Thermodynamic probability usefulness

Chemists learn to use the thermodynamic probability almost instinctively in a qualitative manner it is quantitatively related to entropy through an equation due to Boltzmann ... 

A moment s reflection will convince us that these probabilities can be used as thermodynamic probabilities in Eq. (3.21) to calculate the entropy change on stretching ... 

Using the taken symbols, we get the formulae for configuration energy Ei and thermodynamic probability Wi ... 

Beta values represent the fraction of free atoms present in the hot flame gases of the flame indicated. These values have been taken from various sources and were either experimentally measured or calculated from thermodynamic data using the assumption of local thermodynamic equilibrium in the flame. These values do not have very good agreement within each element however, the values do provide an indication of the probable sensitivity of the particular flame. 

It is probably useful to view the stability constants and the associated thermodynamic parameters in terms of spin-pairing energy or the double-double effect. In the case of... 

It has been contended that the direct attack of oxygen on the double bond has low thermodynamic probability (21, 22), and it has been considered that trace metal contaminants catalyze the initiation of autoxidation by producing free radicals through electron transfer. Alternative pathways are as follows, using cobalt as an... 

In order to determine the statistical thermodynamic probabilities and entropies for the conformational energy surface, a set of "dots" is plotted indicating the angular values of the set of conformers which define the surface. The joystick curser control 1s used to select the set of conformers which occupy a given low energy region. The chosen "dots" are replaced by "asterisks" (to avoid duplication) and the probability and entropy terms are tabulated. Tables of probabilities and entropies may also be produced. 

The 2,2-regioisomer was favoured using the ephedrinium salt instead of TBAB. It is formed from the thermodynamic enolate while the 2,6-regioisomer came from the kinetic enolate. The results obtained show the preference for the thermodynamic enolate using the ephedrinium salt as catalyst, probably as a result of the stabilization of the enolate through a tt-tt interaction between the catalyst and the enolate. 

In addition, there is a relation between entropy and disorder disordered states have higher probabilities than ordered states. In general, the changes that are accompanied by an increase in entropy result in increased molecular disorder. Thus, entropy is also a measure of the molecular disorder of the state. Although disorder may be related to entropy qualitatively, the amount of disorder is a subjective concept and it is much better to relate entropy to probability rather than to disorder. Such concepts can be described in terms of thermodynamic probabilities (Q) in statistical mechanics. The entropy of a system is a function of the probability of the thermodynamic state of this system, S = /( 2). We know from statistical mathematics that only logarithmic functions satisfy probabilistic equations, so that we may use... 

The quantities and (therefore, A, A3, A, -, and appearing in the probability P p are functions of the most probable number density p instead of the average number density p. In the next two sections we are interested in evaluating the cell-pair correlation function and thermodynamic quantities, using p as an independent variable rather than using pg. Hence it is convenient to transform A and A as a function of p at this stage by expanding A2, and A3 around p ... 

Fig. 5 Illustration of a thermodynamic box used to derive a relationship between the rate constants f(P),f (0),f(a) and / (a). A similar box (not shown) that involves the g rates can be drawn. Irrespective of the magnitude of Ap, or of whether there is a load force, the product of the probabilities in the clockwise direction must be equal to the product of the probabilities in the counter-clockwise direction (see (6)), resulting in (7)...

To connect the microscopic and macroscopic descriptions of matter, scientists have developed the field of statistical thermodynamics, which uses the tools of statistics and probability to link the microscopic and macroscopic worlds. Here we show how entropy, which is a property of bulk matter, can be connected to the behavior of atoms and molecules. Because the mathematics of statistical thermodynamics is complex, our discussion will be largely conceptual. 

As a result of the above discussion what can be said about the meaning of entropy if our basic assumptions are correct It is evidently a measure of the mixed-up-ness of a system— a phrase used by Gibbs. It can also be said that high entropy states are those which have a high probability. The quantity Q is sometimes referred to as the thermodynamic probability , because the ratio for the... 

The equation of state of rubber elasticity will now be calculated via statistical thermodynamics, rather than the classical thermodynamics of Section 9.5. Statistical thermodynamics makes use of the probability of finding an atom, segment, or molecule in any one place as a means of computing the entropy. Thus tremendous insight is obtained into the molecular processes of entropic phenomena, although classical thermodynamics illustrates energetic phenomena adequately. 

The concept of entropy was developed so chemists could understand the concept of spontaneity in a chemical system. Entropy is a thermodynamic property that is often associated with the extent of randomness or disorder in a chemical system. In general if a system becomes more spread out, or more random, the system s entropy increases. This is a simplistic view, and a deeper understanding of entropy is derived from Ludwig Boltzmann s molecular interpretation of entropy. He used statistical thermodynamics (which uses statistics and probability) to link the microscopic world (individual particles) and the macroscopic world (bulk samples of particles). The connection between the number of microstates (arrangements) and its entropy is expressed in the Boltzmann equation, S = kin W, where W is the number of microstates and k is the Boltzmann constant, 1.38 x 10 JK L... 

The Boltzmann equation is generally used to obtain an expression for AS of simple mixtures (mixtures of solvent-solvent or solvent-simple solute molecules) from the number of different arrangements ft (or the thermodynamic probabilities) of the solute and solvent molecules in the system For simple systems, the volume elements of solution are modeled by a three-dimensional lattice, where solute or solvent molecules can occupy any cell within the... 

MC simulations aim to realize the probability distributions cmisidered in statistical thermodynamics numerically using random numbers and to calculate the desired averages of various observables in the system using these distributions [18-20]. There exist numerous extensive reviews describing the specific aspects of MC methods for polymers [77, 82, 84, 90, 96], and thus we focus here only on some salient features that are most relevant when one addresses the estimation of the equation of state and phase equilibria of systems containing many polymers. 

In this section, we discuss how one, guided by the principles of nonequilibrium thermodynamics, can use the Monte Carlo technique to drive an ensemble of system configurations to sample statistically appropriate steady-state nonequilibrium phase-space points corresponding to an imposed external field [161,164,193-195]. For simplicity, we limit our discussion to the case of an unentangled polymer melt. The starting point is the probability density function of the generalized canonical... 

Using the Boltzmann distribution function, which weights the possible values of x according to their thermodynamic probability, Kittel calculated the average displacentent x. 

Assuming the form (5.3) for the potential energy cp(Wg), calculate the average displacement and the average force constant by using the Boltzmann distribution function which weights the possible values of Wq and f = 8 cp/3Wq according to their thermodynamic probabilities ... 

In this book I present the theory for enumerating the microscopic states of equilibrium systems and determining their probabilities. I then discuss how to use this knowledge to derive thermodynamic properties, using Eqs. 1.21-1.24, or other similar ones for different ensembles. 

The so-called thermodynamic probability of a macrostate s statistical weights is used in statistical physics. The thermodynamic probability W of a system macrostate is determined by a number of various microstates that can assure the given maCTOstate. 

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