The Boltzmann Distribution

The quantity Slot is typically referred to as the configurational entropy. It can easily be expressed in terms of the probability P by using Stirling s approximation for large N  

There is a standard mathematical tool for solving the problem of maximizing a quantity that has to satisfy constraints, namely the method of Lagrange undetermined 

By using the condition that the sum of all the probabilities must be unity we immediately obtain  

We have now expressed liin terms of 2.2 but we still need to determine what A2 is-We can do this by relating the above equation to something known from thermodynamics. If we do so [see, for example, R.A. van Santen and J.W. Niemantsverdriet, Chemical Kinetics and Catalysis (1995), Plenum, New York, or P.W. Atkins, Physical Chemistry (1998) Oxford University Press, Oxford] we find that I2 equals 1/T and that the above quantity is simply the partition function q. For the interested reader we justify choosing 2.2 equal to 1/T in the following section. 

It is instructive to compare Eq. (22) vith the situation of a particle in a box, because this automatically yields a useful expression for the partition function of translation (Fig. 3.3). 

We take a rather simple approach to the treatment of statistical thermodynamics and the derivations that follow. A rigorous discussion requires introduction of the canonical ensemble and is beyond the scope of our immediate interest. The details can be found, for example, in textbooks on statistical mechanics [269]. The approach taken here yields all of the needed results, and is a compact introduction to the theory. 

In a gas at some temperature, molecules occupy a manifold of many possible energy levels. The Boltzmann distribution quantitatively describes the populations of molecules in the various possible energy levels at a given temperature. This is a well-known result, and is a very important link between a molecular view point of gases and a thermodynamic description. It is possible to derive the Boltzmann distribution through consideration of 

We begin by considering some straightforward results from statistics. Suppose that we have two boxes and a machine that will randomly throw marbles into one or the other. This physical system is depicted in Fig. 8.3. We label the bottom box j = 0 it has an area (which we are going to denote go) of 2 arbitrary units. The other box we label j = 1, and its area gi is taken to be 1. We would like to know the expected probability that a single throw of the marble lands in box j = 0 versus j = 1. Our intuition says that it is twice as likely the marble will land in box 0 than in box 1, and indeed that is the correct result. The likelihood is proportional to g,. 

Suppose that we put a divider in box j = 0, so that it now consists of two sub-boxes, depicted in Fig. 8.4. This time, instead of randomly throwing the marble into the boxes, we get to place the marble into any particular box (or sub-box), and we will count the number of possible unique final states that we can arrange by this procedure. The number of configurations will serve as our new weighting factor. 

Now consider the number of unique possibilities for placing two marbles in this ensemble of boxes. This time we will use one dark marble and one white marble. There turn out to be nine different ways to accomplish this task, each labeled by a roman numeral in Fig. 8.6. The set of weights for each distribution of box populations is W(2,0) = 4 (configurations 1, II, IV, V), W(l, 1) = 4 (configurations III,VI-Vni), and W(0,2) = 1 (configuration IX). 

FIGURE 4.3 Five of the many possible ways twenty coins can be distributed among ten students. Each student is represented by a circle. Thus, for example, the column on the left illustrates the case where each student gets two coins. 

Assume o students end up with no coins, n students with one coin, and so forth. Then the total number of students A (10 in this case) is  

The total number of coins C (20 in this case) serves as the constraint. The number of ways each distribution can be generated, which we will call Q, turns out to be given by what is called the multinomial expansion  

Consider, for example, the second distribution. One student gets all of the coins (no = 9, n2o = 1, and all other n = 0), giving = 10 in Equation 4.23. This makes sense because the big winner could be any one of the ten students, so there are 10 different possibilities for this distribution. On the other hand, the first distribution, which treats everyone exactly the same, gives Q = 1 in Equation 4.23 (n2 = 10, all other n, = 0). Neither distribution is likely. 

Just as with the binomial distribution, calculating factorials is tedious for large N. The binomial distribution converged to a Gaussian for large N (Equation 4.11). The most probable distribution for the multinomial expansion converges to an exponential  

Suppose now that we have an ensemble of N non-interacting particles in a thermally insulated enclosure of constant volume. This statement means that the number of particles, the internal energy and the volume are constant and so we are dealing with a microcanonical ensemble. Suppose that each of the particles has quantum states with energies given by i, 2. and that, at equilibrium there are Ni particles in quantum state Su particles in quantum state 2, and so on. 

