Helmholtz energy differential

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. 

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... 

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... 

We can obtain expressions for the differentials of the enthalpy and the Gibbs and Helmholtz energies from the usual definitions. In such expressions M is an independent variable. Because it is more convenient to use H as an independent variable, the new functions (H — /j0HM), (A — /i0HM), and (G — /i0HM) are used. The differential expressions for these functions are... 

We now return to the definition of the surface excess chemical potential fta given by Equation (2.19) where the partial differentiation of the surface excess Helmholtz energy, Fa, with respect to the surface excess amount, rf, is carried out so that the variables T and A remain constant. This partial derivative is generally referred to as a differential quantity (Hill, 1949 Everett, 1950). Also, for any surface excess thermodynamic quantity Xa, there is a corresponding differential surface excess quantity xa. (According to the mathematical convention, the upper point is used to indicate that we are taking the derivative.) So we may write ... 

The fundamental property relation most intimately connected with statistical mechanics is Eq. (6.9), which expresses the differential of the Helmholtz energy as a function of its canonical variables T and V ... 

Activity coefficient models are equations representing the Gibbs or the Helmholtz energy of solutions. Activity coefficients and related properties are derived form these energy functions by proper differentiation (Equation (1)). 

Differentiating the Helmholtz energy, one obtains other thermodynamic properties, like the compressibility factor, pressure, and internal energy. To clarify, consider the example of a single component with a single acceptor and a single donor. Noting that X = X°, by symmetry in this instance, a simple quadratic equation results in... 

Thus, an equation of state is obtained that functions much like any other equation of state. That is, given a temperature and density, one obtains an estimate of a. from Eq. (2), and consequently an estimate of X and Contributions for disperse attraction and repulsion can be computed from density and temperature in the usual way. Combining the terms provides a complete estimate of the total Helmholtz energy, which can be differentiated to obtain the pressure. The only fundamental difference between this equation of state and those that treat hydrogen bonding implicitly is the intermediate step of computing xX... 

Helmholtz energy is displayed as the natural energy function of its natural variables T and V. The geometry of the equilibrium surface is expressed by the coefficients of the differentials. 

Wertheim found and in his perturbation theory. Upon differentiating the Helmholtz energy with respect to volume, the equation of state is obtained. 

Ideal-gas free energies A and G are, however, p or V dependent. Consider Helmholtz energy first. From the fundamental differential Equation (4.79),... 

We will begin the derivation with Helmholtz energy, as it is the natural energy function for the independent variables T and V of equations of state. By the fundamental differential equation for A, Equation (4.81)... 

We do not stop at this stage, but recall that the total differential of the Helmholtz energy is... 

In the thermodynamics of critical phenomena one prefers alternative thermodynamic variables and alternative variable-dependent thermodynamic potentials, namely, the density of the Helmholtz energy A/V as a function of temperature and molar density p = n/V, or the pressure P as a function of temperature and chemical potential p = G/n = d A/V)/dp r [1, 2]. The corresponding differential equations for the density-dependent potential A/V and for the field-dependent potential P read... 

The thermod5mamic properties of an ideal rubber can be expressed in terms of the Helmholtz energy, A T,a,m). The exact differential of the Helmholtz energy for a closed system is then ... 

This chapter begins with a discussion of mathematical properties of the total differential of a dependent variable. Three extensive state functions with dimensions of energy are introduced enthalpy, Helmholtz energy, and Gibbs energy. These functions, together with internal energy, are called thermodynamic potentials. Some formal mathematical manipulations of the four thermodynamic potentials are described that lead to expressions for heat capacities, surface work, and criteria for spontaneity in closed systems. 

We substitute this relation for dU into the differentials of enthalpy, Helmholtz energy, and Gibbs energy given by Eqs. 5.3.4-5.3.6 to obtain three more relations ... 

To find the spinodal condition, the first step requires the determination of the second differential (5 t, of the segment-molar Helmholtz energy with respect to 5Vs, S J/b and 5[il/B s,Bir)]- The second step is to find that variation function B s sir)] minimizing the second differential S A. In this the condition BWs B(r)]dr = SipB has to be taken into account by using Lagrange s method of undetermined multipliers as minimization procedure. Setting b sM )] the quantity S As SVs,SiJ/Bi [ B s,Bi )T found. 

Each equation of state has been fitted to an equation in the form of a dimensionless Helmholtz energy function using multiproperty fitting of the most accurate experimental data. The Helmholtz function ensures that all the properties are thermodynamically consistent and enables the calculations to be performed by differentiation alone. The properties which can be calculated are ... 

Now, as documented below in Section 1.12, A = A T,V,n is supposed to involve temperature, volume, and composition as the control variables for the Helmholtz energy, whereas Eq. (1.10.3a) involves a mix of quantities. To achieve the desired change, we express the entropy in terms of the appropriate control variables by setting S = S T,V,rij). This leads to the appropriate differential form... 

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