Molecular velocities, probability distribution

The RTD quantifies the number of fluid particles which spend different durations in a reactor and is dependent upon the distribution of axial velocities and the reactor length [3]. The impact of advection field structures such as vortices on the molecular transit time in a reactor are manifest in the RTD [6, 33], MRM measurement of the propagator of the motion provides the velocity probability distribution over the experimental observation time A. The residence time is a primary means of characterizing the mixing in reactor flow systems and is provided directly by the propagator if the velocity distribution is invariant with respect to the observation time. In this case an exact relationship between the propagator and the RTD, N(t), exists... 

This factor is reminiscent of the radial distribution function for electron probability in an atom and the Maxwell distribution of molecular velocities in a gas, both of which pass through a maximum for similar reasons. 

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] ... 

We know that all the gas molecules do not travel with the same velocity. This is because the molecules are colliding with one another quite frequently and so their velocities keep on changing. Maxwell worked out the distribution of molecules between different possible velocities by using probability considerations. According to him, the distribution of molecular velocities is given by,... 

If now we assume that the molecules move completely at random and independently of each other and that the system is at equilibrium and isolated (no exchange of energy between the gas and its environment) then the total energy of the gas is simply the kinetic energy attributable to the random motion of the molecules. This total energy and the volume then fix completely the thermodynamic properties of the gas. If now we could know the probability distribution function for the molecular velocities (at equilibrium), that would determine uniciuely the properties of the system. 

The starting point for the kinetic theory of low density, non-reacting mixtures of mono-atomic gases is the knowledge of the distribution function /s(r,Cg,t). fs r,Cg,t) is defined in such a way that the quantity fs r,Cg,t) dcgdr represents the probable number of molecules of the s-th species which at the time t lie in a unit volume element dr about the point r and which have velocities within the range dCg about c. It is emphasized that denotes the molecular velocity of a species s with respect to a coordinate system fixed in space. 

In gas kinetic theory, the probability density for a component of the molecular velocity is a Gaussian distribution. The normalized probability distribution for Vx, the X component of the velocity, is given by... 

The probability distribution for molecular velocities is the Maxwell-Boltzmann probability distribution ... 

We now seek a formula that represents the probability distribution for the molecular states in our model system. Because there are no forces on the molecules inside the box we assert that all positions inside the box are equally probable and focus on the probability distribution for molecular velocities. We begin with a reasonable (but unproved) assumption The probability distribution of each velocity component is independent of the other velocity components. The validity of this assumption must be tested by comparing our results with experiment. Consider a velocity with components v, Vy, and Uj. Let the probability that Vx lies between (a particular value of Vx) and 4- dvx be given by... 

The first model system designed to represent a dilute gas consists of noninteracting point-mass molecules that obey classical mechanics. We obtained the Maxwell-Boltzmann probability distribution for molecular velocities ... 

In order to accurately resolve the local flow field around a colloid, methods have been proposed which exclude fluid-particles from the interior of the colloid and mimic slip [19,77] or no-slip [78] boundary conditions. The latter procedure is similar to what is known in molecular dynamics as a thermal wall boundary condition fluid particles which hit the colloid particle are given a new, random velocity drawn from the following probability distributions for the normal velocity component, vn, and the tangential component, vt. 

The Maxwell-Boltzmann velocity distribution function resembles the Gaussian distribution function because molecular and atomic velocities are randomly distributed about their mean. For a hypothetical particle constrained to move on the A -axis, or for the A -component of velocities of a real collection of particles moving freely in 3-space, the peak in the velocity distribution is at the mean, Vj. = 0. This leads to an apparent contradiction. As we know from the kinetic theor y of gases, at T > 0 all molecules are in motion. How can all particles be moving when the most probable velocity is = 0 ... 

Molecular mechanics force fields rest on four fundamental principles. The first principle is derived from the Bom-Oppenheimer approximation. Electrons have much lower mass than nuclei and move at much greater velocity. The velocity is sufficiently different that the nuclei can be considered stationary on a relative scale. In effect, the electronic and nuclear motions are uncoupled, and they can be treated separately. Unlike quantum mechanics, which is involved in determining the probability of electron distribution, molecular mechanics focuses instead on the location of the nuclei. Based on both theory and experiment, a set of equations are used to account for the electronic-nuclear attraction, nuclear-nuclear repulsion, and covalent bonding. Electrons are not directly taken into account, but they are considered indirectly or implicitly through the use of potential energy equations. This approach creates a mathematical model of molecular structures which is intuitively clear and readily available for fast computations. The set of equations and constants is defined as the force... 

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