Energy Gap Between 1h Bigger Than 19f Youtube

Proton (1H) is the king of the nuclei (radioactive tritium, 3H, is actually 6.7% stronger) and all other nuclei can be viewed in terms of their magnet strength ( y) relative to proton. 19F is a bit weaker than proton (94%) and 31P is about 40% of the proton frequency. Proton is about four times stronger than 13C, seven times stronger than 2H, and 10 times stronger than 15N. Of all these spin-'/2 nuclei, three have very low natural abundance: 1.11% for 13C, 0.37% for 15N, and 0.015% for 2H. This makes them difficult to observe because the signal strength in NMR is proportional to the number of NMR-active nuclei in the sample: for 13C, only one in every 100 carbon atoms is participating in the NMR experiment. However, we will see that there are advantages to having a "dilute" nucleus—one that is "sprinkled" lightly over the collection of molecules in the sample. We can improve on nature by isotopically labeling or enriching the sample either by synthesis from labeled starting materials or by biosynthesis on labeled growth media. Many compounds can be purchased with nearly 100% abundance of 13C, 2H, or 15N either at one site in the molecule or at all sites ("uniformly labeled"). This can be very costly, but the benefits often justify the cost. For example, with uniform 13 C labelling, the 13 C signal can be increased by a factor of 100, reducing the experiment time by a factor of 10,000. It should be noted that all three of these isotopes are stable, that is, they are not radioactive.

1.4.3 Chemical Shift

Because at any given field strength each nucleus has a characteristic resonant frequency, we can "tune" the radio dial to any nucleus we are interested in observing. We can think of the various NMR-active nuclei in the sample as "radio stations" that we can tune into very accurately, just as stations come into tune in a very narrow range of frequencies on an FM radio. Having chosen a "station" to listen to, what can we learn by observing a particular type of nucleus? The resonant frequency is always, always, always proportional to the magnetic field:

Vo = Y#o/2n but the exact magnetic field experienced by the nucleus may be slightly different than the external magnetic field. The nucleus is located at the center of a cloud of electrons, and we know that electrons are easily pulled away or pushed toward an atom, changing the electron density around that nucleus. Furthermore, electron clouds can begin to circulate under the influence of the laboratory field, creating their own magnetic fields, which subtract from or add to the external field. So the nucleus "feels" a slightly different field, depending on its position within a molecule (its "chemical environment"):

Where a is a "shielding constant" in units of parts per million, which reflects the extent to which the electron cloud around the nucleus "shields" it from the external magnetic field. These differences, which we call "chemical shifts" are really tiny: for a 1H nucleus the "spread" of resonant frequencies around the fundamental frequency is only about 10 ppm. That means that on a 500 MHz NMR instrument, the protons in a molecule might have a range of resonant frequencies between 499.9975 and 500.0025 MHz (0.0025 MHz is 5 ppm of 500 MHz), depending on their location within the molecule. Thus we tune in to a "station" (499.9975-500.0025 MHz) and study the tiny variations (chemical shifts) in resonant frequency to learn something about the chemical structure of the molecule. In this way, physics (and radio electronics) comes to the aid of chemistry in helping us determine a molecule's structure. An NMR spectrum is just a graph of intensity versus frequency for the narrow range of frequencies corresponding to the particular nucleus we are interested in. Each "peak" in this graph corresponds to a particular environment within the molecule, such as a particular hydrogen atom position in an organic structure. When each position in a molecule has a different chemical shift, we can "talk" to these atoms individually in NMR experiments, looking around at the local environment from the point of view of one atom in the structure at a time.

1.4.4 The Energy Diagram

If we consider the energy of a nucleus as it interacts with the external magnetic field, we see that there are two energy levels for a spin-'/2 nucleus. The "aligned" state (or a state) has the nuclear magnet aligned with the laboratory field, giving it a lower energy (more stable) state (Fig. 1.2). The "disaligned" state (or j state) is aligned opposite to the external field, resulting in a higher energy. The energy "gap" between these two levels is:

AE = hvo = hyBo/2n where h is Planck's constant and vo is the Larmor ("resonant") frequency. This relationship between the energy gap between two quantum states and the frequency of electromagnetic radiation ("photons"), which can excite a particle from the lower energy level to the higher one, is fundamental to all forms of spectroscopy. The Larmor frequency, vo, is the same as the rate of precession of the spinning nucleus in the classical model (Fig. 1.1). Note that the size of the energy gap is proportional to the strength of the nuclear magnet (y) and also to the strength of the laboratory magnetic field (Bo). Much effort and expense is put into getting the largest possible energy gap, as we design and build bigger and stronger superconducting magnets for NMR. We will see that a larger energy gap results in a more sensitive NMR experiment and better separation of the resonant frequencies of like nuclei in different chemical environments.

