Thermodynamics differential relations

The equilibrium thermodynamic state of a simple one-component open system can be specified by T, P, and n, the amount of the single component. This gives the differential relation for a general extensive quantity, Y, in a one-component system ... 

From the thermodynamic relation G = U -TS + PV, and the definition of the molecular chemical potential, g, of the system as the partial derivative of G with respect to N (here, the number of particles, not the number of moles), we can express the following differential relation ... 

Pure thermodynamics is developed, without special reference to the atomic or molecular structure of matter, on the basis of bulk quantities like internal energy, heat, and different types of work, temperature, and entropy. The understanding of the latter two is directly rooted in the laws of thermodynamics— in particular the second law. They relate the above quantities and others derived from them. New quantities are defined in terms of differential relations describing material properties like heat capacity, thermal expansion, compressibility, or different types of conductance. The final result is a consistent set of equations and inequalities. Progress beyond this point requires additional information. This information usually consists in empirical findings like the ideal gas law or its improvements, most notably the van der Waals theory, the laws of Henry, Raoult, and others. Its ultimate power, power in the sense that it explains macroscopic phenomena through microscopic theory, thermodynamics attains as part of Statistical Mechanics or more generally Many-body Theory. 

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. 

Additional thermodynamic properties are related to these and arise by arbitrary definition. Multiplication ofEq. (4-11) hy n and differentiation yields the general expression ... 

As we have seen earlier, the thermodynamic variables p, V, T, U, S, H, A, and G (that we will represent in the following discussion as W, X, T, and Z) are state functions. If one holds the number of moles and hence composition constant, the thermodynamic variables are related through two-dimensional Pfaffian equations. The differential for these functions in the Pfaff expression is an exact differential, since state functions form exact differentials. Thus, the relationships that we now give (and derive where necessary) apply to our thermodynamic variables. 

By using relationships for an exact differential, equations that relate thermodynamic variables in useful ways can be derived. The following are examples. 

In classical thermodynamics the (inexact) differential change in heat Bq, is related to the... 

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. 

The quantities appearing in Eq. (16.2) are not independent. They are related by a Gibbs-Duhem equation, which is obtained in the same way as in the ordinary thermodynamics of bulk phases integrating with respect to the extensive variables results in Ua —TSa — pVa + 7Aa + E/if Nf. Differentiating and comparing with Eq. (16.2) gives ... 

By 1984, it was clear that the derivatives that arise in this DFT thermodynamics contain chemically useful information, /a is — 1 times the electronegativity. p(r) is the electron density—fundamental in its own right, but also closely related to the electrostatic potential. If, in analogy to Equation 18.6, one writes the total differential for the chemical potential,... 

The liquid fraction sensitivity is an important parameter for the determination of the semi-solid forming capability. It is defined as the rate of change of the liquid fraction in the alloy with temperature and is related to the relative slopes, in the phase diagram, of the liquidus and solidus curves. It may be determined by differential scanning calorimetry or predicted by thermodynamic modelling. Examples related to various Al alloys have been reported by Maciel Camacho et al. (2003), Dong (2003). See also several papers in Chiarmetta and Rosso (2000). 

Here x and x are isotopomer mole fractions in the binary mixture. Remembering x = 1 — x, differentiating to obtain partial molar free energies (and using the thermodynamic relations p,E(V) = AE(V) — x (dAE(V)/dx ) and xE (V) = AE(V) + x (dAE(V)/dx ) one finds expressions for the excess partial molar free energies, xE(V) and il (V). In the high dilution limit, an important case of practical interest, the excess chemical potential of the trace isotopomer, say the unprimed one, is... 

The Maxwell relations of thermodynamics relate quantities formed by differentiating G once with respect to one variable and once with respect to another (Huang, 1987). Choosing the two variables to be T and n leads to the following relationship ... 

Heat Content or Enthalpy. A thermodynamic property closely related to energy. It is defined by H = E + PV where E is the internal energy of the system, P is the pressure on the system and V is the volume of the system. Often it is used in differential form as in. AH = AE + PAV for a constant pressure process... 

