Entropy, the Thermodynamic Property

In an intermediate case to those discussed above, we place an irreversible (real) heat engine between these two reservoirs. In this case, the work obtained would be less than that provided by the reversible Carnot cycle. Similarly, a real refrigeration cycle would not represent the reverse of the real heat engine but rather would require more work to get a desired level of refrigeration than the corresponding Carnot refrigerator. 

In each example in this section, we can see the driving force for an irreversible process and the direction that each process wants to go. In each example, we see how to make the process reversible so it produces the maximum work (or consumes the minimum). In more complex systems, these effects may not be as obvious. In such cases, we look for answers using the second law of thermodynamics and the related property, entropy. 

We would like to generalize our experience with the directionality of nature (and the limits of reversibility) into a quantitative statement that allows us to do calculations and draw conclusions about what is possible, what is not possible, and whether we are close to or far away from the idealization represented by a reversible process. Indeed, it would be nice if we had a thermodynamic property (i.e., a state function) which would help us to quantify directionality, just as internal energy, it, was central in quantifying the conservation of energy (the first law of thermodynamics). It turns out the thermodynamic property entropy, s, allows us to accomplish this goal. 

Three historical milestones have established three corresponding distinct contextual paradigms for entropy. First, the property entropy was conceived by Rudolph Clausius in 1865, based largely on Sadi Carnot s work on maximizing the efficiency of cycUc 

Gorrespondingly, the thermodynamic property entropy, s, is defined in terms of heat absorbed during a reversible process. In differential form, the change in entropy of a substance undergoing a reversible process is equal to the incremental heat it absorbs divided by the temperature  

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A quantitative theory of rate processes has been developed on the assumption that the activated state has a characteristic enthalpy, entropy and free energy the concentration of activated molecules may thus be calculated using statistical mechanical methods. Whilst the theory gives a very plausible treatment of very many rate processes, it suffers from the difficulty of calculating the thermodynamic properties of the transition state. 

The thermodynamic properties that we have considered so far, such as the internal energy, the pressure and the heat capacity are collectively known as the mechanical properties and can be routinely obtained from a Monte Carlo or molecular dynamics simulation. Other thermodynamic properties are difficult to determine accurately without resorting to special techniques. These are the so-called entropic or thermal properties the free energy, the chemical potential and the entropy itself. The difference between the mechanical emd thermal properties is that the mechanical properties are related to the derivative of the partition function whereas the thermal properties are directly related to the partition function itself. To illustrate the difference between these two classes of properties, let us consider the internal energy, U, and the Fielmholtz free energy, A. These are related to the partition function by ... 

The thermodynamic properties of the solid silicates show the expected entropy change of formation from the constituent oxides of nearly zero, which is typical of the reaction type... 

The earliest hint that physics and information might be more than just casually related actually dates back at least as far as 1871 and the publication of James Clerk Maxwell s Theory of Heat, in which Maxwell introduced what has become known as the paradox of Maxwell s Demon. Maxwell postulated the existence of a hypothetical demon that positions himself by a hole separating two vessels, say A and B. While the vessels start out being at the same temperature, the demon selectively opens the hole only to either pass faster molecules from A to B or to pass slower molecules from B to A. Since this results in a systematic increase in B s temperature and a lowering of A s, it appears as though Maxwell s demon s actions violate the second law of thermodynamics the total entropy of any physical system can only increase, or, for totally reversible processes, remain the same it can never decrease. Maxwell was thus the first to recognize a connection between the thermodynamical properties of a gas (temperature, entropy, etc.) and the statistical properties of its constituent molecules. 

In principle, the second law can be used to determine whether a reaction is spontaneous. To do that, however, requires calculating the entropy change for the surroundings, which is not easy. We follow a conceptually simpler approach (Section 17.3), which deals only with the thermodynamic properties of chemical systems. 

As an example of the application of this procedure, Pitzer3 has calculated the thermodynamic properties for dimethylcadmium, including free rotation. At 298.15 K, he obtained the following result for the entropy ... 

We use a short version of the seven-step method. The problem asks for the entropy and enthalpy changes accompanying a chemical reaction, so we focus on the balanced chemical equation and the thermodynamic properties of the reactants and products. 

The thermodynamic properties of a number of compounds are shown in Appendix D as pressure-enthalpy diagrams with lines of constant temperature, entropy, and specific volume. The vapor, liquid, and two-phase regions are clearly evident on these plots. The conditions under which each compound may exhibit ideal gas properties are identified by the region on the plot where the enthalpy is independent of pressure at a given temperature (i.e., the lower the pressure and the higher the temperature relative to the critical conditions, the more nearly the properties can be described by the ideal gas law). 

In open systems consisting of several components the thermodynamic properties of each component depend on the overall composition in addition to T and p. Chemical thermodynamics in such systems relies on the partial molar properties of the components. The partial molar Gibbs energy at constantp, Tand rij (eq. 1.77) has been given a special name due to its great importance the chemical potential. The corresponding partial molar enthalpy, entropy and volume under the same conditions are defined as... 

Special mention should be made of recently published volumes of the Landolt-Bomstein Tables, references (35) and (51). These contain a large amount of data on aqueous solutions presented in a compact form. Reference (58) cites a new handbook on the thermodynamic properties of inorganic compounds. It gives tables of enthalpy, Gibbs energy, entropy, and heat... 

Criss, C. M. Cobble, J. W. "The Thermodynamic Properties of High Temperature Aqueous Solutions. V. The Calculation of Ionic Heat Capacities up to 200OC. Entropies and Heat Capacities above 200 C" J. An. Chan. Soc., 1964, 86, 5390. 

