Thermodynamics definitions for

Comparison of the combined first/second law (4.28) with (4.30) leads to the more general and rigorous thermodynamic definitions for the intensive properties T, —P respectively conjugate to the extensive properties S, V ... 

If this parameter is assumed to be the same for all vibrations, one can obtain a bulk thermodynamic definition for y. The bulk Griineisen parameter is found to be about 4 for polymers from the effect of pressure on the velocity of sound. The data suggest that for the heat capacity only the interchain contribution should be taken into account. With this assumption, an order of magnitude calculation shows that the bulk Griineisen parameter for proteins is of the same order of magnitude as that of polymers. This suggests that the thermal expansion and the compressibility of proteins reflect primarily the movement between the secondary structures. These movements are reflected in the low frequency part of the... 

It was made clear in Chapter II that the surface tension is a definite and accurately measurable property of the interface between two liquid phases. Moreover, its value is very rapidly established in pure substances of ordinary viscosity dynamic methods indicate that a normal surface tension is established within a millisecond and probably sooner [1], In this chapter it is thus appropriate to discuss the thermodynamic basis for surface tension and to develop equations for the surface tension of single- and multiple-component systems. We begin with thermodynamics and structure of single-component interfaces and expand our discussion to solutions in Sections III-4 and III-5. 

Thermodynamic Properties The variation in solvent strength of a supercritical fluid From gaslike to hquidlike values may oe described qualitatively in terms of the density, p, or the solubihty parameter, 6 (square root of the cohesive energy density). It is shown For gaseous, hquid, and SCF CO9 as a function of pressure in Fig. 22-17 according to the rigorous thermodynamic definition ... 

To extract a desired component A from a homogeneous liquid solution, one can introduce another liquid phase which is insoluble with the one containing A. In theory, component A is present in low concentrations, and hence, we have a system consisting of two mutually insoluble carrier solutions between which the solute A is distributed. The solution rich in A is referred to as the extract phase, E (usually the solvent layer) the treated solution, lean in A, is called the raffinate, R. In practice, there will be some mutual solubility between the two solvents. Following the definitions provided by Henley and Staffin (1963) (see reference Section C), designating two solvents as B and S, the thermodynamic variables for the system are T, P, x g, x r, Xrr (where P is system pressure, T is temperature, and the a s denote mole fractions).. The concentration of solvent S is not considered to be a variable at any given temperature, T, and pressure, P. As such, we note the following ... 

We have seen how to calculate q for the isochoric and isobaric processes. We indicated in Chapter 1 that q = 0 for an adiabatic process (by definition). For an isothermal process, the calculation of q requires the application of other thermodynamic equations. For example, q can be obtained from equation (2.3) if AC and w can be calculated. The result is... 

Fig. 1.5 Schematic representation of the evolution of life from its precursors, on the basis of the definition of life given by the authors. If bioenergetic mechanisms have developed via autonomous systems, the thermodynamic basis for the beginning of the archiving of information, and thus for a one-polymer world such as the RNA world , has been set up. Several models for this transition have been discussed. This phase of development is possibly the starting point for the process of Darwinian evolution (with reproduction, variation and heredity), but still without any separation between genotype and phenotype. According to the authors definition, life begins in exactly that moment when the genetic code comes into play, i.e., in the transition from a one-polymer world to a two-polymer world . The last phase, open-ended evolution, then follows. After Ruiz-Mirazo et al. (2004)...

It is useful to keep in mind that the equilibrium constant is defined in terms of the difference in standard state free energies between products and reactants. Thus, for the reaction written in Equation 4.36 we obtained Equation 4.50, essentially by thermodynamic definition,... 

GENERAL EQUATIONS FOR THE ENTROPY OF GASES 143 conclude from Equation (6.111) that a natural thermodynamic definition of T is... 

In order to derive a thermodynamic Haldane relation for this mechanism, the true dissociation constants for B (i.e., kjks) and for P (i.e., k lkd) are needed. From the definitions for the various kinetic parameters (See Ordered Bi Bi Mechanism) it is readily seen that... 

It is more problematical to define the third law of thermodynamics compared to the first and second laws. Experimental work by Richards (1902) and Nemst (1906) led Nemst to postulate that, as the temperature approached absolute zero, the entropy of the system would also approach zero. This led to a definition for the third law of thermodynamics that at a temperature of absolute zero the entropy of a condensed system would also be zero. This was further refined by Planck (1911) who suggested this be reworded as the entropy of a pure element or substance in a perfect crystalline form is zero at absolute zero. 