The Maxwell relations in thermodynamics are obtained by treating a thermodynamic relation as an exact differential equation. Exact differential equations are of the form... 

Can you prove why this is so ) When x, y, and z are thermodynamic quantities, such as free energy, volume, temperature, or enthalpy, the relationship between the partial differentials of M and N as described above are called Maxwell relations. Use Maxwell relations to derive the Laplace equation for a... 

EXAMPLE 6.1 Laplace Equation for Spherical Surfaces A Thermodynamic Derivation. The Maxwell relations play an important role in thermodynamics. By including the term dA in the usual differential form for cfG, show that (dVfdA)pJ = (dyidp)AJ. Evaluate (dVidA)pJ assuming a spherical surface and, from this, derive the Laplace equation for this geometry. 

Substitution of Equations (36) and (37) into Equation (35) generates a complicated differential equation with a solution that relates the shape of an axially symmetrical interface to y. In principle, then, Equation (35) permits us to understand the shapes assumed by mobile interfaces and suggests that y might be measurable through a study of these shapes. We do not pursue this any further at this point, but return to the question of the shape of deformable surfaces in Section 6.8b. In the next section we examine another consequence of the fact that curved surfaces experience an extra pressure because of the tension in the surface. We know from experience that many thermodynamic phenomena are pressure sensitive. Next we examine the effecl of the increment in pressure small particles experience due to surface curvature on their thermodynamic properties. 

A study of benzocyclobutene polymerization kinetics and thermodynamics by differential scanning calorimetry (DSC) methods has also been reported in the literature [1]. This study examined a series of benzocyclobutene monomers containing one or two benzocyclobutene groups per molecule, both with and without reactive unsaturation. The study provided a measurement of the thermodynamics of the reaction between two benzocyclobutene groups and compared it with the thermodynamics of the reaction of a benzocyclobutene with a reactive double bond (Diels-Alder reaction). Differential scanning calorimetry was chosen for this work since it allowed for the study of the reaction mixture throughout its entire polymerization and not just prior to or after its gel point. The monomers used in this study are shown in Table 3. The polymerization exotherms were analyzed by the method of Borchardt and Daniels to obtain the reaction order n, the Arrhenius activation energy Ea and the pre-exponential factor log Z. Tables 4 and 5 show the results of these measurements and related calculations. 

Dimensional Analysis is.a method by which the variables characterizing a phenomenon may be related. Accdg to Eschbach (Ref 2)> it is fundamentally identical with the analysis of physical equations, and in particular, with the analysis of physical differential equations. Methods of Lord Rayleigh and of E. Buckingham are used in ballistics, thermodynamics and fluid mechanics... 

Overall our objective is to cast the conservation equations in the form of partial differential equations in an Eulerian framework with the spatial coordinates and time as the independent variables. The approach combines the notions of conservation laws on systems with the behavior of control volumes fixed in space, through which fluid flows. For a system, meaning an identified mass of fluid, one can apply well-known conservation laws. Examples are conservation of mass, momentum (F = ma), and energy (first law of thermodynamics). As a practical matter, however, it is impossible to keep track of all the systems that represent the flow and interaction of countless packets of fluid. Fortunately, as discussed in Section 2.3, it is possible to use a construct called the substantial derivative that quantitatively relates conservation laws on systems to fixed control volumes. 

This fundamental relation underlies all thermodynamic descriptions of exact (conserved) differential quantities such as internal energy or entropy, as will be shown in subsequent chapters. 

The Maxwell relations (5.49a-d) are easy to rederive from the fundamental differential forms (5.46a-d). However, these relations are used so frequently that it is useful to employ a simple mnemonic device to recall their exact forms as needed. Sidebar 5.7 describes the thermodynamic magic square, which provides such a mnemonic for Maxwell relations and other fundamental relationships of simple (closed, single-component) systems. 

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