# Inquiries into the Nature of Free Energy and Entropy in Respect to Biochemical Thermodynamics

## Abstract

**:**

## Introduction

#### Energy, Temperature, Work, Work Potential, and Free Energy

## Thermodynamics of Gases

_{1}to a volume V

_{2}, an amount of thermal energy Q equivalent to the work done would be transmitted from the gas to the reservoir in the course of the gas achieving thermal equilibrium with its surroundings during and after the compression. According to the ideal gas equation, if the compression of one mol of the gas were conducted reversibly (i.e., sufficiently slowly that the gas would remain in virtual thermal equilibrium with the reservoir), a minimum amount of work W

_{rev}energetically equivalent to RTln(V

_{1}/V

_{2}) of gravitational potential energy would be required to compress the gas, and, according to the First Law, an equivalent amount of thermal energy Q

_{rev}would be transmitted from the gas to the reservoir during the compression. In the course of this conversion of energy of the highest quality into the ambient thermal energy, the gas would be endowed with a potential for conversion of ambient thermal energy into an amount of work equivalent to that done in compressing it, as is obvious from the specifications that the temperature of the reservoir be constant and that the compression be conducted reversibly. In other words, if one mol of the gas were compressed isothermally (reversibly) from V

_{1}to V

_{2}, an amount RTln(V

_{1}/V

_{2}) of actual energy equivalent in quality to gravitational potential energy would be converted into an amount RTln(V

_{1}/V

_{2}) of ambient thermal energy plus an amount RTln(V

_{1}/V

_{2}) of an energy of the gas which is ordinarily ignored when encountered in this context, but which obviously must be viewed as being free energy. If we were to consider this free energy to be actual energy, the First Law of Thermodynamics would of course appear to be very seriously in error. That it is not in error is evident from the fact that, if the free energy were not used for reversing the degradation of the nonthermal actual energy into ambient thermal energy, it would simply disappear, never to reappear. Thus, if the compressed gas were allowed to expand from V

_{2}to V

_{1}without doing work on the surroundings, as would be the case if the massless and frictionless piston holding the gas in the compressed state (Figure 1) were suddenly released, it would do so rapidly without a net change in actual energy of any kind in either the gas or its surroundings [2].

_{rev}would be imparted even if the piston were moved extremely (infinitely) slowly. The faster the piston is moved the greater the amount of energy required and thus the greater W and Q must be. This implies that the temperature would be determined exclusively by the translational kinetic energies of the particles and that in the course of thermal equilibration with the reservoir, the kinetic energies on average would return to their initial values, leaving only a diminished volume and a consequent augmentation of the pressure to account for the deposition of free energy in the gas. Accordingly, experimental observations on real gases suggest that allowing an ideal gas to expand without doing work (i.e., without the particles expending kinetic energy) would not affect its temperature [2].

_{1}/V

_{2}) for an isothermal compression would be equivalent to RTln(P

_{2}/P

_{1}) and thus for such a compression one could attribute the consequent increase in free energy either to the decrease in the volume or to the increase in the pressure. That it would be appropriate to attribute the increase only to the increase in pressure is evident from the facts (i) that driving forces for change are determined only by intensive variables [4] and (ii) that pressure is the only intensive variable of the system under consideration.

#### Free Energy in Relation to Entropy

_{rev}for an isothermal (reversible) compression or expansion can be determined from the ideal gas equation and Q

_{rev}for such a process can be determined from W

_{rev}by invoking the First Law, neither W

_{rev}nor Q

_{rev}is a difference in a property, or state function, of the gas. W

_{rev}for such a process can be determined from the ideal gas equation because, for conditions of constant T, the state variables consist only of the intensive property P and the extensive property V which correspond to one another such that PV defines an energy having dimensions of mechanical work (i.e., dimensions of force × distance).

