Total differential meaning

In contrast to Clausius, Gibbs did not discuss uncompensated heat, as he started directly with the total differential of entropy. Gibbs presentation appealed very much to De Donder, However, he wanted to find the meaning of this mysterious uncompensated heat. He considered a system whose physical conditions, such as pressure and temperature, were uniform and which was closed to the flow of matter. Chemical reactions, however, could go on inside the system. De Donder first introduced what he called the degree of advancement, , of the chemical reaction so that the reaction rate v is the time derivative of . 

Transform methods are used to solve two-variable linear differential equations essentially by means of the transformation of a partial differential equation into a total differential equation of one independent variable (in general, the number of variables is reduced by one) [1], The major inconvenience of these methods to find analytical solutions is that the inverse transformation is frequently very difficult or cannot be done at all even for not too complex electrochemical processes. In these cases, the solutions have an integral non-explicit form, from which it is not possible to deduce limit behaviors and the influence of the different variables cannot be inferred for a glance. In Electrochemistry, this method has been extensively used to solve the diffusion equation, which is a two-variable partial differential equation. 

The integral of a total differential, however, is the difference between two values of a state function and, therefore, cannot depend on the path of the integral. This holds even if this difference is evaluated by means of partial derivatives. 

As an example, Fig. 6.25a gives the results of the isotherm determination for Troger s base enantiomer on Chiralpak AD (dp = 20 xm) from perturbation measurements (Mihlbachler et al., 2001). Theoretical retention times for the pure components and racemic mixtures (lines) were fitted to the measured data (symbols) by means of Eq. 6.185 to determine the unknown parameter in Eq. 6.186. Total differentials for the mixture (Eq. 6.53) were evaluated using the coherence condition Eq. 6.54, resulting in the isotherm equation Eq. 6.186. Note that the Henry coefficients were independently determined by pulse experiments and were fixed during the fitting procedure. 

Figure 42. Differential scatter of left and right circularly polarized light by poly-L-glutamic acid particles. Curve a, mean residue rotation of PGA reference state plotted on right-hand ordinate. Curve b, contribution of differential scatter term only, (Asi. —Asr) 3300/Cl. Curve c, total change in mean residue ellipticity due to differential scatter, includes differential scatter component of the (Aol— 4or) term. Curve d, comparison of empirical lO " values with total differential scatter. Curve e, approxima-...

Fig. 2,6. A plane tangent to a curved surface, illustrating the meaning of partial derivatives and total differentials.

Thennodynamics commonly deals with continuous changes in multivariable systems. For this rea.son, total differentials are frequently used and it is essential to have a clear idea of their meaning. 

We shall now derive a differential equation for the mean molal property g which is of considerable importance. We consider (7 as a function of r,p, i,. . . , and write its total differential in the form... 

Now we form the difference /(x+Ax, y+Ay) —/(x —Ax, y —Ay), however, with the constraint /tAx + fyAy = 0. This means that we are moving along a tangent line along some point x, y, or in other words, in the limiting case of infinitesimal (Ax, Ay), the total differential of first order, d/(x, y) = 0. Note that the equation of a tangent at point (xq, yo) of some function /(x, y) = C is... 

Think about a change of energy in such a way that the total entropy remains constant. We will exchange a small amount of entropy A5. Initially the energy is (7(5i, Sf). Moreover, we start from an equilibrium position which means that T = T2 = T. After varying the energy we have (7(5i -1- A5, S2 - AS). Inspect the first total differential ... 

The second total derivative along the path is positive. This means that if we are moving on a tangent away from the path, the energy will increase. Thus, the isodynamic path is a minimum with respect to a deviation in direction of any tangent touching the path. The total differential of third order is... 

Then the first-order variation of/merely means the total differential, evaluated at the stationary point. 

Because we have a mathematical model of energy changes, we have a simple mathematical answer to this question. G is a state variable, so dG is an exact differential ( C.2.1). This means that, among other things, we can write the total differential as in Equation (4.41), or in the total energy form as... 

For small linear models, the above steps can be carried out by hand as has been shown by means of the illustrating examples. In any case, no symbolic differentiation of transfer functions has to be performed. The use of an incremental bond graph means that the total differential of constitutive equations has already been taken. 

F is a function of state. This means, in mathematical terms, that it is a total differential. Differentiation gives... 

Hydrolysis of polyamide-based formulations with 6 N HC1 followed by TLC allows differentiation between a-aminocaproic acid (ACA) and hexamethylenedi-amine (HMD) (hydrolysis products of PA6 and PA6.6, respectively), even at low levels. The monomer composition (PA6/PA6.6 ratio) can be derived after chromatographic determination of the adipic acid (AA) content. Extraction of the hydrolysate with ether and derivatisa-tion allow the quantitative determination of fatty acids (from lubricants) by means of GC (Figure 3.27). Further HC1/HF treatment of the hydrolysis residue, which is composed of mineral fillers, CB and nonhydrolysable polymers (e.g. impact modifiers) permits determination of total IM and CB contents CB is measured quantitatively by means of TGA [157]. Acid hydrolysis of flame retarded polyamides allows to determine the adipic acid content (indicative of PA6.6) by means of HPLC, HCN content (indicative of melamine cyanurate) and fatty acid (indicative of a stearate) by means of GC [640]. Determination of ethylene oxide-based antistatic agents... 

Tubular electrodes. Blaedel et al.161 were the first to introduce the TBE, a tubular electrode. It was constructed by melting a Pt cylinder in a glass capillary (see Fig. 5.25), the total length of Pt being 25.5 mm and its diameter 0.75 mm. The authors originally used it for an enzymatic determination of glucose by means of differential amperometry based on the following sequential reactions"12 ... 

The total-chlorine method has been used extensively in the determination of spray residues of the chlorinated hydrocarbons 56). Usually the kind of insecticide applied has been known, and by means of the proper factor the chlorine values could be calculated to the insecticide originally used. This calculation is not entirely valid, as the determinations do not differentiate between the insecticide and its degradation products or other contaminants containing organic chlorine. The values obtained by the total-chlorine method are useful, however, because they indicate the magnitude of the residue and the analysis can be made in a short time with standard laboratory equipment. 

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