How is the edge length related to the length of the rays radius? We also saw an example of inferring a molecular property a bond length from spectroscopic data. In fact, spectroscopy is the principal experimental approach allowing us to study molecules. Our program in this chapter will be to learn about a selected few spectroscopic techniques and what they tell us about molecules.
Instructors in your other courses in chemistry and biochemistry will introduce you to additional techniques as you need them. What I hope you will take away from this chapter is that there are a few basic principles that govern spectroscopy which, once mastered, can be applied to all spectroscopy experiments. Each of these motions has energy associated with it. These motions are not completely independent, but it turns out that we can treat them as if they are for many purposes.
Each of these types of energy has its own energy levels and, as it turns out, these energy levels typically have very distinctive spacings, and so each except for translational energy, as discussed below is associated with its own spectroscopy. These might be of interest in a gas. We usually study the spectroscopy of gases in cylindrical sample cells that are at least 10 cm long. Say we have a cell that is this long, and consider the translational energy levels of a nitrogen molecule in this cell, which will be given by the particle-in-a-box equation 2. We could generalize this argument to three dimensions, but we would conclude much the same thing.
Some simple arguments from statistical mechanics show that the average kinetic energy of a gas molecule in one dimension is 1 2 k B T , where k B is Boltzmanns constant to which we will return shortly. At, say, C, the average kinetic energy works out to 2. Suppose that we have a molecule with this much energy.
What would be its quantum number, n? The molar mass of a nitrogen molecule is Dont forget that the kilogram is the SI unit of mass, not the gram.
This is an extraordinarily small difference in energy, both relative to the average energy of nitrogen molecules, and in terms of the energies of photons Figure 2. Accordingly, no practical spectroscopy experiment could be designed to study the translational energy levels of molecules. For the other types of molecular energy, the energy levels are in fact sufciently spaced for spectroscopy.
For example, if the photons in a spectroscopy experiment have enough energy to cause a vibrational transition, they will normally cause rotational transitions at the same time. Thus you should think of the correspondences in the boxes above as establishing how much energy is required to cause a certain type of transition, with the understanding that transitions of lower energy can occur at the same time. One of the issues that will be important to our understanding of molecular spectroscopy will be the proportion of molecules that can be expected to occupy a given energy level prior to excitation with a light source.
The answer to this question is provided by the Boltzmann distribution.
Deriving the Boltzmann distribution would take more time and space than we can devote to this topic, so well just have to take it as a given. It turns out that there are two factors to consider: the energy of a level, and its degeneracy. The degeneracy is the number of different sets of quantum numbers that give the same energy.
You would have run into this concept in your introductory chemistry course when the energy levels of atoms were discussed, where you would have learned that there are, for example, three 2p orbitals, and that all three are equal in energy in free atoms. The degeneracy of, for example, a boron atom in its ground state 1s 2 2s 2 2p 1 is therefore three. Boltzmanns constant turns out to just be the ideal gas constant expressed on a per-molecule instead of a per-mole basis.
The constant q appearing in the Boltzmann distribution is the normalization constant for the distribution. In other words, it is chosen so that. This may not be immediately obvious. However, if you look back at Equation 2. You can imagine that the three-dimensional generalization of this problem might involve the volume V, and you would be right: in general, q depends on T and V. The key thing to notice here is the nal equation; from basic statistical theory, we know that. It will turn out that all thermodynamic quantities can be calculated from the partition function.
Example 3. At very low temperatures, P E 2 0 and only level 1 is occupied. This would generalize to systems with more energy levels; as the temperature is lowered toward zero, higher-energy states become less and less probable, until we reach a situation where only the ground state is appreciably occupied. At high temperatures, on the other hand, we see that the probabilities approach the values P1: SFK Trim: mm mm Top: Note the logarithmic scale of the temperature axis.
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These probabilities are in direct proportion to the degeneracies. Again, this generalizes to an arbitrary number of states: at very high temperatures, all states are equally probable, so the probability of occupying an energy level is proportional to the number of states of which the energy level consists, i. For convenience, imagine that the ground state has zero energy. We can always arrange for this to be true because the place where we put our zero on an energy scale is arbitrary. Only differences in energy matter in practice.
