The Helmholtz and Gibbs energies

Thus far we have been concerned with holding the entropy of the system constant. 

In practice this cannot be done in any simple manner. On the other hand, we can keep the temperature of a system constant by holding it in a temperature-controlled bath. We are therefore more interested in knowing the conditions for equilibrium when a system is maintained either (a) at constant temperature and volume, or (b) at constant temperature and pressure.  

The first situation is dealt with if we combine equation (5,54) with the relationship 

Therefore, if we maintain a system at constant temperature and volume, i.e., 

Of even greater interest is the condition for chemical equilibrium when we maintain a system at constant temperature and pressure. For example, most biological systems are well thermostatted, and the processes occur at a constant one-atmosphere pressure. This condition is readily obtained from equation (5.69) by combining it with (5.57)  

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The Helmholtz and Gibbs energies on the other hand involve constant temperature and volume and constant temperature and pressure, respectively. Most experiments are done at constant Tandp, and most simulations at constant Tand V. Thus, we have now defined two functions of great practical use. In a spontaneous process at constant p and T or constant p and V, the Gibbs or Helmholtz energies, respectively, of the system decrease. These are, however, only other measures of the second law and imply that the total entropy of the system and the surroundings increases. 

The Helmholtz and Gibbs energies are useful also in that they define the maximum work and the maximum non-expansion work a system can do, respectively. The combination of the Clausius inequality 7dS > dq and the first law of thermodynamics dU = dq + dw gives... 

In this discussion of indifferent states we have always used the entropy, energy, and volume as the possible extensive variables that must be used, in addition to the mole numbers of the components, to define the state of the system. The enthalpy or the Helmholtz energy may also be used to define the state of the system, but the Gibbs energy cannot. Each of the systems that we have considered has been a closed system in which it was possible to transfer matter between the phases at constant temperature and pressure. The differentials of the enthalpy and the Helmholtz and Gibbs energies under these conditions are... 

The differentials of the Helmholtz and Gibbs energies could be written in the usual manner. However, it is convenient to use the functions (A — EP) and (G — EP) in order to change the independent variable from P to E. When this is done we obtain... 

In Chapter 4 the Helmholtz and Gibbs energies of pure components were introduced by the relations... 

These definitions are also valid for mixtures, provided the values of U, H, and 5-used are those for the mixture. That is. the Helmholtz and Gibbs energies of a mixture bear the same relation to the mixture internal energy, enthalpy, and entropy as the pure component Gibbs energies do to the pure component internal energy, enthalpy, and entropy. 

The Helmholtz and Gibbs energies are both extensive, conceptual state functions having dimensions of energy. Unfortunately, only in special cases do the changes AA and AG have physical interpretations. 

The Helmholtz and Gibbs energies are often referred to as free energies. Both are defined using the entropy, S. In Chapter 1, we developed a molecular level definition of the absolute entropy. From that definition, it should be clear that entropy is a thermod3mamic state function, too. Therefore, A and G are state functions. 

Conditions for equilibrium and the definition of Helmholtz and Gibbs energies... 

The first and second laws of thermodynamics and the Helmholtz and Gibbs free energies are rearranged to obtain the relationships between the state functions (i.e., E, H, A, and G) and temperature, pressure, and volume. For an infinitesimal process the first law is given by ... 

The other thermodynamic functions are also easily found. If we confine ourselves to low pressures, of the order of atmospheric pressure, we can neglect the term PV in the Gibbs free energy of liquid or solid. Then the Helmholtz and Gibbs free energies are approximately equal and are given by... 

Before setting out on the exact mean field theory solution to the one-dimensional colloid problem, I wish to emphasize that the existence of the reversible phase transition in the n-butylammonium vermiculite system provides decisive evidence in favor of our model. The calculations presented in this chapter are deeply rooted in their agreement with the experimental facts on the best-studied system of plate macroions, the n-butylammonium vermiculite system [3], We now proceed to construct the exact mean field theory solution to the problem in terms of adiabatic pah-potentials of both the Helmholtz and Gibbs free energies. It is the one-dimensional nature of the problem that renders the exact solution possible. 

To clarify the different roles played by the Helmholtz and Gibbs free energies of ionic solutions, it is relevant to reconsider the derivation of these thermodynamic quantities in the original Debye-Hiickel theory [1—4],... 

TherrnodynarnicaVLy, the interfacial tension is interpreted as the Increase in the Helmholtz or Gibbs energy of the system when the area of the interface under consideration is increased reversibly by an infinitesimal amount dA at constant temperature and composition, and at constant volume or constant pressure, respectively. We can express this as... 

A variety of authors have paid attention to the question of how the charging of a monolayer affects the (Helmholtz or Gibbs) energy, and hence the interfacial pressure. (See for instance refs. ) Thermod3mamics can help to answer some of the basic questions that have given rise to unnecessary confusion in the literature. 

Helmholtz and Gibbs energies are almost indistinguishable). Upon charging, or rather charge-separation, a potential difference is created. The electrical work of withdrawing protons against this potential is... 

We use the terms Helmholtz and Gibbs energies for what has previously been referred to as Helmholtz and Gibbs free energies, respectively. 

Nitrogen is to be isothermally compressed at O C from I bar to 100 bar. Compute the work required for this compre.ssion the change in internal energy, enthalpy Helmholtz and Gibbs energies of the gas and the heat that must be removed to keep the gas at constant temperature if... 

We note that different forms of the first law have been promulgated by various authors. Equation (14-11) is the correct form in the general case in which the dielectric is deformable. We define the Helmholtz and Gibbs free energies of the system by the relations... 

A thermodynamic analysis (2) shows that the Helmholtz and Gibbs free energies of the system are... 

A and G are referred to as the Helmholtz and Gibbs free energy, respectively. Note that terms enthalpy and free energy allude to heat and work, respectively. The terminology subscribes to the circumstances where the potentials are most often directed. For instance ... 

Gibbs-Helmholtz equation This equation relates the heats and free energy changes which occur during a chemical reaction. For a reaction carried out at constant pressure... 

Themodynamic State Functions In thermodynamics, the state functions include the internal energy, U enthalpy, H and Helmholtz and Gibbs free energies, A and G, respectively, defined as follows ... 

Equation 54 implies that U is a function of S and I. a choice of variables that is not always convenient. Alternative fundamental property relations may be formulated in which other pairs of variables appear. They are found systematically through Legendre transformations (1,2), which lead to the following definitions for the enthalpy, H, Helmholtz energy,. 1. and Gibbs energy, G ... 

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