Total differential functions

As we have seen before, exact differentials correspond to the total differential of a state function, while inexact differentials are associated with quantities that are not state functions, but are path-dependent. Caratheodory proved a purely mathematical theorem, with no reference to physical systems, that establishes the condition for the existence of an integrating denominator for differential expressions of the form of equation (2.44). Called the Caratheodory theorem, it asserts that an integrating denominator exists for Pfaffian differentials, Sq, when there exist final states specified by ( V, ... x )j that are inaccessible from some initial state (.vj,.... v )in by a path for which Sq = 0. Such paths are called solution curves of the differential expression The connection from the purely mathematical realm to thermodynamic systems is established by recognizing that we can express the differential expressions for heat transfer during a reversible thermodynamic process, 6qrey as Pfaffian differentials of the form given by equation (2.44). Then, solution curves (for which Sqrev = 0) correspond to reversible adiabatic processes in which no heat is absorbed or released. 

Application of the condition for exactness shows that both derivatives equal zero, so equation (2.45) must be exact. Thus, we have determined that when an ideal gas is involved, T is an integrating denominator for Sqrev, and the right hand side of equation (2.45) is the total differential for a state function that we will represent as dS.cc... 

If equation (2.51) is the total differential for as a function of two variables, 1 and 2, we can expect that its partial derivatives (d E/d Zi) and (<9 /c> 2)5 can be expressed as functions of only those two variables. That is, — ( , 2). Thus, derivatives of (<9 /<9 ) and (d E/d Zi)- with respect to variables other than 1 and 2 should be zero. As we consider the implications of this statement, it is important to note that a change can be made independently in the r variable of one subsystem without affecting that of the other, but a change in 0 will affect both subsystems (since 0 is the same in both subsystems). Therefore, we must consider the implications for c and 0 separately in the analysis that follows. 

The importance of these four equations cannot be overemphasized. They are total differentials for U as f(S, V), H as /(S./ ), A as f V,T), and G as j p,T). Although they were derived assuming a reversible process, as total differentials they apply to both reversible and irreversible processes. They are the starting points for the derivation of general differential expressions in which we express U, H, A and Casa function of p, V, T, Cp and Ci. a These are the relationships that we will now derive. 

Equation (A 1.25) is known as the Maxwell relation. If this relationship is found to hold for M and A in a differential expression of the form of equation (A 1.22), then 6Q — dQ is exact, and some state function exists for which dQ is the total differential. We will consider a more general form of the Maxwell relationship for differentials in three dimensions later. 

That is, if these three relationships are satisfied simultaneously for a given Pfaffian, the Pfaffian is exact, and some function F(.x. y. r) exists such that the total differential dF= bQ. 

The charge density on the electrode a(m) is mostly found from Eq. (4.2.24) or (4.2.26) or measured directly (see Section 4.4). The differential capacity of the compact layer Cc can be calculated from Eq. (4.3.1) for known values of C and Cd. It follows from experiments that the quantity Cc for surface inactive electrolytes is a function of the potential applied to the electrode, but is not a function of the concentration of the electrolyte. Thus, if the value of Cc is known for a single concentration, it can be used to calculate the total differential capacity C at an arbitrary concentration of the surface-inactive electrolyte and the calculated values can be compared with experiment. This comparison is a test of the validity of the diffuse layer theory. Figure 4.5 provides examples of theoretical and experimental capacity curves for the non-adsorbing electrolyte NaF. Even at a concentration of 0.916 mol dm-3, the Cd value is not sufficient to permit us to set C Cc. 

Consider a pool of water in the bed of your pickup truck. If you accelerate from rest, the water will slosh toward the rear, and you want to know how fast you can accelerate (ax) without spilling the water out of the back of the truck (see Fig. 4-4). That is, you must determine the slope (tan ) of the water surface as a function of the rate of acceleration (ax). Now at any point within the liquid there is a vertical pressure gradient due to gravity [Eq. (4-5)] and a horizontal pressure gradient due to the acceleration ax [Eq. (4-23)]. Thus at any location within the liquid the total differential pressure... 

Once the total entropy of a composite system has been formulated as a function of the various extensive parameters of the subsystems, the extrema of this total entropy function may in principle be located by direct differentiation and classified as either maxima, minima or inflection points from the sign of the second derivative. Of these extrema, only the maxima represent stable equilibria. 

Step 5. Look at the total cost function, Equation (c). Observe that the cost function includes a constant term, K3Q. If the total cost function is differentiated, the term K3Q vanishes and thus K3 does not enter into the determination of the optimal value for D. K3, however, contributes to the total cost. 

Just as in classical statistical mechanics, the different pictures of electronic changes are related by Legendre transforms. The state function for closed systems in the electron-following picture is just the electronic ground-state energy, /i v AT The total differential for the energy provides reactivity indicators for describing how various perturbations stabilize or destabilize the system,... 

As previously mentioned, this expression is conceptually clumsy because the number of electrons is a function of the electron density through Equation 19.2. As one cannot vary the number of electrons if the electron density is fixed, the naive expression for the total differential... 

