Distribution law, Boltzmann

Statistical Mechanics Gives Probabihty Distributions for Atoms and Molecules 

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COMPUTER PROJECT 1-4 Maxwell-Boltzmann Distribution Laws... 

The intensity distribution among the rotational transitions is governed by the population distribution among the rotational levels of the initial electronic or vibronic state of the transition. For absorption, the relative populations at a temperature T are given by the Boltzmann distribution law (Equation 5.15) and intensities show a characteristic rise and fall, along each branch, as J increases. 

We need one more equation before we can derive the Boltzmann distribution law. We note that the total energy U - C0 is given by... 

Boltzmann, L. 18. 19 Boltzmann constant 337 Boltzmann distribution law 514-23 bubble-pressure curve in vapor + liquid phase equilibrium 406... 

A second equation linking parameters / and Qy is needed so that they may be calculated individually. To derive such an equation it was assumed that the concentration of ions Cj is determined by the Boltzmann distribution law. 

If the origin of the coordinate system is located in the centre of the central kth ion, the number of particles dNt in the volume dV is given by a distribution function, expressed by Debye and Hiickel in terms of the Boltzmann distribution law (cf. p. 215)... 

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... 

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

Quantitatively, the effect of temperature on solubility can be explained by considering the dissolution process as represented by the population of states of unequal energy. The Boltzmann distribution law relating the populations nx and n2 for states Et and E2 which are of unequal energy can be written as... 

At thermal equilibrium the population of any series of energy levels is described by the Boltzmann distribution law. If N0 molecules are in the ground state then the number Ni in any higher energy level is given by the equation ... 

Calculations based on the Boltzmann distribution law show that, at room temperature, most molecules will be in the v = 0 vibrational state of the electronic ground state and so absorption almost always occurs from So(v = 0) (Figure 2.2). 

Since our solvents are permanent dipoles that develop an orientation under the influence of an outside electric field activity in opposition to the disordering influence of thermal agitation, such an orientation process is governed by the Boltzmann distribution law and results in the dielectric constant being strongly dependent on the absolute temperature. Thus, as the systems become cooler, the random motion of their molecules decreases and the electric field becomes very effective in orienting them the dielectric constant increases markedly with reduction in temperature at constant volume. 

In this article we use transition state theory (TST) to analyze rate data. But TST is by no means universally accepted as valid for the purpose of answering the questions we ask about catalytic systems. For example, Simonyi and Mayer (5) criticize TST mainly because the usual derivation depends upon applying the Boltzmann distribution law where they think it should not be applied, and because thermodynamic concepts are used improperly. Sometimes general doubts that TST can be used reliably are expressed (6). But TST has also been used with considerable success. Horiuti, Miyahara, and Toyoshima (7) successfully used theory almost the same as TST in 66 sets of reported kinetic data for metal-catalyzed reactions. The site densities they calculated were usually what was expected. (Their method is discussed further in Section II,B,7.)... 

Introducing the ni(r) value as given in Eq. (2.10) into Eq. (2.12) one obtains the Boltzmann distribution law in terms of charge densities... 

The set of possible values of a random variable, most typically presented as a plot of frequency of occurrence on the vertical axis versus magnitude of the variable on the horizontal axis. See Boltzmann Distribution Law... 

LINKED EUNCTIONS HEMOGLOBIN ADAIR EQUATION BOLTZMANN DISTRIBUTION LAW CRYOENZYMOLOGY BOLUS... 

When a system is in thermodynamic equilibrium the level population, i.e. the number of atoms A in the excited state, is given by the Boltzmann distribution law ... 

Energy states in solution, 1462 1463 Boltzmann distribution law, 1466 distribution, 1462 1464 ground state, 1464, 1468 number of, 1462 Enthalpy... 

Adsorption. In the simple theory of the space charge inside a semiconductor, it was assumed that all the electrons and holes are free to move up to the surface. Being susceptible to thermal motion, their concentrations from a- = 0 to x — °° were said to be given by the interplay of electrical and thermal forces only, as expressed by the Boltzmann distribution law and Poisson s equation. 

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