Deviations from the Third Law

For linear molecules where p = 2, S = 1.38 cal/deg mole. This is very close to the observed discrepancy in entropy for the molecules cited above. 

Nonconformity with the third law also arises when a substance can exist in more than one state of such low energy that the distribution among these states is not influenced by the falling temperature down to the lowest attainable T. These states are frequently due to interactions of electronic or nuclear magnetic dipoles. An extrapolation to zero using the Debye law would reduce the system to a state of zero vibrational entropy, but the rotational entropy due to nuclear spin or the entropy associated with random orientation of magnetic dipoles in 

Another system in which deviations from the third law appear is a mixture of ortho and para hydrogen. In the lowest energy state all the hydrogen molecules would be in the para state. However, since the rate of conversion from ortho to para is quite slow in the absence of catalysts, several units of entropy are retained in hydrogen as it is cooled. However, if an appropriate catalyst is present for ortho-para conversion, the entropy can be calculated correctly by placing Sq = 0. 

Another source of residual entropy is the presence of isotopic molecules scattered throughout the crystal. In the lowest energy state, the isotopes separate into the component phases. Since the mixture of isotopes is usually present in both products and reactants, their contribution is usually neglected in chemical calculations. 

Some liquids may become glasslike rather than crystalline solids as the temperature is lowered. In this case, the molecules exhibit considerable randomness of arrangement and can be frozen in on further cooling so as to yield a positive entropy at 0°K. Both glycerol and ethanol can be cooled to yield positive entropies in this way. 

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In 1902, T. W. Richards found experimentally that the free-energy increment of a reaction approached the enthalpy change asymptotically as the temperature was decreased. From a study of Richards data, Nernst suggested that at absolute zero the entropy increment of reversible reactions among perfect crystalline solids is zero. This heat theorem was restated by Planck in 1912 in the form The entropy of all perfect crystalline solids is zero at absolute zero.f This postulate is the third law of thermodynamics. A perfect crystal is one in true thermodynamic equilibrium. Apparent deviations from the third law are attributed to the fact that measurements have been made on nonequilibrium systems. 

We conclude this brief discussion of deviations from the third law by stating that, although the cases of nonconformity are frequent, we can usually understand their origin with the aid of molecular concepts and quantum statistics. The latter discipline permits calculation of thermodynamic quantities, thereby providing a useful check on experimental data indeed, it often supplies answers of greater accuracy. In this way, it is possible to use the third law to build up tables of absolute entropies of chemical substances. 

This value agrees with the amount by which carbon monoxide appeared to deviate from the third law. 

Berthelot Equation. This equation is too unwieldy to be used generally as an equation of state. However, it is convenient in calculations of deviations from ideality near pressures of 1 atm hence, it has been used extensively in the determination of entropies from the third law of thermodynamics. This aspect of the equation will receive more attention in subsequent discussions. 

Apparently the zero field splitting within the ground state is very weak since XmT does not exhibit any deviation from the Curie law below 10 K. This is confirmed by the EPR spectrum at 4.2 K where no fine structure is detected. This spectrum exhibits the AMS = 1 allowed transition at g = 1.991 and the AMS = 2, 3 and 4 forbidden transitions of decreasing intensity at half, third and quarter-field respectively. The theoretical expression for the magnetic susceptibility has been established from the spin state structure of Fig. 41 and the Zeeman perturbation expressed as ... 

It is certainly clear from an examination of the experimental differential entropy curve for argon on rutile (9) that the entropy terms dropped from Eq. (35) are not negligible in this case, at least up to 6 = 4.5. In fact, in the notation of Eq. (35), dSs/dN — (SL/N) is about one-half of R In x at 6 = 2.4 and these two quantities are about equal at 6 = 4.5. One may hope that the refinements discussed above will make some contribution to understanding such entropy effects as well as deviations from the third power law (the two problems are of course related). 

The third approach is called the thermodynamic theory of passive systems. It is based on the following postulates (1) The introduction of the notion of entropy is avoided for nonequilibrium states and the principle of local state is not assumed, (2) The inequality is replaced by an inequality expressing the fundamental property of passivity. This inequality follows from the second law of thermodynamics and the condition of thermodynamic stability. Further the inequality is known to have sense only for states of equilibrium, (3) The temperature is assumed to exist for non-equilibrium states, (4) As a consequence of the fundamental inequality the class of processes under consideration is limited to processes in which deviations from the equilibrium conditions are small. This enables full linearization of the constitutive equations. An important feature of this approach is the clear physical interpretation of all the quantities introduced. 

