Boltzman term

The DE were determined by titrimetry, and the galacturonic acid (GalA) content was measured by the m-hydroxybiphenyl method [25]. The structural charge density parameter introduced by Lifson and Katchalsky [26], is given by = e2/bDkT(l-DE/100) were e is the electron charge, kT the Boltzman term, b... 

The reaction has an associated equilibrium constant K, which is understood to carry Boltzman terms accounting for electrostatic effects of any surface complexa-tion reactions embedded within the redox reaction. The reaction s activity product at any point in the simulation is given by,... 

Let us consider an V-component fluid in a volume V, at temperature T, and at chemical potentials /r = mi, > Mv - The fluid is in contact with an impermeable solid surface. We assume that the fluid particles interact between themselves via the pair potential denoted by u pir), and interact with the confining surface via the potential (a,f3= 1,2,. ..,V). The potential v ir) contains a hard-wall term to ensure that the solid surface is impermeable. For the sake of convenience, the hard-wall term is assumed to extend into the bulk of the solid [46,47], such that the Boltzman factor (r), and the local density Pa r) are cutoff at a certain distance z = z, ... 

Our final form for the Boltzman equation is then obtained by substituting these last two expressions for the gain and loss terms into equation 9.29 and adding the term Vijf x, v, t) to the LHS for the case where there is an external force F Vjj is the gradient operator with respect to v and m is the hard-sphere mass) ... 

Taking the lowest order terms 6 and 5t, we have the basic collisionless form of the discrete Boltzman equation ... 

In equation (2) Rq is the equivalent capillary radius calculated from the bed hydraulic radius (l7), Rp is the particle radius, and the exponential, fxinction contains, in addition the Boltzman constant and temperature, the total energy of interaction between the particle and capillary wall force fields. The particle streamline velocity Vp(r) contains a correction for the wall effect (l8). A similar expression for results with the exception that for the marker the van der Waals attraction and Born repulsion terms as well as the wall effect are considered to be negligible (3 ). 

Entropy is a measure of the degree of randomness in a system. The change in entropy occurring with a phase transition is defined as the change in the system s enthalpy divided by its temperature. This thermodynamic definition, however, does not correlate entropy with molecular structure. For an interpretation of entropy at the molecular level, a statistical definition is useful. Boltzmann (1896) defined entropy in terms of the number of mechanical states that the atoms (or molecules) in a system can achieve. He combined the thermodynamic expression for a change in entropy with the expression for the distribution of energies in a system (i.e., the Boltzman distribution function). The result for one mole is ... 

In Equations 6-10 d0 is the density of the solvent which was measured at each solvent composition at each temperature, a is the ion-size parameter, z is an empirical constant, N is Avagadro s number, D is the dielectric constant of the solvent, k is the Boltzman constant, and e is the electronic charge. Other terms have been defined already. 

To extend this expression to any temperature means that we have to include an explicit ho), the Boltzman constant ks and absolute temperature (T) in the final Gaussian term for the Franck-Condon factor. The resulting expression is harder to solve in one s head, but can be useful. 

Here, N is the number of molecules (monomers) per mole, k is the Boltzman constant, T is absolute temperature, h is Planck s constant, AGj is a term denoting the mobility of the hpid molecules, j is interfacial tension, Tf is melting temperature, and Hf is latent heat of fusion. The first exponential term in Equation 1 has been related to the ability of a lipid molecule to attain the necessary conformation to become attached to the crystal lattice, and it is often given as (20)... 

In this expression k is the Boltzman constant and T the temperature of the system in degrees Kelvin. The (Uqm-Ucl) term gives the differences between the energies of the quantum and classical systems at a particular configuration. 

The first term, Gexp is a constant characteristic of the experimental set-up used, while the remaining terms are predicted theoretically. B o), T) is a temperature function related to the Boltzman population factor n co)... 

In previous publications, we have presented a formal treatment of the steady-state distribution of charges in terms of Boltzman s law with two allowed energy states. It... 

Velocities are allocated randomly to atoms to produce a Maxwell-Boltzman distribution (Figure 9.4). If a low temperature is chosen (for instance 100 K, blue curve) then most atoms will have roughly the same velocity. In terms of the model at the... 

Boltzman Superposition Principle (3,10). This principle states that the stress at a given point in the polymer is a function of the entire strain history at that point. Therefore, to each strain term in equation 9 is added an integral that represents contributions to the stress at a given time from strain increments at earlier times. 

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