Helmholtz function defined

As U, T, and S are state functions, the quantity (U— TS) also must be a state function. This quantity is sufficiently important that it is given the name Helmholtz function, or Helmholtz free energy, which is defined as... 

As the procedure is the same for both the Gibbs function and the Helmholtz function, we shall consider in detail only the derivation for the Gibbs function G. After Equation (7.14), we obtained the differential of the function, which was later to be defined as G, as... 

Application to Macromolecular Interactions. Chun describes how one can analyze the thermodynamics of a particular biological system as well as the thermal transition taking place. Briefly, it is necessary to extrapolate thermodynamic parameters over a broad temperature range. Enthalpy, entropy, and heat capacity terms are evaluated as partial derivatives of the Gibbs free energy function defined by Helmholtz-Kelvin s expression, assuming that the heat capacities integral is a continuous function. 

Since we will ultimately deal with equilibrium in constant volume systems it is convenient to define a modified Helmholtz free energy function Fg(z) = U(z) - T(z)S(z) which differs from the normal Helmholtz function by inclusion of the gravitational potential energy term. We now obtain... 

Helmholtz function (Helmholtz free energy) Symbol F A thermodynamic function defined by... 

The 1981 article summarizes the development of a thermodynamic surface for water with which all thermodynamic properties for the fluid states can be calculated from the freezing line to 1000 K and up to 1 GPa in pressure. The discussion is very brief, but gives references to earlier work and indicates that a more detailed publication Is forthcoming. Given are coefficients of the Helmholtz function which define the surface. Plots of heat capacity, enthalpy, and speed of sound are included. 

Having obtained the internal energy and the entropy, we can conclude that the canonical partition function of the system can completely define the system in the thermodynamic plane. We can therefore ejqtress any thermodynamic function using the canonical partition function, expressed as variables amount of matter N), volume (V) and temperature (7), i.e. the canonical variables associated with the Helmholtz function 7. ... 

However, the energy of the chemical input and output is not so easily defined. At a simple level we could say that it is the chemical energy of the H2, O2, and H2O that is in question. The problem is that chemical energy is not simply defined - and terms such as enthalpy, Helmholtz function, and Gibbs free energy are used. In recent years the useful term exergy has become quite widely used, and the concept is particularly useful in high-temperature fuel cells. There are also older (but still useful) terms such as calorific value. 

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 (2.33) now defines the double layer in the final model of the structure of the electrolyte near the electrode specifically adsorbed ions and solvent in the IHP, solvated ions forming a plane parallel to the electrode in the OHP and a dilfuse layer of ions having an excess of ions charged opposite to that on the electrode. The excess charge density in the latter region decays exponentially with distance away from the OHP. In addition, the Stern model allows some prediction of the relative importance of the diffuse vs. Helmholtz layers as a function of concentration. Table 2.1 shows... 

For convenience, two new thermodynamic functions are defined, the Helmholtz (A) and Gibbs (G) energies ... 

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 problem at hand is the evaluation of the activity coefficient defined in Eq. (76). It will be assumed that only pairwise interactions between the defects need be considered at the low defect concentrations we have in mind. (The theory can be extended to include non-pairwise forces.23) Then the cluster function R(n) previously defined in Eq. (78) is the sum of all multiply connected diagrams, in which each bond represents an /-function, which can be drawn among the set of n vertices, the /-function being defined by Eqs. (66), (56), and (43). The Helmholtz free energy of interaction of two defects appearing in this definition can be written as... 

Define, now, a new thermodynamic function, the Helmholtz free energy A,... 

Protein stability has generally been defined in terms of the free energy change between the native and the unfolded states (AGu). This parameter of unfolding can be used to decipher stability of the native state across a wide temperature range.43 The relation that incorporates all of the terms described above is the modified Gibbs-Helmholtz equation and is a function of temperature. 

The application of the second law to chemical equilibrium is facilitated by the introduction of two more state functions. These are defined as (a) the Helmholtz free energy... 

The Green s function of the modified Helmholtz equation is defined in analogy to Eq. (B.2),... 

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... 

It is also useful to define the chemical potential in terms of the partition function. By the definition of p,k in Eq. 9.24 and the Helmholtz free-energy expression of Eq. 8.114,... 

The Helmholtz free energy, which was called the psi function, and which is perhaps most commonly known as the wurk function. It is defined by the equation... 

Rh is the hydrodynamic radius of the analyte, k is the inverse of the Debye length, r is the viscosity of the separation buffer, e is the fundamental unit of charge, and ft is a function that describes the effect of the molecule (or particle) on the electric field and is defined between two limits (i) the Htickel limit,/ = 1 when k,Rh < 1 (when the hydrodynamic radius is lower than the Debye length) and (ii) the Helmholtz-Smoluchovski limit, fi= /2 when k,Rh > 10 (when the hydrodynamic radius is higher than the Debye length). Between the limits / is calculated from the following equation ... 

The only two functions actually required in thermodynamics are the energy function, obtained from the first law of thermodynamics, and the entropy function, obtained from the second law of thermodynamics. However, these functions are not necessarily the most convenient functions. The enthalpy function was defined in order to make the pressure the independent variable, rather than the volume. When the first and second laws are combined, as is done in this chapter, the entropy function appears as an independent variable. It then becomes convenient to define two other functions, the Gibbs and Helmholtz energy functions, for which the temperature is the independent variable, rather than the entropy. These two functions are defined and discussed in the first part of this chapter. 

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