Thermodynamic partial derivatives

More About Thermodynamic Partial Derivatives (Optional) 259... 

MORE ABOUT THERMODYNAMIC PARTIAL DERIVATIVES (OPTIONAL)... 

The procedure would then require calculation of (2m+2) partial derivatives per iteration, requiring 2m+2 evaluations of the thermodynamic functions per iteration. Since the computation effort is essentially proportional to the number of evaluations, this form of iteration is excessively expensive, even if it converges rapidly. Fortunately, simpler forms exist that are almost always much more efficient in application. 

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

In our thermodynamic derivations, we will routinely make use of equations of the type represented by equation (A1.3) to replace AX7jdYz with (dX/dY)z. We may also represent this ratio of differentials as (dA/d Y)z in which the direct equality with the partial derivative (dX/0Y)Z is more immediately evident. That is... 

Partial derivatives, as introduced in Section 2.12 are of particular importance in thermodynamics. The various state functions, whose differentials are exact (see Section 3.5), are related via approximately 1010 expressions involving 720 first partial derivatives Although some of these relations are not of practical interest, many are. It is therefore useful to develop a systematic method of deriving them. Hie method of Jacobians is certainly the most widely applied to the solution of this problem. It will be only briefly described here. For a more advanced treatment of the subject and its application to thermodynamics, die reader is referred to specialized texts. 

In the case of reciprocal systems, the modelling of the solution can be simplified to some degree. The partial molar Gibbs energy of mixing of a neutral component, for example AC, is obtained by differentiation with respect to the number of AC neutral entities. In general, the partial derivative of any thermodynamic function Y for a component AaCc is given by... 

Importantly, the contribution from thermodynamics is not restricted to finite interval. With y c [0,1], the normalized partial derivative may attain any (absolute) value between zero and infinity. In particular, for reactions close to equilibrium y 1, we obtain... 

The two partial derivatives in Equation 4.34, which are equal to each other, are very important in the thermodynamics of chemical equilibrium, and are referred to as the chemical potential, p, . 

The most important property of a liquid-gas interface is its surface energy. Surface tension arises at the boundary because of the grossly unequal attractive forces of the liquid subphase for molecules at its surface relative to their attraction by the molecules of the gas phase. These forces tend to pull the surface molecules into the interior of the liquid phase and, as a consequence, cause liquids to minimize their surface area. If equilibrium thermodynamics apply, the surface tension 7 is the partial derivative of the Helmholtz free energy of the system with respect to the area of the interface—when all other conditions are held constant. For a phase surface, the corresponding relation of 7 to Gibbs free energy G and surface area A is shown in eq. [ 1 ]. 

Because most chemical, biological, and geological processes occur at constant temperature and pressure, it is convenient to provide a special name for the partial derivatives of all thermodynamic properties with respect to mole number at constant pressure and temperature. They are called partial molar properties, and they are defined by the relationship... 

When the partial derivative reaches zero, the two potentials coincide and component i is present unmixed as a pure term. If unmixing takes place in solvus conditions, the thermodynamic activity of component i remains constant for the entire solvus field. However, in the case of spinodal decomposition, the activity of i within the spinodal field plots within a maximum and minimum (cf sections 3.11 and 3.12). 

Actually, the various equations listed in this section are insufficient to perform the complete calculation since one would first calculate the density of H2O through eq. 8.12 or 8.14. Equation 8.14 in its turn involves the partial derivative of the Helmholtz free energy function 8.15. Moreover, the evaluation of electrostatic properties of the solvent and of the Bom functions (o, Q, Y, X involve additional equations and variables not given here for the sake of brevity (eqs. 36, 40 to 44, 49 to 52 and tables 1 to 3 in Johnson et ah, 1991). In spite of this fact, the decision to outline here briefly the HKF model rests on its paramount importance in geochemistry. Moreover, most of the listed thermodynamic parameters have an intrinsic validity that transcends the model itself... 

Application to Macromolecular Interactions. Chun describes how one can analyze the thermodynamics of a particular biological system as well as the thermal transition taking place. Briefly, it is necessary to extrapolate thermodynamic parameters over a broad temperature range. Enthalpy, entropy, and heat capacity terms are evaluated as partial derivatives of the Gibbs free energy function defined by Helmholtz-Kelvin s expression, assuming that the heat capacities integral is a continuous function. 

Because partial derivatives are used so prominently in thermodynamics (See Maxwell s Relationships), we briefly consider the properties of partial derivatives for systems having three variables x, y, and z, of which two are independent. In this case, z = z(x,y), where x and y are treated as independent variables. If one deals with infinitesimal changes in x and y, the corresponding changes in z are described by the partial derivatives ... 

The final thermodynamic quantity for review is the chemical potential, which is represented with the Greek letter mu, ji. The chemical potential can be defined in terms of the partial derivative of any of the previous thermodynamic quantities with respect to the number of moles of species i, Ui, at constant Uj (where j indicates all species other than i) and thermodynamic quantities as indicated ... 

Partial Derivatives Are You Kidding Teaching Thermodynamics Using Virtual Substance... 

The benefit of this course is that it provides all students taking the physical chemistry lecture course with the same mathematical foundation. In the physical chemistry lecture we can discuss the relationship between different thermodynamic functions without stopping to review partial derivatives. We can talk about the difference between work, heat, and energy without stopping to teach the difference between path functions and work functions. We can write... 

Using eight thermodynamic potentials introduced, 24 Maxwell relations containing certain partial derivatives can be obtained easily. These relations together with the corresponding specific heats Cy z = T(dS/dT)y z (where y represents either V or P, and z represents either E, or x) permit to describe phenomenological relationships between the deformation (or stress) in solids and the accompanying thermal effects. 

Note that this identity clearly shows that (dz/dx)y (dz/dx)e, i.e., that the variable held constant matters in these derivatives (Strictly speaking, a lazy notation such as dz/dx has no meaning whatsoever ) Although the inconvenient notation of partial derivatives makes it somewhat tedious to keep the inactive (constant) background variables in mind, it is important from a physical and pedagogical standpoint that this be done as carefully as possible. (The tedium of this notation is avoided in the geometrical thermodynamics to be presented in Part III.)... 

The Maxwell relations are powerful tools of thermodynamic derivation. With the help of these relations and derivation techniques analogous to those illustrated in Sidebars 5.3-5.6, the skilled student of thermodynamics can (in principle ) re-express practically any partial derivative in terms of a small number of base properties involving only PVT variables. Consider, for example, the eight most common variables... 

It may finally be remarked that equations such as (12.14), (12.16), (12.18), which represent partial derivatives as simple ratios of geometrical projections, make it easy (as well as largely superfluous) to recover various identities among partial derivatives that are often taken as starting points in thermodynamic manipulations (cf. Section 1.2). 

As this method is pre-eminently suitable to fast reactions, only the case of a d.c. reversible system will be considered. Then immediately, the thermodynamic relationships between the partial derivatives as derived by Reinmuth [24] can be introduced. These are [7, 24]... 

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