According to Boltzmann s law, the average fraction of particles in quantum state i with energy with energy e, is 

The sum in the denominator relates to the quantum states. The formula is often written in terms of energy levels rather than quantum states in the case that some of the energy levels are degenerate, with degeneracy factors gi then the formula can be modified to refer to energy-level populations directly  

The numbers iVj and N- are only equal if there are no degeneracies. The sum in the denominator runs over all available molecular energy levels and it is called the molecular partition function. It is a quantity of immense importance in statistical thermodynamics, and it is given the special symbol q (sometimes z). We have 

If we deal with N isolated non-interacting entities such as the molecules in a gas at low density, we can further divide up molecular energies with reasonable accuracy into their electronic, vibrational and rotational contributions 

The inverse argument shows how a chemical potential field imposes a particular energy distribution on units of matter. 

Consider a quantized field with discrete energy levels, each occupied by a characteristic number of molecules. These numbers can be thought of as representing relative activities during the course of a hypothetical reaction that starts with n molecules at the initial energy level Ui and reaching equilibrium with rij molecules of the final product at the level Uj. Intermediate levels are occupied by secondary products. As before 

The quantized energy ej can be of electronic, vibrational, rotational or translational type, readily calculated from the quantum laws of motion. In a macrosystem the sum over all the quantum states for the complete set of molecules, the sum over states defines the canonical partition function  

One of the most important expressions in science, the Boltzmann distribution, helps to elucidate the concept of temperature as well as underlying virtually all the bulk properties and reactions of matter and their variation with temperature. 

In preparation for the large number of occurrences of exponential functions throughout the text, it will be useful to know the shape of exponential functions. Here we deal with two types, and eAn exponential function of the form e starts off at 1 when X = 0 and decays toward zero, which it reaches as x approaches infinity (see the illustration). This function approaches zero more rapidly as a increases. The Boltzmann distribution is an example of an exponential function. The function e is called a Gaussian 

The exponential function, e , and the bell-shaped Gaussian function, e Note that both are equal to 1 atx=0, butthe exponential function rises to infinity as x — -o°. 

The apparently rjuidom motion that molecules undergo at T 0 is called thermal motion. The energy associated with this motion is the energy of thermal motion, but is commonly called simply thermal energy. A useful rule of thumb is 

Thermal motion ensures that molecules will be found spread over the energy levels available to them such that their mean energy is of order kT. The population of each energy level depends on the temperature, and a very important result is that in a system at a temperature T, the ratio of populations Nj and N, in states with energies Ei and 2 is given by the Boltzmann distribution, one form of which is 

We now see that the equilibrium configuration of molecules in a system will have the maximum probability as calculated by equation (6.4). It will have the greatest number of microstates, the most random distribution of molecules, and the highest entropy permitted by the constraints (T, P, V, etc.) of the system. Recall also that entropy is related to the probability or degree of disorder, W, by equation (6.3). 

The mathematical solution finds that set of energy-level occupation numbers N, N2,. .., Nj which maximizes the probability fU of a configuration as given in (6.4), subject to the following two constraints the total number of particles and the total energy (E) must remain constant. 

This is a problem in constrained extremals that can be solved using Lagrange s method. The constraining equations (6.6) and (6.7) are multiplied by two arbitrary constraints a and / , added to the logarithm of (6.4) and the desired maximum is given by 

This is the most general form of the Boltzmann distribution. It tells us that the fraction of molecules in an energy level e increases exponentially with temperature and decreases exponentially with the energy of that level. This is as important for macroscopic systems as the Schrodinger equation is for individual atoms or molecules. The Schrodinger equation, for example, shows that energy levels (Is, 2s, 2p.) are 

---

In this case, a spin A that was coupled to the a orientation of the B spin may end up, after the exchange, coupled to either a or (3. Because of the Boltzmann distribution, the amounts of a and P orientation are each... 

The Boltzmann distribution is fundamental to statistical mechanics. The Boltzmann distribution is derived by maximising the entropy of the system (in accordance with the second law of thermodynamics) subject to the constraints on the system. Let us consider a system containing N particles (atoms or molecules) such that the energy levels of the... 

The Boltzmann distribution gives the number of particles n, in each energy level e, as ... 

There are two additive terms to the energy, POP and TORS, that have not been mentioned yet because they are zero in minimal ethylene. The POP term comes from higher-energy conformers. If the energy at the global minimum is not too far removed from one or more higher conformational minima, molecules will be distributed over the conformers according to the Boltzmann distribution... 

