SOME CONCEPTS IN ENERGETICS

Introduction.

Energetics (or thermodynamics) is the branch of physical chemistry that describes, on the macroscopic level, matter, its properties and the physical and chemical changes it undergoes. It has great power because the entire building of relationships that have been developed rests on only two principles that have never been invalidated by experimental evidence. Therefore, its conclusions are valid as long as they are properly derived. At the same time, energetics have important limitations. It is based on a simplified model, or representation, of physical reality which totally ignores the microscopic structure of matter. Hence, the theory cannot yield direct information about the microscopic realm. Only when additional assumptions about molecules are introduced can it contribute to our understanding of processes at the molecular level. Still, precisely because their reliability is not weakened by their dependence on these assumptions, the results obtained are considered as the most dependable in the whole realm of physics. Another limitation of classical thermodynamics is that its theory has nothing to say about the rates at which processes take place; it only predicts the direction of the change. We can theoretically establish that glucose spontaneously reacts with oxygen to form carbon dioxide and water. (In fact, we can induce this reaction to proceed explosively.) Yet, we may keep glucose in the presence of oxygen for years without observing any changes.  Time is not a factor considered in thermodynamics.

Living systems continuously transform energy through both physical and chemical processes. Being alive implies the necessity too acquire energy from the surroundings in some form and to process it in numerous different ways so that this energy appears in different forms. The chemical energy contained in potential form in the carbon compounds that serve as nutrients to a cell, or the radiant energy trapped by chlorophyll, eventually appear partly as the work done by the cell in synthesizing many new molecules, in transporting substance across the membrane, in movement, etc., and partly as heat. The theory of thermodynamics constitutes the fundamental unifying conceptual framework of all of these processes when considered as energy transformations. Each of these processes is, then, a special case of a general principle. Consequently, an understanding of this theory provides us with the perspective that guides our thinking along reliable paths from the general to the particular, and the tools to proceed down those paths with scientific precision. An example may clarify this. In order to determine whether a certain chemical reaction will proceed in a given direction or not, we need to apply the same concepts and the same equations, both derived from thermodynamics, as when considering whether a specific active transport across the plasma membrane of a cell is feasible, or whether certain ionic species will spontaneously cross the membrane, etc.

Thermodynamic theory depicts the world in a highly simplified fashion. Each object is considered to be characterized by a small number of measurable properties (temperature, pressure, energy, etc.) whose values are uniform throughout the object. Any portion of the universe that one wishes to consider with exclusion of the rest is called the system. The excluded portion constitutes the surroundings of the system, and the real or imaginary surface that separates them is the system’s boundary. A particular cell, a mitochondrion, or a protein solution could be systems. These would be real systems, but a system could also be a mental construct, a purely imaginary one whose properties are arbitrarily defined. These ideal systems are extremely useful in deriving the theoretical edifice of thermodynamics as well as in analyzing some practical problems.

Systems interact with their surroundings through their boundaries, and may be classified according to the boundary’s properties. In an open system, heat, work and matter can cross the boundary. In a closed one, matter cannot cross. In an adiabatic one, matter and heat cannot cross  In an isolated one, nothing can cross. Strictly speaking, adiabatic and isolated systems are ideal, since no real boundary can perfectly prevent the flow of heat.

Characterization of the state of a system.

In mechanics, a system is completely specified at a given instant if the position and velocity of each mass-point are given. Six variables are needed for each mass-point, three to specify the position and three for the velocity. When a rigid macroscopic body is considered and only its overall state of motion is of interest, the position and velocity of its center of mass are sufficient to specify the state of the system. This simplified view ignores the state of thermal motion of the material particles that constitute the body and, consequently, cannot deal with heat exchanges. Still, it is useful in dealing with certain problems. A full description of the dynamical state of the body would require the specification of six variables for each material particle in the body. This is clearly not practical.

