The Maxwell-Boltzmann distribution

Under this heading we are concerned with the question what is the average number of molecules in a sample of a perfect gas which are in some particular energy state To start with, it will perhaps be clearest if we deal with the quantum states of the molecules, rather than with the energy states, since the latter may be degenerate. Let the quantum states be numbered 0,1,2. t,y,. 

Over any large period of time let pn bo the fraction of the time that there is just one molecule in the s unple of gas in the tth quantum state, let Pis be the fraction of the time that there are two molecules in the ith 

as discussed already in 12 l,t the number of quantum states per molecule is enormously larger than the nximber of molecules in the sample. (This is the basis of the Maxwell-Boltzmcuin statistics and applies under all conditions of temperature and pressure at which a gas is effectively perfect.) In any sample of gas it is therefore very improbable that there will be more than one molecule in a particular quantum state — in fact, the mean number,, will be very much less than unity. In the above equation we can therefore neglect all p s except pn and write 

For simplicity consider a gas sample containing just three identical molecules. A quantum state of the whole system will be specified by saying how many moleciiles there are in each of the molecular quantum states, but without seeking to distinguish between these molecules, which is impossible. An expression for the probability of this state hets already been obtained in equation (11 42). For example, the probability that one molecule is the zeroth molecular quantum state, a second also in the zeroth, and the third in the ith is 

The total probability that there is at least one molecule in the ith state is obtained by summing (all expressions of the above kind in which at least one of the 6 s is an C. Thus 

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Clearly, G = A +. S in this example. The entropy matrix can be obtained from the Maxwell-Boltzmann distribution... 

The initial velocities may also be chosen from a uniform distribution or from a simp Gaussian distribution. In either case the Maxwell-Boltzmann distribution of velocities usually rapidly achieved. 

The Maxwell-Boltzmann distribution function (Levine, 1983 Kauzmann, 1966) for atoms or molecules (particles) of a gaseous sample is... 

The second assumption is that the concentration c, (particles per unit volume) of type,/ ions in the electrical Field is related to c°, the concentration at zero field, by the Maxwell-Boltzmann distribution function, ... 

In Chapter 10, we will derive the Maxwell-Boltzmann distribution function and describe its properties and applications. 

To understand how collision theory has been derived, we need to know the velocity distribution of molecules at a given temperature, as it is given by the Maxwell-Boltzmann distribution. To use transition state theory we need the partition functions that follow from the Boltzmann distribution. Hence, we must devote a section of this chapter to statistical thermodynamics. 

Often, we will be interested in how the velocities of molecules are distributed. Therefore we need to transform the Boltzmann distribution of energies into the Maxwell-Boltzmann distribution of velocities, thereby changing the variable from energy to velocity or, rather, momentum (not to be confused with pressure). If the energy levels are very close (as they are in the classic limit) we can replace the sum by an integral ... 

The relative velocity between the molecules not only determines whether A and B collide, but also if the kinetic energy involved in the collision is sufficient to surmount the reaction barrier. Velocities in a mixture of particles in equilibrium are distributed according to the Maxwell-Boltzmann distribution in spherical coordinates ... 

Figure 2. Comparison of the simulated velocity distribution (histogram) with the Maxwell— Boltzmann distribution function (solid line) for kgT —. The system had volume V — 1003 cells of unit length and N = 107 particles with mass m = 1. Rotations (b were selected from the set Q — tt/2, — ti/2 about axes whose directions were chosen uniformly on the surface of a sphere.

One may also show that MPC dynamics satisfies an H theorem and that any initial velocity distribution will relax to the Maxwell-Boltzmann distribution [11]. Figure 2 shows simulation results for the velocity distribution function that confirm this result. In the simulation, the particles were initially uniformly distributed in the volume and had the same speed v = 1 but different random directions. After a relatively short transient the distribution function adopts the Maxwell-Boltzmann form shown in the figure. 

Consider, as an example, the calculation of the mean-square speed of an ensemble of molecules which obey the Maxwell-Boltzmann distribution law. This quantity is given by... 

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

Equation (30) is the Maxwell-Boltzmann distribution function in rectangular coordinates. Thus, in a system of N total molecules, the fraction of molecules, dN/ N, with velocity components in the ranges x component, vx to vx + dvx y component, vy to Vy + dvy, and z component, vz to vz + dvz is given by... 

