Partial differential coefficient

Together with Eos. (4-6) and (4-7), this definition allows ehmination of all the partial differential coefficients from the preceding equation ... 

The obvious next step is to ehminate the partial-differential coefficients in favor of measurable quantities. 

Since the entropy is the partial differential coefficient of with r constant, or with p constant, with respect to T, the magnitudes and thermodynamic potentials at constant volume and at constant pressure respectively. 

The differentials are partial, so when the differentiation is done with respect to one quantity, all the others are considered constant. The variable dy/dxt is known as the sensitivity coefficient (sometimes written cfj with respect to , and it tells how changes as xt changes. For the most simple case where = constant x x, the sensitivity coefficient is just the constant. 

If u =f(x, y) be a continuous function of two independent variables x and y, then the differential cofficient w.r.t. x (taking y as constant) is called the partial derivative or partial differential coefficient of u w.r.t. x and is represented by different symbols such as... 

If it exists then, it is called the partial derivative or partial differential coefficient of u w.r.t. jc. It is denoted by,... 

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. 

The suffixes to the partial differential coefficient indicate that temperature, pressure, and the concentrations of all charged and uncharged components are unchanged when the potential is altered. 

A further thermodynamic expression for l is possible.4 Since the electrocapillary equation (Eq. 8) is a total differential equation, the second cross-partial-differential coefficients of y are equal ... 

For the special case of reaction systems that are in a steady state, n,right-hand side of Eg. (17) must be zero. What were hitherto partial differential coefficients of r) with respect to become ordinary differential cmef-ficients. Dividing through by the factor I — then gives... 

Kinetic studies of ion exchange on partially ion-exchanged type A zeolites of Mg Ca and Mn " revealed that mini-mums and maximums characterize the differential coefficients of internal diffusion for every exchange of 2 Na " ions for one divalent cation per unit cell of the zeolite. On the basis of these observations, assuming definite interactions between the cations and the zeolite lattice, predictions can be made concerning the distribution and arrangement of cations in the unit cells of a type A zeolite. Research on liquid phase adsorption of n-alkanes on partially ion-exchanged type A zeolites indicated that the differential diffusion coefficients for alkane adsorption are influenced likewise by cation distribution in the unit cells of the zeolite. 

As the system is monovariant, we may write the complete instead of the partial differential coefficient. 

The expressions for T and -P of the preceding paragraph and the definition of 4, allow replacement of the partial differential coefficients in the preceding equation by T, -P, and ij. The result is Eq. (4-6) of Table 4-1, where inmortant equations of this section are collected. Equation (4-6) is the mndamental property relation for single-phase PVT systems, from which all other equations connecting properties of... 

U, H, and S as Functions of T and P or T and V At constant composition, molar thermodynamic properties can be considered functions of T and P (postulate 5). Alternatively, because V is related to T and P through an equation of state, V can serve rather than P as the second independent variable. The useful equations for the total differentials of U, H, and S that result are given in Table 4-1 by Eqs. (4-22) through (4-25). The obvious next step is substitution for the partial differential coefficients in favor of measurable quantities. This purpose is served by definition of two heat capacities, one at constant pressure and the other at constant volume ... 

If there are more equations than unknowns, an arbitrary symbol w (or several such symbols) may be used as a multiplier of the constant terms, when the Jacobian becomes the determinant of the coefficients of the variables. Then w is put equal to unity after the partial differential coefficients are found. For example, let ... 

I have just explained the meanings of the partial derivatives of u with respect to x and y. Let me again emphasize the distinction between the partial differential coefficient du/ dx, and the differential coefficient du/dx. In bufbx, y is treated as a constant in du/dx, y is treated as a function of x. The partial derivative denotes the rate of change of u per unit change in the value of x when the other variable or variables remain constant du/dx represents the total rate of change of u when all the variables change simultaneously. 

A differential equation is ordinary or partial, according as there is one or more than one independent variables present. Ordinary differential equations will be treated first. Equations like (2) and (3) above are said to be of the first order, because the highest derivative present is of the first order. For a similar reason (4) and (6) are of the second order, (5) of the third order. The order of a differential equation, therefore, is fixed by that of the highest differential coefficient it contains. The degree of a differential equation is the highest power of the highest order of of differential coefficient it contains. This equation is of the second order and first degree ... 

From (16), 162, page 530, when the effect of each observation on the final result is the same, the partial differential coefficients... 

The finite difference method The finite difference method is replaced differential coefficient by difference quotient. So do the boundary conditions and initial conditions, turning the problem for determining solution to a algebraic equation set. Its essence is replacing the partial differential equation of groundwater movement by the corresponding differential equation approximately, and then solving the differential equation. 

Miller s first major independent application of Gibbs theories came with his published explanation of the unexpected observation, made by one of his students, that the addition of salt to aqueous alcohol raised the partial vapor pressure of the alcohol. He was able to explain by thermodynamic reasoning how this phenomenon came about in a paper published in 1897, "On the Second Differential Coefficient of Gibbs Function (jL) This title provides a clue... 

The present section is concerned mainly with the first two groups above, and in particular it will be shown how the various partial differential coefficients of the previous section may be expressed in terms of experimental magnitudes. 

In order to compute the change of, say, the enthalpy it is desirable to use temperature and pressure as the independent variables and also to express the various partial differential coefficients in terms of experimental magnitudes. This may be illustrated as follows. 

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