Enthalpies and Entropies

We need to understand the concepts of Gibbs free enei gy (G) and chemical potential i/i) in order to know the direction of spontaneous change of a reaction or system. These concepts can also be used to define or predict the most stable (equilibrium) assemblage and gas, fluid, or rock compositions expected in a system at a given pressure and temperature. Some phases and aqueous species in a system may be out of equilibrium with that system. Free-energy calculations permit us to decide which substances are out of equilibrium, and, therefore, which concentrations may be governed by chemical kinetics. 

The concepts of enthalpy (H) and entropy (S) are basic to the definitions of Gibbs free energy, chemical potential, and the equilibrium constant (K ), and allow us to predict the effect of temperature on G and Kg. We will also see that entropy is a measure of the state of order-disorder in solids and solid solutions, and helps explain the stabilities of aqueous complexes. 

In sections to follow we make no attempt to present a rigorous, detailed development of thermodynamic principles. Such a rigorous approach can be found in most textbooks on physical chemistry and thermodynamics (cf. Lewis and Randall 1961 Reid 1990 Anderson and Crerar 1993 Nordstrom and Munoz 1994). We will limit ourselves to principles essential to an understanding of equilibrium concepts. 

Enthalpy (H) is defined as the heat content of a substance at constant pressure. We cannot know absolute values of H, only differences in the enthalpies of substances. The enthalpy of formation, AHf of a substance at 25°C (298.15 K) and 1 bar pressure is its heat of formation from the elements in their most stable forms at that temperature and pressure. Here, and generally, the superscript degree symbol to the right of the state function denotes that the function is for 1 bar pressure. AHf for the elements in their most stable forms is taken as zero by definition at any temperature and 1 bar pressure. For example, AH° = 0 for rhombic sulfur, the most stable form of sulfur at 1 bar and 25°C. Monoclinic sulfur, with AHf = 0.071 kcal/mol at 1 bar and 25°C is unstable relative to the rhombic form. 

The enthalpy of a reaction, 6B°, is the heat transfer between a system and its surroundings for a process at constant pressure, but not at constant temperature and volume (VO. For example, consider the formation of liquid water from gaseous hydrogen and oxygen at 25°C, which, with respective volumes and AHf values given beneath it is written 

In determining the course of enthalpy and entropy of a substance with temperature it is usual to start from very accurate specific heat measurements. Enthalpy and entropy may then be calculated by integration  

If this method is applied to thermodynamic data of polymers, the same difficulty arises as mentioned in Sect. 5.1 for the determination of the specific heat most polymer samples are partly crystalline, only. The thermodynamic quantities have values somewhere between those for purely crystalline and purely amorphous polymer. A large number of measurements are needed to derive the data for these two idealized states. Only for a limited number of polymers have data of this kind been published. 

As an example, in Figs. 5.5 and 5.6 enthalpy and entropy as a function of temperature are plotted for polypropylene, according to the data of Gee and Melia (1970), Dainton et al. (1962) and Passaglia and Kevorkian (1963). 

The corresponding data for some other polymers may be found in a series of articles by Dainton et al. (1962). 

As appears from Fig. 5.5, the enthalpy curves for crystalline and amorphous polypropylene run parallel up to the glass transition temperature. The distance between these curves is called AH(0) = the enthalpy of the amorphous polymer at 0 K. From the glass transition temperature on the curve for the amorphous polymer gradually approaches the curve for the melt, while the curve for the crystalline polymer shows a discontinuity at the melting point. The distance between the curves for crystal and liquid at the melting point is the latent heat of fusion, AHm. 

The standard relationship between Gibbs energy, enthalpy and entropy is 

Further, the Gibbs energy is related to the stability constant of a reaction, log/C, via Eq. (2.80)  

If the change in heat capacity, ACp, of a reaction is zero, the enthalpy and entropy of a reaction do not change with temperature and are constants. Thus, plotting logic values at different temperatures against the reciprocal of absolute temperature, l/T, can be used to determine the values of AH° and AS . 