Figure 10 A graphical illustration of the Maxwell-Boltzmann distribution laws. Normalized speed---is vlvp, and normalized energy - is EUcT.

A study of the Maxwell-Boltzmann distribution of molecular energies at temperatures T Kelvins and at (T + 10) K illustrates this. 

The Maxwell-Boltzmann distribution of molecular energies can be used to explain how a catalyst works at constant temperature. 

The distribution function (24) for an ideal gas, shown in figure 6 is known as the Maxwell-Boltzmann distribution and is specified more commonly [118] in terms of molecular speed, as... 

The frequency with which the transition state is transformed into products, iT, can be thought of as a typical unimolecular rate constant no barrier is associated with this step. Various points of view have been used to calculate this frequency, and all rely on the assumption that the internal motions of the transition state are governed by thermally equilibrated motions. Thus, the motion along the reaction coordinate is treated as thermal translational motion between the product fragments (or as a vibrational motion along an unstable potential). Statistical theories (such as those used to derive the Maxwell-Boltzmann distribution of velocities) lead to the expression ... 

As stated earlier, in the state of thermal equilibrium at room temperature, dihydrogen (H2) contains 25.1% parahydrogen (nuclear singlet state) and 74.9% orthohydrogen (nuclear triplet state) [19]. This behavior reflects the three-fold degeneracy of the triplet state and the almost equal population of the energy levels, as demanded by the Maxwell-Boltzmann distribution. At lower temperatures, different ratios prevail (Fig. 12.5) due to the different symmetry of the singlet and the triplet state [19]. 

The thermodynamic temperature is the sole variable required to define the Maxwell-Boltzmann distribution raising the temperature increases the spread of energies. 

Figure 1.9 Molecular energies follow the Maxwell-Boltzmann distribution energy distribution of nitrogen molecules (as y) as a function of the kinetic energy, expressed as a molecular velocity (as x). Note the effect of raising the temperature, with the curve becoming flatter and the maximum shifting to a higher energy...

Equation (8.55) comes ultimately from the Maxwell - Boltzmann distribution in Equation (1.16). 

D) Whether you can answer this question depends on whether you are acquainted with what is known as the Maxwell-Boltzmann distribution. This distribution describes the way that molecular speeds or energies are shared among the molecules of a gas. If you missed this question, examine the following figure and refer to your textbook for a complete description of the Maxwell-Boltzmann distribution. 

Transition State Theory [1,4] is the most frequently used theory to calculate rate constants for reactions in the gas phase. The two most basic assumptions of this theory are the separation of the electronic and nuclear motions (stemming from the Bom-Oppenheimer approximation [5]), and that the reactant internal states are in thermal equilibrium with each other (that is, the reactant molecules are distributed among their states in accordance with the Maxwell-Boltzmann distribution). In addition, the fundamental hypothesis [6] of the Transition State Theory is that the net rate of forward reaction at equilibrium is given by the flux of trajectories across a suitable phase space surface (rather a hypersurface) in the product direction. This surface divides reactants from products and it is called the dividing surface. Wigner [6] showed long time ago that for reactants in thermal equilibrium, the Transition State expression gives the exact... 

The rate constant is measured in units of moles dnr3 sec /(moles dnr3)", where n = a + b. Time may also be in minutes or hours. It should be noted that in case where the reaction is slow enough, the thermal equilibrium will be maintained due to constant collisions between the molecules and k remains constant at a given temperature. However, if the reaction is very fast the tail part of the Maxwell-Boltzmann distribution will be depleted so rapidly that thermal equilibrium will not be re-established. In such cases rate constant will not truly be constant and it should be called a rate coefficient. 

To determine the average number of photons per mode, we should first determine the mean energy, W, for the particular mode at v = 5 x 10 " s at 300 K. According to the Maxwell-Boltzmann distribution, this is given by... 

MSN.l. I. Prigogine, Sur la perturbation de la distribution de Maxwell-Boltzmann par des reactions chimiques (On the perturbation of the Maxwell-Boltzmann distribution by chemical reactions), Suppl Nuovo Cimento 6, 289-295 (1949). 

If we are going to relate the properties of our system to a physical situation, we need to be able to characterize the system s temperature, T. In a macroscopic collection of atoms that is in equilibrium at temperature T, the velocities of the atoms are distributed according to the Maxwell -Boltzmann distribution. One of the key properties of this distribution is that the average kinetic energy of each degree of freedom is... 

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