Problems are associated with quantitative analysis using IR. First, deviation from Beer s law affect quantitative analyses profoundly, especially those deviations resulting from saturation effects. Variations in the path length that are not accounted for can also cause problems. Second, specific interactions between components in the sample can influence the quantitation, especially those interactions that are temperature and pressure sensitive. Third, if the quantitation is based on the peak being due to only one absorbance when in reality it is a result of overlapping bands, then there will be a bias in data that is not necessarily linear. Currently available IR spectrometers have software packages containing matrix methods that simplify the operations associated with multicomponent... 

We can now discuss certain systems for which we expect the third law to be valid, other systems for which the law is not valid at all, and some specific systems that appear to deviate from the law. 

Data regressions based on the law of mass action are generally adequate for most situations. However, this model only retains validity in liquid-phase reactions at equilibrium without cooperativity. Reactions that involve solid-phase, multiple cooperative binding, and not reaching equilibrium, deviate from the model. Therefore, empirical equations that are not based on the law of mass action have been used for curve fitting also. Among these, polynomial (205) and spline functions are often used (206-209). Polynomial regression can be a second-order (parabolic) or third-order (cubic) function ... 

The existence of an isosbestic point is not proof of the presence of only two components. There may be a third component with e = 0 at this particular wavelength. The absence of an isosbestic point, however, is definite proof of the presence of a third component, provided the possibility of deviation from Beer s law in the two-component system can be dismissed. For a two-component system, the isosbestic point is a unique wavelength for quantitative determination of the total amount of two absorbing species in mutual equilibrium. 

The third law of thermodynamics, like the first and second laws, is a postulate based on a large number of experiments. In this chapter we present the formulation of the third law and discuss the causes of a number of apparent deviations from this law. The foundations of the third law are firmly rooted in molecular theory, and the apparent deviations from this law can be easily explained using statistical mechanical considerations. The third law of thermodynamics is used primarily for the determination of entropy constants which, combined with thermochemical data, permit the calculation of equilibrium constants. 

The relative standard deviation of the results from the mean value is less than 1-2% (Table 16.32), confirming the high precision of determinations by the third-law method. When averaging the results, the results of determination in a vacuum, and also the less reliable values of corresponding... 

Deviations from the linearity of Hooke s law are represented to the next higher order by the third-order stiffnesses compliances Th se are... 

The use of thermodynamic functions corrected for the electronic excitation of RQ determined from the excitation energy of the free R ion leads to smaller deviations from the experimental data (Sapegin, 1984) not only in calculations according to the third law but also for those made according to the second law. 

The deviation from linearity was explained by including a third term due to the catalyst in the rate law equation (equation 20) and the results are given in Table 14. 

Based on experimental results and a model describing the kinetics of the system, it has been found that the temperature has the strongest influence on the performance of the system as it affects both the kinetics of esterification and of pervaporation. The rate of reaction increases with temperature according to Arrhenius law, whereas an increased temperature accelerates the pervaporation process also. Consequently, the water content decreases much faster at a higher temperature. The second important parameter is the initial molar ratio of the reactants involved. It has to be noted, however, that a deviation in the initial molar ratio from the stoichiometric value requires a rather expensive separation step to recover the unreacted component afterwards. The third factor is the ratio of membrane area to reaction volume, at least in the case of a batch reactor. For continuous opera-... 

The rate data for individual runs can be used to derive independent estimates of k7 + kg, and these are shown in Table II. Both rate constants for an assumed second-order and third-order rate law are shown. The second-order rate constants show the smaller deviation from constancy, but the total change in concentration of the reactants is relatively small so that the order cannot be definitely proved at present. 

A plot of 2 vs. -t2 for symmetrical systems (i.e., ii vo) is shown in Fig. 1 for a series of values of the heat lerm, It shows how the partial vapor pressure of a component of a binary solution deviates positively from Raoult s law more and mure as the components become more unlike in their molecular attractive forces. Second, the place of T in die equation shows that tlic deviation is less die higher the temperature. Third, when the heat term becomes sufficiently large, there are three values of U2 for the same value of ay. This is like the three roots of the van der Waals equation, and corresponds to two liquid phases in equilibrium with each other. The criterion is diat at the critical point the first and second partial differentials of a-i and a are all zero. 

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