When the change in heat capacity is a non-zero constant, the enthalpy and entropy of a reaction at a particular temperature can be related to that at 25 °C (298.15 K) through the following equations  

Substituting these last two expressions into Eq. (2.79) gives AG = A-BT - Cr In r 

The change in the free enthalpy of a chemical reaction (i.e., its AG) depends on a number of factors—e. g the concentrations of the reactants and the temperature (see p. 18). Two further factors associated with molecular changes occurring during the reaction are discussed here. 

All chemical reactions involve heat exchange. Reactions that release heat are called exothermic, and those that consume heat are called endothermic. Heat exchange is measured as the enthalpy change AH (the heat of reaction). This corresponds to the heat exchange at constant pressure. In exothermic reactions, the system loses heat, and AH is negative. When the reaction is endothermic, the system gains heat, and AH becomes positive. 

In many reactions, AH and AG are similar in magnitude (see B1, for example). This fact is used to estimate the caloric content of foods. In living organisms, nutrients are usually oxidized by oxygen to CO2 and H2O (see p. 112). The maximum amount of chemical work supplied by a particular foodstuff (i. e., the AG for the oxidation of the utilizable constituents) can be estimated by burning a weighed amount in a calorimeter in an oxygen atmosphere. The heat of the reaction increases the water temperature in the calorimeter. The reaction heat can then be calculated from the temperature difference AT. 

Entropy is a physical value that describes the degree of order of a system. The lower the degree of order, the larger the entropy. Thus, when a process leads to increase in disorder—and everyday experience shows that 

The dissolution of salt in water (2) is endothermic (AH 0)—i. e., the liquid cools. Nevertheless, the process still occurs spontaneously, since the degree of order in the system decreases. The Na and Cl ions are initially rigidly fixed in a crystal lattice. In solution, they move about independently and in random directions through the fluid. The decrease in order (AS 0) leads to a negative -T AS term, which compensates for the positive AH term and results in a negative AG term overall. Processes of this type are described as being entropy-driven. The folding of proteins (see p. 74) and the formation of ordered lipid structures in water (see p. 28) are also mainly entropy-driven. 

Enthalpy is the prime quantitative descriptor of chemical energetic stability enthalpy variations reflect the energetic gain cashed in, or the energetic price paid, when the nuclei and electrons of a chemical system break apart from one structure and coalesce into another structure. Enthalpy differences can be measured by calorimetric experiments, either directly or through specific heat measurements, or calculated by 

The enthalpy change of reaction is the heat exchanged with the surroundings (at constant pressure) in a chemical reaction. This represents the difference in stability (bond strength) of the reagents and products. 

The entropy change of reaction provides a measure of the change in molecular disorder or randomness caused by a reaction. 

For a negative value of AG° (in which products are favoured over reactants at equilibrium), we require low positive, or high negative, values of AH° and high positive values of TAS°. 

Two factors contribute to the change in free energy the change in enthalpy and the change in entropy multiplied by the temperature. 

AG° = (free energy of products) — (free energy of reactants) 

At low temperatures, the enthalpy term (AH°) is usually much larger than the entropy term —TAS°), and the entropy term is sometimes ignored. 

The change in enthalpy (A//°) is the heat of reaction—the amount of heat evolved or consumed in the course of a reaction, usually given in kilojoules (or kilocalories) per mole. The enthalpy change is a measure of the relative strength of bonding in the products and reactants. Reactions tend to favor products with the lowest enthalpy (those with the strongest bonds). 

If weaker bonds are broken and stronger bonds are formed, heat is evolved and the reaction is exothermic (negative value of AH°). In an exothermic reaction, the enthalpy term makes a favorable negative contribution to AG°. If stronger bonds are broken and weaker bonds are formed, then energy is consumed in the reaction, and the reaction is endothermic (positive value of AH°). In an endothermic reaction, the enthalpy term makes an unfavorable positive contribution to AG°. 

