublets. The frequency distance between the two peaks of the doublet in Hz is the coupling constant J and is independent of the magnitude of B0. The actual chemical shifts of HA and HX will be at the middle of their multiplicities respectively. Iz (HX) = +1/2 Iz (HX) = -1/2 Iz (HA) = +1/2 JAX JAX d HA Iz (HA) = -1/2 ?? d HX In general the chemical shift of a simple first order multiplet will reside at the center of the multiplet. A multiplet will be a simple first order multiplet if the ratio of ??/J = 8, where ?? is the distance between the midpoints of the coupled multiplets in Hz. Note that ?? is dependent on the field strength of the spectrometer. The consequences are that when high magnetic field spectrometers are used, the spectra is more likely to have simple first order charcteristics. The number of peaks in the multiplet is n + 1, where n is the number of neighboring protons with the same coupling constant. In a simple first order multiplet all the neighboring protons have the same coupling constant, therefore the distance between the individual peaks of the multiplet in Hz will be the coupling constant. The intensities of the individual peaks of the multiplet are given by Pascal’s triangle. n n+1 Relative line Multiplet name intensities 0 1 1 singlet 1 2 1:1 doublet 2 3 1:2:1 triplet 3 4 1:3:3:1 quartet 4 5 1:4:6:4:1 pentet 5 6 1:5:10:10:5:1 sextet 6 7 1:6:15:20:25:6:1 septet Pople notation A spin system is defined as a set of multiplets coupled to one another but do not couple outside the spin system. Proton spin systems are often separated from one another by heteroatom bonds or quaternary carbons. In the Pople notation system, multiplets are now called sets. Each set represents a group of chemically equivalent protons and is designated by a capital letter A, B, C …..Z. If ??/J = 8, the coupled sets are weakly coupled and designated by the well separated letters AX. If ??/J < 8, The coupled sets are strongly coupled and are designated by the neighboring letters of the alphabet AB. In general weakly coupled sets can be analyzed bya first order method and strongly coupled sets can not, however if an AB system has ??/J = 3, first order analyses can often be applied without significant error. If there is more than one chemically equivalent proton contained within the set the letters are give subscripts that identify the number of protons in each set. An ethyl group (CH3CH2-) would be given the notation A3X2. If we introduce another weakly coupled set into the spin system, we use the letter M to denote another set well separated from A and X. A substituted propane (CH3CH2CH2-X) would be designated A3M2X2.