Helmholtz thermodynamic potential

Then the Helmholtz thermodynamic potential (AF) is proportional to the product of zFA(p and Avrami equations will be expressed by the following formulae in accordance with the above models, one of them can be as follows ... 

The Helmholtz free energy, A, which is the thermodynamic potential, the natural independent variables of which are those of the canonical ensemble, can be expressed in terms of the partition function ... 

The potential I speak of is usually called the potential of average force. Insofar as it is to be identified to a thermodynamic potential it is a local Helmholtz free energy as a function of the coordinate positions of all the atoms (or radicals) that must change relative positions in the reaction it may be defined by... 

OTHER THERMODYNAMIC POTENTIALS GIBBS AND HELMHOLTZ FREE ENERGY... 

When the pH is specified, we enter into a whole new world of thermodynamics because there is a complete set of new thermodynamic properties, called transformed properties, new fundamental equations, new Maxwell equations, new Gibbs-Helmholtz equations, and a new Gibbs-Duhem equation. These new equations are similar to those in chemical thermodynamics, which were discussed in the preceding chapter, but they deal with properties of reactants (sums of species) rather than species. The fundamental equations for transformed thermodynamic potentials include additional terms for hydrogen ions, and perhaps metal ions. The transformed thermodynamic properties of reactants in biochemical reactions are connected with the thermodynamic properties of species in chemical reactions by equations given here. 

In order to determine a system thermodynamically, one has to specify some independent parameters (e.g. N, T, P or V) besides the composition of the system. The most common choice in MC simulation is to specify N, V and T resulting in the canonical ensemble, where the Helmholtz free energy A is the natural thermodynamical potential. However, MC calculations can be performed in any ensemble, where the suitable choice depends on the application. It is straightforward to apply the Metropolis MC algorithm to a simple electric double layer in the iVFT ensemble. It is however, not so efficient for polymers composed of more than a few tens of monomers. For long polymers other algorithms should be considered and the Pivot algorithm [21] offers an efficient alternative. MC simulations provide thermodynamic and structural information, but time-dependent properties are not accessible. If kinetic or time-dependent properties are of interest one has to use molecular dynamic or brownian dynamic simulations. 

NVT systems that minimize Helmholtz free energy maximize Q. Therefore A is a useful thermodynamic potential for NVT systems NVT systems spontaneously move down gradients in Helmholtz free energy. 

When the adsorptive gas is brought into contact with the clean adsorbent, part of it leaves the gas phase and becomes adsorbed (i.e. dna > 0). If the adsorption takes place at constant T, V, A and n, the Helmholtz energy FT V A n is the thermodynamic potential of the adsorption system since this potential attains the minimum value at equilibrium ... 

At equilibrium, the extensive properties U, S, V, Nh and the linear combination of them are functions of state. Such combinations are the Helmholtz free energy, the Gibbs free energy, and enthalpy, and are called the thermodynamic potentials. Table 1.13 provides a summary of the thermodynamic potentials and their differential changes. The thermodynamic potentials are extensive properties, while the ordinary potentials are the derivative of the thermodynamic potentials and intensive properties. 

A thermodynamic potential reaches an extremum value toward equilibrium under various conditions. The Helmholtz free energy A is particularly useful for systems at constant volume and temperature. Combining Eq. (1.76) and Eq. (1.244) at constant temperature yields... 

Electrostriction. By the thermodynamics of dielectrics, the thermodynamic potential (Helmholtz function 10 of a didectric in a uniform external fidd Eo at constant pressure and temperature is ... 

All three derivations of the Clausius equation (3) are identical in principle, as they all make use of the second law of thermodynamics. In giving them all in detail we merely wished to show in what diverse ways the second law can be made to lead to concrete experimental results. The most diverse methods have been employed by various investigators in deriving such results. The choice of method depends partly on the nature of the problem and partly also on the task of the investigator. Van t Hoif, for example, generally used reversible cycles in his classical researches. Other physical chemists prefer Helmholtz s equation or the thermodynamic potential, while partial differential equations, as used in the first of the above derivations, are generally found in physical papers. 

Equation (3) is often called the Helmholtz equation on account of its close relationship with the original Helmholtz equation (Chapter VI. p. 186(22)). It can be derived directly from equation (21) (p. 186), since — is the change in the thermodynamic potential and —Qp the change in the heat content H produced by the isothermal interaction of unit mass in mols. of each of the reacting substances. 

Thus, an irreversible change at constant entropy and pressure is accompanied by a decrease in the enthalpy we say that the enthalpy is the thermodynamic potential associated with the physical variables 8 and p. We now define the Helmholtz free energy F) and the Gibbs free energy [0) by the relations... 

The thermodynamic potential of the canonical ensemble, the Helmholtz free energy, is the first thermodynamic potential g=F, which is a function of the variables of state u 1 = T, x2=V, x3=N, and x4=z. It is obtained from the fundamental thermodynamic potential / =E (the energy) by the Legendre transform (Eq. (7)), exchanging the variable of state x1 =S of the fundamental thermodynamic potential with its conjugate variable u 1 = / . In the canonical ensemble, the first partial derivatives (Eq. (1)) of the fundamental thermodynamic potential are defined asu2=-p, u3=p, and u 4 = - S. The entropy (Eq. (46)) for the Tsallis and Boltzmann-Gibbs statistics in the canonical ensemble can be rewritten as... 

The system is said to be at equilibrium when the grand thermodynamic potential is a minimum. To perform this minimization, we need to determine the molecular Helmholtz free energy, and this is the crucial part of the DFT method as we shall show below. The molecular Helmholtz free energy/(r) may be expressed as a sum of four contributions ... 

The small additions to all thermodynamic potentials are the same when expressed in terms of appropriate variables. Thus the first-order correction term when expressed in terms of V and p is the correction term for the Helmholtz free energy 4 ... 

The canonical ensemble corresponds to a system of fixed N and V, able to exchange energy with a thermal bath at temperature T, which represents the effects of the surroundings. The thermodynamic potential is the Helmholtz free energy, and it is related to the partition function Q ryT as follows ... 

A ol,P) is the Gibbs thermodynamic potential associated with 3, F is the usual Helmholtz free energy and N is the average number of particles in the system compatible with the value of a ... 

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