Boltzmann distribution dependences

Due to the simple product form of the Maxwell-Boltzmann distribution, the derivations given above are easily generalized to the expression for the relative velocity in three dimensions. Since the integrand in Eq. (2.18) (besides the Maxwell-Boltzmann distribution) depends only on the relative speed, we can simplify the expression in Eq. (2.18) further by integrating over the orientation of the relative velocity. This is done by introducing polar coordinates for the relative velocity. The full three-dimensional probability distribution for the relative speed is... 

Not to be forgotten is the assumption that neither the presence of the electrolyte nor the interface itself changes the dielectric medium properties of the aqueous phase. It is assumed to behave as a dielectric continuum with a constant relative dielectric permittivity equal to the value of the bulk phase. The electrolyte is presumed to be made up of point charges, i.e. ions with no size, and responds to the presence of the charged interface in a competitive way described by statistical mechanics. Counterions are drawn to the surface by electrostatic attraction while thermal fluctuations tend to disperse them into solution, surface co-ions are repelled electrostatically and also tend to be dispersed by thermal motion, but are attracted to the accumulated cluster of counterions found near the surface. The end result of this electrical-thermodynamic conflict is an ion distribution which can be represented (approximately) by a Boltzmann distribution dependent on the average electrostatic potential at an arbitrary point multiplied by the valency of individual species, v/. 

In the modified model, as illustrated in Fig. 4, the proportional depletion of the Boltzmann distribution depends on the energy. At high energies k2 E) > these energies are heavily depleted and therefore do not contribute much to the rate. On the other hand, just above the reaction threshold the rate coefficient is small, k2 E) < and the population is barely depleted at... 

Nagel, Oppenheim and Putnam saw the explanatory appheation of physical laws to chemistry as the paradigm example of reduction, and it is stiU cited as such. So how accurately does classical reductionism portray the imdoubted explanatory success of physical theory within chemistry Two main examples are cited in the literature (i) the relationship between thermodynamics and statistical mechanics and (ii) the explanation of chemical valence and bonding in terms of quantum mechanics. The former reduction is widely presumed to be unproblematic because of the identification of temperature with mean molecular kinetic energy, but Needham [2009] points out that temperature can be identified with mean energy only in a molecular population at equilibrium (one displaying the Boltzmann distribution), but the Boltzmann distribution depends on temperature, so any reduction of temperature will be circular (for a survey of the issues see [van Brakel, 2000, Chapter 5]. 

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... 

For most purposes only the Stokes-shifted Raman spectmm, which results from molecules in the ground electronic and vibrational states being excited, is measured and reported. Anti-Stokes spectra arise from molecules in vibrational excited states returning to the ground state. The relative intensities of the Stokes and anti-Stokes bands are proportional to the relative populations of the ground and excited vibrational states. These proportions are temperature-dependent and foUow a Boltzmann distribution. At room temperature, the anti-Stokes Stokes intensity ratio decreases by a factor of 10 with each 480 cm from the exciting frequency. Because of the weakness of the anti-Stokes spectmm (except at low frequency shift), the most important use of this spectmm is for optical temperature measurement (qv) using the Boltzmann distribution function. 

Methods of disturbing the Boltzmann distribution of nuclear spin states were known long before the phenomenon of CIDNP was recognized. All of these involve multiple resonance techniques (e.g. INDOR, the Nuclear Overhauser Effect) and all depend on spin-lattice relaxation processes for the development of polarization. The effect is referred to as dynamic nuclear polarization (DNP) (for a review, see Hausser and Stehlik, 1968). The observed changes in the intensity of lines in the n.m.r. spectrum are small, however, reflecting the small changes induced in the Boltzmann distribution. 

The origin of postulate (iii) lies in the electron-nuclear hyperfine interaction. If the energy separation between the T and S states of the radical pair is of the same order of magnitude as then the hyperfine interaction can represent a driving force for T-S mixing and this depends on the nuclear spin state. Only a relatively small preference for one spin-state compared with the other is necessary in the T-S mixing process in order to overcome the Boltzmann polarization (1 in 10 ). The effect is to make n.m.r. spectroscopy a much more sensitive technique in systems displaying CIDNP than in systems where only Boltzmann distributions of nuclear spin states obtain. More detailed consideration of postulate (iii) is deferred until Section II,D. 

Such conformational dependence presents challenges and an opportunity. The challenges he in properly accounting for its consequences. In many cases, exact conformational energetics and populations in a sample may be unknown, and the nature of the sample inlet may sometimes also mean that a Boltzmann distribution cannot be assumed. Introducing this uncertainty into the data modeling process produces some corresponding uncertainty in the theoretical interpretation of data... 

Boltzmann distribution, time-dependent mechanical work, transition probability, 55-57... 

We consider a biological macromolecule in solution. Let X and Y represent the degrees of freedom of the solute (biomolecule) and solvent, respectively, and let U(X, Y) be the potential energy function. The thermal properties of the system are averages over a Boltzmann distribution P(X, Y) that depends on both X and Y. To obtain a reduced description in terms of the solute only, the solvent degrees of freedom must be integrated out. The reduced probability distribution P is... 

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. 

At room temperature, most of the molecules are in the lowest vibrational level of the ground state (according to the Boltzmann distribution see Chapter 3, Box 3.1). In addition to the pure electronic transition called the 0-0 transition, there are several vibronic transitions whose intensities depend on the relative position and shape of the potential energy curves (Figure 2.4). 

By reference to Fig. 18 and by assuming a Boltzmann distribution of electron spins among the three states, the temperature dependence of the signal intensity for Tj gave AEqj = 61 cm ( = 175 cal mol ) and J was calculated to be -f-16cm ... 

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.)... 

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