Free Energy Change Of Reaction Research Articles

Mineralization of Trace Nitro/Chloro/Methyl/Amino-Aromatic Contaminants in Wastewaters by Advanced Oxidation Processes

It has great significance to investigate the degradation of aromatic contaminants as they are highly carcinogenic and nondegradable pollutants in drinking water. In this paper, the mineralization orders of the representative nitro/chloro/methyl/amino-aromatic contaminants with oxidants (·OH, H2O2, ·O−, O2,O3) in advanced oxidation processes (AOPs) are investigated based on the calculated standard molar Gibbs free energy changes of reaction (ΔrGm0) and the results are consistent with those from previous experimental results, electrophilic substitution orientation rules of the Hammett equation, and predicted results with a quantitative structure−activity relationship (QSAR). In addition, the quantitative function relationship between the degradation rate (r) of the aromatic contaminants and the thermodynamic driving force (ΔrGm0) is analyzed in order to investigate the degradation kinetics more rigorously.

Entropy-favored human antibody binding reactions with a non-infectious antigen

Little is known about the thermodynamic properties of human antibodies directed against ‘natural’ antigen surfaces, possibly due to the complex interactions that are involved. Using an affinity distribution method, we have previously characterized the binding reactions between a major allergen from ragweed, Amb a 1, and serum Amb a 1-specific IgE as a model system. We determined the temperature dependence of these interactions using serum samples from people with established allergic sensitivity to ragweed pollen. Each sample provided evidence for three epitope-specific IgE reactions with extremely high equilibrium binding affinities @ 37 °C (10 8 to 10 11 M −1). Determining the affinities over a range of temperatures (4–41 °C) revealed a favorable exothermic Gibbs free energy change, Δ G ≈ −17.59 (± 5.04) kcal/M, comparable to previous reports using monoclonal antibodies produced against well-defined artificial antigens. In contrast to previous studies, in this system there was minimal input from enthalpy: Δ H ≈ −2.41 (± 2.32) kcal/M. However, a significant contribution was found from entropic changes: Δ S ≈ 48.98 (± 9.20) cal/K M. Human ‘secondary antibodies’ such as IgE, produced after exposure to ‘natural’ antigens, are optimized in terms of their high equilibrium binding constants with the antigen (allergen) that induced their production. Thermodynamically this is exemplified by minimal enthalpic (bond formation) concomitant with significant entropic (alignment) contributions to the total free energy change of reaction. These results suggest a high degree of ‘complementarity’ between the antibody and antigen surfaces in this experimental system, and may be a general guiding principle in the evolution of antibody repertoires by the adaptive immune system.

Electrolyte-Concentration and Ion-Size Dependence of Excited-State Intramolecular Charge-Transfer Reaction in (Alkylamino)benzonitriles: Steady-State Spectroscopic Studies

Steady-state spectroscopic studies have been performed with three intramolecular charge-transfer molecules, 4-(1-azetidinyl)benzonitrile (P4C), 4-(1-pyrrolidinyl)benzonitrile (P5C), and 4-(1-piperidinyl)benzonitrile (P6C), in ethyl acetate and acetonitrile in presence of lithium perchlorate (LiClO(4)) at room temperature to investigate the effects of electrolytes on excited-state intramolecular charge-transfer reaction. Electrolyte-concentration and ion-size dependences of several spectroscopic properties such as quantum yield, absorption and emission transition moments, radiative and nonradiative rates, and changes in reaction free energies associated with LE --> CT conversion have been determined for these molecules and reported. For P4C, quantum yield decreases by a factor of approximately 7 at the highest electrolyte concentration relative to that in pure ethyl acetate whereas it is a factor of approximately 4 for both P5C and P6C. However, in acetonitrile with 1.0 M LiClO(4), quantum yield reduces to almost half of that in the pure solvent. Formation of a charge-transfer (CT) state is found to be strongly favored over the locally excited (LE) state as the electrolyte (LiClO(4)) concentration is increased, electrolyte effects being more pronounced in ethyl acetate than in acetonitrile. Relative to pure ethyl acetate, reaction free energy change (-DeltaG(r)) increases by a factor of approximately 5, approximately 4, and approximately 2 for P4C, P5C, and P6C, respectively, at 2.5 M LiClO(4) in this solvent. -DeltaG(r) for P4C exhibits a change in sign (from negative to positive) upon addition of electrolyte in ethyl acetate. In acetonitrile, however, these changes are within a few percent, except for P4C where it is about 4 times greater at 1.0 M LiClO(4) than that in pure acetonitrile. The electrolyte-induced total red shift of the CT band of these TICT molecules is 3 times higher in ethyl acetate than in acetonitrile. Although both the quantum yield and CT emission peak frequency decrease linearly with the increase in ion size, -DeltaG(r) remains largely insensitive. Further studies using a nonreactive probe (coumarin 153) in concentrated electrolyte solutions also show qualitatively similar results.

