Partial Derivatives with Respect to T, p, and

Since Cv is positive, we see from Eqs. 7.4.2 and 7.4.7 that heating a phase at constant volume causes both U and S to increase. 

We may derive relations for a temperature change at constant pressure by the same methods. From Cp = dH/dT)p (Eq. 1.3.2), we obtain 

From dS = dq/T and Eq. 7.3.2 we obtain for the entropy change at constant pressure 

The Gibbs energy changes according to dG/dT)p = —S (Eq. 5.4.11), so heating at constant pressure causes G to decrease. 

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CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES 7.5 PARTIAL Derivatives with Respect to T, p, and V... 

PARTIAL DERIVATIVES WITH RESPECT TO T, p, AND V 7.5.1 Tables of partial derivatives... 

We can solve the second integral on the right-hand side of Equation (5.36) with the appropriate PvT data or with data for the thermal expansion coefficient and the isothermal compressibility. For example, if an equation of state is available of the form P = f T, v), we can take the partial derivative with respect to T at constant v, multiply by T, subtract P, and integrate. If no equation is available, we could solve graphically. 

A. Partial Derivatives and Polarizability Coefficients Expansion of (8) yields a polynomial, the characteristic or secular polynomial, whose roots are determined by the values of the parameters , vw- The ground state energy (12) is likewise a function of the (a,j3) parameter values, as are all quantities such as AO coefficients in the MO s, charges q bond orders p t, etc. It is possible, therefore, to specify the h partial derivative with respect to any or at an arbitrary point defined by a set of values (a,j8) in the parameter space, and to make expansions such as... 

As composition variables we use molal solute concentrations m. The piolality nts of solute s is the ratio of the number of moles of s to the number of kilograms of solvent. For a single electrolyte with ions of species a and b the independent thermodynamic variables are T, P, nta, and mi,. In any partial derivative with respect to one of them, e.g., dG/dnia, all of the others are held fixed. 

Figure 2 Chemical potential and chemical potential partial derivative with respect to composition for a model mixture of two spherical molecules at conditions T, P where the two components are miscible (plots a and c), and at conditions T2

In experimental work it is usually most convenient to regard temperature and pressure as die independent variables, and for this reason the tenn partial molar quantity (denoted by a bar above the quantity) is always restricted to the derivative with respect to Uj holding T, p, and all the other n.j constant. (Thus iX = [right-hand side of equation (A2.1.44) it is apparent that the chemical potential... 

The frozen-core (fc) approach is not restricted to spin-independent electronic interactions the spin-orbit (SO) interaction between core and valence electrons can be expressed by a sum of Coulomb- and exchange-type operators. The matrix element formulas can be derived in a similar way as the Sla-ter-Condon rules.27 Here, it is not important whether the Breit-Pauli spin-orbit operators or their no-pair analogs are employed as these are structurally equivalent. Differences with respect to the Slater-Condon rules occur due to the symmetry properties of the angular momentum operators and because of the presence of the spin-other-orbit interaction. It is easily shown by partial integration that the linear momentum operator p is antisymmetric with respect to orbital exchange, and the same applies to t = r x p. Therefore, spin-orbit... 

The Elementary Partial Derivatives.—We can set up a number of familiar partial derivatives and thermodynamic formulas, from the information which we already have. We have five variables, of which any two are independent, the rest dependent. We can then set up the partial derivative of any dependent variable with respect to any independent variable, keeping the other independent variable constant. A notation is necessary showing in each case what are the two independent variables. This is a need not ordinarily appreciated in mathematical treatments of partial differentiation, for there the independent variables are usually determined in advance and described in words, so that there is no ambiguity about them. Thus, a notation, peculiar to thermodynamics, has been adopted. In any partial derivative, it is obvious that the quantity being differentiated is one of the dependent variables, and the quantity with respect to which it is differentiated is one of the independent variables. It is only necessary to specify the other independent variable, the one which is held constant in the differentiation, and the convention is to indicate this by a subscript. Thus (dS/dT)P, which is ordinarily read as the partial of S with respect to T at constant P, is the derivative of S in which pressure and temperature are independent variables. This derivative would mean an entirely different thing from the derivative of S with respect to T at constant V, for instance. 

We now proceed to find the partial derivatives of K T, p) with respect to T and p. From the definition (7.27) we have immediately... 

The chemical potentials are the key partial molar quantities. The pi s determine reaction and phase equilibrium. Moreover, all other partial molar properties and all thermodynamic properties of the solution can be found from the pi s if we know the chemical potentials as functions of T, P, and composition. For example, the partial derivatives of p with respect to T... 

For example, consider the series MSS of two components with m = 3. The structure function of this system is define as system components by (6) need to compute Direct Partial Logic Derivatives of the structure function with respect to variable X and X2. These derivates are in Table 1 (derivates of this structure function that have only zero values in this Table aren t shown). 

We may derive the general expressions as follows. We are considering differentiation with respect only to T, p, and V. Expressions for dV/dT)p, dV/dp)r, and dp/dT)v come from Eqs. 7.1.1, 7.1.2, and 7.1.7 and are shown as functions of a and Kj. The reciprocal of each of these three expressions provides the expression for another partial derivative from the general relation... 

In a phase containing species i, either pure or in a mixture, the partial derivative of in / T with respect to T at constant p and a fixed amount of each species is given by ... 

Solution First, note that the equation of state. Equation 2.33, does not involve S. So direct differentiation of the equation of state does not yield the desired first derivative. Instead, we need an equivalent derivative, and the thermodynamic compass is an aid in finding it. S, P, and T are arranged clockwise on the compass, and counterclockwise from the same starting point are V, T, and P. Thus, the partial derivative of V with respect to T when P is constant is -1 times the derivative we seek. To find this partial derivative, divide the equation of state by V ... 

Of particular importance are the partial derivatives of G with respect to T and P ... 

The partial derivative of G with respect to the mole number n, at constant T and P and mole numbers rij m is defined as... 

We have defined the chemical potential of a component as the partial derivative of the Gibbs free energy of the system (or, for a homogeneous system, of the phase) with respect to the number of moles of the component at constant P and T—i.e.,... 

Novozhilov (Ref 9) noted other instances where the instability criterion of Zel dovich is not satisfied. He also noted ZeTdovich s assertion that the form of the stability criterion may change if the variation in the surface temperature and the inertia of the reaction layer of the condensed phase are taken into account, and stability criteria obtained under the assumption that the chemical reaction zone in the condensed phase and all of the processes in the gas phase are without inertia. Novozhilov used a more general consideration of the problem to show that the stability region is determined by only two parameters Zel dovich s k and the partial derivative r of the surface temperature with respect to the initial temperature at constant pressure t=(dTi/(fro)p. Combustion is always stable if k 1, combustion is stable only when r >(k — l) /(k +1)... 

The first relation follows directly by taking the partial derivative of G = H — TS with respect to nt (holding P, T, and the other n- s constant). The latter two relations can be proved by reversing the order of differentiating with respect to , and with T and P respectively. 

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