The Helmholtz energy

One differential expression for the energy of a system is given by Equation (2.26)  

Although Equation (4.1) is useful, particularly for isentropic changes of state for which dS = 0, the entropy appears as an independent variable. This circumstance is inconvenient from an experimental point of view, because no meters or devices exist that measure entropy. A new function in which the temperature is an independent variable rather than the entropy is obtained by subtracting 

1 If the product TS is not uniform throughout the system, the system can be divided into parts in which TS is uniform. Then TS, the sum being taken over the individual parts of the system, may be substituted for TS. If the volumes in which TS is uniform are infinitesimal, the TS may be substituted by J (TS/V) d V, the integral being taken over the entire volume of the system. 

---

The chemical potential p, of the adsorbate may be defined, following standard practice, in terms of the Gibbs free energy, the Helmholtz energy, or the internal energy (C/,). Adopting the last of these, we may write... 

To obtain the Helmholtz energy of the system we require the entropy, which is S — k nW, where W is the number of different realizations of the system. The number of ways in which N molecules of type 1 and N2 = N — N molecules of type 2 can be distributed onto N sites is ... 

If the entropy of the system decreases some of the energy must escape as heat in order to produce enough entropy in the surroundings to satisfy the second law of thermodynamics. Hence the maximum work is less than IAU I. AA is the part of the change in internal energy that is free to use for work. Hence the Helmholtz energy is in some older books termed the (isothermal) work content. 

The variation of the Helmholtz energy of the van der Waals equation of state for H2O with volume can be calculated by... 

All thermodynamic properties can be derived from the partition function. It can be shown that the Helmholtz energy, A, is related to Zby the simple expression... 

The contribution from the partition function (right-hand side of eq. (9.3)) should be interpreted as the value of the Helmholtz energy relative to its value in the lowest energy state, or in other words at the absolute zero of temperature, A(0). In the further discussion we will only consider the values relative to 0 K. Thus... 

The Helmholtz energy thus plays a key role. In the quasiharmonic approximation it is assumed that at temperature T this can be written as the sum of static and vibrational contributions... 

It is often more convenient to control the temperature than to control the entropy, and therefore it is more convenient to switch to the Helmholtz energy F=U TS, for which we can write ... 

The quantity in square brackets is the Helmholtz energy change for the process of bringing a ligand from the reservoir at a given chemical potential l onto a specific site, here a, of an empty molecule. The process is carried out at constant temperature T. 

Similarly, the quantity k C is related to the Helmholtz energy change for the process... 

In Section 9.2 we have defined the Gibbs energy of solvation AGo in the T, P, N ensemble. In the T, V, N (canonical) ensemble the appropriate quantity is A4a, the Helmholtz energy of solvation. It can be shown that the two are equal for macroscopic systems, provided the volume V in the T, V, N ensemble is equal to the average volume of a system in the T, P, N ensemble. 

A thermodynamic function, symbolized by /, equal to the negative of the Helmholtz energy divided by the absolute temperature thus, J = -AIT. The SI units are joules per kelvin. See also Planck Function Helmholtz Energy... 

For example, from the right (G) edge, the arrow from natural variable T ends up at — S (taken minus because it lies at the tail of the arrow), whereas that from variable P ends up at +V (taken plus because it lies at the head of the arrow), giving the differential form dG = (—S)dT + (+V)dP. Similarly, from the top (A) edge, we can read the differential form for the Helmholtz energy as dA = ( S) dT + (— P) dV, because both coefficients fall at arrow tails. 

The new function A is called the Helmholtz energy.2 Since E, T, and S are functions of the state of the system, A is also a function of the state of the system, and its differential is exact. The change in the value of the Helmholtz energy in going from some initial state to some final state is independent of the path. However, the determination of this change must be obtained by the integration of Equation (4.3) along any reversible path between the two states. 

Any physical interpretation of the Helmholtz energy must be based on interpreting Equation (4.3). Thus, for an isothermal change of state, the equation becomes... 

Gibbs used ifr for this function G. N. Lewis called it the work function, with the symbol A and in many European countries it was called the Helmholtz free energy, with the symbol F. By international agreement the accepted name is the Helmholtz energy and the symbol is A. 

In this discussion of indifferent states we have always used the entropy, energy, and volume as the possible extensive variables that must be used, in addition to the mole numbers of the components, to define the state of the system. The enthalpy or the Helmholtz energy may also be used to define the state of the system, but the Gibbs energy cannot. Each of the systems that we have considered has been a closed system in which it was possible to transfer matter between the phases at constant temperature and pressure. The differentials of the enthalpy and the Helmholtz and Gibbs energies under these conditions are... 

Thus, the enthalpy is a function of the entropy and the Helmholtz energy is a function of the volume, and each function may be used in place of the other variable. However, the Gibbs energy is a constant for any closed system at constant temperature and pressure, and therefore its value is invariant with the transfer of matter within the closed system. 

Every coefficient in Equation (5.112) except that of (ST)2 can be expressed more simply as a second derivative of the Helmholtz energy. We take only the coefficient of (SV)2 as an example. Both (dE/dV)Sn and (cM/5K)r are equal to — P. The differential of (dE/dV)Sn is expressed in terms of the entropy, volume, and mole numbers and the differential of (dA/dV)Tmole numbers, so that at constant mole numbers... 

The allowed variations are then the volume and mole numbers at constant temperature for the Helmholtz energy, and only the mole numbers at constant temperature and pressure for the Gibbs energy. 

The stability of the critical phase can be discussed most easily in terms of the Helmholtz energy and the condition expressed in Equation (5.154). By use of the method used in Section 5.15, the second-order variation at constant temperature is expressed as... 

The subject of partial molar quantities needs to be developed and understood before considering the application of thermodynamics to actual systems. Partial molar quantities apply to any extensive property of a single-phase system such as the volume or the Gibbs energy. These properties are important in the study of the dependence of the extensive property on the composition of the phase at constant temperature and pressure e.g., what effect does changing the composition have on the Helmholtz energy In this chapter partial molar quantities are defined, the mathematical relations that exist between them are derived, and their experimental determination is discussed. 

Many equations have been suggested to express the behavior of real gases. In general, there are those equations that express the pressure as a function of the volume and temperature, and those that express the volume as a function of the pressure and temperature. These cannot usually be converted from one into the other without obtaining an infinite series. The most convenient thermodynamic function to use for those in which the volume and temperature are the independent variables is the Helmholtz energy. The... 

The change of the Helmholtz energy, AAmix, is the same as that for the Gibbs energy. The change of the entropy, ASmix, is given by c c... 

---