Thermodynamic calculation on reduction of tungsten oxide in H2 atmosphere

Thermodynamic calculation on reduction of tungsten oxide in H2 atmosphere is carried out in this paper. The general calculation model of the standard free energy changes for reactions are established. Accurate calculation and plotting of the standard free energy changes, equilibrium constants and gas composition for preparing tungsten by reduction of tungsten oxide in H2 atmosphere are realized using the developed general computer program, especially for the polynomial integral and subsection integral operations.

Unified Interpretation of Exciplex Formation and Marcus Electron Transfer on the Basis of Two-Dimensional Free Energy Surfaces

The mechanism of exciplex formation proposed in a previous paper has been refined to show how exciplex formation and Marcus electron transfer (ET) in fluorescence quenching are related to each other. This was done by making simple calculations of the free energies of the initial (DA*) and final (D+A-) states of ET. First it was shown that the decrease in D-A distance can induce intermolecular ET even in nonpolar solvents where solvent orientational polarization is absent, and that it leads to exciplex formation. This is consistent with experimental results that exciplex is most often observed in nonpolar solvents. The calculation was then extended to ET in polar solvents where the free energies are functions of both D-A distance and solvent orientational polarization. This enabled us to discuss both exciplex formation and Marcus ET in the same D-A pair and solvent on the basis of 2-dimensional free energy surfaces. The surfaces contain more information about the rates of these reactions, the mechanism of fluorescence quenching by ET, etc., than simple reaction schemes. By changing the parameters such as the free energy change of reaction, solvent dielectric constants, etc., one can construct the free energy surfaces for various systems. The effects of free energy change of reaction and of solvent polarity on the mechanism and relative importance of exciplex formation and Marcus ET in fluorescence quenching can be well explained. The free energy surface will also be useful for discussion of other phenomena related to ET reactions.

Thermodynamics of the alkaline transition in phytocyanins

The thermodynamics of the alkaline transition which influences the spectral and redox properties of the type 1 copper center in phytocyanins has been determined spectroscopically. The proteins investigated include Rhus vernicifera stellacyanin, cucumber basic protein and its Met89Gln variant, and umecyanin, the stellacyanin from horseradish roots, along with its Gln95Met variant. The changes in reaction enthalpy and entropy within the protein series show partial compensatory behavior. Thus, the reaction free energy change (hence the pK (a) value) is rather variable. This indicates that species-dependent differences in reaction thermodynamics, although containing an important contribution from changes in the hydrogen-bonding network of water molecules in the hydration sphere of the protein (which feature enthalpy-entropy compensation), are to a large extent protein-based. The data for axial ligand variants are consistent with the hypothesis of a copper-binding His as the deprotonating residue responsible for this transition.

