Velocities, molecular, Maxwellian distribution

Maxwell found that he could represent the distribution of velocities statistically by a function, known as the Maxwellian distribution. The collisions of the molecules with their container gives rise to the pressure of the gas. By considering the average force exerted by the molecular collisions on the wall, Boltzmann was able to show that the average kinetic energy of the molecules was... 

However, because of the Maxwellian distribution function for molecular speeds, not all H2 molecules will be faster than all 02 molecules and some H2 molecules will have velocities near 0 m/s. Some 02 molecules will be moving faster than the average speed of H2 molecules. 

If one follows the approach of Landau and Teller [11], who in dealing with vibrational relaxation developed an expression by averaging a transition probability based on the relative molecular velocity over the Maxwellian distribution, one can obtain the following expression for the recombination rate constant [6] ... 

In this section we shall be concerned with a molecular theory of the transport properties of gases. The molecules of a gas collide with each other frequently, and the velocity of a given molecule is usually changed by each collision that the molecule undergoes. However, when a one-component gas is in thermal and statistical equilibrium, there is a definite distribution of molecular velocities—the well-known Maxwellian distribution. Figure 1 shows how the molecular velocities are distributed in such a gas. This distribution is isotropic (the same in all directions) and can be characterized by a root-mean-square (rm speed u, which is given by... 

The theory of effusion (Knudsen flow) is quite straightforward since only molecular flow is involved i.e., in the process of effusing, the molecules act independently of each other. For a Maxwellian distribution of velocities it can be shown that the number Zof molecular impacts on a unit area of wall surface in unit time is... 

We have made the additional approximation of assuming that the number of collisions Z at any point is independent of d, the distance between plates. This is justifiable if the mean speed 5 F L/d, where F L/d is the difference in velocity between two layers of gas separated by a mean free path. Und( r such conditions the molecular density in each layer is constant and most collisions then take place between molecules that have essentially the same relative Maxwellian distribution. When this condition is not satisfied, there will be important density gradients and thermal gradients, so that the entire analysis does not apply. This condition is the equivalent of saying that the velocity of the moving plate is small compared to the velocity of sound. 

As might be expected, the model leads to a great simplification over the calculations required for molecules with a continuous potential energy function, as it enables the analysis to be confined to binary collisions and permits the definition of a collision frequency. Because there is no molecular interaction between collisions, the velocity distributions of two colliding molecules may be assumed to be re-established by the time a second collision occurs between them. Thus a Maxwellian distribution around the local mass velocity may be postulated for the calculation of the mean frequency of collision and the average momentum and energy transported per collision in the nonuniform state of the liquid. 

In a hybrid method, molecules are displaced in time according to conventional molecular dynamics (MD) algorithms, specifically, by integrating Newton s equations of motion for the system of interest. Once the initial coordinates and momenta of the particles are specified, motion is deterministic (i.e., one can determine with machine precision where the system will be in the near future). In the context of Eq. (2.1), the probability of proposing a transition from a state 0 to a state 1 is determined by the probability with which the initial velocities of the particles are assigned from that point on, motion is deterministic (it occurs with probability one). If initial velocities are sampled at random from a Maxwellian distribution at the temperature of interest, then the transition probability function required by Eq. 

A similar situation is observed for the Ramsey fringes. Since the velocities of the molecules in the molecular beam are not equal but follow a Maxwellian distribution, the phase differences (coq — co)Llv show a corresponding distribution. The interference pattern is obtained by integrating the contributions to the signal from all molecules N(v) with the velocity v... 

For flow with high Knudsen number, the number of molecules in a significant volume of gas decreases, and there could be insufficient number of molecular collisions to establish an equilibrium state. The velocity distribution function will deviate away from the Maxwellian distribution and is non-isotropic. The properties of the individual molecule then become increasingly prominent in the overall behavior of the gas as the Knudsen number increases. The implication of the larger Knudsen number is that the particulate nature of the gases needs to be included in the study. The continuum approximaticui used in the small Knudsen number flows becomes invalid. At the extreme end of the Knudsen number spectrum is when its value approaches infinity where the mean free path is so large or the dimension of the device is so small that intermolecular collision is not likely to occur in the device. This is called collisionless or free molecular flows. 

