Thermodynamics probability and

These ideas were expressed mathematically by L. Boltzmann and J. W. Gibbs in terms of a quantity ft, called the thermodynamic probability and defined as the number of ways that microscopic particles can be distributed among the states accessible to them. It is given by the general formula... 

Data analysis options include the preparation of isoenergy contour maps (CMAPS), linear energy vs. rotation angle plots (LINPLOT) and tables of local energy minima, statistical thermodynamic probabilities, and entropy terms to enable the reporting of "free" energies in addition to "conformational" energies. 

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. 

If the polymer system was able to exist in an equilibrium state only, then a strictly defined correlation between (a, ph) and (a, ph) would exist in particular conditions, according to minimum of free energy of system formation. Consequently, there would occur only one temperature at which process initiation is thermodynamically probable. In rare ca.ses there may occur different correlations between ( ph, a) and ( ph, a ), which display one and the same value of free energy minimum of system formation. 

To examine the situation with alloys in a little more detail, the Cu-Ni alloys will first be considered. Here the mutual solubility of the two oxides NiO and CU2O can probably be neglected, and these are the only two possible oxidation products. Assume for simplicity that the alloy is thermodynamically ideal, and let and Xn be the mole fractions in the alloy. Consider the reactions... 

In equation (1.17), S is entropy, k is a constant known as the Boltzmann constant, and W is the thermodynamic probability. In Chapter 10 we will see how to calculate W. For now, it is sufficient to know that it is equal to the number of arrangements or microstates that a molecule can be in for a particular macrostate. Macrostates with many microstates are those of high probability. Hence, the name thermodynamic probability for W. But macrostates with many microstates are states of high disorder. Thus, on a molecular basis, W, and hence 5, is a measure of the disorder in the system. We will wait for the second law of thermodynamics to make quantitative calculations of AS, the change in S, at which time we will verify the relationship between entropy and disorder. For example, we will show that... 

Soil water flow is decidedly episodic. During dry times the water solutions in the soil are probably fairly concentrated and not very reactive. Time-averaged reaction rates should be roughly proportional to the fraction of time reacting minerals are in contact with thermodynamically imdersaturated (and reactive) water. In a study of the relationship between denudation rate and runoff for rivers draining igneous and metamorphic rock in Kenya, Dunne (1978) obtains the relationship of (denudation rate in tons/km per year) = 0.28 (runoff in mm/ year)°. ... 

PP is probably the most thoroughly investigated system in the nanocomposite field next to nylon [127-132]. In most of the cases isotactic/syndiotactic-PP-based nanocomposites have been prepared with various clays using maleic anhydride as the compatibilizer. Sometimes maleic anhydride-grafted PP has also been used [127]. Nanocomposites have shown dramatic improvement over the pristine polymer in mechanical, rheological, thermal, and barrier properties [132-138]. Crystallization [139,140], thermodynamic behavior, and kinetic study [141] have also been done. 

The present theory can be placed in some sort of perspective by dividing the nonequilibrium field into thermodynamics and statistical mechanics. As will become clearer later, the division between the two is fuzzy, but for the present purposes nonequilibrium thermodynamics will be considered that phenomenological theory that takes the existence of the transport coefficients and laws as axiomatic. Nonequilibrium statistical mechanics will be taken to be that field that deals with molecular-level (i.e., phase space) quantities such as probabilities and time correlation functions. The probability, fluctuations, and evolution of macrostates belong to the overlap of the two fields. 

That this should be so is a corollary of the Second Law of Thermodynamics which is concerned essentially with probabilities, and with the tendency for ordered systems to become disordered a measure of the degree of disorder of a system being provided by its entropy, S. In seeking their most stable condition, systems tend towards minimum energy (actually enthalpy, H) and maximum entropy (disorder or randomness), a measure of their relative stability must thus embrace a compromise between H and S, and is provided by the Gibb s free energy, G, which is defined by,... 

Function 0)(s) determines the probability that the RW trajectory of the random walk is simultaneously also by SARJV trajectory and by its own last step realizes the state M(s). Hence, it is numerically equal to part of these SARW trajectories among general number (Idf of RW trajectories which realize the state M(s). Number L(s) of such SARJV trajectories determines the thermodynamical probability of the realization M(s) ... 

The choice of a given database as source of auxiliary values may not be straightforward, even for a thermochemist. Consistency is a very important criterion, but factors such as the publication year, the assignment of an uncertainty to each value, and even the scientific reputation of the authors or the origin of the database matter. For instance, it would not be sensible to use the old NBS Circular 500 [22] when the NBS Tables of Chemical Thermodynamic Properties [17], published in 1982, is available. If we need a value for the standard enthalpy of formation of an organic compound, such as ethanol, we will probably prefer Pedley s Thermodynamic Data and Structures of Organic Compounds [15], published in 1994, which reports the error bars. Finally, if we are looking for the standard enthalpy of formation of any particular substance, we should first check whether it is included in CODATA Key Values for Thermodynamics [16] or in the very recent Active Thermochemical Tables [23,24],... 

Traditional thermodynamics gives a clear definition of entropy but unfortunately does not tell us what it is. An idea of the physical nature of entropy can be gained from statistical thermodynamics. Kelvin and Boltzmann recognised diat there was a relationship between entropy and probability (cf., disorder) of a system with the entropy given by... 

---