In practice, then, it is fairly straightforward to convert the potential energy determined from an electronic structure calculation into a wealth of thennodynamic data - all that is required is an optimized structure with its associated vibrational frequencies. Given the many levels of electronic structure theory for which analytic second derivatives are available, it is usually worth the effort required to compute the frequencies and then the thermodynamic variables, especially since experimental data are typically measured in this form. For one such quantity, the absolute entropy 5°, which is computed as the sum of Eqs. (10.13), (10.18), (10.24) (for non-linear molecules), and (10.30), theory and experiment are directly comparable. Hout, Levi, and Hehre (1982) computed absolute entropies at 300 K for a large number of small molecules at the MP2/6-31G(d) level and obtained agreement with experiment within 0.1 e.u. for many cases. Absolute heat capacities at constant volume can also be computed using the thermodynamic definition... 

The requirements for the thermodynamic definition of solubility are 1) well defined initial solid and final solution states, and 2) establishment of equilibrium between these two states. Under these conditions, the solubility at a given temperature and pressure is the concentration of the sample in solution. At a given temperature and pressure, the solid state is the stable crystalline state and the solution state is... 

What are the empirical inductive laws on which thermodynamics rests For future reference, Table 2.1 lists the six general statements IL-l-IL-6 of observational experience on which the present exposition will be based. Several of these require additional definitions or explanations before they can be properly understood. Each will be introduced explicitly in the text as its definitional basis is properly laid and its logical role in the formal construction of the theory becomes apparent. 

Definition For thermodynamic purposes, a property of the system is the reproducible resultb of a definite measurement on the system, verifiable by an independent observer0 and capable of changed... 

The distinction between reversible and irreversible work is one of the most important in thermodynamics. We shall first illustrate this distinction by means of a specific numerical example, in which a specified system undergoes a certain change of state by three distinct paths approaching the idealized reversible limit. Later, we introduce a formal definition for reversible work that summarizes and generalizes what has been learned from the path dependence in the three cases. In each case, we shall evaluate the integrated work w 2 from the basic path integral,... 

What is the physical nature of the Gibbs free energy, and what is free about it We can consider this question first from the viewpoint of fundamental thermodynamic definitions, with no microscopic molecular connotations. For a reversible change of state carried out under conditions of constant T and P, we can write... 

The Buckingham statement fares no better in this regard, for the concept of a true equilibrium state is no less tautological than that of a perfect crystal. Moreover, the implied restriction to true equilibrium states (presumably, those for which no kinetic conversion is possible on any timescale) is even more strongly at odds with fundamental thermodynamic definitions, as outlined in Sections 2.10 and 2.11. Indeed, such a restriction, if enforced zealously, would preclude application of thermodynamics to any chemical system—past, present, or future—except for the final universal Warmetod state.]... 

Finally, let us note some interesting identities for other thermodynamic potentials that follow from Equation (6.31). From the energy identity U = TS — PV + ix tj and basic thermodynamic definitions, we can readily infer that... 

Equations 27 and 28 permit a simple comparison to be made between the actual composition of a chemical system in a given state (degree of advancement) and the composition at the equilibrium state. If Q K, the affinity has a positive or negative value, indicating a thermodynamic tendency for spontaneous chemical reaction. Identifying conditions for spontaneous reaction and direction of a chemical reaction under given conditions is, of course, quite commonly applied to chemical thermodynamic principle (the inequality of the second law) in analytical chemistry, natural water chemistry, and chemical industry. Equality of Q and K indicates that the reaction is at chemical equilibrium. For each of several chemical reactions in a closed system there is a corresponding equilibrium constant, K, and reaction quotient, Q. The status of each of the independent reactions is subject to definition by Equations 26-28. 

In this chapter we will review some of the principles of thermochemistry, with particular attention to the air-water vapor system. Basic definitions in thermodynamics are reviewed along with important physical properties and definitions for gaseous mixtures. It is important that these definitions be learned early on. Note, however, that this chapter is only meant as a review. The references listed at the end of this chapter should be consulted for a detailed treatment of these subjects. Further, example problems are included at the end of the chapter to stress principles discussed. 

However, it is useful, to provide a thermodynamic definition of a first-order transition. Specifically, it is one in which there is a discontinuity in a first derivative of the Gibbs free energy. The advantage of this definition is the guidance it provides for the experimental study of phase transitions. A useful expression for the free energy in this regard is... 

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 ... 

Let us consider the concept of "relaxation in more detail since no accurate definition for it has been given previously. The term "relaxation is often used for the process by which either an equilibrium or a steady state is achieved in the system, and the relaxation time is treated as the time to achieve complete or partial thermodynamic equilibrium. It is evident that, in this context, the difference between "equilibrium and "steady state is insignificant. The concept of "relaxation time is often used for the time during which a certain function characterizing the deviation from the equilibrium or the steady state diminishes by e ( 2.718) times compared with its initial value. It is evident, however, that this definition is only correct for one-dimensional linear systems. For multi-dimensional linear systems, a spectrum of relaxation times must be used. For non-linear systems, the application of these definitions is correct only in the neighbourhood of a singular point. 

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