Nucleophilic reactivity of aniline derivatives towards the nitroso group

AbstractThis study investigates the reactivity of the electrophilic nitroso group towards nucleophilic aniline derivatives from various nitrosating agents. A relationship between the rate of nitrosation and Edwards' nucleophilic parameter (En) is disclosed, indicative of a transition state that is mostly product‐like in nature with respect to cleavage of the nitroso bond in the nitrosating agent. A similar relationship based on the work of Marcus provides a considerably better explanation of the data. Through application of the Marcus equation, the free energy change of reaction is calculated for the nitrosation reactions studied, which in turn is applied to develop an equation linking the free energy of formation of a nitrosamine and its corresponding protonated amine. This equation accounts for the often‐observed Brønsted relationships in nitrosation reactions. The intrinsic barrier to reaction is estimated to be 10 kJ mol−1, indicating that the main impedance to nitrosation arises from the unfavourable reaction thermodynamics. However, for the nitrosation of aniline derivatives substituted with π‐electron withdrawing groups, an unbalancing of the transition state results in an increased intrinsic barrier, of the order of 19 kJ mol−1. For electronic‐effect aniline derivatives, a More O'Ferrall–Jencks diagram shows the nitrosation transition state to be synchronously well balanced between reactants and products. This diagram also confirms that the reaction follows a concerted mechanism. The nitrosation of resonance‐stabilised aniline derivatives is somewhat less synchronous, however, due to delocalisation of the lone electron pair on the amino group, induced by π electron withdrawing substituents. Transition states were located according to the theory of harmonic parabolic wells. The results of these calculations agreed with transition state locations predicted using linear free energy relationship techniques. A method is developed which approximates free energy profiles by treating the product and reactant free energy wells as harmonic parabola, while the free energy profile around the transition state is taken as the parabolic barrier to the product and reactant energy wells. Applying this technique, free energy profiles for electronic‐effect and resonance‐stabilised aniline nitrosation are predicted, utilising measured kinetic values and predicted thermodynamic properties. We couple the simulated free energy profiles with their corresponding More O'Ferrall–Jencks diagrams in order to elucidate the three‐dimensional reaction path traversed between reactants and products, in terms of both structure and energy. We also demonstrate that the nitrosonium ion behaves as a soft acid, and should therefore undergo covalent frontier‐orbital controlled bonding. Copyright © 2007 John Wiley & Sons, Ltd.

Thermodynamics-Based Metabolic Flux Analysis

A new form of metabolic flux analysis (MFA) called thermodynamics-based metabolic flux analysis (TMFA) is introduced with the capability of generating thermodynamically feasible flux and metabolite activity profiles on a genome scale. TMFA involves the use of a set of linear thermodynamic constraints in addition to the mass balance constraints typically used in MFA. TMFA produces flux distributions that do not contain any thermodynamically infeasible reactions or pathways, and it provides information about the free energy change of reactions and the range of metabolite activities in addition to reaction fluxes. TMFA is applied to study the thermodynamically feasible ranges for the fluxes and the Gibbs free energy change, Δ r G′, of the reactions and the activities of the metabolites in the genome-scale metabolic model of Escherichia coli developed by Palsson and co-workers. In the TMFA of the genome scale model, the metabolite activities and reaction Δ r G′ are able to achieve a wide range of values at optimal growth. The reaction dihydroorotase is identified as a possible thermodynamic bottleneck in E. coli metabolism with a Δ r G′ constrained close to zero while numerous reactions are identified throughout metabolism for which Δ r G′ is always highly negative regardless of metabolite concentrations. As it has been proposed previously, these reactions with exclusively negative Δ r G′ might be candidates for cell regulation, and we find that a significant number of these reactions appear to be the first steps in the linear portion of numerous biosynthesis pathways. The thermodynamically feasible ranges for the concentration ratios ATP/ADP, NAD(P)/NAD(P)H, and H extracellular + / H intracellular + are also determined and found to encompass the values observed experimentally in every case. Further, we find that the NAD/NADH and NADP/NADPH ratios maintained in the cell are close to the minimum feasible ratio and maximum feasible ratio, respectively.

The Planck‐Benzinger thermal work function: Determining the thermal set point in interacting biological systems

AbstractOne of the more interesting discoveries in recent studies of biological thermodynamics concerns the way in which application of the Planck–Benzinger methodology allows one to evaluate how the thermal set point of a biological system is established and maintained on a molecular level. Macromolecular interactions will always exhibit a negative minimum value of the Gibbs free energy change at a well‐defined temperature, <TS>, which may be defined as the thermal set point. Each interacting biological system will have its own unique value of <TS> where the bound unavailable energy TΔS0 = 0. In the human body, that thermal set point is 37°C. At the thermal set point, the maximum work can be accomplished and the system is at its most stable. Each interacting biological system examined by the Chun method confirms the existence of a thermodynamic molecular switch wherein a change of sign in ΔCp0(T) leads to true negative minimum in the Gibbs free energy change of reaction, and hence a maximum in the related equilibrium constant, Keq. Application of the Planck–Benzinger methodology has demonstrated that there is a lower cutoff point, <Th>, where entropy is favorable but enthalpy is unfavorable, and an upper cutoff, <Tm>, above which enthalpy is favorable but entropy unfavorable. Only between these two limits, where ΔG0(T) = 0, is the net chemical driving force favorable for such biological processes as protein folding, protein–protein or protein–nucleic acid interaction and protein self‐assembly. In examining interacting biological systems, the thermal set point will vary from one organism to another, but TΔS0, the bound unavailable energy, will always be equal to zero. It is apparent from Chun's extensive studies of biological interactions that the origins of life in any chosen system are inevitably linked to a single, unique thermal set point. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