1.4.5 Populations

In an NMR sample there are a very large number of identical spins, a number approaching Avogadro's number. Even though there may be different types of spins (*H, 13C, 15N, etc.)

within a molecule and different environments (Hi, H2, H3, etc.) within a molecule for each type of spin, we can view each molecule in a sample of a pure compound as identical and experiencing the same magnetic field. This is because the magnetic field has a very high degree of spatial homogeneity (on the order of parts per billion variation in Bo) and each molecule is tumbling very rapidly and has no preferred orientation in the magnetic field. Let us focus on one type of nucleus (1H) and one position within the molecule (H2). If there are Nmolecules in the sample (e.g., for a 1 mM sample, N = 3 x 1017), then we can talk about the N 1H nuclei at position H2 in the molecule: each one will be either aligned with the Bo field (lower energy or a state) or disaligned with the Bo field (higher energy or j state). At thermal equilibrium, there will be a tendency for the spins to prefer the lower energy state, but because the energy difference (AE = hyBo/2n, where h is Planck's constant) is small compared to the average energy available at room temperature (kT), the populations are very nearly equal in the a and j states. The population of the more stable a state is N/2 + 8, and the population of the less stable j state is N/2 - 8, where 8 is a very small number roughly equal to NAE/4kT.

For example, at 7.05 T magnetic field (a 300 MHz NMR instrument) and 25 °C, the population difference for protons is 0.00064% of the number of nuclei N. This equilibrium population difference is a constant throughout the NMR experiment and, as we perturb the equilibrium, the spins will always try to return to this equilibrium population distribution. Because the measureable signal from a nucleus in the j state is exactly cancelled by the signal from a nucleus in the a state, it is this population difference that is the only material we have to work with and to detect in the NMR experiment. Because the difference is so small, the sensitivity of NMR is in many orders of magnitude lower than all other analytical techniques; so low, in fact, that NMR is not considered a branch of "analytical chemistry" but rather a tool used by organic chemists and biologists.

1.4.6 Net Magnetization at Equilibrium

At thermal equilibrium, the Boltzmann distribution determines the populations in various energy levels. For any two quantum states, the ratio of populations between the higher energy state and the lower energy state at equilibrium will always be:

where k is the microscopic gas constant, Tis the absolute temperature in kelvin (K), and AE is the difference in energy between the two states—the "energy gap." We can think of kT as the average amount of total energy that a molecule has—analogous to the amount of money the average person is carrying in his or her pocket. AE is analogous to the price difference between a hamburger and a cheeseburger. If the amount of money the average person has (kT) is very small and the price difference (AE) is large, then nearly everyone will take the hamburger. But if the average person is carrying around a lot of money and the price difference is very small, there will be only a very slight preference for the hamburger. Just how big is kT compared to the energy difference in NMR? At 25 °C (298 K), kT is equal to 2478 J/mol. For a proton (1H) in a 7.05 T magnetic field (vo = 300 MHz), the energy gap is:

So the energy gap is very, very small compared to the average energy that a molecule has at room temperature. Another way of saying this is that AE/kT is a number much, much less than 1. The exponential function can be simplified by approximation if the argument is a very small number compared to 1:

We can now simplify the Boltzmann equation:

The population difference, Pa — Pp, is the most interesting thing for us because the magnetism of every "up" nuclear magnet cancels the magnetism of every "down" nuclear magnet, and it is only the difference in population that results in a "net magnetization" of the sample.

