The Third Law and Absolute Entropy Measurements

In the previous chapter, we saw that entropy is the subject of the Second Law of Thermodynamics, and that the Second Law enabled us to calculate changes in entropy AS. Another important generalization concerning entropy is known as the Third Law of Thermodynamics. It states that  

Every substance has a finite positive entropy, but at the absolute zero of temperature the entropy may become zero, and does so become in the case of a perfect crystalline substance  

As with the first and second laws, the Third Law is based on experimental measurements, not deduction. It is easy, however, to rationalize such a law. In a perfectly ordered3 crystal, every atom is in its proper place in the crystal lattice. At T— 0 Kelvin, all molecules are in their lowest energy state. Such a configuration would have perfect order and since entropy is a measure of the disorder in a system, perfect order would result in an entropy of zero.b Thus, the Third Law gives us an absolute reference point and enables us to assign values to S and not just to AS as we have been restricted to do with U, H, A, and G. 

To obtain S, we start with equation (3.15) from the previous chapter 

Equation (4.2) can be used to determine the entropy of a substance. A pure crystalline sample is placed in a cryogenic calorimeter and cooled to low temperatures. Increments of heat, q, are added and the temperature change, AT, is measured, from which the heat capacity can be calculated from the relationship 

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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. 

This is an expression of Nernst s postulate which may be stated as the entropy change in a reaction at absolute zero is zero. The above relationships were established on the basis of measurements on reactions involving completely ordered crystalline substances only. Extending Nernst s result, Planck stated that the entropy, S0, of any perfectly ordered crystalline substance at absolute zero should be zero. This is the statement of the third law of thermodynamics. The third law, therefore, provides a means of calculating the absolute value of the entropy of a substance at any temperature. The statement of the third law is confined to pure crystalline solids simply because it has been observed that entropies of solutions and supercooled liquids do not approach a value of zero on being cooled. 

The symbol 9 is called the characteristic temperamre and can be calculated from an experimental determination of the heat capacity at a low temperature. This equation has been very useful in the extrapolation of measured heat capacities [16] down to OK, particularly in connection with calculations of entropies from the third law of thermodynamics (see Chapter 11). Strictly speaking, the Debye equation was derived only for an isotropic elementary substance nevertheless, it is applicable to most compounds, particularly in the region close to absolute zero [17]. 

So far, we have been able to calculate only changes in the entropy of a substance. Can we determine the absolute value of the entropy of a substance We have seen that it is not possible to determine absolute values of the enthalpy. However, entropy is a measure of disorder, and it is possible to imagine a perfectly orderly state of matter with no disorder at all, corresponding to zero entropy an absolute zero of entropy. This idea is summarized by the third law of thermodynamics ... 

Over the years, many experiments have been carried out which confirm the third law. The experiments have generally been of two types. In one type the change of entropy for a change of phase of a pure substance or for a standard change of state for a chemical reaction has been determined from equilibrium measurements and compared with the value determined from the absolute entropies of the substances based on the third law. In the other type the absolute entropy of a substance in the state of an ideal gas at a given temperature and pressure has been calculated on the basis of statistical mechanics and compared with those based on the third law. Except for well-known, specific cases the agreement has been within the experimental error. The specific cases have been explained on the basis of statistical mechanics or further experiments. Such studies have led to a further understanding of the third law as it is applied to chemical systems. 

We thus can obtain a consistent set of absolute values of the entropy function for pure substances from thermal measurements alone on the practical basis of assigning the value of zero to the entropy function at 0 K with the exclusion of nuclear and isotopic effects, within the understanding of the third law as discussed in Section 15.4. The calculation of the entropy function of pure substances in the ideal gas state by the methods of statistical mechanics must be consistent with the practical basis. In addition to obtaining absolute values by the methods that have been discussed, values can also be obtained from equilibrium measurements from which ASe can be determined for some change of state. If all but one of the absolute values in the equivalent sum VjSP are known, then the value of that one can be calculated. 

We have seen that absolute values of the entropy function based on the third law can be obtained from measurements of the heat capacity and heats of transitions. A more general equation than Equation (15.9) may be written as... 

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. 

Measurements of heat capacities at very low temperatures provide data for the calculation from Eq. (5.11) of entropy changes down to 0 K. When these calculations are made for different crystalline forms of the same chemical species, the entropy at 0 K appears to be the same for all forms. When the form is noncrystalline, e.g., amorphous or glassy, calculations show that the entropy of the more random form is greater than that of the crystalline form. Such calculations, which are summarized elsewhere,t lead to the postulate that the absolute entropy is zero for all perfect crystalline substances at absolute zero temperature. While the essential ideas were advanced by Nemst and Planck at the beginning of the twentieth century, more recent studies at very low temperatures have increased our confidence in this postulate, which is now accepted as the third law. 

The hydration entropy can also be deduced experimentally (Latimer 18) as the difference between the standard entropy of the hydrated ions (deduced from measurements of the specific heat on the basis of Nernst s Heat Theorem or the Third Law of Thermodynamics) and the theoretically calculated absolute entropy of the gaseous ion, both reckoned per unit volume at constant concentration. This entropy can also be calculated (Eley and Evans18). 

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. 

It is of importance to note that, except for hydrogen and deuterium molecules, the entropy derived from heat capacity measurements, i.e., the thermal entropy, as it is frequently called, is equivalent to the practical entropy in other words, the nuclear spin contribution is not included in the former. The reason for this is that, down to the lowest temperatures at which measurements have been made, the nuclear spin does not affect the experimental values of the heat capacity used in the determination of entropy by the procedure based on the third law of thermodynamics ( 23b). Presumably if heat capacities could be measured right down to the absolute zero, a temperature would be reached at which the nuclear spin energy began to change and thus made a contribution to the heat capacity. The entropy derived from such data would presumably include the nuclear spin contribution of R In (2i + 1) for each atom. Special circumstances arise with molecular hydrogen and deuterium to which reference will be made below ( 24n). 

In some databases—for example, in the very extensive NIST Chemistry Web-Book " —the data reported for each substance are the the standard state heat of formation AfW" and the absolute entropy both at 25°C. Here by absolute entropy is meant entropy based on the third law of thermodynamics as defined in Sec. 6.8. The reason for reporting these two quantities is that they are determined directly by thermal or calorimetric measurements, unlike the Gibbs energy of formation, which is obtained by measuring chemical equilibrium constants. 

As is not the case with energy and enthalpy, it is possible to determine the absolute value of entropy of a system. To measure the entropy of a substance at room temperature, it is necessary to add up entropy from the absolute zero up to 25°C (77°F). However, the absolute zero is unattainable in practice. This dilemma is resolved by applying the third law of thermodynamics, which states that the entropy of a pure, perfect crystalline substance is zero at the absolute zero of temperature. The increase in entropy from the lowest reachable temperature upward can then be determined Ifom heat capacity measurements and enthalpy changes due to phase transitions. 

Entropy, alone among the thermodynamic functions, is therefore measured on an absolute scale having the reference point zero at 0 K for pure, perfectly crystalline materials. As we shall see, the other functions (G, H, A, U) are measured against an arbitrary standard state and are assigned relative, rather than absolute values. (That is why entropy is designated by an absolute symbol, S°, and all others by relative symbols, e.g., AjG°, in thermochemical tables, as we shall see in Chapter 7). Apart from this, most of thermodynamics as presented here would survive intact if the Third Law had never been discovered. 

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