From statistical mechanics, it follows that temperature is weU defined when the velocity distribution is Maxwellian. Systems for which this condition is fulfilled are complex reactions where the rate of elastic collisions is larger than the rate of reactive collisions. This is generally true for reactions in not too rarefied media and for many biological and transport processes. It may be noted that molecular collisions are responsible for attainment of Maxwellian distribution. Normally, significant deviations from the Maxwellian distribution are observed only under extreme conditions. Distribution is perturbed when physical processes are very rapid. Thus for a gas, local equilibrium assumption would not be valid when the relative variation of temperature is no longer small within a length equal to mean free path. 

As discussed in Chap. 4, under equilibrium conditions the function fi becomes a gaussian or maxwellian distribution function for the molecular velocities, and g becomes an equilibrium radial distribution function, i.e.. 

The physical condition of the kinetic theory of gases can be described by elastic collisions of monodispersed spheres with the Maxwellian velocity distribution in an infinite vacuum space. Therefore, for an analogy between particle-particle interactions and molecular interactions to be directly applicable, the following phenomena in gas-solid flows should not be regarded as significant in comparison to particle-particle interactions the gas-particle... 

If a molecule travels 1 cm, it sweeps out an imaginary volume of ira2(l). With n molecules per unit volume, the number of molecules struck per centimeter is tto n, and the mean free path is then the reciprocal, or l/(mra2). If it is assumed that the molecular velocities are distributed according to maxwellian theory rather than having a sin-... 

From the preceding discussions it is evident that at least four different temperatures have to be considered which under laboratory conditions are all equal the excitation temperature Tex of the molecule, defined by the relative populations of the levels, the kinetic temperature Tk, corresponding to the Maxwellian velocity distribution of the gas particles, the radiation temperature Traa, assuming a (in some cases diluted) black body radiation distribution, and the grain temperature 7, . With no thermodynamic equilibrium established, as is common in interstellar space, none of these temperatures are equal. These non-equilibium conditions are likely to be caused in part by the delicate balance between the various mechanisms of excitation and de-excitation of molecular energy levels by particle collisions and radiative transitions, and in part by the molecule formation process itself. Table 7 summarizes some of the known distribution anomalies. The non-equilibrium between para- and ortho-ammonia, the very low temperature of formaldehyde, and the interstellar OH and H20 masers are some of the more spectacular examples. 

In order to calculate the flow we must know something about the distribution of molecular velocities in the gas. Since the gas is not at equilibrium but only in a steady state, we cannot say that we have an equilibrium distribution. However we can make the approximation of assuming that the velocity distribution is flocally Maxwellian, i.e., that the molecules at any given point distant Z from the fixed plate have the normal distribution of velocities with respect to an average which is not zero but is given by the macroscopic stream velocity at that point. Thus at a point Z from the fixed plate the distribution is to be taken as... 

Miller and Kusch (3 ) determined the molecular composition of KI vapor by measurement of the velocity distribution of the molecules in the beam produced as the vapor effused through a small slit in a source. The analysis was based on an assumption that the velocity distribution within the oven is Maxwellian and that the vapor effuses through the ideal slit of kinetic theory. The velocity distributions of potassium and thallium atomic beams were found to be in excellent agreement with the theoretical distributions so the determination of the molecular composition of KI beams was tried. Using the derived equilibrium constants, we calculate the enthalpy change of the dissociation reaction by the 2nd and 3rd law methods. The results are presented in the following table. 

FYom this formula it is seen that to calculate /i we need to determine the mean molecular speed ( ci )m- For real systems the average molecular speed is difficult to determine. Assuming that the system is sufficiently close to equilibrium the velocity distribution may be taken to be Maxwellian. For molecules in the absolute Maxwellian state the peculiar velocity equals the microscopic molecular velocity, i.e., Ci = ci, because the macroscopic velocity... 

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