In thermodynamics, a simpler concept of the sate of a system is introduced based on bulk properties: volume, pressure, temperature, composition, state of aggregation, etc., which are called variables of state or properties. A characteristic of a system is a variable of state or property only if its value does not depend on the past history of the system, but only on the conditions at the time of measurement. A property is either intensive or extensive depending on whether its value remains the same or is reduced when a part of the system is removed. Table I contains a list of some properties. A system is said to be in a specified state when all its variables or properties have specified values. A hot cup of coffee with a floating ice cube in it is not in a specified state because neither the temperature nor the concentration of coffee can be properly specified since they have different values in different parts of the system and they are changing. A defined state is a state of thermodynamic equilibrium if the values of its properties are independent of time and if there is no flow of matter or energy. A cell that is receiving nutrients and oxygen, metabolizing them, and releasing waste and heat, all at constant rates such that its temperature, volume, pressure and the concentrations of all its components remain constant in time, is not at equilibrium, but in a steady state, because matter and heat are flowing through its boundary. It is evident that the knowledge of the thermo dynamical state alone is not sufficient for the determination of its microscopic dynamical state. Given a certain thermo dynamical state of a homogeneous fluid, defined by any two of the variables of state, we observe that there is an infinite number of states of molecular motion that correspond to it. With increasing time, the system exists successively in all these dynamical states. Then we may say that a thermo dynamical state is the ensemble of all the dynamical states through which, as a result of molecular motion, the system is rapidly passing while the values of the sate variables remain constant.

A mathematical expression that relates some properties of a system is called an equation of state. These are not derivable from thermodynamic principles, but may be obtained empirically or derived from a molecular theory. PV = nRT is an equation of state, and so is V = a + bT + cT² + dT³. On the other hand, a function of state is a mathematical expression (an equation of state) that relates the value of a property to the values of as many other properties as necessary to specify the state of the system. P = P (n, V, T) indicates the existence of a function of state for an ideal gas even though its precise form is not stated. Its precise form,   PV = nRT, was obtained through experimentation. V = a + bT + cT² + dT³ , on the other hand, is not a function of state because it does not include enough properties to specify the state of the system. Gibbs established in 1878 the phase rule, which states that, if a system consists of a single phase and a single component, its state is fully defined (specified) when the values of one extensive and of two intensive properties are known. For each additional component in the system, the value of another intensive property needs to be known; and for each additional phase, the value of one more extensive property must be known. When these requirements are fulfilled, the values of all other properties are determined.

TABLE I

Intensive and Extensive Properties

Process.

A process is an event in which the value of at least one property of the system changes in time. Therefore, the state of the system changes, and the new state is characterized by the new values of the properties. The occurrence of a process implies that the initial state of the system was not a state of equilibrium or that an external influence is causing something (heat, work or matter) to cross the system’s boundary. A system at equilibrium will not spontaneously undergo a process, but when an external influence causes a process to occur, the effect of that influence has been to transform the state of the system into a nonequilibrium state and the process will then happen spontaneously.

Consider the following two situations. The first one consists of a gaseous mass in a cylinder covered with a frictionless piston with added weight that compresses the gas. If we define our system as the gas, it will be at equilibrium as long as its pressure, temperature and composition remain constant (the volume will be, of course, determined by the other properties). If now some of the weight on top of the piston is removed, the pressure of the gas will momentarily be higher than the external pressure and the gas will expand doing work on the surroundings by lifting the piston with the remaining weight. Work crosses the boundary. We could have applied a heat source to increase the temperature with the consequent expansion. Then, both heat and work would cross the boundary. The second example consists of a Zn electrode immersed in a solution of ZnCl₂ and a copper electrode immersed in a solution of CuSO₄ , with both solutions electrically communicated by a salt bridge forming an electrochemical cell. A potentiometric device is connected to the electrodes to impose a variable electrical potential difference to the cell. When the applied potential difference exactly opposes the one generated by the call, the system is at equilibrium, i.e., no electrical current will flow and no chemical reaction will take place. If the potential difference applied to the cell is reduced, electrical current will flow throughout he external circuit, i.e., electrical work will be done on the surroundings, a chemical reaction will happen in the cell so the composition of the system changes, and heat will be produced in the cell and flow to the surroundings. In both examples, an intervention on the surroundings has disturbed the initial state of equilibrium of the system.

A reversible process is one that, starting from equilibrium, proceeds through a succession of equilibrium states, each differing from the preceding one by only an infinitesimal change in at least on variable of state. This seems to contradict the definition of equilibrium and, in fact, reversible processes do not occur in nature; only irreversible processes happen spontaneously. Yet, it is possible to conceive such a process. Imagine that, in the above examples, the weight on the piston or the potential difference applied by the potentiometer are reduced by one infinitesimal. The system will not be essentially away from equilibrium, but it will undergo an infinitesimal change in state. The gas, for example, will suffer an increase in volume and a decrease in pressure both of an infinitesimal magnitude. If this is repeated an infinite number of times, a finite change in volume and pressure, and consequently in the state of the system, will take place.

