Absolute entropy The

If the heat capacity can be evaluated at all temperatures between 0 K and the temperature of interest, an absolute entropy can be calculated. For biological processes, entropy changes are more useful than absolute entropies. The entropy change for a process can be calculated if the enthalpy change and free energy change are known. 

Two points concerning the evaluation of the first integral in Equation (15.9) require further discussion. In most experimental determinations of absolute entropies, the lowest temperature attained ranges from 1 to 15 K ... 

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. 

Standard absolute entropy. The entropy of a substance compared to its entropy when crystalline and at 0 K. 

The third iaw of thermodynamics—the entropy of a perfect crystal is zero at 0 K—allows us to calculate absolute entropies. The standard molar entropy (S°) of a substance is influenced by temperature, physical state, dissolution, and atomic size or molecular complexity. 

The third law is used for the calculation of absolute entropies. The differential change in entropy in a reversible process is given by... 

All previous conclusions about equilibria in gases and in dilute solutions retain their applicability, but the convention now adopted will affect assessments of the absolute entropy. The distribution law itself is not appreciably affected. When the density is such that g can no longer be regarded as large compared with N further complications enter which will not be dealt with at this stage. 

Equation 17.61 is one form of what is called the Sackur-Tetrode equation. It provides what is probably the best example of how well statistical thermodynamics applies to gaseous systems, because we can measure absolute entropies. The following example illustrates. 

The third law of thermodynamics asserts that if the entropies of all samples of pure perfect crystalline elements are taken as zero, the entropies of all samples of pure perfect crystalline compounds can also consistently be taken as zero. Entropies relative to the entropy at zero temperature are called absolute entropies. The values of these absolute entropies can be used to calculate entropy changes of chemical reactions. 

Figure A2.1.10. The impossibility of reaching absolute zero, a) Both states a and p in complete internal equilibrium. Reversible and irreversible paths (dashed) are shown, b) State P not m internal equilibrium and with residual entropy . The true equilibrium situation for p is shown dotted.

We shall assume that Cp = 0 at 7 = 0 K. We wish to obtain the absolute entropy of solid lead at 298 K. Each entry in Table 1-2 leads to a value of CpjT. The... 

Draw the curve of Cp vs. T and Cp/T vs. T from the following heat capacity data for solid chlorine and determine the absolute entropy of solid chlorine at 70.0 K... 

Figure 5-11 Pitrtiiil MMII Output Shinvirig Entropy and Heat Capacities for Water. E. perimerital values are 4. i. IS eal K iiiol for the absolute entropy and H.OS eal K iiiol i or the heat eaitaeity.

Ideal gas absolute entropies of many compounds may be found in Daubert et al.,"" Daubert and Danner," JANAF Thermochemical Tables,TRC Thermodynamic Tables,and Stull et al. ° Otherwise, the estimation method of Benson et al. " is reasonably accurate, with average errors of 1-2 J/mol K. Elemental standard-state absolute entropies may be found in Cox et al." Values from this source for some common elements are listed in Table 2-389. ASjoqs may also be calculated from Eq. (2-52) if values for AHjoqs and AGJoqs are known. 

The third law of thermodynamics states that the entropy of any crystalline, perfectly ordered substance must approach zero as the temperature approaches 0 K, and at T = 0 K entropy is exactly zero. Based on this, it is possible to establish a quantitative, absolute entropy scale for any substance as... 

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. 

In summary, the absolute entropies we calculate and tabulate are, in fact, not so absolute, since they do not include isotopic entropies of mixing nor nuclear spin alignment entropies. The entropies we tabulate are sometimes called practical absolute entropies. They can be used to correctly calculate AS for a chemical process, but they are not true" absolute entropies. 

A cryogenic calorimeter measures Cp,m as a function of temperature. We have seen that with the aid of the Third Law, the Cp,m data (along with AHm for phase changes) can be integrated to give the absolute entropy... 

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