Pp/Pa = 1 — AE/kT; 1 — Pp/Pa = AE/kT; Pa/Pa — Pp/Pa = AE/kT

(Pa — Pp)/Pa = AE/kT; Pa — Pp = PaAE/kT = NAE/2kT

The last equality is obtained by substituting N/2 for Pa because both Pa and Pp are very close to half the total number of spins in the sample. Finally, substituting hyBo for AE we obtain:

Thus the population difference is proportional to the total number of spins in the sample and to the strength of the nuclear magnet (y) and inversely proportional to the absolute temperature (T). If we add together all of the nuclear magnets, each spin in the p state cancels one in the a state and we end up with only Pa — Pp spins in the a state, aligned with the magnetic field. These add together to give a net magnetization, which is equal to the net number of spins pointing "up" times the magnet strength of each individual spin, y. The magnitude of this net magnetization is called Mo,

The net magnetization of the sample at equilibrium is proportional to the amount of sample (N), the square of the nuclear magnet strength (y2), and the field strength (Bo), and inversely proportional to the absolute temperature (T).

1.4.7 Absorption of Radio Frequency Energy

In order to measure the resonant frequency of each nucleus within a molecule, we need to have some way of getting the nuclei to absorb or emit RF energy. If we subject the sample to an oscillating magnetic field provided by a coil (the equivalent of a radio transmitter's antenna), a spin in the lower energy state can be "bumped" into the higher energy state if the radio frequency is exactly equal to the Larmor frequency, vo. Formally, one spin jumps up to the higher energy level and one "photon" of electromagnetic radiation (energy hvo) is absorbed. Unfortunately, there is another process that is equally likely, called "stimulated emission," in which one photon (hvo) is absorbed by a spin in the upper (ft) energy state, kicking it down to the lower state with the emission of two photons. So as long as our RF energy is applied at the resonant frequency, spins are jumping up (absorption of one photon) and down (emission of one photon) constantly. The rate of these processes is proportional to the population of spins in each of the two states: absorption occurs at a rate proportional to the number of spins in the sample that are in the lower energy state, and emission occurs at a rate proportional to the number of spins in the upper energy state.

In order to understand the net behavior of this system, we have to think about the populations (number of spins in the sample) in each of the two states. At thermal equilibrium, there will be a slight preference for the lower energy state according to the Boltzmann distribution. For now we will only think about this preference qualitatively; it turns out to be very small indeed at room temperature—a population difference of about 1 in 106 spins. But as long as there are more spins in the lower energy state, we will see a net absorption of RF energy when we turn on an RF energy source at the Larmor frequency. As there is a net migration of spins from the lower energy state to the upper energy state (absorption exceeds emission), we will quickly see the two populations become equal:

where N is the total number of identical spins in the sample and 5 is a very small fraction of this number. With the equal populations, the rate of absorption equals the rate of emission and we no longer have any net absorption of RF energy. This condition is called saturation. If there were no other way for the spins to drop down to the lower energy state, this would be the end of the NMR experiment: a quick burst of absorption and then nothing. But there is a pathway to reestablish the Boltzmann distribution: spins can drop down from the higher energy state to the lower energy state with the energy appearing as thermal energy (molecular motion) instead of in the form of a photon. This process is called relaxation and is an extremely important phenomenon that will be discussed in detail. If our source of RF energy is weak enough, we can reach a steady state in which the absorption of RF energy is exactly equal to the rate of relaxation. The amount of energy absorbed is very small, and the heating of the sample resulting from relaxation is not even noticeable.

