## Maximal work which the system can perform on its environment Characteristic functions or thermodynamic potentials

Consider a thermally isolated system consisting of different particles, which are neither in thermal nor chemical equilibrium. Let the energy of the system at some initial time be E0, and the energy in equilibrium (i.e. at the final state) as a function of the supersystem entropy in this state be E(S). Since the supersystem is closed, then E0 = lAl + E(S) where l A l is the absolute value of work. Differentiating l A l with respect to the entropy at the final state we have d lAl/9S = — (dE/dS)V = — T where T is the temperature at this state. Since this derivative is negative then lAl increases as S decreases. On the other hand, the entropy of a closed system cannot decrease. Therefore, the value of lA l will be maximal if only the total entropy S remains constant in the course of the entire transition, i.e. when the transition to an equilibrium state is a reversible one. For any irreversible process lAl ^ lAlmax.

Since the full differential of internal energy is known (see Eq. (2.5)), then the internal energy is written as the following function of variables S, V and nm:

This is the first thermodynamic identity, and the internal energy is the first characteristic function (or thermodynamic potential). (Sometimes we shall omit the term y m Mm dnm describing the effect of chemical reaction.) If to differentiate formally the function U, assuming that it depends on the variables T, S, p and V, then we obtain dU = T dS + S dT — p dV — V dp + ^ mm dnm + ^ nm dMm or mm dU + V dp — S dT = T dS — p dV. (4.2)

In order to get the Gibbs equation we have to assume that the temperature, pressure and chemical potentials remain constant during the transition. But this means that the process of transition has to be quasi-stationary. This has already been required when we deduced the Gibbs equation. Since the right-hand side of equality (4.2) is a full differential (as the right-hand side of the Gibbs equation), then the left-hand side can be written as the full differential of the function F, so that dF = dU + V dp — S dT = d(U + pV — TS). (4.3)

If the volume and temperature of the system remain constant during the transition then p and S have to be the variables. From Eq. (4.3) for dT = 0 we obtain dF = dU + V dp = d(U + pV). (4.4)

The function

is called the heat function or enthalpy of the system. Its change during a process with a constant pressure is equal to the heat gained by the system. By taking into account the Gibbs equation, its full differential can be rewritten as dH = T dS + V dp. (4.6)

Since the transformation of the first identity into the other identities, defined with respect to the other characteristic functions (thermodynamic potentials), for instance enthalpy, does not affect the variables nm (moreover, they could be not only molecules but any virtual particles, for instance biological individuals), it is clear that terms which are proportional to the differentials dnm may be added to any thermodynamic identities. Then dH = T dS + V dp + ^ Mm dnm. (4.7)

Let some chemical reaction occur in the system with a constant temperature. As a result, the initial mixture of the reacting substance transforms into some final products. This reaction could go through many stages, some of which either do not go to completion or are wholly unobserved. Since in this case the heat is a function of state, then the total heat released or absorbed in the process of chemical reaction (the heat of reaction) does not depend on paths from the initial mixture of reactants to the final products; it is only determined by the differences in their enthalpies. This is the Hess law, which is the main law of thermodynamic chemistry and a special corollary of the First Law of Thermodynamics.

The heat of reaction has biochemical significance. If heat is produced, the process is named exothermic. Decomposition of the organic matter in food is an exothermic process, which provides heterotrophic organisms with the energy that is needed for the maintenance of life. Endothermic processes require the addition of heat. A typical endothermic process is the formation of adenosine triphosphate (ATP) from phosphate and adenosine diphosphate (ADT), expressed in biochemistry as the following equation:

The opposite process, formation of P and ADP from ATP, similarly yields 42 kJ/mol. ATP is applied by the organisms as a suitable unit of energy wherever it is needed to carry out the necessary biochemical processes.

The heat of reaction may be either at a constant pressure or constant volume, which implies that the heat of reaction becomes, respectively, the change in enthalpy and the change in internal energy. In reactions involving only liquids, or when solid DV is negligible, there is practically no difference between the two heats of reaction. A convenient standard state for a substance may be taken to be the state when it is stable at 25°C and atmospheric pressure. The standard enthalpy of any compound is the heat of reaction, by which it is formed from its elements, reactants and products all being in the standard state. A superscript of zero indicates that the standard heat of formation with reactants and products is at a pressure of one atmosphere.

Before we introduce new characteristic functions, we shall define some special transition processes, so-called adiabatic ones. Strictly speaking, a process is named adiabatic if during the process the system entropy is conserved, i.e. dS = 0. Any adiabatic process is reversible. The process can be realised if the system is thermally isolated, and its environment changes rather slowly.

If the pressure remains constant during the adiabatic process then T and Vhave to be the variables. From Eq. (4.3) for dp = 0 we obtain dF = dU — S dT = d(U — TS). (4.8)

The function

is called the free energy or Helmholtz's free energy of the system. Its change during an adiabatic process at constant pressure is equal to the work done on the system. The full differential of F (with regard to the differentials dnm) is equal to dF = — S dT — p dV + ^ Mm dnm. (4.10)

This is the third thermodynamic identity.

At last, we have to get the characteristic function related to the variables T and p, i.e. for an adiabatic process at constant volume. In this case, as follows from Eq. (4.3), the function, which is called the thermodynamic potential (in the narrow sense of the word) or the Gibbs free energy or the Gibbs potential is defined as

G(T, p)= F = U + pV — TS = F + pV = H — TS (4.11)

and its full differential is equal to dG = — S dT + V dp + ^ Mm dnm. (4.12)

This is the fourth thermodynamic identity.

Note that, if only the numbers of virtual particles change slightly, then the changes of characteristic functions that are also small will always be equal to one another if every one of them is considered with the appropriate pair of quantities constant, i.e.

(dU V = (dH)sp = (dF)T,v = (dG)Tp = £ Mm dnm. (4.13)

Under these conditions, chemical potentials depend only on the composition of the mixture consisting of n1, n2,... particles, i.e. on their frequencies, and they do not depend on their total number.