Perfect crystal, entropy

Thermodynamics is concerned with the relationship between heat energy and work and is based on two general laws, the 1st and 2nd laws of thermodynamics, which both deal with the interconversion of the different forms of energy. The 3rd law states that at the absolute zero of temperature the entropy of a perfect crystal is zero, and thus provides a method of determining absolute entropies. 

The entropies of all perfect crystals approach zero as the absolute temperature approaches zero. 

That is, S —> 0 as T - 0. The perfect crystal part of this statement of the third law refers to a substance in which all the atoms are in a perfectly orderly array, and so there is no positional disorder. The T— 0 part of the statement implies the absence of thermal motion-—thermal disorder vanishes as the temperature approaches zero. As the temperature of a substance is raised from zero, more orientations become available to the molecules and their thermal disorder increases. Thus we can expect the entropy of any substance to he greater than zero above T = 0. 

SOLUTION (a) Because T = 0, all motion has ceased. We expect the sample to have zero entropy, because there is no disorder in either location or energy. This value is confirmed by the Boltzmann formula because there is only one way of arranging the molecules in the perfect crystal, W = 1. 

In the case, however, of solid solutions where variability of composition is possible we may expect increased entropy owing to the possibilities of increased randomness of arrangement just as with liquid and gaseous solutions. Also, other special cases might arise where a greater degree of randomness would be possible than in pure, perfect crystals, but such cases could be given a special treatment and would cause no confusion. 

The entropy of a perfect crystal at the absolute zero is not dependent on the complexity of the unit of crystal structure. 

To reach W = 1 and S = 0, we must remove as much of this vibrational motion as possible. Recall that temperature is a measure of the amount of thermal energy in a sample, which for a solid is the energy of the atoms or molecules vibrating in their cages. Thermal energy reaches a minimum when T = 0 K. At this temperature, there is only one way to describe the system, so — 1 and — 0. This is formulated as the third law of thermodynamics, which states that a pure, perfect crystal at 0 K has zero entropy. We can state the third law as an equation, Equation perfect crystal T=0 K) 0... 

The third law of thermodynamics establishes a starting point for entropies. At 0 K, any pure perfect crystal is completely constrained and has S = 0 J / K. At any higher temperature, the substance has a positive entropy that depends on the conditions. The molar entropies of many pure substances have been measured at standard thermodynamic conditions, P ° = 1 bar. The same thermodynamic tables that list standard enthalpies of formation usually also list standard molar entropies, designated S °, fbr T — 298 K. Table 14-2 lists representative values of S to give you an idea of the magnitudes of absolute entropies. Appendix D contains a more extensive list. 

The third law of thermodynamics states that the entropy of a perfect crystal is zero at a temperature of absolute zero. Although this law appears to have limited use for polymer scientists, it is the basis for our understanding of temperature. At absolute zero (-273.14 °C = 0 K), there is no disorder or molecular movement in a perfect crystal. One caveat must be introduced for the purist - there is atomic movement at absolute zero due to vibrational motion across the bonds - a situation mandated by quantum mechanical laws. Any disorder creates a temperature higher than absolute zero in the system under consideration. This is why absolute zero is so hard to reach experimentally ... 

In a perfect crystal at 0 K all atoms are ordered in a regular uniform way and the translational symmetry is therefore perfect. The entropy is thus zero. In order to become perfectly crystalline at absolute zero, the system in question must be able to explore its entire phase space the system must be in internal thermodynamic equilibrium. Thus the third law of thermodynamics does not apply to substances that are not in internal thermodynamic equilibrium, such as glasses and glassy crystals. Such non-ergodic states do have a finite entropy at the absolute zero, called zero-point entropy or residual entropy at 0 K. 

Statistical mechanics affords an accurate method to evaluate ArSP, provided that the necessary structural and spectroscopic parameters (moments of inertia, vibrational frequencies, electronic levels, and degeneracies) are known [1], As this computation implicitly assumes that the entropy of a perfect crystal is zero at the absolute zero, and this is one of the statements of the third law of thermodynamics, the procedure is called the third law method. 

There is a third law of thermodynamics. It can be stated in the following way The entropy of a perfect crystal at 0 K is zero. A perfect crystal is one with no lattice defects. The third law gives rise to the concept of absolute entropy. There will be no further mention of the third law in this book. 

The third law of thermodynamics states that, for a perfect crystal at absolute zero temperature, the value of entropy is zero. The entropy of a molecule at other temperatures can be computed from the heat capacities and heats of phase changes using... 

The defect concentration comes from thermodynamics. While we will discuss thermodynamics of solids in more detail in Chapter 2, it is useful to introduce some of the concepts here to help us determine the defect concentrations in Eq. (1.43). The free energy of the disordered crystal, AG, can be written as the free energy of the perfect crystal, AGq, plus the free energy change necessary to create n interstitials and vacancies (n, =n-o = n), Ag, less the entropy increase in creating the interstitials ASc at a temperature T ... 