Although reversible processes can only be approximated in practice, the concept is of extraordinary importance because some properties that involve heat may be defined and calculated only for this type of process. Since, by definition, the values of functions of state depend only on the state of the system, the change that these values undergo during a process depend only on what the initial and final states are, not on the route followed in going from one state to the other. In the example of the expansion of a compressed gas, let a portion of the external weight be removed suddenly thereby reducing the external pressure. The gas will rapidly expand from V₁ to V₂ and its pressure and temperature will drop from P₁ and T₁ to P₂ and T₂. Now, let heat flow from the hotter surroundings until the temperature equilibrates at T₁ and the pressure and volume readjust to P ₃ and V ₃. This process consisted in the change in the state of the gas from V₁ P₁ T₁ to V ₃ P ₃ T₁ . A different way to conducting the same change in state would be to reversibly allow the same expansion by removing the weight in infinitely small amounts at a time. Then, the process will happen isothermically because heat will continuously flow into the gas at an infinitely slow rate and the temperature will remain equal to that of the surroundings. Therefore, when the volume reaches V ₃ the pressure will be P ₃ since the temperature is T₁ and the equation of state does not allow any other value of P. So, we have achieved exactly the same change in state along a different path and, since the initial and final states are identical, the changes in the values of all properties are equal for both paths. This means that a calculation of any such change made for the reversible process will yield a result that is valid for any other process that starts at the same initial state and ends at the same final state.

The laws of thermodynamics

The entire edifice of thermodynamics is based on two principles (or laws) that cannot be proven or logically deduced. They are empirically derived and, because many attempts to prove them wrong have always failed, they are placed in a privileged category of scientific laws. Since only these two principles support its conclusions and no other assumptions regarding the structure of matter or of any other kind are invoked in deriving its results, these results are given greater validity than those of any other branch of physics.

1^st Law - Conservation of Energy

Early in the study of physics one becomes acquainted with the concepts of energy and work and with the notion that a system that has a certain amount of potential energy, if properly controlled, can lose that energy while doing work and releasing heat. The demonstration by Joule of the quantitative equivalence between heat and work lead to the conclusion that the loss of internal energy by the system is equal to the sum of the work performed and the heat evolved.

                                )E = Q - W                    (1)

Consider the following reaction

                Pyruvate (1M) + H₂ (1 atm)     6   Lactate (1M)

It is convenient to define the quantity Q - W, in which Q is heat absorbed and W is work done by the system, in order to evaluate the change in energy of the system. If the reaction takes place reversibly in a suitable electrochemical cell, it can perform as much as 47,865 joules of electrical work while releasing 42, 677 joules as heat per mole of pyruvate transformed (W = 47,865 j /mol; Q = - 42,677 j /mol). Then, Q - W = - 90,542 j /mol, which is the amount of energy lost by the system when 1 mole of pyruvate and one mole of hydrogen are transformed into 1 mol of lactate. The same reaction could take place in a closed vessel while not doing any work. Then, if the entire process is conducted so that the final temperature and pressure are the same as in the reversible process, Q = - 90,542 j /mol. Notice that in both processes the system has undergone exactly the same change in state and lost the same amount of energy, but W and Q are different for the two processes. Many other examples in which the same is observed justify the conclusion that the energy change that accompanies a change of state of a system only depends on the initial and final states of the system and not on the way in which the change happened. Therefore, E is a property of the system whereas Q and W are not.

The above relationship may be stated in differential form for any process in which a system exchanges heat and work with its surroundings

                                dE = *Q - *W                                 (1')

where “d” indicates an exact differential and “*” an inexact differential. Properties are represented by functions of state, so, their differentials are exact (for a discussion of exact and inexact differentials, see Appendix I ). Consequently, for a finite change from state 1 to state 2

                                                  (2)

When the only kind of work that the system can perform is limited to expansion work by the  way its relationship to the surroundings is structured, (see Table II for various forms of thermodynamic work) the change in E at constant volume is

                                dE = *Q_V  - PdV

where the subscript in Q means that the change happens at constant volume. Then, if the process happens at constant volume, the change in Q becomes equal to the change in E, which is a property, so its differential becomes exact.

