Molecules Maxwell-Boltzmann distribution

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

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 ... 

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... 

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 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...

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. 

The area under a Maxwell-Boltzmann distribution graph represents the distribution of the kinetic energy of collisions at a constant temperature. At a given temperature, only a certain fraction of the molecules in a sample have enough kinetic energy to react. 

The Maxwell-Boltzmann distributions in one and three dimensions will be used next to find the frequency with which molecules undergo collisions, both with other gas-phase molecules as well as with a wall. 

Begin by considering the translational motion of molecules in a container that has a total concentration of molecules per unit volume [c]. The distribution of velocities in the x, y, and z directions is given by the one-dimensional Maxwell-Boltzmann distribution of... 

The one-dimensional Maxwell-Boltzmann distribution gives the fraction of molecules in the velocity range vx - vx + dvx, so the product [c] P(vx) gives the number of molecules per unit volume in that velocity range. The total number of molecules passing through the plane in a time At that have velocities in the range vx - vx + dvx is... 

In the general approach to classical statistical mechanics, each particle is considered to occupy a point in phase space, i.e., to have a definite position and momentum, at a given instant. The probability that the point corresponding to a particle will fall in any small volume of the phase space is taken proportional to die volume. The probability of a specific arrangement of points is proportional to the number of ways that the total ensemble of molecules could be permuted to achieve the arrangement. When this is done, and it is further required that the number of molecules and their total energy remain constant, one can obtain a description of the most probable distribution of the molecules in phase space. Tlie Maxwell-Boltzmann distribution law results. 

The Maxwell-Boltzmann distribution defines the most probable route. Here the Maxwell-Boltzmann distribution is being used in terms of a probability, rather than a fraction of molecules with energy at least a certain critical energy. The probability that a molecule has a given potential energy at any point on the surface is proportional to exp(—s/kT), where e is the PE at the point. 

In the kinetic theory of gases, the molecules are assumed to be smooth, rigid, and elastic spheres. The only kinetic energy considered is that from the translational motion of the molecules. In addition, the gas is assumed to be in an equilibrium state in a container where the gas molecules are uniformly distributed and all directions of the molecular motion are equally probable. Furthermore, velocities of the molecules are assumed to obey the Maxwell-Boltzmann distribution, which is described in the following section. 

Specialized to thermal equilibrium, the velocity distributions for the molecules are the Maxwell-Boltzmann distribution (a special case of the general Boltzmann distribution law). The expression for the rate constant at temperature T, k(T), can be reduced to an integral over the relative speed of the reactants. Also, as a consequence of the time-reversal symmetry of the Schrodinger equation, the ratio of the rate constants for the forward and the reverse reaction is equal to the equilibrium constant (detailed balance). 

We now proceed to develop a specific expression for the rate constant for reactants where the velocity distributions /a( )(va) and /B(J)(vB) for the translational motion are independent of the internal quantum state (i and j) and correspond to thermal equilibrium.4 Then, according to the kinetic theory of gases or statistical mechanics, see Appendix A.2.1, Eq. (A.65), the velocity distributions associated with the center-of-mass motion of molecules are the Maxwell-Boltzmann distribution, a special case of the general Boltzmann distribution law ... 

That is, the Maxwell-Boltzmann distribution for the two molecules can be written as a product of two terms, where the terms are related to the relative motion and the center-of-mass motion, respectively. After substitution into Eq. (2.18) we can perform the integration over the center-of-mass velocity Vx. This gives the factor y/2iVksTjM (IZo eXP( —ax2)dx = sjnja) and, from the equation above, we obtain the probability distribution for the relative velocity, irrespective of the center-of-mass motion. 

The well-known Maxwell-Boltzmann distribution for the velocity or momentum associated with the translational motion of a molecule is valid not only for free molecules but also for interacting molecules in a liquid phase (see Appendix A.2.1). The average kinetic energy of a molecule at temperature T is, accordingly, (3/2)ksT. For the molecules to react in a bimolecular reaction they should be brought into contact with each other. This happens by diffusion when the reactants are dispersed in a solution, which is a quite different process from the one in the gas phase. For fast reactions, the diffusion rate of reactant molecules may even be the limiting factor in the rate of reaction. 

Example A.4 The Maxwell Boltzmann distribution for N interacting molecules... 

---