The MEP is defined as the path of steepest descent in mass-weighted Cartesian coordinates. This is also called intrinsic reaction coordinate (IRC). In reality, we know that many other paths close to the IRC path would also lead to a reaction and the percentage of the time each path is taken could be described by the Boltzmann distribution. 

Standardizing the Method Equation 10.34 shows that emission intensity is proportional to the population of the excited state, N, from which the emission line originates. If the emission source is in thermal equilibrium, then the excited state population is proportional to the total population of analyte atoms, N, through the Boltzmann distribution (equation 10.35). 

The intensity distribution among rotational transitions in a vibration-rotation band is governed principally by the Boltzmann distribution of population among the initial states, giving... 

The intensity distribution among the rotational transitions is governed by the population distribution among the rotational levels of the initial electronic or vibronic state of the transition. For absorption, the relative populations at a temperature T are given by the Boltzmann distribution law (Equation 5.15) and intensities show a characteristic rise and fall, along each branch, as J increases. 

For most purposes only the Stokes-shifted Raman spectmm, which results from molecules in the ground electronic and vibrational states being excited, is measured and reported. Anti-Stokes spectra arise from molecules in vibrational excited states returning to the ground state. The relative intensities of the Stokes and anti-Stokes bands are proportional to the relative populations of the ground and excited vibrational states. These proportions are temperature-dependent and foUow a Boltzmann distribution. At room temperature, the anti-Stokes Stokes intensity ratio decreases by a factor of 10 with each 480 cm from the exciting frequency. Because of the weakness of the anti-Stokes spectmm (except at low frequency shift), the most important use of this spectmm is for optical temperature measurement (qv) using the Boltzmann distribution function. 

To obtain thermodynamic averages over a canonical ensemble, which is characterized by the macroscopic variables (N, V, T), it is necessary to know the probability of finding the system at each and every point (= state) in phase space. This probability distribution, p(r, p), is given by the Boltzmann distribution function. 

Relaxation refers to all processes which regenerate the Boltzmann distribution of nuclear spins on their precession states and the resulting equilibrium magnetisation along the static magnetic field. Relaxation also destroys the transverse magnetisation arising from phase coherenee of nuelear spins built up upon NMR excitation. 

Spin-lattice relaxation is the steady (exponential) build-up or regeneration of the Boltzmann distribution (equilibrium magnetisation) of nuelear spins in the static magnetic field. The lattice is the molecular environment of the nuclear spin with whieh energy is exchanged. 

Figure 5-8. Plot of Eq. (5-17), the Boltzmann distribution, n, is the number of molecules having energy e,.

Finally, the probability factor rj, which is taken to be coverage-independent in the model of a homogeneous surface with no lateral interactions between adsorbed particles, will be expressed by means of the Arrhenius formalism based on the Boltzmann distribution, viz. 

We need one more equation before we can derive the Boltzmann distribution law. We note that the total energy U - C0 is given by... 

Figure 10.6 Graph of the Boltzmann distribution function for the CO molecule in the ground electronic state for (a), the vibrational energy levels and (b), the rotational energy levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations.

Methods of disturbing the Boltzmann distribution of nuclear spin states were known long before the phenomenon of CIDNP was recognized. All of these involve multiple resonance techniques (e.g. INDOR, the Nuclear Overhauser Effect) and all depend on spin-lattice relaxation processes for the development of polarization. The effect is referred to as dynamic nuclear polarization (DNP) (for a review, see Hausser and Stehlik, 1968). The observed changes in the intensity of lines in the n.m.r. spectrum are small, however, reflecting the small changes induced in the Boltzmann distribution. 

The intensity of a non-degenerate n.m.r. transition between the nuclear Zeeman levels n and m is proportional to the difference in population of levels n and m as given by the Boltzmann distribution. This can be expressed by equation (35). 

The effect of temperature upon the situation in Fig. 5-5 is to modify the Boltzmann distribution. Lowering the temperature depopulates the higher-lying energy levels in favour of the lower. Therefore, susceptibility increases with decreasing temperature. Quantitative studies of the simple (first-order )... 

For all known cases of iron-sulfur proteins, J > 0, meaning that the system is antiferromagnetically coupled through the Fe-S-Fe moiety. Equation (4) produces a series of levels, each characterized by a total spin S, with an associated energy, which are populated according to the Boltzmann distribution. Note that for each S level there is in principle an electron relaxation time. For most purposes it is convenient to refer to an effective relaxation time for the whole cluster. 

---