Problem 6.8 Given each of the following values, is the starting material or product favored at equilibrium  

Problem 6.9 Given each of the following values, is the starting material or product lower in energy a. AG° = 2.0 kcal b. K = 0 c. AG° = -3.0 kcal 

These equations can be used for any process with two states in equilibrium. As an example, monosubstituted cyclohexanes exist as two different chair conformations that rapidly interconvert at room temperature, with the conformation having the substituent in the roomier equatorial position favored (Section 4.13). Knowing the energy difference between the two conformations allows us to calculate the amount of each at equilibrium. 

For example, the energy difference between the two chair conformations of phenylcyclohexane is -2.9 kcal/mol, as shown in the accompanying equation. Using the values in Table 6.3, this corresponds to an equilibrium constant of 100, meaning that there is approximately 100 times more B (equatorial phenyl group) than A (axial phenyl group) at equilibrium. 

The equilibrium constant for the conversion of the axial to the equatorial conformation of methoxycyclohexane is 2.7. 

/3-galactoside transport is coupled to an exergonic system providing 4132 cal/mole. 

Suppose a mole of ATP is hydrolyzed at 25 C under standard conditions where AG = —7700cal/mole. The hydrolysis is not coupled to any group transfer reaction. Does this mean then that the entire AG appears as heat The first and second laws of thermodynamics relate the AG of a reaction to the heat evolved in the following way  

AH is called enthalpy change, and represents the quantity of heat released (or absorbed) at constant temperature, pressure, and volume. AS is the entropy change and is a measure of the change in the randomness of the system. If we measure or calculate AH for the hydrolysis of ATP, we obtain a value of about — 4000 cal/mole. The remaining 3700 cal/mole is not released as heat. This amount of energy is retained by the products in the form of 

The sign of AH gives no indication of the spontaneous direction of a reaction. We see from Equation 45 that even if AH is positive, AG can still be negative if AS is positive enough. For example, if we mix certain solid salts with water, we observe a marked decrease in the temperature as the salt dissolves. Heat has been absorbed. AH is positive (heat must be added to keep the temperature of the system constant). The salt spontaneously dissolves, therefore AG is negative. The TAS term is positive—the ions of the salt that were originally in a highly ordered crystalline array are now randomly distributed in solution. 

Entropy changes are not usually reported in terms of TAS, but rather, in terms of AS. 

In introducing the Gibbs energy directly, we have short-circuited a great deal of the usual methodical derivation. It will be sufficient for our purposes to say simply that the formal definition of G is 

S is the molar entropy, a measure of the disorder in the system, and ArS and ArS° are the entropy changes in a reaction, using either the real entropies, or the standard state entropies. All these A terms use the products—reactants convention. 

Finally, although neither G nor H can be measured in absolute terms, so that we are forced always to use differences of these quantities, absolute values of S can be measured calorimetrically. Thus, tables of thermodynamic data for compound i contain values of A/G°, A fH°, and Sf, where S is the entropy of i. If we want a value of A fS°, we must calculate it from the tabulated values of S for the compound and its constituent elements. 

A typical table of thermodynamic data looks like Table 3.1. Note that A/H° and A fG° for the element Ca are both zero. This does not mean that Hfa = 0, or G a = 0. We don t know what these values are. A/G° = 0 just means that G ,a — G a = 0. 

The free energy change (AG°) depends on the enthalpy change (AH°) and the entropy change (AS°). AH° indicates relative bond strength, bnt what does AS° measure  

Entropy (S°) is a measure of the randomness in a system. The more freedom of motion or the more disorder present, the higher the entropy. Gas molecules move more freely than liquid molecules and are higher in entropy. Cychc molecules have more restricted bond rotation than similar acyclic molecules and are lower in entropy. 

The entropy change (AS°) is the change in the amount of disorder between reactants and products. AS° is positive (-e) when the products are more disordered than the reactants. AS° is negative (-) when the products are less disordered (more ordered) than the reactants. 

For example, when a single starting material forms two products, as in the homolytic cleavage of a bond to form two radicals, entropy increases and favors formation of the products. In contrast, entropy decreases when an acyclic compound forms a ring, because a ring has fewer degrees of freedom. In this case, therefore, entropy does not favor formation of the product. 

