Thermodynamic coordinates

Agmon N, Levine RD. Energy, entropy, and the reaction coordinate thermodynamic-like relations in chemical kinetics. Chem Phys Lett 1977 52(2) 197-201. 

Thermodynamic coordinates, thermodynamic variables, and thermodynamic degrees of freedom All three designate linearly independent experimental macroscopic variables that are employed to characterize the state of a system. 

The free energy differences obtained from our constrained simulations refer to strictly specified states, defined by single points in the 14-dimensional dihedral space. Standard concepts of a molecular conformation include some region, or volume in that space, explored by thermal fluctuations around a transient equilibrium structure. To obtain the free energy differences between conformers of the unconstrained peptide, a correction for the thermodynamic state is needed. The volume of explored conformational space may be estimated from the covariance matrix of the coordinates of interest, = ((Ci [13, lOj. For each of the four selected conform-... 

Unfortunately, the number of mechanistic studies in this field stands in no proportion to its versatility" . Thermodynamic analysis revealed that the beneficial effect of Lewis-acids on the rate of the Diels-Alder reaction can be primarily ascribed to a reduction of the enthalpy of activation ( AAH = 30-50 kJ/mole) leaving the activation entropy essentially unchanged (TAAS = 0-10 kJ/mol)" . Solvent effects on Lewis-acid catalysed Diels-Alder reactions have received very little attention. A change in solvent affects mainly the coordination step rather than the actual Diels-Alder reaction. Donating solvents severely impede catalysis . This observation justifies the widespread use of inert solvents such as dichloromethane and chloroform for synthetic applications of Lewis-acid catalysed Diels-Alder reactions. 

Olefin synthesis starts usually from carbonyl compounds and carbanions with relatively electropositive, redox-active substituents mostly containing phosphorus, sulfur, or silicon. The carbanions add to the carbonyl group and the oxy anion attacks the oxidizable atom Y in-tramolecularly. The oxide Y—O" is then eliminated and a new C—C bond is formed. Such reactions take place because the formation of a Y—0 bond is thermodynamically favored and because Y is able to expand its coordination sphere and to raise its oxidation number. 

Using the coordinates of special geometries, minima, and saddle points, together with the nearby values of potential energy, you can calculate spectroscopic properties and macroscopic thermodynamic and kinetic parameters, such as enthalpies, entropies, and thermal rate constants. HyperChem can provide the geometries and energy values for many of these calculations. 

Enterobactin (ent), the cycHc triester of 2,3-dihydroxy-A/-benzoyl-l-serine, uses three catecholate dianions to coordinate iron. The iron(III)-enterobactin complex [62280-34-6] has extraordinary thermodynamic stabiUty. For Fe " +ent , the estimated formal stabiUty constant is 10 and the reduction potential is approximately —750 mV at pH 7 (23). Several catecholate-containing synthetic analogues of enterobactin have been investigated and found to have lesser, but still impressively large, formation constants. 

Thermodynamic properties (71,72), force constants (73), and infrared absorption characteristics (74) are documented. The coordinatively unsaturated species, Ni(CO)2 and Ni(CO)2, also exist and the bonding and geometry data have been subjected to molecular orbital treatments (75,76). 

The coordinates of thermodynamics do not include time, ie, thermodynamics does not predict rates at which processes take place. It is concerned with equihbrium states and with the effects of temperature, pressure, and composition changes on such states. For example, the equiUbrium yield of a chemical reaction can be calculated for given T and P, but not the time required to approach the equihbrium state. It is however tme that the rate at which a system approaches equihbrium depends directly on its displacement from equihbrium. One can therefore imagine a limiting kind of process that occurs at an infinitesimal rate by virtue of never being displaced more than differentially from its equihbrium state. Such a process may be reversed in direction at any time by an infinitesimal change in external conditions, and is therefore said to be reversible. A system undergoing a reversible process traverses equihbrium states characterized by the thermodynamic coordinates. 

The a-rhombohedral form of boron has the simplest crystal stmcture with slightly deformed cubic close packing. At 1200°C a-rhombohedral boron degrades, and at 1500°C converts to P-rhombohedral boron, which is the most thermodynamically stable form. The unit cell has 104 boron atoms, a central B 2 icosahedron, and 12 pentagonal pyramids of boron atom directed outward. Twenty additional boron atoms complete a complex coordination (2). 

Stability. The thermodynamic stabiUty of coordination compounds in solution has been extensively studied (12). The equiUbrium constants may be reported as stabiUty or formation constants. 

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. 

The fundamental thermodynamic properties that arise in connection with the first and second laws of thermodyuamics are internal energy and entropy These properties, together with the two laws for which they are essential, apply to all types of systems. However, different types of systems are characterized by different sets of measurable coordinates or variables. The type of system most commonly... 

Temperature, pressure, and composition are thermodynamic coordinates representing conditions imposed upon or exhibited by the system, andtne functional dependence of the thermodynamic properties on these conditions is determined by experiment. This is quite direct for molar or specific volume which can be measured, and leads immediately to the conclusion that there exists an equation of. state relating molar volume to temperature, pressure, and composition for any particular homogeneous PVT system. The equation of state is a primaiy tool in apphcations of thermodyuamics. 

Although the T-s diagram is veiy useful for thermodynamic analysis, the pressure enthalpy diagram is used much more in refrigeration practice due to the fact that both evaporation and condensation are isobaric processes so that heat exchanged is equal to enthalpy difference A( = Ah. For the ideal, isentropic compression, the work could be also presented as enthalpy difference AW = Ah. The vapor compression cycle (Ranldne) is presented in Fig. H-73 in p-h coordinates. 

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

To obtain thermodynamic perturbation or integration formulas for changing q, one must go back and forth between expressions of the configuration integral in Cartesian coordinates and in suitably chosen generalized coordinates [51]. This introduces Jacobian factors... 

For the other broad category of reaction conditions, the reaction proceeds under conditions of thermodynamic control. This can result from several factors. Aldol condensations can be effected for many compounds using less than a stoichiometric amount of base. Under these conditions, the aldol reaction is reversible, and the product ratio will be determined by the relative stability of the various possible products. Conditions of thermodynamic control also permit equilibration among all the enolates of the nucleophile. The conditions that permit equilibration include higher reaction temperatures, protic solvents, and the use of less tightly coordinating cations. 

Molar diagram A plot of the thermodynamic properties of a substance that has specific enthalpy as one of its coordinates. 

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