State Helmholtz free energy

The standard state Helmholtz free energy difference, 8AA°, was introduced in Equations 5.9 and 5.11 to show the connection between VPIE and molecular structure and dynamics. Molecular properties are conveniently expressed using standard state canonical partition functions for the condensed and vapor phases, Qc° and Qv° remember A0 = —RT In Q°. The Q s are 3nN dimensional, n is the number of atoms per molecule and N is Avogadro s number. For convenience we have now dropped the superscript o s on the Q s. The o s specify standard state conditions, now to be implicitly understood. For VPIE and a respectively, not too close to the critical region,... 

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) ... 

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 ... 

Consider a thermodynamic system with an external parameter (or constraint) A that can be used to control the state of the system. When changing the control parameter A a certain amount of work is performed on the system. According to the second law of thermodynamics the average work necessary to do that is smaller than the Helmholtz free energy difference between the two equilibrium states corresponding to the initial and final values of the constraint [33]... 

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...

In addition to the intermolecular potential, there is an intramolecular portion of the Helmholtz free energy. Cheetah uses a polyatomic model to account for this portion including electronic, vibrational, and rotational states. Such a model can be expressed conveniently in terms of the heat of formation, standard entropy, and constant-pressure heat capacity of each species. 

Wilson and Cunningham developed the PFGC equation of state on the basis of the following definition of the Helmholtz free energy... 

Equation 5.19 relates the molecular energy states of the primed and unprimed isotopomers in condensed and vapor phase to VPIE. The correction terms account for the difference between the Gibbs and Helmholtz free energies of the condensed phase, and vapor nonideality. The comparison is between separated isotopomers at a common temperature, each existing at a different equilibrium volume, V or V, and at a different pressure, P or P, although AV = (V — V) and AP = (P — P) are small. Still, because condensed phase Q s are functions of volume, Q = Q(T,V,N), rigorous analysis requires knowledge of the volume dependence of the partition function, and thus MVIE, since the comparisons are made at V and V. That point is developed later. 

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... 

An even more precise treatment, based on the assumption that the vibrational Helmholtz free energy of the crystal, divided by temperature, is a simple function of the ratio between T and a characteristic temperature dependent on the volume of the crystal, leads to the Mie-Gruneisen equation of state (see Tosi, 1964 for exhaustive treatment) ... 

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... 

Here Fe(t) and Fg(t) are the time-dependent nonequilibrium Helmholtz free energies of the e and g states, respectively. The energy difference A U(t) can be replaced by a free energy difference due to the fact that the entropy is unchanged in a Franck-Condon transition [51]. Free energies in Eq. (3) can be represented [54] by a sum of an equilibrium value Fcq and an additional contribution related to nonequilibrium orientational polarization in the solvent. Thus for the free energy in the excited state Fe(t) we have... 

A is a state function of the system because U, T, and S are all state functions of the system. We have thus remarkably transformed our criteria for real processes (at constant Tand V) so that we do not have to consider the surroundings at all We can say that a system with constant Tand V will spontaneously undergo a process if it lowers the system s Helmholtz free energy. 

The condition of phase stability for such a system is closely related to the behavior of the Helmholtz free energy, by stating that the isothermal compressibility yT > 0. The positiveness of yT expresses the condition of the mechanical stability of the system. The binodal line at each temperature and densities of coexisting liquid and gas determined by equating the chemical potential of the two phases. The conditions expressed by Eq. (115) simply say that the gas-liquid phase transition occurs when the P — pex surface from the gas... 

Equation (1.77) implies that the maximum work done by the system under isothermal conditions can be carried out by the change of Helmholtz free energy. The state function A is known as the work function. The Helmholtz free energy is not useful for most chemical and biological processes since these processes occur at constant pressure rather than at constant volume. 

For a system at constant temperature, this tells us that the work done is less than or equal to the decrease in the Helmholtz free energy. The Helmholtz free energy then measures the maximum work which can be done bv the system in an isothermal change For a process at constant temperature, in which at the same time no mechanical work is done, the right side of Eq. (3.5) is zero, and wo see that in such a process the Helmholtz free energy is constant for a reversible process, but decreases for an irreversible process. The Helmholtz free energy will decrease until the system reaches an equilibrium state, when it will have reached the minimum value consistent with the temperature and with the fact that no external work can be done. 

For a vstem at constant pressure and temperature, we see that the Gibbs free energy is constant for a reversible process but decreases for an irreversible process, reaching aminimum value consistent with the pressure and temperature for the equilibrium state just as for a system at constant volume the Helmholtz free energy is constant for a reversible process but decreases for an irreversible process. As with A, we can get the equation of state and specific heat from the derivatives of <7, in equilibrium. We have... 

The result of Eq. (5.17) isLfor practical purposes, the most important, result of statistical mechanics. For it gives a perfectly direct and stmiprhtfnrwRrd wav of deriving the Helmholtz free, energy, and henee the equation of state and specific heat, of anv system, if we know its energy as a function of coordinates aufl mniTirntnr- The sum of Eg. 

To carry out any calculations with the equation of state, we wish to approximate it in some analytic way. First, let us consider the most convenient variables to use. The results of experiment are usually expressed by giving the volume as a function oTprcs rcTand temperature. Thus theThermal expansion Is investigated as a function of temperature at atmospheric pressure, and in measurements of compressibility the volume is found as a function of pressure at certain fixed temperatures. On the other hand, for deriving results from statistical mechanics, it is convenient to find the Helmholtz free energy, and hence the pressure, as a function of volume and temperature. We shall express the equation of state in both forms, and shall find the relation between the two. We let V0 be the volume of our solid at no pressure and at the absolute zero of temperature. Then we shall assume... 

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