                                dE = dQ_V

                                )E = )Q_V                         (3)

The change in E for only expansion work and at constant pressure (but variable volume) is

                                dE = *Q_P - PdV

so                            dQ_P = dE + PdV                   (4)

which is an exact differential because it is a function of properties of the state of the system and may be used to define a new property, enthalpy: H = E + PV. Then, in general,

                                dH = *Q - *W + PdV + VdP               (5)

For expansion work only,

                                dH = *Q + VdP

and, at constant pressure,

                                dH = dQ_P                                    (6)

which shows that a measurement of the heat exchanged by a system during a process conducted at constant pressure and while doing only expansion work is a measurement of the change in enthalpy of the system for that process. Measurement of changes in enthalpy are the basis for the applications of the first law in chemistry and biochemistry. Through these thermochemical studies it is possible to determine the relative energies of compounds and the energies of chemical bonds.

TABLE II

Various Forms of Thermodynamic Work

An important goal in thermodynamics is to find a reliable criterion for spontaneity. Will a process being considered happen spontaneously or not? The minimization of E is an appropriate criterion for changes in the mechanical state of macroscopic systems in which heat is not a factor, e.g., movement of masses in a gravitational field. But many other spontaneous processes have  )E  $ 0. When two masses of the same metal at different temperatures are placed in contact with each other while insulated from their surroundings, heat will transfer spontaneously from the hotter mass to the cooler one. Still, for this process )E = 0, since heat has not been exchanged with the surroundings and the expansion of one mass (work) is canceled  by the contraction of the other. When ice is placed in surroundings at 1^oC, the ice will spontaneously melt while absorbing heat from the environment at constant temperature. Since there is also a decrease in volume (ice is less dense than water at that temperature),
)E = Q - W > 0.

These examples demonstrate that another property whose change is a reliable criterion of spontaneity needs to be sought. Work can be totally converted into heat, but the reverse is not possible. The energy obtained from a source of heat can only be partially converted into work, and then, only under certain conditions. Work is the concerted movement of many particles of matter under a macroscopic force, while heat seems more as chaotic molecular motions. Once work has been “degraded” to heat, only a fraction of the energy can be converted back to work in a cyclic process, (one in which the initial and final states are the same). It follows that the amount of heat in the universe increases as energy conversions take place. It seems, then, it might be possible to derive a property related to heat such that it would change in a given direction as processes spontaneously happen. A process in which the system releases heat to the surroundings is exothermic. If conducted at constant pressure, Q_P = )H. Since by the convention used in stating the first law heat released by the system has a negative sign, the )H of an exothermic process is negative. For a long time it was considered that this was a characteristic of spontaneous processes, but a consideration of the processes discussed in the previous paragraph in connection with )E and others shows that this is not the case. The dissolution of, e.g., magnesium sulfate in water, which is obviously spontaneous, is endothermic, as can be readily established by observing the drop in temperature of the system as the dissolution proceeds.

2^nd Law - Entropy.

Since Q is not a property, the heat Q  for any change in state may have any value depending on the path followed by the process and all differentials involving *Q would be inexact, but *Q_REV is defined because the path is defined. It can be shown that, although *Q_REV is an inexact differential, *Q_REV / T is exact. Hence, this ratio can be used to define a new property. This new property is called entropy (S) and defined as

                                                (7)

If the definition used Q / T and *Q / T, the values of these fractions would not be specified, i.e.,  Q and *Q could have any values. Then the definition would be useless. Notice that *Q_REV now has a specified value because the path of the process is being specified, but does not become an exact differential. Earlier, with *Q_V and  *Q_P, when the paths were specified (only expansion work with constant volume or constant pressure, respectively) these differentials became not just specified in value, but also functions of only properties of the state of the system and, therefore, functions of state and exact differentials.

When a system and its surroundings are enclosed by a boundary that prevents any exchange of matter, heat or any kind of work, the larger system composed of the system of interest and its surroundings is said to be isolated. The properties of entropy are such that, if a reversible process is conducted within such an isolated enclosure, the total change in entropy is zero, whereas if the process is spontaneous, the total change in entropy is positive:

                dS_TOT = dS_SYS + dS_SUR $ 0                                         (8)

Since the system undergoing a reversible process is always at equilibrium, the equal sign in (8) also applies to a state of equilibrium. So, for reversible processes dS_TOT = 0 and dS_SYS = - dS_SUR.

As indicated before, the fact that the definition of entropy involves Q_REV and not Q does not invalidate its application to spontaneous (irreversible) processes because S is a property. This is one reason why the concept of reversible processes is useful: it allows us to express the change in entropy in a completely defined way, and this value is true for any process that accomplishes the same change in state.