Thermodynamic Calculation on the Reduction of Iron Oxide in an H2 Atmosphere

Thermodynamic calculation on the reduction of iron oxide in H2 atmosphere is carried out in this paper. The general calculation model of the standard free energy changes for reactions are established. Accurate calculation and plotting of the standard free energy changes, equilibrium constants and gas composition for preparing iron by reduction of iron oxide in H2 atmosphere are realized using the developed general computer program.

Thermodynamic properties of LaCrO 3

The nano powders of LaCrO 3 were prepared by a sol-gel route. The heat capacity of LaCrO 3 nano powders from 350 to 550 K was measured by DSC method and expressed as: C p(LaCrO3) (±0.112) = 166.844 − 8.500 × 10 −3 T − 1.022 × 10 6 T −2 (J/(mol·K)) (350–550 K). An EMF measurement assembly was developed with CaF 2 as an electrolyte for the galvanic cell. From measured EMF data of the reversible cell, (−) Pt, La 2O 3, LaF 3, O 2 (1 atm)|CaF 2|O 2(1 atm), LaF 3, LaCrO 3, Cr 2O 3, Pt(+), and the relevant value of Gibbs free energy, the Gibbs free energy of formation of LaCrO 3 was calculated from 700 to 885 K: Δ G f,LaCrO 3 ▪ = −1555.364 + 0.354 T (kJ/mol) (700–885 K). And the Gibbs free energy change of reaction from simple oxides La 2O 3 and Cr 2O 3 was calculated to be: Δ G f,ox(LaCrO 3) ▪ = −94.758 + 8.530 × 10 −2 T(kJ/mol) (700–885 K).

The characterization of the fluorescence of l-histidine in simulated body fluid

In addition to being an essential natural amino acid, l-histidine is biologically important in the dismutation of superoxide radical (O 2 − ) by superoxide dismutase (SOD). In this work, fluorescence and absorptiometric techniques were used to characterize the photo-phenomenon and optical properties of this compound in a simulated body fluid (SBF). l-Histidine fluoresces at 360 nm when excited at 220 nm. Its molar absorptivity, ɛ, is 4.8 × 10 3 M −1 cm −1. The observed bimolecular quenching rate constant, k q, of 7.5 × 10 8 M −1 s −1, by hydrogen peroxide, suggests a non-diffusional activation-controlled mechanism with a rate constant, k a, of 8.55 × 10 8 M −1 s −1 and an electron transfer rate constant, k ET of 6.06 × 10 8 s −1. The determined radiative and non-radiative rate constants, 4.73 × 10 7 and 2.9 × 10 8 s −1, respectively, suggests that the deactivation of the thermally excited l-histidine is by non-radiative route rather than by normal fluorescence, which accounts for the low quenching constant, K SV, of 2.22 M −1 that was obtained. The solvent reorganization energy, λ s, and the reaction free energy change, Δ G, of 1.48 and −5.62 eV, respectively, suggest that the electron transfer reaction in the l-histidine–H 2O 2 reaction is through a solvent separated mechanism.

CALCULATION AND PLOTTING OF STANDARD FREE ENERGY CHANGES AND EQUILIBRIUM CONSTANTS FOR CVD HARD COATINGS REACTIONS

CVD (chemical vapor deposition) coatings such as TiC , TiN , Al 2 O 3 and so on are widely used as tools' surface preserved materials with corrosion resistance and wear resistance. Standard Gibbs free energy changes for reactions are widely used to approximately analyze the trends of substance reactions and phase transitions in chemical reactions, metallurgy processes, materials synthesis and processing. Hard coatings and thin films can be prepared by CVD process. Accurate calculation and plotting of the standard free energy changes and equilibrium constants for CVD TiC , TiN , Al 2 O 3 coatings reactions are realized using the developed general computer program. Only inputting the basic thermodynamic data tabulated in data books into the computer program, the relationships of the standard free energy changes and equilibrium constants for most reactions with the temperatures can be gained in the same diagrams.

