Total differentiation

At constant temperature and pressure a small change in the surface free energy of the system shown in Fig. IV-1 is given by the total differential... 

The total differential cross-section (equation (B1,3,A11 + equation (B1,3,A211 is then... 

Parker G A and Pack R T 1978 Rotationally and vibrationally inelastic scattering in the rotational lOS approximation. Ultra-simple calculation of total (differential, integral and transport) cross sections for nonspherical molecules J. Chem. Phys. 68 1585... 

In Figure 2, we show the total differential cross-section for product molecules in the vibrational ground state (no charge bansfer) of the hydrogen molecule in collision with 30-eV protons in the laboratory frame. The experimental results that are in aibitrary units have been normalized to the END... 

Figure 2. Total differential cross-section versus laboratory scattering angle for vibrational ground state of hydrogen molecules in single collisioins with 30-eV protons.

The total differentials of these three equations in combination with equation 54 yield the following ... 

Equipment The geometiy of the pump or compressor is veiy important in seal effectiveness. Different pumps with the same shaft diameter and the total differential head can present different sealing problems. 

The Hugoniot can be described with a differential equation by taking the total differential of the Rankine-Hugoniot equation (2.4)... 

Since this equation came directly from differentiation of the Rankine-Hugoniot equation, it only holds true on the Hugoniot. We can also write T dS as a total differential in terms of dP and d V... 

Determine total differential head across pump. 

The Stern model predicts that the total differential capacitance C will consist of two terms representing two capacitors in series... 

The variation with temperature and pressure of the composition of the equilibrium clathrate is given by the total differential of Eq. 25, ... 

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. 

For the Gibbs free energy, the total differential given above as equation (5.1) can be written as... 

A1.2 Total Differentials and Relationships Between Partial Derivatives... 

The quantities dX and d Y are called differentials, the coefficients in front of dX and dT are called partial derivatives,11 and dZ is referred to as a total differential because it gives the total change in Z arising from changes in both X and Y. If Z were to depend upon additional variables, additional terms would be included in equation (A 1.1) to represent the changes in Z arising from changes in those variables. For much of our discussion, two variables describe the processes of interest, and therefore, we will limit our discussion to two independent variables, with the exception of the description of Pfaffian differentials in... 

The total differential given by equation (A 1.1) is a useful starting point for many thermodynamic derivations. If we consider a process in which X and Y are changed such that Z remains constant, then dZ = 0, and equation (A 1.1) can be rearranged to yield... 

Occasionally, we will find it necessary to consider derivatives of the form (dZ/dZJg where Q = Q(X, Y). This derivative can be obtained by starting with the total differential for dZ given by equation (A 1.1), dividing by dX, and specifying constant Q to get... 

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 model more generally accepted for metal/electrolyte interfaces envisages the electrical double layer as split into two parts the inner layer and the diffuse layer, which can be represented by two capacitances in series.1,3-7,10,15,32 Thus, the total differential capacitance C is equal to... 

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