Applying the definition (7) to (8) we get

                                                   (9)

since the heat absorbed by the system must be equal to that released by the surroundings when they together constitute an isolated system. The temperatures of the system and surroundings need not be equal. In fact, if they were equal we would have equilibrium and dS_TOT = 0. This may present another conceptual difficulty. If the temperatures were different, a spontaneous transfer of heat would take place and the process would be irreversible. How could we, then, conceive a reversible transfer of heat? This problem could be obviated as follows: imagine a reversible process, such as the displacement of a phase equilibrium (like the freezing of water at 0^oC), to go on in the surroundings in a manner that allows the reversible transfer of the reversibly generated heat to the system which is at a lower temperature. This way it is possible to unambiguously express the change in entropy for the surroundings and the system even when their temperatures are different and the process taking place in the system is irreversible.

An illustration of how the change in entropy is used to show the direction of spontaneous processes is given by the following use of equation (9) to predict the direction of the spontaneous flow of heat. Since the total change in entropy is greater than zero for a spontaneous process,

                             (10)

If  T_SYS > T_SUR,   1/ T_SYS < 1/ T_SUR  and   (1/T_SYS - 1/ T_SUR) is negative. So,  in order for dS_TOT to be >0,  *Q_REV must be negative, meaning that the system loses heat to the surroundings. If  T_SYS < T_SUR,  *Q_REV must be positive for dS_TOT to be positive, i.e., heat flows from the surroundings to the system. This, of course, is the same as saying that heat flows from the hot region to the cooler one. But this is predicted by the second law, not by the first.

The change in entropy of a system plus its surroundings, when both together form an isolated system, constitutes an effective criterion of spontaneity and of equilibrium. Entropy has the inconvenient, at times serious, that its change for the surroundings needs to be evaluated. It would be advantageous to find another property that, as entropy, always changes in the same direction for spontaneous processes, but that, unlike entropy, shows this behavior when its change is evaluated for the system with exclusion of the surroundings. There are two such functions: Helmholtz free energy (A = E - TS) and Gibbs free energy (G = H - TS). We will deal only with Gibbs free energy whose differential is

                dG = dH -TdS - SdT                                                 (11)

Remembering that H = E + PV and that its differential is dH = dE + PdV + VdP and substituting 

                dG = dE +  PdV + VdP -T dS - SdT

Next, we use the first law for a reversible change

                dE = *Q_REV - *W_REV

                dG = *Q_REV - *W_REV + PdV + VdP - TdS - SdT         (12)

and use equations (8) and (9), and the definition of entropy from the second law to get

                dS_SYS = *Q_REV /T

and assume that only expansion work is performed

                *W_REV = PdV

Applying these two last results to the definition of dG,

                dG = TdS - PdV + PdV + VdP -TdS - SdT

                dG = VdP + SdT                                                                                   (13) 

for a reversible process with only expansion work.

Notice that all the properties and functions used here are system properties and functions; there is no one that refers to the surroundings. If the system could be doing work other than expansion work, the work term in the statement of the first law needs to be split into expansion work and other work. Then,

                *W_REV = PdV + *W’_REV

and, at constant temperature and pressure, dP and dT are zero and the change in G is equal to the maximum amount of work other than expansion that the system can do, i.e., if performed reversibly.

                dG =  - *W’_REV   

and, after integration between the initial and final states,

                )G = G₂ - G₁ = W’_REV                                     (14)

Since most processes in biology take place at constant temperature and pressure, calculation of their free energy change in undergoing a specified finite change in state yields a measure of the maximum amount of non expansion work that the system can perform on the surroundings while undergoing that change. Maximum, because the work is being done reversibly. In spontaneous natural processes where the work is done irreversibly, some of the free energy lost by the system in doing the work would have to be spent against the resistances or frictions encountered in any real change and will appear as heat. The change in free energy would be the same whether the process is reversible or not. (Why?). If  W’_REV = 0, i.e., if no “other” work is done under conditions in which it could be done during a reversible process at constant P and T, )G = 0 and the system is at equilibrium. For example, if the Zn/Cu electrochemical cell used above in the section on Process is not doing any electrical work, then the system is at equilibrium.

If P and T are not kept constant

                dG = * W’_REV + VdP - SdT                    (15)

and if only expansion work is done, we get again equation (13)

                dG = VdP + SdT.