Why Does the Human Body Maintain a Constant 37-Degree Temperature?: Thermodynamic Switch Controls Chemical Equilibrium in Biological Systems

Applying the Planck-Benzinger methodology to biological systems, we have established that the negative Gibbs free energy minimum at a well-defined stable temperature, ⟨TS⟩, where the bound unavailable energy TΔS° = 0, has its origin in the sequence-specific hydrophobic interactions. Each such system we have examined confirms the existence of a thermodynamic molecular switch wherein a change of sign in [ΔCp°]reaction leads to a true negative minimum in the Gibbs free energy change of reaction, and hence a maximum in the related equilibrium constant, Keq. At this temperature, ⟨TS⟩, where ΔH°(TS)(-) = ΔG°(TS)(-)min, the maximum work can be accomplished in transpiration, digestion, reproduction or locomotion. In the human body, this temperature is 37°C. The ⟨TS⟩ values may vary from one living organism to another, but the fact that the value of TΔS°(T) = 0 will not. There is a lower cutoff point, ⟨Th⟩, where enthalpy is unfavorable but entropy is favorable, i.e. ΔH°(Th)(+) = TΔS°(Th)(+), and an upper limit, ⟨Tm⟩, above which enthalpy is favorable but entropy is unfavorable, i.e. ΔH°(Tm)(−) = TΔS°(Tm)(−). Only between these two temperature limits, where ΔG°(T) = 0, is the net chemical driving force favorable for such biological processes as protein folding, protein-protein, protein-nucleic acid or protein-membrane interactions, and protein self-assembly. All interacting biological systems examined using the Planck-Benzinger methodology have shown such a thermodynamic switch at the molecular level, suggesting that its existence may be universal.

Review of the synthesis of layered double hydroxides: a thermodynamic approach

The synthesis of layered double hydroxides (LDHs) by hydrothermal-LDH reconstruction and coprecipitation methods is reviewed using a thermodynamic approach. A mixture model was used for the estimation of the thermodynamics of formation of LDHs. The synthesis and solubility of LDHs are discussed in terms of standard molar Gibbs free energy change of reaction. Data for numerous divalent and trivalent metals as well as for some monovalent and tetravalent metals that may be part of the LDH structure have been compiled. Good agreement is found between theoretical and experimental data. Diagrams and tables for the prediction of possible new LDH materials are provided.

Planck–Benzinger thermal work function in biological systems

AbstractThe Planck–Benzinger methodology provides a means of determining the innate temperature‐invariant enthalpy and thermal agitation energy, or the heat capacity integrals, ∫ ΔCp°(T)dT, and allows precise determination of 〈TCp〉, 〈Th〉, 〈Ts〉, and 〈Tm〉. It is a method for evaluating [ΔH − ΔH°(T0)], the heat of reaction for biological molecules at room temperature. The results imply that the negative Gibbs free energy change minimum at a well‐defined stable temperature, 〈Ts〉, where the bound unavailable energy TΔS° = 0, has its origin in the sequence‐specific hydrophobic interactions. Each case confirms the existence of a thermodynamic molecular switch wherein a change of sign in ΔCp°(T)reaction leads to true negative minimum in the Gibbs free energy change of reaction and, hence, a maximum in the related equilibrium constant, Keq. At this temperature, 〈TS〉, ΔH°(TS)(−) = ΔG°(TS)(−)min, the maximum work can be accomplished in biological systems. Application of the Planck–Benzinger methodology to biological systems has demonstrated a basic rule for life processes, in that there is a lower cutoff point, 〈Th〉, where entropy is favorable but enthalpy is unfavorable, that is, ΔH°(Th)(+) = TΔS°(Th)(+), and an upper cutoff, 〈Tm〉, above which enthalpy is favorable but entropy unfavorable, that is, ΔH°(Tm)(−) = TΔS°(Tm)(−). Only between these two limits, that is, where ΔG°(T) = 0, is the net chemical driving force favorable for such biological processes as protein folding; protein–protein, protein–nucleic acid or protein‐membrane interactions; and protein–self‐assembly. Indeed, all interacting biological systems examined using the Planck–Benzinger methodology have shown such a thermodynamic switch at the molecular level, suggesting that its existence may be universal. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Kinetic Study of the Phthalimide N-Oxyl (PINO) Radical in Acetic Acid. Hydrogen Abstraction from C−H Bonds and Evaluation of O−H Bond Dissociation Energy of N-Hydroxyphthalimide