_{rev}being located in the surroundings rather than in the gas, Q

_{rev}/T for an isothermal change in the state of an ideal gas corresponds to a change in a property of the gas. Clausius named this property ‘entropy’ and represented it with the symbol S. With respect to gases and other chemical substances, the symbol S is now ordinarily used to express entropy as an intensive quantity having the same dimensions as R, the universal gas constant (i.e., dimensions of energy per mol per degree Kelvin). As a result, nST defines an amount of energy just as does nRT of the ideal gas equation, n being the number of mols and nS the extensive component of the energy. However, whereas nRT refers to an amount of pressure-volume work energy, nST refers to an amount of thermal energy that is somehow hidden with respect to temperature. Of course if the thermal energy of which entropy is the capacity factor were not hidden with respect to temperature, the appropriate comparable capacity factor would be the molar heat capacity of the substance.

_{P}and C

_{V}, respectively. According to the classical kinetic theory of gases, C

_{P}and C

_{V}for an ideal gas would be constants having the values $\frac{5}{2}R$ and $\frac{3}{2}R$, respectively, regardless of the magnitudes of the state properties temperature, pressure, and volume at which they are determined [6]. C

_{P}exceeds C

_{V}by R because an amount of pressure-volume work equivalent to R would be done on the surroundings if the heat capacity were determined at constant pressure. Thus, only C

_{V}is a heat capacity that refers exclusively to energy possessed by the gas.

_{2}/V

_{1}) or by −TΔS, ΔS being equivalent to Rln(V

_{2}/V

_{1}), a negative quantity in the case of compression, indicating a decrease in the entropy of the gas. Of course the reason why the entropy of the gas would decrease in this case is that the thermal energy on which the entropy is based would be located in the surroundings rather than in the gas. Entropy is universalized by using this volume-dependent entropy in a Carnot cycle to define an entropy in terms of the thermal energy possessed by the gas, in which case the free energy that is subject to loss would have temperature and heat capacity as its intensive and extensive components. Since both C

_{V}and C

_{P}would be constants, any changes in this ‘derivative’ entropy must necessarily be expressed in terms of changes in the temperature and thermal energy of the gas.

**Appendix**).

_{2}/V

_{1}) for a difference in the volume-dependent entropy on a per mol basis simply by changing a sign. The kind of entropy that would depend only on the temperature and thermal energy of the gas shall be referred to as characteristic entropy, and the corresponding kind of free energy shall be referred to as characteristic free energy. It is important to note that these ‘concentration’ and ‘characteristic’ kinds of entropy and free energy are very similar to but not entirely identical with the ‘cratic’ and ‘unitary’ kinds defined by Gurney.

_{2}/V

_{1}) or RTln(P

_{2}/P

_{1}) and thus would be directly proportional to temperature, it is evident that changes in free energy of concentration, although not involving net changes in the characteristic free energy of the gas, would require that the gas possess characteristic free energy if the changes are to be finite, the characteristic free energy being that associated with the thermal energy at the temperature of the two states. Of course the higher the temperature, the greater the change in free energy of concentration for a given change in concentration of the particles possessing the thermal energy and free energy.

_{V}, the molar heat capacity determined under conditions of constant volume, meets this requirement and, for an ideal gas, would not depend on the magnitudes of the state properties at which it is determined. This being the case, the total energy for one mol of such a gas would be given by C

_{V}T and finite changes in total energy on a per mol basis by C

_{V}ΔT, C

_{V}being a constant. Since reversible processes do not expend free energy, C

_{V}ΔT for any reversible adiabatic compression or expansion would be equivalent to the change in characteristic free energy as well as to the change in total energy. Thus, if the quality of thermal energy were judged on the basis of the predicted changes in the characteristic free energy of an ideal gas, it would appear to be equivalent to that of nonthermal actual energy. However, as outlined below, a proper assessment of the quality of thermal energy in terms of changes in the properties of a gas can be achieved only by means of the Carnot cycle.

