Partial derivatives interpretation

The rate-controlling step is the elementary reaction that has the largest control factor (CF) of all the steps. The control factor for any rate constant in a sequence of reactions is the partial derivative of In V (where v is the overall velocity) with respect to In k in which all other rate constants (kj) and equilibrium constants (Kj) are held constant. Thus, CF = (5 In v/d In ki)K kg. This definition is useful in interpreting kinetic isotope effects. See Rate-Determining Step Kinetic Isotope Effects... 

Canonical analysis achieves this geometric interpretation of the response surface by transforming the estimated polynomial model into a simpler form. The origin of the factor space is first translated to the stationary point of the estimated response surface, the point at which the partial derivatives of the response with respect to all of the factors are simultaneously equal to zero (see Section 10.5). The new factor... 

This example is used in Frame 5 to illustrate the meaning and interpretation of partial derivatives. A is a function depending on two variables A = f(x,y) and this has implications for the meaning and interpretation of the differential coefficients corresponding to the various slopes which are represented as partial derivatives (3A/dx)y and (3A/3y)x rather than as ordinary derivatives d r/dx etc. as were discussed earlier in this Frame and for which only one variable is involved. 

While Equation (4B-2) permits any light scattering data to be interpreted as partial derivatives of chemical potential or activity, dilute solution measurements are conventionally presented in terms of a virial coefficient expansion of the chemical potential. 

In an engineering view the ensemble of system points moving through phase space behaves much like a fluid in a multidimensional space, and there are numerous similarities between our imagination of the ensemble and the well known notions of fluid dynamics [35]. Then, the substantial derivative in fluid dynamics corresponds to a derivative of the density as we follow the motion of a particular differential volume of the ensemble in time. The material derivative is thus similar to the Lagrangian picture in fluid d3mamics in which individual particles are followed in time. The partial derivative is defined at fixed (q,p). It can be interpreted as if we consider a particular fixed control volume in phase space and measure the time variation of the density as the ensemble of system points flows by us. The partial derivative at a fixed point in phase space thus resembles the Eulerian viewpoint in fluid dynamics. 

However, the partial differentiations here are not at constant T and p consequently, pn is not partial molar U, nor partial molar H, nor partial molar A. The conditions of the partial differentiation for U and H cannot be physically interpreted, making the partial derivatives of U and H of little interest for the description of real systems. But the partial differentiation of A is at a physically meaningful condition that makes the partial derivative useful for the calculation of chemical potential. 

This equation contains several things that look like ordinary derivatives. However, we must interpret them as partial derivatives, since we have specified that we want to have V and n constant. We change the symbols to the symbols for partial derivatives and add the appropriate subscripts to indicate the variables that are being held fixed. These variables must be the same in all four of the derivatives to keep a valid equation. We want V and n to be constant, so we write... 

This equation delivers the value of dy corresponding to arbitrary infinitesimal changes in x and z, so it is still correct if we choose values of dz and dx such that dy vanishes. We now divide nonrigorously by dx, and interpret the quotients of differentials as partial derivatives, remembering that y is held fixed by our choice that dy vanishes. 

But these definitions can lead to ambiguities, especially when we must interpret certain partial derivatives that often arise in thermodynamics. For example, is the system pressure P extensive Some definitions suggest that P does not change with N, and for a pure substance it is true that... 

Basic relations among thermodynamic variables are routinely stated in terms of partial derivatives these relations include the fundamental equations from the first and second laws, as well as innumerable relations among properties. Here we define the partial derivative and give a graphical interpretation. Consider a variable z that depends on two independent variables, x and y,... 

Figure A.l The partial derivative in (A.1.2) can be interpreted as the slope of a y-level curve on...

Obviously, the scalar product between the nabla operator and the partial derivative of has to be interpreted as the divergence of the corresponding vector field. These equations of motion determine the actual form of the fields. Their solution is, however, much more involved than for discrete systems. The canonical momentum field conjugate to the field (pi is analogously defined by... 

We interpret the quotients as partial derivatives, since we have specified that we want to have V and n constant. The same variables must be held fixed in all four of the partial derivatives to keep a valid equation. We now write... 

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