Boltzmann condition

It is clear that, in extension of Eq. (39), under Brouwer and Boltzmann conditions, the result for any carrier concentration must be... 

The phenomenon of chemically induced dynamic nuclear polarization (CIDNP) consists of the manifestation of unusual line intensities and/or phases of signals of radical reaction products in the NMR spectrum when reaction takes place directly in the probe of the spectrometer. These anomalous lines (enhanced absorption or emission of NMR signals), which reflect the populations of nuclear spin states deviating from the Boltzmann condition, are observed within the time range of nuclear relaxation times of the diamagnetic molecules (T, ), which are as a rule, several seconds to tens of seconds. Subsequently, the NMR spectrum re-acquires its usual form. In 1967, two research groups in Europe (J. Bargon, H. Fischer, and U. Johnson) and the USA (H. Ward and R. Lawler) discovered independently that this phenomenon is directly associated with the free radicals involved in the process. Later on, it was shown that this also pertains to radical ions and triplet excited states of molecules. 

The velocity distribution/(v) depends on the conditions of the experiment. In cell and trap experiments it is usually a Maxwell-Boltzmann distribution at some well defined temperature, but /(v) in atomic beam experiments, arising from optical excitation velocity selection, deviates radically from the nonnal thennal distribution [471. The actual signal count rate, relates to the rate coefficient through... 

The Boltzmann equation (Equation 18.2) shows that, under equilibrium conditions, the ratio of the number (n) of ground-state molecules (A ) to those in an excited state (A ) depends on the energy gap E between the states, the Boltzmann constant k (1.38 x 10" J-K" ), and the absolute temperature T(K). 

If the spectrum is observed in emission it is the rotational populations in the upper state which determine relative intensities. They may or may not be equilibrium Boltzmann populations, depending on the conditions under which the molecule got into the upper state. 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

Both entropic and coulombic contributions are bounded from below and it can be verified that the second variation of is positive definite so that the above equations correspond to a minimum [27]. Using conditions in the bulk we can eliminate //, from the equations. Then we get the Boltzmann equation in which the electric potential verifies the Poisson equation by construction. Hence is equivalent within MFA to the... 

Since the equilibrium probability Ed.s, t) contains the Boltzmann factor with an energy Tid.s, ), the condition (12) leads to the ratio of transition probabilities of the forward and backward processes as... 

Object in this section is to review how rheological knowledge combined with laboratory data can be used to predict stresses developed in plastics undergoing strains at different rates and at different temperatures. The procedure of using laboratory experimental data for the prediction of mechanical behavior under a prescribed use condition involves two principles that are familiar to rheologists one is Boltzmann s superposition principle which enables one to utilize basic experimental data such as a stress relaxation modulus in predicting stresses under any strain history the other is the principle of reduced variables which by a temperature-log time shift allows the time scale of such a prediction to be extended substantially beyond the limits of the time scale of the original experiment. 

In its more advanced aspects, kinetic theory is based upon a description of the gas in terms of the probability of a particle having certain values of coordinates and velocity, at a given time. Particle interactions are developed by the ordinary laws of mechanics, and the results of these are averaged over the probability distribution. The probability distribution function that is used for a given macroscopic physical situation is determined by means of an equation, the Boltzmann transport equation, which describes the space, velocity, and time changes of the distribution function in terms of collisions between particles. This equation is usually solved to give the distribution function in terms of certain macroscopic functions thus, the macroscopic conditions imposed upon the gas are taken into account in the probability function description of the microscopic situation. 

The probability of an ensemble distribution in classical statistics is maximized under the condition of given total energy in the ensemble, to yield the familiar Boltzmann distribution ... 

Bohr frequency condition, 13 Bohr radius, 23 boiling, 314 boiling point, 314 alkanes, 737 anomalous, 184 boiling point prediction, 180 boiling point trends, 183 boiling-point elevation, 332 Boltzmann, L., 276 Boltzmann formula, 276 bomb calorimeter, 224... 

The Boltzmann equation, S = kXxiW, establishes the zero point for entropies. Because ln(l) = 0, this equation predicts that the entropy will be zero for a system with only one possible description. Figure 14-1 la shows a system with W =, nine identical marbles placed in nine separate compartments. The figure also shows two types of conditions where > 0. Figure 14-llZ) shows a condition when the marbles are not identical. If one marble is a different color than the other eight, there are nine different places to place the different marble. Figure I4-IIc shows a condition where there are more compartments than marbles. If there are more places to put the marbles than there are marbles, there are multiple ways to place the marbles in the compartments. 

Direct dynamics trajectory calculations at the MP2/6-31-FG level of theory were then used to explore the reaction dynamics of this system [63]. Sixty-four trajectories were started from the central barrier shown at A in Fig. 11, with initial conditions sampled from a 300 K Boltzmann distribution. Of the 31 trajectories that moved in the direction of products, four trajectories followed the MEP and became trapped in the hydrogen-bonded [CH3OH ... 

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