Energy free. Helmholtz

In this section we study closed systems (closed to mass transport but not energy transfer) held at constant temperature. In statistical mechanics these systems are referred to as NVT systems (because the thermodynamic variables N, V, and T are held fixed). We shall see that the Helmholtz free energy represents the driving force for NVT systems. Just as an isolated system (an NVE system) evolves to increase its entropy, an NVT system evolves to decrease its Helmholtz free energy. 

Since a system of constant volume and mass held at constant temperature exchanges energy with its surroundings, we can no longer define a fixed total internal 

3 Closed isothermal systems 1.3.1 Helmholtz free energy 

the internal energy U = A + TS can be expressed (through a clever manipulation [156]) as  

We can equate Equations (1.9) and (1.11) by recalling that = l/keT. Thus, the Helmholtz free energy can be calculated directly from the quantity Q in Equation (1.7)  

Processes that occur according to Equation 5.25 are, in principle, reversible. According to Equation 5.19, 8q = dU - 5w. We first assume that 6w = 0, meaning that no pressure-volume work is carried out. Equation 5.24 may be written as 

Equation 5.26 suggests that a new state function may be defined as 

With the help of Equations 5.23 and 5.27 A may be expressed using the partition function Z in a very simple way  

Clearly, A is a state function, referred to as Helmholtz free energy. 

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Helmholtz free energy The maximum amount of energy available to do work resulting from changes in a system at constant volume. See free energy and Gibbs-Helmholtz equation. 

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, > )... 

Thus, for spontaneous processes at constant temperature and volume a new quantity, the Helmholtz free energy A, decreases. At equilibrium under such restrictions cL4 = 0. 

If there are other kinds of work, similar expressions apply. For example, with electromagnetic work (equation (A2.1.8)1 instead of pressure-volume work, one can write for the Helmholtz free energy... 

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...

Figure A2.5.9. (Ap), the Helmholtz free energy per unit volume in reduced units, of a van der Waals fluid as a fiinction of the reduced density p for several constant temperaPires above and below the critical temperaPire. As in the previous figures the llill curves (including the tangent two-phase tie-lines) represent stable siPiations, the dashed parts of the smooth curve are metastable extensions, and the dotted curves are unstable regions. See text for details.

The molar Helmholtz free energy of mixing (appropriate at constant volume) for such a synnnetrical system of molecules of equal size, usually called a simple mixture , is written as a fiinction of the mole fraction v of the component B... 

If one assumes that tlie singular part A of the Helmholtz free energy is such a fimction... 

The canonical ensemble corresponds to a system of fixed and V, able to exchange energy with a thennal bath at temperature T, which represents the effects of the surroundings. The thennodynamic potential is the Helmholtz free energy, and it is related to the partition fiinction follows ... 

In Equation (5.58) the outer summation is over the p points q which are used to sample the Brillouin zone, is the fractional weight associated with each point (related to the volume of Brillouin zone space surrounding q) and vi are the phonon frequencies. In addition to the internal energy due to the vibrational modes it is also possible to calculate the vibrational entropy, and hence the free energy. The Helmholtz free energy at a temperature... 

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 ... 

Our discussion so far has considered the calculation of Helmholtz free energies, which a obtained by performing simulations at constant NVT. For proper comparison with expe inental values we usually require the Gibbs free energy, G. Gibbs free energies are obtaini trorn a simulation at constant NPT. 

Helmex Helmezine Helmholtz energy Helmholtz free energy... 

The definitions of enthalpy, H, Helmholtz free energy. A, and Gibbs free energy, G, also give equivalent forms of the fundamental relation (3) which apply to changes between equiUbrium states in any homogeneous fluid system ... 

Theoretically, several aspects of the Thommes-Findenegg experiment can be analyzed at the mean-field level [157]. A key quantity of a mean-field theory of confined fluids is the (Helmholtz) free energy, given by... 

The last equation comes from the condition that both the compressi-bihty and the energy route lead to the same value of the Helmholtz free energy. 

In the case of an associating fluid with the repulsive-attractive reference system potential, the attractive van der Waals forces between molecules may also be considered in a perturbational manner [114]. The Helmholtz free energy can be written as a sum of three terms... 

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

N, Number of particles P, Pressure V, Volume T, Temperature E, Energy fi. Chemical potential A, Helmholtz free energy S, Entropy G, Gibbs free energy. 

As noted above, it is very difficult to calculate entropic quantities with any reasonable accmacy within a finite simulation time. It is, however, possible to calculate differences in such quantities. Of special importance is the Gibbs free energy, as it is the natoal thermodynamical quantity under normal experimental conditions (constant temperature and pressme. Table 16.1), but we will illustrate the principle with the Helmholtz free energy instead. As indicated in eq. (16.1) the fundamental problem is the same. There are two commonly used methods for calculating differences in free energy Thermodynamic Perturbation and Thermodynamic Integration. 

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 ... 

In the absence of specific interactions of the receptor - ligand type the change in the Helmholtz free energy (AFadj due to the process of adsorption is AFads = yps - ypi - Ysi, where Yps, YPi and ys, are the protein-solid, protein-liquid and solid-liquid interfacial tensions, respectively [5], It is apparent from this equation that the free energy of adsorption of a protein onto a surface should depend not only of the surface tension of the adhering protein molecules and the substrate material but also on the surface tension of the suspending liquid. Two different situations are possible. 

The analog of Eq. (8-211) for the Helmholtz free energy is the grand potential... 

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