Entropy, absolute definition

The energy difference provided by MO calculations, with or without the inclusion of solvent, serves as a direct measure of the anomeric energy, AE(AE3), when its absolute definition by Eq. 4 is used. However, the more-frequent use of a relative definition by Gibbs energy difference in Eq. 1 warrant an attempt to recalculate the AE(AE3) data to the values AG(AE 1). Such a procedure is, of necessity, an approximation, because the assumption that AG° = AE(AE3) neglects the entropy and volume changes of conformers owing to absence of suitable information, and the cyclohexane-based and solvent-independent A values must be used. 

The third law allows the calculation of an absolute value for entropy. By definition the entropy of a perfect crystalline substance is zero at zero absolute temperature. For a pure component ideal gas at the temperature T we may write ... 

In 1877, the Austrian physicist Ludwig Boltzmann proposed a molecular definition of entropy that enables us to calculate the absolute entropy at any temperature (Fig. 7.6). His formula provided a way of calculating the entropy when measurements could not be made and deepened our insight into the meaning of entropy at the molecular level. The Boltzmann formula for the entropy is... 

Does this make sense in this case Absolutely. We are first going from a system containing 2 moles of liquid fuel to 2 moles of gaseous fuel - a big increase in entropy. Then before reaction we have 27 moles of gas, and after reaction we have a system containing 34 moles of gas. Entropy involves an increase in the relative positions of the molecules with respect to each other and the energies they can have. The entropy of this system has definitely increased after the combustion reaction has occurred. 

As was the case with energy, the definition of entropy permits only a calculation of differences, not an absolute value. Integration of Equation (6.48) provides an expression for the finite difference in entropy between two states ... 

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. 

The third law of thermodynamics says that the entropy of pure, perfect crystalline substance is zero at absolute zero. But, in actual practice, it has been found that certain chemical reactions between crystalline substance, do not have DS = 0 at 0°K, which indicates that exceptions to third law exist. Such exceptional reactions involve either ice, CO, N2O or H2. It means that in the crystalline state these substances do not have some definite value of entropy even at absolute zero. This entropy is known as Residual Entropy. At 0°K the residual entropies of some crystalline substances are... 

The differential change in entropy for a closed system from one state to another is, by definition, directly proportional to the change in reversible heat, dQrev, and inversely proportional to the absolute temperature, T ... 

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... 

Let us ask what the randomness that we associated with entropy in Chap. I means in terms of the assembly. A random system, or one of large entropy, is one in which the microscopic properties may be arranged in a great many different ways, all consistent with the same large-scale behavior. Many different assignments of velocity to individual molecules, for instance, can be consistent with the picture of a gas at high temperatures, while in contrast the assignment of velocity to molecules at the absolute zero is definitely fixed all the molecules are at rest. Then... 

The entropy is most easily determined as a function of volume and temperature from the equation (dS/dT)v = Cv/T. At the absolute zero of temperature, the entropy of a solid is zero independent of its volume or pressure. The reason goes back to our fundamental definition of entropy... 

Equation (5.20) is the basis for calculation of absolute entropies. In the case of an ideal gas, for example, it gives the probability ft for the equilibrium distribution of molecules among the various quantum states determined by the translational, rotational, and vibrational energy levels of the molecules. When energy levels are assigned in accord with quantum mechanics, this procedure leads to a value for the energy as well as for the entropy. From these two quantities all other thermodynamic properties can be evaluated from definitions (of H. G,... 

Calculate the entropy of vaporization ASvap. By definition, ASV = A Hnp/Tmp, where the numerator is the heat of vaporization and the denominator is the absolute temperature at which the vaporization takes place. Since the heat of vaporization is given on a weight basis, it must be multiplied by the molecular weight to obtain the final result on a molar basis. Thus... 

The science of thermodynamics has evolved from the mathematical treatment of common observations related to heat and work. The terms thus evolved have primarily mathematical definition and to find physical significance have been a somewhat tortuous process. We have seen that Entropy, which is the mathematical ratio of energy per unit absolute temperature, has been found to represent the extent of disorder. 

The entropy >S, like the energy C7, is a function of volume and temperature. It is not, however, possible without further assumption to predict the nature of the S-T curve in the neigh-bom hood of the absolute zero. By the definition of entropy... 

Provisionally T will be identified with the temperature on the ideal gas scale. By assuming the above property as a definition of the absolute scale, a crucial experiment can be devised (Joule and Thomson, 1862) which exactly verifies this conclusion. If a standard entropy, under chosen conditions, is lenoted by 5° then the entropy is generally... 