#### The Carnot Cycle

_{char}= dE/T or its equivalent. Since dE = C

_{V}dT and C

_{V}would be a constant, this equation can be expressed in the forms dS

_{char}= C

_{V}dT/T = C

_{V}dlnT and integrated between two specific temperatures to yield ΔS

_{char}= C

_{V}ln(T

_{2}/T

_{1}). The comparable expression Rln(V

_{2}/V

_{1}) for a finite change in the entropy of concentration on a per mol basis is obtained essentially in the same manner. In this case, however, it is the pressure and volume that would be incapable of changing independently of one another. In consequence of this and of the fact that the volume must change if the entropy is to change, a change in S

_{conc}can be expressed properly in terms of volume and pressure only by means of the differential equation dS

_{conc}= PdV/T, which can be modified by substituting RT/V for P to obtain the equation in the integrable form dS

_{conc}= RdlnV. Of course the scaling of ΔS

_{char}to ΔS

_{conc}in the adiabatic steps is accomplished through the facts (i) that a net change in entropy would not be possible in a reversible adiabatic process and (ii) that, in consequence, the ratios of the initial and final temperatures and volumes in the adiabatic steps of a reversible Carnot cycle would be constrained to agree with one another according to the relationship: T

_{2}/ T

_{1}= (V

_{1}/ V

_{2})

^{R/ CV}.

_{char}for the adiabatic steps is based on the thermal energy transmitted isothermally to the surroundings at the relatively low temperature is evident from the facts (i) that the maximum efficiency η

_{max}of a Carnot heat engine is equivalent to 1 − T

_{low}/T

_{high}and (ii) that ΔS

_{char}for the adiabatic steps of the reversible cycle is therefore equivalent to ±C

_{V}ln(1 − η

_{max}), the positive and negative signs referring to the expansion and compression steps, respectively. By noting that η

_{max}refers to the available (work) fraction of the thermal energy absorbed at T

_{high}, one can readily see that ΔS

_{char}refers to the unavailable (thermal) fraction. For any given finite value of T

_{high}, the unavailable fraction is a linear function of T

_{low}and decreases to zero as T

_{low}approaches zero.

**Appendix**). In any particular case, the fundamental process giving rise to the finite rate would be the net conversion of relatively high-quality actual energy, both thermal and nonthermal, into ambient thermal energy, the spontaneous and unidirectional nature of which is the basis for the Second Law of Thermodynamics, which, unlike the First Law, acknowledges the existence of free energy and says in effect that if thermodynamic work is to be done at a finite rate, free energy must be expended. Also unlike the First Law, the Second Law, owing to the fact that the individually mobile particulate constituents of macroscopic amounts of matter at finite temperatures vary widely as to translational kinetic energy, is a statistical law appropriate for application only to macroscopic phenomena. This means that the Second Law is obeyed only on average over time in processes at the microscopic level and thus that conversions of ambient thermal energy into nonthermal actual energy in chemically active substances can occur at the molecular level. Of significance in this regard is the fact that translational thermal energy at the molecular level is kinetic energy of the directed variety, a consequence of which is that no distinction can be made between this form of actual energy and nonthermal actual energy at the molecular level. Accordingly, energy transfer at the molecular level occurs without expenditure of free energy, and irreversibility, like temperature and pressure in respect to a gas, is a concept applicable only to macroscopic phenomena.

#### Real Gases

_{high}to the amount rejected isothermally at T

_{low}would have the same value regardless of the nature of the working substance.

_{V}for an ideal gas would be a true constant implies that the amount of thermal energy possessed by a given amount of such a gas would be directly proportional to temperature. This in turn implies that an ideal gas can be considered to provide linear absolute scales for actual energy and free energy as well as for temperature. Although based on predicted properties of a fictitious gas that differs appreciably from real gases in that its particulate constituents possess only translational kinetic energy, these scales are very important in that they are commonly used with remarkable success as a framework for characterization of the thermodynamic properties of all real substances, a fact which accords with the universality of the universal gas constant R. As will in effect be suggested below, the widespread success of the ideal gas model is likely due in large part to the above-noted prediction of the equipartition principle being applicable to liquids and solids as well as to gases and to temperature in liquids and solids being determined by the average kinetic energies of individually mobile particles consisting of clusters of attractively bound molecules, the average size of which tends to increase with decrease of temperature.

