Boltzmann relationship

The potential of mean force is a useful analytical tool that results in an effective potential that reflects the average effect of all the other degrees of freedom on the dynamic variable of interest. Equation (2) indicates that given a potential function it is possible to calculate the probabihty for all states of the system (the Boltzmann relationship). The potential of mean force procedure works in the reverse direction. Given an observed distribution of values (from the trajectory), the corresponding effective potential function can be derived. The first step in this procedure is to organize the observed values of the dynamic variable, A, into a distribution function p(A). From this distribution the effective potential or potential of mean force, W(A), is calculated from the Boltzmann relation ... 

We look at the simple gas laws to explore the behaviour of systems with no interactions, to understand the way macroscopic variables relate to microscopic, molecular properties. Finally, we introduce the statistical nature underlying much of the physical chemistry in this book when we look at the Maxwell-Boltzmann relationship. 

A 0f is the standard potential difference between phases a and p for this ion, kf is the standard rate constant for transfer of ion / and a is the charge-transfer coefficient. Concentrations c (a) and c (/3) correspond to the immediate vicinity of the phase boundary and are functions of the potential differences in the diffuse double layers according to the Boltzmann relationship... 

The ions are distributed according to a Boltzmann relationship, and the well-known Poisson-Boltzmann (P-B) equation can be derived ... 

Similarly, when a voltage V is applied across a semiconductor/metal junction, the total voltage drop in the semiconductor depletion region is Vbi -F V, so we obtain an analogous Boltzmann relationship away from eqnihbrinm ... 

This equation represents the physical sitnation that the electron concentration at the semiconductor surface can be either increased or decreased through the use of an additional voltage. This applied voltage controls the surface carrier concentration in the same fashion as the built-in voltage, so the same Boltzmann relationship applies. 

To describe the conductivity of an intrinsic semiconductor sample quantitatively, we need to calculate the concentrations of both types of charge carriers in the solid. The key quantity that controls the equilibrium concentration of electrons and holes in an intrinsic semiconductor is the band gap. Because the thermal excitation energy required to produce an electron and a hole is equal to Eg, the intrinsic carrier concentrations can be related to Eg using the Boltzmann relationship ... 

In this equation, k is the Boltzmann constant, T is the absolute temperature, and and pi are the electron and hole concentrations at equilibrium in the intrinsic semiconductor, respectively. Both and pi are expressed as particles cm , which is often abbreviated as cm . The constant in equation (5) is actually a function of the band structure of the solid and is numerically equal to the product of and Av. It is clear that this type of Boltzmann relationship should correctly describe the concentrations of electrons in the conduction band and of holes in the valence band that exist as a result of thermal excitation across the band gap at any given temperature. 

Thus, the frequency of the nuclear spin transition depends on the strength of the applied magnetic field and magnitude of Y for the nucleus. The transition frequency of a particular nucleus is one of the important parameters in NMR and is related to the chemical shift as explained in the next section. The value of y also determines sensitivity of a given isotope, as AE (equation 4) determines the ratio of the number of spins in the upper (ti ) and lower (ti2) energy levels according to the Boltzmann relationship ... 

At equilibrium, the relative population of excited and ground state atoms at a given temperature can be considered using the Boltzmann relationship. N and N0 of any two states is given by ... 

At thennal equilibrium, the defect fraction, , in a perfect crystal is given by a Boltzmann relationship ... 

Atom-ion equilibria in flames create a number of important consequences in flame spectroscopy, b or example, intensities ol atomic emission or absorption lines for the alkali metals, particularly potassium, rubidium, and cesium, are affected by leniperalure in a complex way. Increased temperature cause an increase in the population of excited atoms, according lo the Boltzmann relationship (Kqualion S-l). Counteracting this effect, however, is a decrease in concentration of atoms resulting from ionization. Thus, under some circumstances a decrease in emission or abst>rp-lion may be observed in hotter flames. It is or this reason that lower e.xciialion Icmperaliircs are usually spcciliod for the deierminaiion of alkali metals. 

If the two processes are in thermodynamic equilibrium these rates will be equal. The ratio of the populations of the two levels N /N will equate to the Boltzmann relationship and conform to Planck s radiation law. From this can be derived the relationship... 

The loss of entropy of a single polymer chain attached to a colloidal particle, assumed to be a plate, on compression is found from the Boltzmann relationship to be... 

We can now consider the passage of ions through cell membranes. Firstly, we note that since cell membranes possess a negative potential at physiological pH, arising from one or a combination of the various processes already described in this section, then as predicted by the Boltzmann relationship of Eq. (9) only cations will tend to diffuse towards... 

Within this Gaussian distribution function (r )o applies to the network chains both in the unstretched and stretched state. The free energy of such a chain is described by a Boltzmann relationship [23] ... 

On the other hand, according to the Boltzmann relationship and the Gaussian distribution of chain conformations, the conformational entropy loss is proportional to kT(RIRof. The total conformation entropy loss of the single coil is thus scaled as... 

Here A5f is the entropy of fusion per segment. The entropy change in the second step arises from the transformation of the chain length distribution in the amorphous portion given by Eq. (8.38) to that given by Eq. (8.2). From the Boltzmann relationship 5 = In IT this entropy change can be expressed as... 

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