This is an appropriate point to remark on some of the thermodynamic aspects of the complicated random network structure envisaged for the liquid. Now, the thermodynamic properties of ices II and III are very similar 1h The ice II ice III transition (249 K, 3.4 kbar) involves only a very small change in volume, namely 0.26 cm3/mole, 1.6% of the molar volume), a small change in entropy 1.22 cal/° mole, and a small change in enthalpy, 304 cal/mole. Similarly, the ice I ice II,... 

Andon, R.J.E., Counsell, J.F., Tees, E.B., Martin, J.F., and Mash, MJ. Thermodynamic properties of organic oxygen compounds. Part 17. Tow-temperature heat capacity and entropy of the cresols, Trans. Faraday Soc., 63 1115-1121,1967. Andon, R.J.E., Cox, J.D., and Herington, E.F.G. Phase relationships in the pyridine series. Part V. The thermodynamic properties of dilute solutions of pyridine bases in water at 25 °C and 40 °C, J. Chem. Soc. (London), pp. 3188-3196, 1954. Andrades, M.S., Sanchez-Martin, M.J., and Sanchez-Camazano, M. Significance of soil properties in the adsorption and mobility of the fungicide metalaxyl in vineyard soils, J. Agric. Food Chem., 49(5) 2363-2369, 2001. 

A summary of developments in physical adsorption during the period from 1943 to 1955 has been given recently by Everett 94). The chief difference between the approach used by Brunauer in his book published in 1943 and that in vogue in 1955 is in the great development of the thermodynamic aspects of the subject. Prior to 1943, the main effort was in developing theories to predict the shape of adsorption isotherms. Since then, emphasis has shifted towards the thermodynamic properties of the adsorbed phase, particularly its entropy. 

The normal vibrations and structural parameters of Sg S, S, and Sjj have been used to calculate several thermodynamic functions of these molecules in the gaseous state. Both the entropy (S°) and the heat capacity (C°) are linear functions of the number of atoms in the ring in this way the corresponding values for Sj, Sg, Sjo and can be estimated by inter- and extrapolation For a recent review of the thermodynamic properties of elemental sulfur see Ref. 

A phase diagram is often considered as something which can only be measured directly. For example, if the solubility limit of a phase needs to be known, some physical method such as microscopy would be used to observe the formation of the second phase. However, it can also be argued that if the thermodynamic properties of a system could be properly measured this would also define the solubility limit of the phase. The previous sections have discussed in detail unary, single-phase systems and the quantities which are inherent in that sjrstem, such as enthalpy, activity, entropy, etc. This section will deal with what happens when there are various equilibria between different phases and includes a preliminary description of phase-diagram calculations. 

Interestingly, the standard entropies (and in turn heat capacities) of both phases were found to be rather similar [69,70]. Considering the difference in standard entropy between F2(gas) and the mixture 02(gas) + H2(gas) taken in their standard states (which can be extracted from general thermodynamic tables), the difference between the entropy terms of the Gibbs function relative to HA and FA, around room temperature, is about 6.5 times lower than the difference between enthalpy terms (close to 125 kJ/mol as estimated from Tacker and Stormer [69]). This indicates that FA higher stability is mostly due to the lower enthalpy of formation of FA (more exothermic than for HA), and that it is not greatly affected by entropic factors. Jemal et al. [71] have studied some of the thermodynamic properties of FA and HA with varying cationic substitutions, and these authors linked the lower enthalpy of formation of FA compared to HA to the decrease in lattice volume in FA. 

Most of the studies on thermodynamics of mixed micellar systems are based on the variation of the critical micellar concentration (CMC) with the relative concentration of both components of the mixed micelles (1-4). Through this approach It Is possible to obtain the free energies of formation of mixed micelles. However, at best, the sign and magnitude of the enthalpies and entropies can be obtained from the temperature dependences of the CMC. An Investigation of the thermodynamic properties of transfer of one surfactant from water to a solution of another surfactant offers a promising alternative approach ( ), and, recently, mathematical models have been developed to Interpret such properties (6-9). 

Thermochemical attention in this chapter is directed towards compounds with carbon—zinc bonds, i.e. species that are usually labeled organometallic. The thermodynamic properties that we discuss are restricted to the enthalpy of formation (often called the heat of formation ), enthalpy of vaporization and carbon—zinc bond energies. We forego discussion of other thermochemical properties such as entropy, heat capacity or excess enthalpy. The energy units are kJmoU where 4.184 kJ is defined to equal 1 kcal. 

J. W. Cobble, High-temperature aqueous solutions. Science, 152, 1479-1485 (1966) C. M. Criss and J. W. Cobble, The thermodynamic properties of high temperature aqueous solutions. IV. Entropies of the ions up to 200 °C and the correspondence principle. V. The calculation... 

Other features of this behavior are illustrated concretely using the Hg-Te system. The calculations were made using values for the thermodynamic properties that are slightly different than those shown in Table VII and used in the calculations described in the main text. Thus the calculations are not quite correct for Hg-Te but are still valid for illustrative purposes. The enthalpy of fusion is taken as 8,680 cal/mol and the enthalpy and entropy of formation of HgTe(s) at 943. as -13,933 cal/mol and —9.538 cal/°K mol, respectively. After applying the constraints of Eqs. (109)—(111) and setting / 13 = / 14 = / 34 = 0, the independent model parameters are z, SlR, and SlA. The former... 

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