Consider the free energy of an ideal gas. At constant T, and Using VP = nRT,

                dG = VdP = nRT(dP/P)

and

                             (16)

It is frequently convenient to express the free energy content itself rather than its change. Since the absolute value of G cannot be measured or calculated because, based only on the first and second principles of thermodynamics it is not possible to define an absolute zero for G, it becomes necessary to define a baseline state at which a mole of a substance i would have a free energy content symbolized by G_i⁺ and called its standard free energy per mole. This baseline or standard state for gases is defined as P = P⁺ = 1 atmosphere. Then,

                G_i = n_i  G_i⁺ + ) G_i

in which

                G_i = free energy of gas i at any pressure

) G_i = free energy change for going from the standard state to any other pressure at constant T

                n_i = number of moles of gas i present

                G_i⁺ = standard molar free energy, or free energy of one mole of i at the standard state.

Using equation (16),

                G_i = n_i  G_i⁺ + n_i RT ln (P_i / P_i⁺)                       (17)

The partial pressure of an ideal gas is

                p_i = n_i (RT/V).

In a mixture of gases, the total pressure P_T = G p_i = G n_i (RT/V). So,

                p_i / P_T = n_i / G n_i = X_i   which is the molar fraction of i.

An ideal gas in an expandable container at pressure P_T and temperature T may be made to expand isothermically by addition of other gases at constant total pressure until its partial pressure is p_i . Then, the free energy change for this dilution is

) G_i = n_i RT ln ( p_i / P_T) = n_i RT ln (X_i /1)             (18)

Since the properties of an ideal gas are not altered by the presence of other ideal gases, the first equality in equation (18) is also valid for the isothermic expansion of the gas achieved by reduction of the external pressure from P_T to p_i.

Ideal aqueous solutions are modeled as ideal gases and the thermodynamic equations are the same for both. Instead of assuming no interactions at all between molecules, they are assumed to interact equally with solvent and other solute molecules. So, expression (18) for  ) G_i can be applied to aqueous solutions, but now this represents the free energy change of diluting i from       X = 1 to any lower value. In other words, the standard state has changed from P = 1 to X = 1, or pure substance, and the standard molar free energy is represented by G_iⁿ. Then,

                G_i = n_i  G_iⁿ + n_i RT ln (X_i / X_iⁿ)                        (19)

in which X_iⁿ = 1.

Usually we work with concentrations expressed in molarity units. For a binary solution (solvent A plus solute B),

                X_B  = n_B / n_A + n_B

and, since for dilute solutions n_A >> n_B,    X_B   ( n_B / n_A )

The molar concentration of B is C_B = n_B /L of solution, but, from the above, n_B / L (n_A / L) · X_B. The number of moles n_A in one liter of water at standard T and P is a constant (55.56), so             C_B = 55.56 ·  X_B molar. Then, substituting these values in (19) and indicating the corresponding change in standard state (C ^o = 1M),

                G_i = n_i  G_i^o + n_i RT ln (C i /C ^o)                    (20)

For a chemical reaction, the change in free energy ()G) is given by

For the reaction A + B 6 C

                )G = G_C - (G_A +G_B).

Using (20),

                     (21)

and, if the stoichiometry is as indicated in the reaction,

                          (22)

For convenience, the symbols for the concentrations in the standard state of the various reactants and products are usually suppressed in this equation, since their value is one. This practice should not lead us to forget their implied presence. If not, the argument of the logarithmic function may seem to have units of M^-1, while it really is, as it must, dimensionless. Then equation (22) may be written

                                        (23)

When the concentrations of reactants and products and the value of  )G⁰ are known, the change in free energy for the reaction may be readily calculated.

At equilibrium )G = 0, thus,

                                           (24)

where the concentrations inside the parenthesis are concentrations at equilibrium. Therefore, that fraction is equal to the equilibrium constant for the reaction. This equation is very useful, since it allows the calculation of the equilibrium constant when the )G⁰ is known and vice versa.

In dealing with transport across a membrane, the equations are very similar. For example, for the diffusion of substance A, instead of having different reactants and products we have a single substance at different concentrations across the membrane. In the same way we use reactants and products to indicate the direction being considered for the reaction, in diffusion we one side of the membrane as the initial state and the other as the final state. We could, for example, consider diffusion from inside the cell to the outside. Then, )G = G_OUT - G_IN, and, since the value of the standard molar free energy is the same for a given substance no matter where it is, Equation (23) would lose the )G⁰ term. The equation will look as follows

                )G/mole = RT ln ( [A]_OUT / [A]_IN )

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