The reactions of the phthalimide N-oxyl (PINO) radical with several hydrocarbons having different C−H bond dissociation energies (BDEs) were investigated in HOAc at 25 °C. The slope of the Evans−Polanyi plot is 0.38, which indicates that the reactions are mildly exothermic or almost thermoneutral. This finding is supported by the O−H BDE of N-hydroxyphthalimide (NHPI), 375 ± 10 kJ mol-1, obtained by means of a thermodynamic cycle. The observed kinetic isotope effects (KIEs) are related to the reaction free-energy changes, which can be explained by a Marcus-type equation. The comparison of the PINO radical reactions with analogous hydrogen atom abstraction reactions of the dibromide radical implies that HBr2• reactions are more exothermic than the PINO radical reactions. This factor is put forth to explain the different KIEs.

Gibbs free energy of formation of the Nd 2BaCuO 5 phase determined by the e.m.f. method

The Gibbs free energies of formation of the solid CuNdO 2 and CuNd 2O 4 phases in the (copper + neodymium + oxygen) system were determined by employing electrochemical cells with zirconia solid electrolyte. The results obtained in this study were used to derive the Gibbs free energy changes of reactions of their formation from the respective oxides. Then, stability domains of the ternary oxides as a function of temperature and p(O 2) were calculated. Next, employing electrochemical cells with the single-crystal calcium fluoride electrolyte: air, CaO + CaF 2| CaF 2| CuNd 2 O 4 + Nd 2 BaCuO 5 + Nd 2 O 3 + BaF 2, air, the Gibbs free energy change of the reaction of formation of the solid Nd 2CuBaO 5 phase was determined in the temperature range (973 to 1173) K. The results obtained in this study were used to derive the Gibbs free energy change of the reaction of formation of this phase from the respective oxides in the following form: ( Δ f G 0 m ±4450)/( J· mol −1)=−137 880 + 65.18·(T/ K) . It was found that the calculated enthalpy value followed the trend established so far for the lanthanide 2,1,1 “green phases”.

Molecular-Level Thermodynamic Switch Controls Chemical Equilibrium in Sequence-Specific Hydrophobic Interaction of 35 Dipeptide Pairs

Applying the Planck-Benzinger methodology, the sequence-specific hydrophobic interactions of 35 dipeptide pairs were examined over a temperature range of 273–333 K, based on data reported by Nemethy and Scheraga in 1962. The hydrophobic interaction in these sequence-specific dipeptide pairs is highly similar in its thermodynamic behavior to that of other biological systems. The results imply that the negative Gibbs free energy change minimum at a well-defined stable temperature, 〈 T s〉, where the bound unavailable energy, TΔ S o = 0, has its origin in the sequence-specific hydrophobic interactions, are highly dependent on details of molecular structure. Each case confirms the existence of a thermodynamic molecular switch wherein a change of sign in Δ Cp o( T) reaction (change in specific heat capacity of reaction at constant pressure) leads to true negative minimum in the Gibbs free energy change of reaction, Δ G o( T) reaction, and hence a maximum in the related equilibrium constant, K eq. Indeed, all interacting biological systems examined to date by Chun using the Planck-Benzinger methodology have shown such a thermodynamic switch at the molecular level, suggesting its existence may be universal.

Rigorous calculation of electric field effects on the free energy change of the electron transfer reaction

We theoretically investigate the effect of an external electric field on the free energy change of electron transfer reaction in polar solvents. The external electric field produces polarization both on the solutes and in the solvent. Since the polarization produced on the solute differs from that in the solvent, apparent surface charge is created on the surface of the solutes. The polarization charge on the surface of the solutes interacts with the charge associated with the electron transfer. The free energy change of the reaction including such effect is calculated rigorously. A simple formula is derived and compared to the exact result in the case of spherical solutes in the dielectric continuum media. Only slight deviations are observed for any values of the solvent polarity and of the ratio between the radii of the donor and the acceptor molecules. In addition, we also applied the same method to evaluate the reorganization energy rigorously: The Marcus expression for the reorganization energy is an approximate one. The accuracy of the Marcus expression is assessed by comparing it with the exact result.