The partition function therefore contains a term for this energy level approximately equal to g i , the degeneracy of the level. On the other hand, if E i k B T , the Boltzmann factor is approximately 0. If we exclude the case where some of the energies are similar to k B T , we get the following: q. If we consider the case where some of the energies are similar to k B T , we can interpret the partition function similarly, but the number has to be interpreted as a rough indication of the number of accessible states with energies similar to or less than k B T rather than as an actual count.
If you want an example, look back at the molecular partition function 3. In between, we get values between 1 and 4, which tells us that there is one low-lying state the ground state , as well as some states for which E i k B T.
The Boltzmann distribution is often used to compare the populations of two energy levels. From Equation 3. Lets apply this formula to the relative populations of typical ground and excited states of molecules at room temperature. UV is an abbrevia- tion for ultraviolet. Lets take a wavelength in that range, say nm, and see what the corresponding energy level spacing implies about the relative populations of the ground and excited state. Note that assuming a higher degeneracy for the excited state than for the ground state would not have altered this conclusion, given the tiny value of the Boltzmann factor.
We can repeat this calculation for typical vibrational infrared, IR and rotational microwave energy level spacings. The ubiquitous appearance of such equations in physical chemistry is not a coincidence. It derives from the Boltzmann distribution which governs the statistical distribution of molecular energies.
Exercise group 3. For a proton in a 7. What percentage of proton nuclear spins would you expect to be in the higher energy opposing the eld state at room temperature C at m the altitude of my house in Lethbridge? At m La Paz, Bolivia? Note: Dont forget that the SI unit of mass is the kg. The dark circles represent hydrophobic residues, while the light circles represent hydrophilic residues.
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Assume a uniform temperature of 0. A contact between a hydrophobic residue and a water molecule reduces the opportunities for the water to form hydrogen bonds or other energetically favorable contacts. There is therefore an energetic cost to a water hydrophobic contact. Similarly, there is an energetic cost to a hydrophobichydrophilic contact because the hydrophilic group loses opportunities for more favorable contacts with other polar groups including the solvent. Flipping these statement on their heads, hydrophobichydrophobic contacts in a protein, which reduce the number of hydrophobic water and hydrophobichydrophilic contacts, are energetically favorable.
One very simple model for protein folding treats residues as beads connected by bonds of unit length placed on a simple lattice. We represent hydrophobic residues by dark circles, while light circles represent hydrophilic residues. The bonds between residues are placed along edges of the lattice, and the amino acid residues are placed on lattice points.
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We only count 1 D. Grifn, Aerobiologia 24, 19 Dill et al. You should be able to convince yourself that there are just ve distinct conformations of the peptide on the lattice, illustrated in Figure 3. You might object that conformations 2b and 2c look the same.
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Because the two ends of a protein are different there is an amino group at one end of the protein and a carboxy group at the other we can tell residues A and D apart, even if they are derived from the same amino acid, and so 2b bent at B, which is nearest to A and 2c bent at C, which is nearest to D are actually different.
Suppose that the energetic payoff for a hydrophobichydrophobic contact is. The latter conformations are unfolded or, as they say in biochemistry, denatured. Note that these curves generally look best when plotted against a logarithmic temperature axis. Emission methods: In emission spectroscopy, we measure the amount of light emitted by a sample, typically after the sample has been energized in some way with photons, heat, electricity, etc.
Scattering methods: Scattering experiments involve the study of photons that have collided with molecules in the sample and been deected, possibly with some loss or gain of energy. In any given energy range, we can carry out all three types of experiments. Each type of spectroscopy gives us different information about the material studied. The light produced by a source e.
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Most commonly, a monochromator would be a diffraction grating, and different wavelengths are selected by rotating the grating. The monochro- matic light passes through the sample, and the transmitted light is detected. If the sample absorbs light at a given wavelength, then fewer photons reach the detector than would otherwise be the case. A spectrometer of this design would be called a single-beam spec- trometer.
Single-beam spectrometers are simple, so they are inexpensive and relatively sturdy. There are two basic types of experiments that can be carried out by a single-beam absorption spectrometer. The simplest type of experiment involves just taking readings at a xed wavelength. This can be useful in quantifying the amount of substance in a solution, as we will see shortly.
The second type of experiment is the measurement of the light intensity as the wavelength is varied. A graph of intensity vs. The problem with single-beam spectrometers is that changes in the signal observed at the detector can have a number of causes.