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The relations which permit us to express equilibria utilize the Gibbs free energy, to which we will give the symbol G and which will be called simply free energy for the rest of this chapter. This thermodynamic quantity is expressed as a function of enthalpy and entropy. This is not to be confused with the Helmholtz free energy which we will note sF (L" j (j, > )... 

A general prerequisite for the existence of a stable interface between two phases is that the free energy of formation of the interface be positive were it negative or zero, fluctuations would lead to complete dispersion of one phase in another. As implied, thermodynamics constitutes an important discipline within the general subject. It is one in which surface area joins the usual extensive quantities of mass and volume and in which surface tension and surface composition join the usual intensive quantities of pressure, temperature, and bulk composition. The thermodynamic functions of free energy, enthalpy and entropy can be defined for an interface as well as for a bulk portion of matter. Chapters II and ni are based on a rich history of thermodynamic studies of the liquid interface. The phase behavior of liquid films enters in Chapter IV, and the electrical potential and charge are added as thermodynamic variables in Chapter V. 

If the dependence on temperature as well as on composition is known for a solution, enthalpies and entropies of adsorption may be calculated from the appropriate thermodynamic relationships [82]. Neam and Spaull [147] have, for example, calculated the enthalpies of surface adsorption for a series of straight-chain alcohols. They find an increment in enthalpy of about 1.96 kJ/mol per CH2 group. 

In a fiormal analogy to the expressions for the thenuodynamical quantities one can now defiine the standard enthalpy // and entropy ofiactivation. This leads to the second Eyring equation. ... 

Figure B2.4.2. Eyring plot of log(rate/7) versus (1/7), where Jis absolute temperature, for the cis-trans isomerism of the aldehyde group in fiirfiiral. Rates were obtained from tln-ee different experiments measurements (squares), bandshapes (triangles) and selective inversions (circles). The line is a linear regression to the data. The slope of the line is A H IR, and the intercept at 1/J = 0 is A S IR, where R is the gas constant. A and A are the enthalpy and entropy of activation, according to equation (B2.4.1)...

Figure 1.4. Temperature dependence of the change in Gihhs energy, enthalpy and entropy upon transfer of ethane and butane from the gas phase to water. The data refer to transfer from the vapour phase at 0.101 MPa to a hypothetical solution of unit mole fraction and are taken from ref. 125.

NMR signals of the amino acid ligand that are induced by the ring current of the diamine ligand" ". From the temperature dependence of the stability constants of a number of ternary palladium complexes involving dipeptides and aromatic amines, the arene - arene interaction enthalpies and entropies have been determined" ". It turned out that the interaction is generally enthalpy-driven and counteracted by entropy. Yamauchi et al. hold a charge transfer interaction responsible for this effect. 

Molecular enthalpies and entropies can be broken down into the contributions from translational, vibrational, and rotational motions as well as the electronic energies. These values are often printed out along with the results of vibrational frequency calculations. Once the vibrational frequencies are known, a relatively trivial amount of computer time is needed to compute these. The values that are printed out are usually based on ideal gas assumptions. 

Vibrational frequencies Rotational enthalpy and entropy Vibrational enthalpy and entropy Translational enthalpy and entropy... 

In a parallel study Goursot and Wadso (322) determined calorimetri-cally the free energies, enthalpies, and entropies of dissociation of the conjugate acids of thiazoles in aqueous media (Table 1-51). 

Hq and Sq = enthalpy and entropy of the same stream at equiUbrium with the surroundings and Tq = temperature of the surroundings (sink). 

Molecular Nature of Steam. The molecular stmcture of steam is not as weU known as that of ice or water. During the water—steam phase change, rotation of molecules and vibration of atoms within the water molecules do not change considerably, but translation movement increases, accounting for the volume increase when water is evaporated at subcritical pressures. There are indications that even in the steam phase some H2O molecules are associated in small clusters of two or more molecules (4). Values for the dimerization enthalpy and entropy of water have been deterrnined from measurements of the pressure dependence of the thermal conductivity of water vapor at 358—386 K (85—112°C) and 13.3—133.3 kPa (100—1000 torr). These measurements yield the estimated upper limits of equiUbrium constants, for cluster formation in steam, where n is the number of molecules in a cluster. 