Entropy also plays a role in the Third Law of Thermodynamics, which states that the entropy of a perfect crystal is zero at zero absolute temperature. 

Let s calculate the entropy of a tiny solid made up of four diatomic molecules of a binary compound such as carbon monoxide, CO. Suppose the four molecules have formed a perfectly ordered crystal in which all molecules are aligned with their C atoms on the left. Because T = 0, all motion has ceased (Fig. 7.5). We expect the sample to have zero entropy, because there is no disorder in either location or energy. This value is confirmed by the Boltzmann formula because there is only one way of arranging the molecules in the perfect crystal, W = l and S = k In 1 =0. Now suppose thar the molecules can lie with their C atoms on either side yet still have the same total energy (Fig. 7.6). The total number of ways of arranging the four molecules is... 

The Third Law of Thermodynamics postulates that the entropy of a perfect crystal is zero at 0 K. Given the heat capacity and the enthalpies of phase changes, Eq. (12-3) allows the calculation of the standard absolute entropy of a substance, S° = AS for the increase in temperature from 0 K to 298 K. Some absolute entropies for substances in thermodynamic standard states are listed in Table 12-1. 

What is complexity There is no good general definition of complexity, though there are many. Intuitively, complexity lies somewhere between order and disorder, between regularity and randomness, between perfect crystal and gas. Complexity has been measured by logical depth, metric entropy, information content (Shannon s entropy), fluctuation complexity, and many other techniques some of them are discussed below. These measures are well suited to specific physical or chemical applications, but none describe the general features of complexity. Obviously, the lack of a definition of complexity does not prevent researchers from using the term. 

Equation (16-2) allows the calculations of changes in the entropy of a substance, specifically by measuring the heat capacities at different temperatures and the enthalpies of phase changes. If the absolute value of the entropy were known at any one temperature, the measurements of changes in entropy in going from that temperature to another temperature would allow the determination of the absolute value of the entropy at the other temperature. The third law of thermodynamics provides the basis for establishing absolute entropies. The law states that the entropy of any perfect crystal is zero (0) at the temperature of absolute zero (OK or -273.15°C). This is understandable in terms of the molecular interpretation of entropy. In a perfect crystal, every atom is fixed in position, and, at absolute zero, every form of internal energy (such as atomic vibrations) has its lowest possible value. 

There are two types of macroscopic structures equilibrium and dissipative ones. A perfect crystal, for example, represents an equilibrium structure, which is stable and does not exchange matter and energy with the environment. On the other hand, dissipative structures maintain their state by exchanging energy and matter constantly with environment. This continuous interaction enables the system to establish an ordered structure with lower entropy than that of equilibrium structure. For some time, it is believed that thermodynamics precludes the appearance of dissipative structures, such as spontaneous rhythms. However, thermodynamics can describe the possible state of a structure through the study of instabilities in nonequilibrium stationary states. 

A finitE number of point defects (e.g. vacancies, impurities) can be found in any crystalline material as the configirrational entropy term, TAS, for a low point defect concentration, outweighs the positive formation enthalpy in the free-energy expression, AG = AH — TAS. Thus, introduction of a small number of point defects into a perfect crystal gives rise to a free energy minimum, as illustrated in Figure 2.6a. Further increases in the point defect concentration, however, will raise the free energy of the system. Point defects in crystals are discussed in Sections 3.5.1 and 6.4.1. 

If, at ambient temperature, the slow growth of a crystal is attempted, then, against a background of thermal agitation, the systematic assembly of a perfect crystal turns out to be impossible. Dislocations and interstitial atoms are unavoidable. The second law is obeyed, in that entropy growth is unavoidable. Hence fuel cell materials are real and imperfect, and will need careful optimisation. 

However, we can assign absolute entropy values. Consider a solid at 0 K, at which molecular motion. virtually ceases. If it is a perfect crystal, its internal arrangement is absolutely regular [see Fig. 10.11(a)]. There is only one way to achieve this perfect order every particle must be in its place. For example, with N coins there is only one way to achieve the state of all heads. Thus a perfect crystal represents the lowest possible entropy that is, the entropy of a perfect crystal at 0 K is zero. This is a statement of the third law of thermodynamics. 

As the temperature of a perfect crystal is increased, the random vibrational motions increase, and disorder increases within the crystal [see Fig. 10.11(b)]. Thus the entropy of a substance increases with temperature. Since S is zero for a perfect crystal at 0 K, the entropy value for a substance at a particular temperature can be calculated if we know the temperature dependence of entropy. 

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