Helmholtz function/free energy

Before turning to the surface enthalpy we would like to derive an important relationship between the surface entropy and the temperature dependence of the surface tension. The Helmholtz interfacial free energy is a state function. Therefore we can use the Maxwell relations and obtain directly an important equation for the surface entropy ... 

However, the specific entropy s is not a convenient independent variable as it is intuitively difficult to comprehend and practically difficult to control. The classical approach consists of introducing another state function, the specific Helmholtz s free energy, via the Legendre transform ... 

The most successful statistical theory of liquids is that derived by Simha and Somcynsky. The model considers liquids to be mixtures of voids dispersed in solid matter, i.e., a lattice of unoccupied and occupied sites. The occupied volume fraction, y (or its counterpart the free volume fraction f = 1 - y), is the principal variable y = P, T). From die configurational partition function the configurational contribution to the Helmholtz molar free energy of liquid i was expressed as [3] ... 

The relations which permit us to express equilibria utilize the Gibbs free energy, to which we will give the symbol G and which will be called simply free energy for the rest of this chapter. This thermodynamic quantity is expressed as a function of enthalpy and entropy. This is not to be confused with the Helmholtz free energy which we will note sF (L" j (j, > )... 

Figure A2.2.1. Heat capacity of a two-state system as a function of the dimensionless temperature, lc T/([iH). From the partition fimction, one also finds the Helmholtz free energy as...

LS now consider the problem of calculating the Helmholtz free energy of a molecular 1. Our aim is to express the free energy in the same functional form as the internal that is as an integral which incorporates the probability of a given state. First, we itute for the partition function in Equation (6.21) ... 

A consequence of writing the partition function as a product of a real gas and an ideal g part is that thermod)mamic properties can be written in terms of an ideal gas value and excess value. The ideal gas contributions can be determined analytically by integrating o the momenta. For example, the Helmholtz free energy is related to the canonical partitii function by ... 

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 ... 

Once the partition function is known, thermodynamic functions such as the internal energy U and Helmholtz free energy A may be calculated according to... 

A = work function (Helmholtz free energy), Btu/lb or Btu C = heat capacity, Btu/lb °R Cp = heat capacity at constant pressure = heat capacity at constant volume F= (Gibbs) free energy, Btu/lb or Btu g = acceleration due to gravity = 32.174 ft/s ... 

Work function or maximum work (Helmholtz free energy)... 

Debye Helmholtz free energy (Am - U, m)/JiRT as a function of 9ofT... 

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 ... 

When calculating free energies, one generates, either by molecular dynamics or MC, configuration space samples distributed according to a probability distribution function (e.g., the Boltzmann distribution in the case of the Helmholtz free energy). 

Histograms for two runs at different chemical potentials are presented in Fig. 10.2. There is a range of N over which the two runs overlap. In Fig. 10.3 we show the function lnp(iV) — f S/iN for the data of Fig. 10.2. Rearranging (10.15) and taking the logarithm, we see that this function is related to the Helmholtz free energy... 

Fig. 11.1. The Helmholtz free energy as a function of /3 for the three free energy models of the harmonic oscillator. Here we have set h = uj = 1. The exact result is the solid line, the Feynman-Hibbs free energy is the upper dashed line, and the classical free energy is the lower dashed line. The classical and Feynman-Hibbs potentials bound the exact free energy, and the Feynman-Hibbs free energy becomes inaccurate as the quantum system drops into the ground state at low temperature...

From the normalization condition the relationship between Helmholtz free energy and the canonical partition function... 

In order to fix ideas, we will consider in this section a canonical ensemble [38], In this ensemble, the Helmholtz free energy is straightforwardly related with the canonical partition function through... 

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... 

The discussion will be limited to systems at temperatures which are high enough so that we need only consider the classical partition function Q in calculating the Helmholtz free energy A, where, of course... 

---