1.4.8 A Continuous Wave Spectrometer

So now we have a way to construct a simple NMR spectrometer: We have a weak source of RF energy (a transmitter) and we gradually decrease the frequency, with the magnetic field strength (Bo) remaining constant. A detector in the transmitter circuit monitors the amount of RF energy absorbed, and this signal is applied to a pen, which moves up and down. The pen moves from left to right across the paper as the frequency is gradually decreased, and when we reach the Larmor frequency (vo), there is a net absorption of energy and the pen moves up. As we pass through the Larmor frequency, the resonance condition is no longer met and absorption stops, so the pen moves back down. The spins never reach the saturated state because the RF energy level is very low, and after passing through the resonance condition they quickly reestablish the equilibrium energy difference through the process of relaxation. The result is an NMR spectrum: a graph of absorption of RF energy (vertical axis) versus frequency (horizontal axis). The range of frequencies "scanned" by the spectrometer is very narrow—for example, from 500.0025 MHz down to 499.9975 MHz, and the position of the absorption peak on the spectrum (its "chemical shift") tells us something about the chemical environment of the spin within the molecule. This technique is called "continuous wave" (CW) NMR because the radio frequency energy is applied continuously as the frequency is gradually varied. The first commercial NMR spectrometers (e.g., the Varian T-60 operating at 60 MHz) were all continuous wave. In the earliest CW instruments, the radio frequency was held constant and the field (Bo) was gradually changed ("swept"). This gave the same result because the absorption of RF energy led to a peak when the field reached a value that satisfied the resonance condition (vo = yBo/2n). The left-hand side of the spectrum was called "low field" and the right-hand side was called "high field." The chemical shift scale was in ppm units of t (t = 10 - 5), which increased from left to right. To this day we use the terms "downfield" and "upfield" to refer to the left-hand and right-hand side of the spectrum, respectively, and the frequency scale runs from right-hand to left-hand side, contrary to all other graphical scales. This is because a higher frequency in the frequency-swept spectrum corresponds to a lower field ("downfield") in the old field-swept instruments.

1.4.9 Pulsed Fourier transform NMR

All modern spectrometers now use a "pulsed Fourier transform" method, which is much faster and allows repeating the experiment many times and summing the resulting data to increase sensitivity. A very brief pulse of high-power radio frequency energy is used to excite all of the nuclei in the sample of a given type (e.g., 1H). Immediately after the pulse is over, the nuclei are organized in such a way that their precessing magnets sum together to form a net magnetization of the sample, which rotates at the Larmor frequency. The coil that was used to transmit RF is now used as a receiver, and a signal is observed at the precise Larmor frequency, vo. This signal, which oscillates in time at the Larmor frequency, is recorded by a computer and a mathematical calculation called the Fourier transform converts it to a spectrum, a graph of intensity versus frequency. Essentially the Fourier transform measures the frequency of oscillation of the signal. If there are a number of slightly different Larmor frequencies, corresponding to different positions within a molecule, their signals add together to give the recorded signal, and the Fourier transform can sort out all the signals into a spectrum with many peaks at different frequencies. The whole experiment (pulse followed by recording the "echo" signal) takes only a few seconds and can be repeated as many times as desired, summing the data to get a stronger signal.

1.4.10 Sensitivity of the NMR Experiment

Although techniques like mass spectrometry require only nanograms (10-9 gram) of sample, NMR requires milligrams (10-3 gram) of a typical organic molecule. This insensitivity stems primarily from the fact that only the difference in population at thermal equilibrium is active in the experiment. That means that only approximately one spin in 106 is actually detected. We saw this in the CW experiment, where absorption of RF energy is almost completely cancelled by stimulated emission. Another important aspect is the relative sensitivity of different nuclei: because of the inherent differences between different nuclei in the strength of the nuclear magnet (y ), the signal strength received can be very much weaker than a proton signal. There are three ways in which y affects the sensitivity of the experiment ("the three gammas"):

1. The population difference at thermal equilibrium is proportional to the energy gap, which is in turn proportional to yBo, and inversely proportional to absolute temperature. This population difference is the only thing we can observe by NMR.

2. As the nuclear magnet precesses, it induces a signal in the receiver coil. The amplitude of this signal is proportional to the strength of the rotating magnet, which is the magnetogyric ratio y.

3. The rate at which the nuclear magnet precesses (vo) is also proportional to yBo. As with any electrical generator, if you turn the crank faster you get a higher voltage out of the generator.

Factors 1 and 2 taken together give the net magnetization at equilibrium, Mo, so we can also think of a large magnet of strength Mo rotating in the x-y plane when we consider the final factor, the rate of rotation (3). Either way we can say that the amplitude of the NMR signal (sensitivity) for a spin-/ nucleus is proportional to:

where N is the number of identical spins in the sample. This tells us that sensitivity depends on the third power of y as well as the square of Bo. So it is worth a lot of money to build larger and more powerful magnets, and we will pay a big price in sensitivity to study nuclei with relatively small y. Consider some of the most useful nuclei for organic chemistry and biological research:

0 0

Post a comment