## Thermodynamics of Liquids and Solids

## Third Law of Thermodynamics

## Thermodynamics of Biochemical Reactions

_{1}and k

_{2}are the forward and backward rate constants, and the arrows preceding the reactant and following the product are meant to indicate that a and p, the concentrations of A and P, are maintained constant as a result of the reaction being in a long sequence of reactions of the sort that one might expect to find in living systems. Thus, Reaction (1) as presented is considered to be a stationary-state reaction in an open system of more or less constant volume and to have a net velocity in the direction indicated regardless of whether the equilibrium constant is favorable or unfavorable. Although the system as presented is considered to be open and therefore relevant primarily to biological systems, in most of what follows it will be considered to be closed (i.e., capable of exchanging only thermal energy with the surroundings).

#### Determination of ΔG° by the Equilibrium Method

^{−G/RT}, where V = $\frac{1}{c}$ = molar volume and A = e

^{C/RT}, this foundational relationship of the classical approach to chemical thermodynamics has much in common with those of the Maxwell-Boltzmann statistical approach. In consequence of C being unknown, G must be determined in relation to a reference Gibbs potential G° and its concentration component RTlnc°, which by convention are subtracted from G and RTlnc. Thus, G − G° = RTlnc − RTlnc° and G = G° + RT ln(c/c°). By convention, the difference in Gibbs potential between a reactant and a product, such as A and P of Reaction (1), is obtained by subtracting the Gibbs potential of the reactant from that of the product. Thus, for Reaction (1):

#### Nature of Characteristic Chemical Potential Energy

^{+}and A

^{−}which possess binding energy only by virtue of their net charges. Since unlike charges attract one another, binding energy for a binding interaction between A

^{+}and A

^{−}must necessarily be viewed as being attractive. On the other hand, since like charges repel, binding energy for an interaction between A

^{+}and A

^{+}, for example, must necessarily be regarded as being repulsive. Since the attractive binding interaction would clearly involve a net consumption of binding energy, the repulsive one must by implication involve a net production of binding energy. Since both attractive and repulsive binding energy must be forms of nonthermal actual energy, it appears from this example that the binding energies of molecules possessing particularly large amounts of bond energy are likely to be of the repulsive kind. However, as is evident from the fact that the bulk of local matter occurs naturally as liquids and solids consisting of attractively bound molecules, most binding interactions between atoms are net attractive. In view of this and of the large amount of energy required to force like charges into close proximity to one another, it seems likely that most of the molecules we ordinarily think of as possessing relatively large amounts of bond energy are molecules in which the constituent atoms are relatively unattractive to one another and/or are constrained (bonded) in such a way as to prevent optimal neutralization of their attractive forces. If we were to consider this invariably to be the case, we would be ignoring the fact that, by virtue of the existence of energy barriers to the making and breaking of chemical bonds, it is possible for kinetically stable bonds to be formed between atoms despite the interaction being net repulsive.

#### Determination of ΔG° by the Calorimetric Method

_{2}⇌ 2A, a net increase in the total free energy of concentration due to a chemical change under the conditions assumed here will be accompanied by an increase in a latent thermal energy that will be indistinguishable from if not identical with the above-described extramolecular kind.

#### Enthalpy-Entropy Compensation

_{conc}) for solvation of the solute from the high solvent activity alone. Thus, one could expect some solvation of the solute to occur even if the attractive forces between the solute and solvent molecules should be very weak. If the attractive forces between molecules of the solute and solvent should be weaker than those between molecules of the solvent, one could expect solvation of the solute to be accompanied by net conversion of thermal energy into attractive binding energy due to hindrance of bond formation between solvent molecules more or less as proposed by Hildebrand and coworkers [67,68], resulting in a temperature-sensitive tension in the solvent at the surfaces of the solute molecules similar to that which occurs at the interface of the solvent and its vapor. As a result of this tension, one could expect the existence of a force for minimization of the amount of space occupied by the solute. Accordingly, surface tension in a liquid results from molecules at the surface possessing relatively large amounts of attractive binding energy which in turn results in there being a relatively large attractive (contractive) force among molecules at the surface that tends to minimize surface area [69].