Equation for the Total Differential. Let us consider a specific example the volume of a pure substance. The molar volume is a function/only of the temperature T and pressure P of the substance thus, the relationship can be written in general notation as... 

General Formulation. To understand the notation for exact differentials that generally is adopted, we shall express the total differential of a general function L(x, y) to indicate explicitly that the partial derivatives are functions of the independent variables (x and y), and that the differential is a function of the independent variables and their differentials (dx and dy). That is. 

For an ideal gas we will show later that the molar entropy is a function of the independent variables, molar volume Vm and temperature T. The total differential dSra is given by the equation... 

The partial derivative (dU/dT)p is not Cy, but if it could be expanded into some relationship with (dU/dT)y, we would have succeeded in introducing Cy into Equation (4.58). The necessary relationship can be derived by considering the internal energy U as sl function of T and V and setting up the total differential ... 

It frequently is necessary to express the Joule-Thomson coefficient in terms of other partial derivatives. Considering the enthalpy as a function of temperature and pressure H T, P), we can write the total differential... 

In deriving an equation for the entropy of a real gas we can start with Equation (6.110), which is general and not restricted to ideal gases. A suitable substitution for dU in Equation (6.110) can be obtained from the total differential of t/ as a function of y and T [Equation (4.59)] ... 

The entropy S also can be considered to be a function of V and T thus, a second equation for the total differential dS is... 

By an analogous procedure, it can be shown that the total differential of the Helmholtz function is given by the expression... 

Derive the total differential for the Massieu function J, with natural variables T and V determine dJ/dT)v and (dJ/dV)f, and show that... 

For a closed system of fixed composition, the extensive thermodynamic properties such as y, U, S, A, Y, and G are functions of any pair of convenient independent variables. For example. Equation (7.38) suggests that G is a natural function of T and P. That is G =f(T, P). The total differential of G would be... 

As at constant pressure and temperature / is a function of two variables,/(ui, n2), the following equation is valid for the total differential ... 

Thus, the total differential of the free energy function is... 

Note 2 The relationships between the total differentials of the functions g, define how particles of the material move relative to each other. Thus, if two particles are at small distances dx,, dx, d apart at time f and dxi, dx2, dx3 at time t then... 

Taking into consideration that the solvents are practically immiscible and that 1-octadecanol and dodecylaimmonium chloride are soluble only in hexane and water respectively, the total differential of the interfacial tension Y can be expressed as a function of temperature T, pressure p, molality m, and molality as follows (4) ... 

When the values of Cdipole calculated from Eq. (6.162) are plotted as a function of qM (Fig. 6.83), it turns out that the values of the dipole capacity are extremely large compared with the experimental values of the capacity. What does this imply Consider the complete expression for the total differential capacity of the interface ... 

To this end, we consider the thermodynamic functions of a homogeneously stressed solid, e.g., [L.D. Landau, E.M. Lifshitz (1989) W. W. Mullins, R. Sekerka (1985)]. In contrast to the unstressed solid, the internal energy of which is U(S, K ,), the internal energy of a stressed solid is given as U(S, VuJk,nj). For the total differential of the internal energy one has1... 

The practical applications of the theory just outlined divide themselves into two broad classes (1) Those which are based on the existence and properties of the functions U and S and some others related to them—all "thermodynamic identities being merely the integrability condition for the total differentials of these functions and (2) those which aie based on die Principle of Increase of Entropy the entropy of the actual state of an adiabatically enclosed system being greater than that of any neighboring virtual state. 

The Dimensionless Parameter is a mathematical method to solve linear differential equations. It has been used in Electrochemistry in the resolution of Fick s second law differential equation. This method is based on the use of functional series in dimensionless variables—which are related both to the form of the differential equation and to its boundary conditions—to transform a partial differential equation into a series of total differential equations in terms of only one independent dimensionless variable. This method was extensively used by Koutecky and later by other authors [1-9], and has proven to be the most powerful to obtain explicit analytical solutions. In this appendix, this method will be applied to the study of a charge transfer reaction at spherical electrodes when the diffusion coefficients of both species are not equal. In this situation, the use of this procedure will lead us to a series of homogeneous total differential equations depending on the variable, v given in Eq. (A.l). In other more complex cases, this method leads to nonhomogeneous total differential equations (for example, the case of a reversible process in Normal Pulse Polarography at the DME or the solutions of several electrochemical processes in double pulse techniques). In these last situations, explicit analytical solutions have also been obtained, although they will not be treated here for the sake of simplicity. 

Another concise strategy for the synthesis of alkaloids related to crinine features the application of a general and useful procedure for the elaboration at a carbonyl center of a quaternary carbon atom bearing differentially functionalized alkyl substituents. The application of this methodology to the total syntheses of ( )-crinine (359) and ( )-buphanisine (361) (Scheme 46) commenced with the... 

When U is expressed as a function of S, V, and n , calculus requires that the total differential of U is given by... 

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