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The classic thermodynamics has introduced two more energy-based functions, named the work function, A, and the Gibbs free energy, G. They have been introduced because it is desirable to obtain criteria for thermodynamic equilibrium under practical conditions, which means that the temperature is approximately constant in addition to either constant volume in bomb calorimeters or constant pressure in chemostates.

Most chemical processes in the laboratory or in organisms are carried out at constant volume, pressure and also temperature. As follows from the results of Section 2.4 under these conditions, the Helmholtz and Gibbs free energies, F and G, play the most important role; moreover, they practically coincide, dF < dG. This implies that dF < dG = — SA. At thermodynamic equilibrium, all gradients are eliminated according to the definition. This means that no work can be performed. This implies that at constant temperature and pressure (dG)Tp = Xm Mm dnm = 0, which is an important consequence of the energy conservation law. A general dynamic equilibrium, but not a thermodynamic equilibrium, is possible by equalising process rates in opposite directions to ensure that the steady state is maintained.

Consider the system consisting of one ith sort of particles distributed between two phases, a and b. The transfer of dni number of particles from a into b is accompanied by the change of Gibbs potential (or free energy), dG. Since — dna = dnb then at p, T = const (dG)p T = M dna + Mb dnb = dna(M — Mb), and in the equilibrium (dG)p T = dna(M — Mb) = 0 so that M = Mb, i.e. the equality of chemical potentials is the condition of equilibrium for the process of transfer of ith sort of particles between two phases. Moving to the thermodynamic equilibrium, the system is always moving toward the state with lower chemical potential.

It was shown in standard text-books on chemical thermodynamics that the chemical potential:

where R is the gas constant, T is the temperature, [ci] is the concentration (or frequency) of ith particles, M° is a constant depending on the temperature and the origin of matter (substance).

Standard free energies (or thermodynamic potentials) of formation of chemical compounds are very important for calculation of chemical equilibriums, since their knowledge allows us to estimate the "useful" work, various energetic effects and possible directions of the chemical evolution of the system. Let there be a chemical reaction:

where A1; A2,..., A'1; A'2,... are reacting substances, and v1; v2, ..., v'1; v'2,... are their stoichiometric coefficients. At the thermodynamic equilibrium:

(dG)Ptr = £Mi Vi = Xm(0) Vi + RT^v Vi ln cf1 = 0, (5.3)

where cèq are equilibrium concentrations and the summation is produced over all substances, both products and reactants. After simple transformation, we obtain from Eq. (5.3):

eq where K(T) is the so-called equilibrium constant depending only on the temperature.

Using these formulas, we can define a so-called standard value of the change of thermodynamic (Gibbs) potential AG0, which corresponds to the temperature 25°C and the conditions when the concentrations (activities) of all products and reactants are equal to 1. Indeed, after denoting AG0 = Y.MPvi = —RT ln K(T), we have

AG = AG0 + RT^ Vi ln ct = —RT ln K(T) + R^ vt ln q.

There are different methods for the determination of AG0 in biochemical reactions. For instance, if we know the equilibrium constant then AG0 = —RT ln K(T). The other method is based on the Hess law when we sum the partial changes of Gibbs potential corresponding to all intermediate stages of reaction:

The thermodynamic potential describes the chemical affinity under conditions of constant temperature and pressure: AG = G(products) — G(reactants).

When the potential is zero there is no net work obtainable by any change or reaction at constant temperature and pressure. The system is in a state of thermodynamic equilibrium. When the change of Gibbs potential is positive for a proposed process, net work must be put into the system to effect the reaction, otherwise it cannot take place. When it is negative, the reaction can proceed spontaneously by providing useful net work.

As an example we consider the reaction of glucose oxidation in the process of respiration: C6H12O6 + 6O2 = 6CO2 + 6H2O. The value of AG0 was found by the Hess method: AG0 = — 2840 kJ/mol. It is known that the general direction of photosynthesis is opposite to the respiration; therefore, the photosynthetic process of formation of one molecule of glucose from water and carbon dioxide demands an increase of thermodynamic potential by the value of DG0 = 2840 kJ/mol > 0. Therefore, it cannot occur spontaneously and demands an additional energy inflow. The latter is provided by solar photons.

Note that for real systems, when the interaction between particles is rather strong, it is convenient to introduce a new function for the considered substances, called fugacity, f. Fugacity is defined from the following equation:

A standard state may, however, be defined as the state of unit fugacity, as the standard state for ideal gases was the state of unit pressure. It is now possible to set up an expression for the equilibrium constant which is true in general not only for real (non-ideal) gases but for substances in any state of aggregation. It can be shown that the fugacity can be replaced by concentrations (pressures for gases) in many calculations with a good approximation. For solutions, it is possible to find the fugacity by multiplying the concentration with an activity coefficient that can be found by empirical equations. For an aquatic solution, the fugacity coefficient is close to 1.00 with a total concentration of dissolved matter of less than 1 g/l.

It is emphasised in this context that these thermodynamic calculations of equilibrium constant and standard heat and free energy are also valid for biochemical processes that are the reactions of interest in an ecosystem.

We distinguish different forms of energy. All forms can be described as some product of "quantitative" (extensive) and "qualitative" (intensive) variables. These pairs of extensive and intensive variables for different energy forms are summarised in Table 2.1. Work can be performed when the extensive variable is changed from one level of the intensive variable to another, and the work = the extensive variable X the difference of the intensive variable between the two levels. As energy is conserved, work implies that one energy form is transferred to another energy form. The energy of a considered system is often defined as "the ability to do work". It is presumed that the system is transferred to the level where the intensive variable is zero, whereby the work performed becomes equal to the energy content of the system.

The change in the number of moles of any chemical substance caused by a chemical reaction within the system is proportional to its stoichiometric coefficient: dinm = vm dJ where J is called the co-ordinate of reaction or the degree of its completeness. From this formula it follows that din1 _ din2 _ _ din1 _ din2 _ ^ ^ ^

where the sign " — " corresponds to reactants and " + " to products, dn1 < 0, dn2 < 0,..., dn'j > 0, dn'2 > 0, Let V = const be the volume of the considered homogenous system. Then the rate of reaction is defined as ch 1 di nm 1 d£

Vvm dt V dt

If the mth substance simultaneously participates in several reactions, then the total change in the number of moles, dinm, is

where k is the index of reaction and K is the total number of simultaneously resulting reactions.

Since the system is homogenous, we can take into consideration such functions of density as the density of entropy production and the specific concentrations (or, simply, concentrations) of substance:

Analogously, we define the local function of dissipation:

V dt

If to take into account the matter exchange between the system and its environment, then dc d c K

The main hypothesis of the theory of irreversible processes (Glansdorff and Prigogine, 1971) is that the fundamental Gibbs equation (see Section 2.2) is also valid at any local time. Since Eq. (3.4) is one form of fundamental equation then its local form is written as (we set Nm = and (SAirrev/St) = (d'Airrev/dt))

The chemical affinity of kth reaction is defined as

Then

This is a very important relation in the thermodynamics of irreversible processes, which will be used in Chapter 3.

Finally, we would like to clarify the concept of affinity. Consider as an example a simple reaction of the transformation of mth substance into (m + 1 )th one: cm ! cm+l. The stoichiometric coefficients and chemical potentials are vm, vm+1 and /xm, Mm+1 ; respectively.

In accordance with Eq. (5.12) A = — (—VMm + Vm+Mm+1) = VmMm — Vm+Mm+1. In the special case, vm = vm+1 = 1, A = /xm — Mn+1, i.e. the value of affinity is equal to the difference of chemical potentials.

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