#### Role of Free Energy of Concentration in Free Energy Transfer and Conservation

## Acknowledgement

## Appendix

_{1}at a temperature T

_{1}of the reservoir were compressed to a volume V

_{2}adiabatically, the temperature of the gas would increase by an amount depending on the heat capacity and on the amount of energy ΔE imparted to the gas. This amount of energy would be equivalent to the amount of work W done and thus would depend on the rate of the compression. Regardless of the rate, W and ΔE would be equivalent to C

_{V}ΔT. Since S, T, and V are state functions and C

_{V}would be a constant, the expression for the net change in entropy

_{net}= ΔS

_{char}+ ΔS

_{conc}= C

_{V}ln(T

_{2}/T

_{1}) + R ln(V

_{2}/V

_{1})

_{char}would be equivalent to −ΔS

_{conc}and T

_{2}could be determined from the relationship T

_{2}= T

_{1}(V

_{1}/V

_{2})

^{R}

^{/CV}. If the gas were compressed at a finite rate, T

_{2}would be relatively high and ΔS

_{net}for the compression would be finite and positive as a result of the increase in characteristic entropy exceeding the decrease in entropy of concentration. In this ‘irreversible’ case, T

_{2}could be determined only by experimental means. Although calculation of ΔS

_{net}could be easily achieved if T

_{2}for the irreversible compression were known, doing so would not be helpful in respect to precise quantification of the amount of free energy consumed. However, since the work potential of the gas would increase by an amount equivalent to the amount of work done regardless of the rate of the compression, doing so could serve to indicate in a semiquantitative fashion that work potential has been rendered unavailable. As indicated below, if T

_{2}were known, the precise amount rendered unavailable could be determined theoretically by considering the gas to undergo reversible expansion to its original volume.

_{2}and the lower temperature that would exist upon return of the gas to its original volume V

_{1}. Since ΔS

_{char}for the expansion would be equivalent to −ΔS

_{conc}, this lower temperature would be equivalent to T

_{2}(V

_{2}/V

_{1})

^{R/CV}and would of course be higher than the temperature T

_{1}of the reservoir by an amount depending on how fast the gas was compressed. In addition to the difference in temperature, there would remain differences in the pressure, thermal energy, and characteristic entropy, the difference in characteristic entropy being equivalent to that existing at the end of the compression phase. Elimination of these differences would require removal of the adiabatic constraint so that an amount of thermal energy C

_{V}ΔT could undergo transmission to the reservoir, C

_{V}ΔT in this case being equivalent to the amount of characteristic free energy consumed as a result of the gas being compressed at a finite rate. Since the amount of the gas was specified to be one mol, this amount would be equivalent to C

_{V}[T

_{2}(V

_{2}/V

_{1})

^{R/CV}− T

_{1}]. The value thus obtained divided by the temperature of the reservoir would be the net increase in entropy of the reservoir and for the overall process. As in any cyclic process, the net changes in free energy and entropy would be changes only in the characteristic kinds. Since the increase in entropy for the overall process would differ from that incurred by the gas in the compression phase and could be determined quantitatively only through determination of the amount of thermal energy transmitted to the surroundings, quantification of the net change in entropy for the cyclic process would in effect require prior quantification of the net change in free energy.

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**Figure 1.**Compression and expansion of an ideal gas in a rigid-cylinder, piston system. The piston is assumed to be massless and frictionless and the temperature of the thermal reservoir is assumed to be constant. W = an amount of mechanical work; Q = an amount of thermal energy; V = volume of the gas; P = pressure of the gas.

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**MDPI and ACS Style**

Stoner, C.D.
Inquiries into the Nature of Free Energy and Entropy in Respect to Biochemical Thermodynamics. *Entropy* **2000**, *2*, 106-141.
https://doi.org/10.3390/e2030106

**AMA Style**

Stoner CD.
Inquiries into the Nature of Free Energy and Entropy in Respect to Biochemical Thermodynamics. *Entropy*. 2000; 2(3):106-141.
https://doi.org/10.3390/e2030106

**Chicago/Turabian Style**

Stoner, Clinton D.
2000. "Inquiries into the Nature of Free Energy and Entropy in Respect to Biochemical Thermodynamics" *Entropy* 2, no. 3: 106-141.
https://doi.org/10.3390/e2030106