The third law, like the two laws that precede it, is a macroscopic law based on experimental measurements. It is consistent with the microscopic interpretation of the entropy presented in Section 13.2. From quantum mechanics and statistical thermodynamics, we know that the number of microstates available to a substance at equilibrium falls rapidly toward one as the temperature approaches absolute zero. Therefore, the absolute entropy defined as In O should approach zero. The third law states that the entropy of a substance in its equilibrium state approaches zero at 0 K. In practice, equilibrium may be difficult to achieve at low temperatures, because particle motion becomes very slow. In solid CO, molecules remain randomly oriented (CO or OC) as the crystal is cooled, even though in the equilibrium state at low temperatures, each molecule would have a definite orientation. Because a molecule reorients slowly at low temperatures, such a crystal may not reach its equilibrium state in a measurable period. A nonzero entropy measured at low temperatures indicates that the system is not in equilibrium. 

By definition, the entropy change in a reaction is equal to the heat ihange, when the process is carried out reversibly, divided by the absolute emperature. If the amount of heat liberated in a reversible cell when it >perates could be measured, the entropy change for the reaction could be letermined. Because of experimental difficulties this method does not ippear to have been used. 

The chemical potential provides the fundamental criteria for determining phase equilibria. Like many thermodynamic functions, there is no absolute value for chemical potential. The Gibbs free energy function is related to both the enthalpy and entropy for which there is no absolute value. Moreover, there are some other undesirable properties of the chemical potential that make it less than suitable for practical calculations of phase equilibria. Thus, G.N. Lewis introduced the concept of fugacity, which can be related to the chemical potential and has a relationship closer to real world intensive properties. With Lewis s definition, there still remains the problem of absolute value for the function. Thus,... 

Rubber Company Handbook (Weast, 1987) is one of the more commonly available sources. More complete sources, including some with data for a range of temperatures, are listed in the references at the end of the chapter. Note that many tabulations still represent these energy functions in calories and that it may be necessary to make the conversion to Joules (1 cal = 4.1840J). Because of the definition of the energy of formation, elements in their standard state (carbon as graphite, chlorine as CI2 gas at one bar, bromine as Br2 liquid, etc.) have free energies and enthalpies of formation equal to zero. If needed, the absolute entropies of substances (from which AS may be evaluated) are also available in standard sources. 

The constant of integration is independent of the particular state in which the system may happen to exist at any time (The constant necessanly vanishes when we write down a definite integral, for m this case we subtract the values characteristic of the initial state from those characteristic of the final state, with the result that the integration constant disappears ) The term now stands for the entropy possessed by the system under the conditions considered, ie we have assumed that the lower limit of temperature from which the integration is carried out is absolute zero Denoting the term iS by the symbol S and the integration constant by the symbol S, we can write—... 

The Third Law of Thermodynamics relates the change of entropy to temperature, stating that the limiting value of the entropy of a system can be taken as zero as the absolute value of temperature approaches zero. Thus, the absolute entropy is zero for all perfect crystalline substances at absolute zero temperature, and from this definition it is clear that entropy has a universal reference state, while enthalpy and free energy quantities do not. The Third Law allows us to calculate the absolute value of entropy by integrating Equation (111) so that... 

Because the concept of entropy is generally not familiar to hydrologists, a brief introduction is probably in order. A thorough and rigorous explanation can be obtained from standard works such as those by Fast (I), Fitts (2), Katchalsky and Curran (3), Klotz (4), Lewis and Randall (5), and Prigogine (6). A statement of the second law of thermodynamics is generally used as a definition of entropy of a system as follows dS DQ/T, where dS is an infinitesimal change in entropy for an infinitesimal part of a process carried out reversibly, DQ is the heat absorbed, and T is the absolute temperature at which the heat is absorbed. In one sense, entropy is a mathematical function for the term... 

The concept of entropy is a difficult one. An understanding of it comes with repeated encounters in different contexts. Although a formal definition will be given later, it is sufficient for the time being to associate it pictorially with the degree of randomness of a material, as exemplified in the processes discussed above. It is possible to assign absolute numerical values to this entropy, 5°, which refer to individual materials in a particular physical state. In Table 4.1 values are given for a few compounds. 

The present author has the impression from the literature on the stability of diazomethane relative to diazirine that two different physico-chemical phenomena were called (thermal) stability in some of the publications, namely the thermodynamic stability, as defined by the free energy of formation AGf and the free enthalpy of formation A//f for the (hypothetical) formation of a compound from the elements in a gas phase reaction under standardized conditions (298 K, 1 mol). AGf and A//f are related to one another by the free entropy A5f in the Gibbs-Helmholtz equation AGf = A/ff-TASf. The absolute values of AGf, A/ff and ASf do not give definite information on the stability of a compound, as this word is used in the everyday language of a chemist, because it is related to an unrealistic chemical process, namely the formation from the elements. If A/ff is known, however, for a given compound and for a real product of one of its reactions, the difference in magnitude of the two free enthalpies tells us whether this reaction is likely to take place, but we cannot depict at all, at least in principle, the half-life of such a reaction. 

Actually, the ideal gas law in Eq. (2.46) is sound when the gas is in contact with a thermal reservoir. The thermal reservoir has by definition an infinite capacity. Here, we want to focus that interest to the case of finite entropy and the consequences for heat capacity, when the absolute temperature approaches zero, as important in statements on the third law of thermodynamics. 

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