Entropy function, defined

The final postulated property of the entropy function is that it vanishes in the state for which (dU/dS)v,Nj = 0, i.e. at zero temperature. An immediate implication of this postulate is that S, unlike U has a uniquely defined zero. 

The most important new concept to come from thermodynamics is entropy. Like volume, internal energy and mole number it is an extensive property of a system and together with these, and other variables it defines an elegant self-consistent theory. However, there is one important difference entropy is the only one of the extensive thermodynamic functions that has no obvious physical interpretation. It is only through statistical integration of the mechanical behaviour of microsystems that a property of the average macrosystem, that resembles the entropy function, emerges. 

That we again have a quantity whose sum over a closed cycle is zero suggests that Q/T a thermodynamic property, even though we know that Q is not a thermodynamic property. Acting on this suggestion, we define the entropy function, as Clausius did, by the equatifon... 

Application to Macromolecular Interactions. Chun describes how one can analyze the thermodynamics of a particular biological system as well as the thermal transition taking place. Briefly, it is necessary to extrapolate thermodynamic parameters over a broad temperature range. Enthalpy, entropy, and heat capacity terms are evaluated as partial derivatives of the Gibbs free energy function defined by Helmholtz-Kelvin s expression, assuming that the heat capacities integral is a continuous function. 

Figure 21. LCT computations for the reduced configurational entropy 5ic defined by Eq. (53) as a function of the reciprocal of the reduced pressure SP = P — Po /P (where P denotes the Vogel pressure) for high molar mass (M = 40001) F-S polymer fluid at fixed temperature T = 388 K. The inset illustrates the temperature dependence of P (symbols) and the line represents a fit to the data points, using P = a + bT with a = —378.6 MPa and b = 1.0825 MPa/K. (Used with permission from J. Dudowicz, K. F. Freed, and J. F. Douglas, Journal of Chemical Physics 123, 111102 (2005). Copyright 2005 American Institute of Physics.)...

In the development of the second law and the definition of the entropy function, we use the phenomenological approach as we did for the first law. First, the concept of reversible and irreversible processes is developed. The Carnot cycle is used as an example of a reversible heat engine, and the results obtained from the study of the Carnot cycle are generalized and shown to be the same for all reversible heat engines. The relations obtained permit the definition of a thermodynamic temperature scale. Finally, the entropy function is defined and its properties are discussed. 

However, the cyclic integral of an exact differential is zero and therefore QJT is an exact differential of some function. The notation dQ, is used to emphasize that the process is reversible. The new function is called the entropy function and is defined in terms of its differential, so... 

Having defined the entropy function, we must next determine some of its properties, particularly its change in reversible and irreversible processes taking place in isolated systems. (In each case a simple process is considered first, then a generalization.)... 

The only two functions actually required in thermodynamics are the energy function, obtained from the first law of thermodynamics, and the entropy function, obtained from the second law of thermodynamics. However, these functions are not necessarily the most convenient functions. The enthalpy function was defined in order to make the pressure the independent variable, rather than the volume. When the first and second laws are combined, as is done in this chapter, the entropy function appears as an independent variable. It then becomes convenient to define two other functions, the Gibbs and Helmholtz energy functions, for which the temperature is the independent variable, rather than the entropy. These two functions are defined and discussed in the first part of this chapter. 

The energy and entropy functions have been defined in terms of differential quantities, with the result that the absolute values could not be known. We have used the difference in the values of the thermodynamic functions between two states and, in determining these differences, the process of integration between limits has been used. In so doing we have avoided the use or requirement of integration constants. The many studies concerning the possible determination of these constants have culminated in the third law of thermodynamics. 

Suppose now that we decide to combine a secondfactor, in addition to the enthalpy change, AH0 in the search for an overall parameter (G) the change in which (AG°) will indicate to us whether a given reaction or process is likely to take place spontaneously. Suppose we call this second factor, the entropy change ( AS0) which will be based on the change in a function defined as entropy (S). Then, from the evidence above, it seems likely that ... 

In contradistinction to entropy Boltzmann defines a certain one-valued function of the instantaneous state distribution of the molecules, which he calls the 27-func-tion. 3 Consider a distribution, which may be arbitrarily different from the Maxwell-Boltzmann distribution, and let us denote by /At the number of those molecules whose state lies in the small range At of the state variables.54 Then we define the ff-function as... 

In thermodynamics the entropy is defined only for states of equilibrium. Indeed, Boltzmann by actually computing the H-function has shown for a very general class of gas models ([6, Chap. VI 10, Chap. V], also Qtutheorie, I, 139) that this function, is the same, apart from an additive constant, as the negative entropy if we consider states of equilibrium. For states of nonequilibrium, —If is a generalization of the thermodynamical entropy. For the combinatorial meaning of the quantity H, see Section 12d. 

Now we will connect entropy and probability quantitatively by defining the entropy function S as follows ... 

The function S, which, like the energy U, has a characteristic value for each state of the system, is called the entropy of the system. (This term was invented by Clausius, and is derived from the Greek word evTpoireiVj to change.) The entropy is defined by the above equations. The change in entropy is equal to the sum of all the quantities of heat which the system has been made to absorb reversibly and isothermally, each divided by the temperature at which the absorption of heat took place. 

This section defines the entropy function in terms of measurable macroscopic quantities to provide a basis for calculating changes in entropy for specific processes. The definition is part of the second law of thermodynamics, which is... 

The meaning of this equation is quite clear. If we define the ideal entropy functional Sideal according to... 

In previous sections we have shown how the structures of elemental materials may be rationalized on the basis of microscopic considerations. Our idea was to build up the free energy of a discrete set of competitors and then to make a free energy comparison between these competitors as a function of temperature and perhaps pressure. We next turn to the analysis of alloy phase diagrams. Here we will have to expand the scope of our previous analysis in order to account for the fact that the presence of more than one chemical constituent will at the very least alter the configurational and vibrational entropy. We define alloy in the present context to include any system in which there is more than one... 

He designed a function which he called "entropy" and defined it... 

The theorem not only allowed the calculation of chemical equilibria, it was also soon recognized as an independent third law of general thermodynamics with many important consequences. One such consequence was that it is impossible to reach the absolute zero. Another consequence was that one could define a reference point for entropy functions, such that the entropies of all elements and all perfect crystalline compounds were taken as zero at the absolute zero. 

AE and Ah describe the energy changes, but tell nothing about the favored direction for a process. To do this, one must take into account the degree of randomness or disorder of a system. The degree of randomness or disorder of a system is measured by a state function called the Entropy (S). Entropy is defined as S = kln(W), where k is the Boltzmann constant (the gas constant R divided by Avogadro s number) and W is the number of thermodynamic substates of equal energy. 

The physical process of protein folding involves a phase transition from a statistical coiled state to a uniquely compact native state. A powerful approach to define these systems is to make use of the microcanonical entropy function [18-27]. The entropy function S(E) is related to the... 

From the entropy function, one can directly define the Helmholtz free energy F as a function of energy and temperature ... 

Entropy (5) - A thermodynamic function defined such that when a small quantity of heat dQ is received by a system at temperature T, the entropy of the system is increased by dQ/ T, provided that no irreversible change takes place in the system. [1]... 

Helmholz energy (A) - A thermodynamic function defined byA = E-TS, where E is the energy, S the entropy, and Tthe thermodynamic temperature. [2]... 

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