Various equations of state have been developed to treat association ia supercritical fluids. Two of the most often used are the statistical association fluid theory (SAET) (60,61) and the lattice fluid hydrogen bonding model (LEHB) (62). These models iaclude parameters that describe the enthalpy and entropy of association. The most detailed description of association ia supercritical water has been obtained usiag molecular dynamics and Monte Carlo computer simulations (63), but this requires much larger amounts of computer time (64—66). 

Available data on the thermodynamic and transport properties of carbon dioxide have been reviewed and tables compiled giving specific volume, enthalpy, and entropy values for carbon dioxide at temperatures from 255 K to 1088 K and at pressures from atmospheric to 27,600 kPa (4,000 psia). Diagrams of compressibiHty factor, specific heat at constant pressure, specific heat at constant volume, specific heat ratio, velocity of sound in carbon dioxide, viscosity, and thermal conductivity have also been prepared (5). 

The solvophobic model of Hquid-phase nonideaHty takes into account solute—solvent interactions on the molecular level. In this view, all dissolved molecules expose microsurface area to the surrounding solvent and are acted on by the so-called solvophobic forces (41). These forces, which involve both enthalpy and entropy effects, are described generally by a branch of solution thermodynamics known as solvophobic theory. This general solution interaction approach takes into account the effect of the solvent on partitioning by considering two hypothetical steps. Eirst, cavities in the solvent must be created to contain the partitioned species. Second, the partitioned species is placed in the cavities, where interactions can occur with the surrounding solvent. The idea of solvophobic forces has been used to estimate such diverse physical properties as absorbabiHty, Henry s constant, and aqueous solubiHty (41—44). A principal drawback is calculational complexity and difficulty of finding values for the model input parameters. 

The basicity of pyrazole and its relation with imidazole basicity (due both to enthalpy and entropy changes (77MI40403)) have been discussed on theoretical grounds (Section 4.04.1.2.1). The pK values of 90 pyrazoles have been determined by Gonzalez et al. (68BSF707,68BSF5009) and it is essentially his work that will be discussed below. A selection of pK values are shown in Table 28. The pK values for some other pyrazoles have been measured in connection with nitration studies (Section 4.04.2.1.4(ii)) (71JCS(B)2365). 

Studies on metal-pyrazole complexes in solution are few. The enthalpy and entropy of association of Co(II), Ni(II), Cu(II) and Zn(II) with pyrazole in aqueous solution have been determined by direct calorimetry (81MI40406). The nature of the nitrogen atom, pyridinic or pyrrolic, involved in the coordination with the metal cannot be determined from the available thermodynamic data. However, other experiments in solution (Section 4.04.1.3.3(i)) prove conclusively that only the N-2 atom has coordinating capabilities. 

Enthalpy and Entropy as Eunctions of T and P At constant composition the molar thermodynamic properties are functions of temperature and pressure (Postulate 5). Tmns... 

Equations (4-34) and (4-35) are general expressions for the enthalpy and entropy of homogeneous fluids at constant composition as functions of T and P. The coefficients of dT and dP are expressed in terms of measurable quantities. 

The enthalpy and entropy are simple sums of the ideal gas and residual properties, which are evaluated separately. 

Given saturated-liquid enthalpies and entropies, the calculation of these properties for pure compressed hquids is accomplished by integration at constant temperature of Eqs. (4-34) and (4-35) ... 

The free energy is the most important equilibrium thennodynamic function, but other quantities such as the enthalpy and entropy are also of great interest. Thermodynamic integration and permrbation fonnulas can be derived for them as well. For example, the derivative of the entropy can be written [24]... 

The temperature dependence of reaction rates permits evaluation of the enthalpy and entropy components of the free energy of activation. The terms in Eq. (4.4) corresponding to can be expressed as... 

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