Reaction quantity molar differential

In dealing with the change of an extensive property X as changes, we must distinguish between molar integral and molar differential reaction quantities. 

AfX is a molar differential reaction quantity. Equation 11.2.16 shows that A X is the rate at which the extensive property X changes with the advancement in a closed system at constant T and p. The value of A X is in general a function of the independent variables T, p, and... 

The notation for a molar differential reaction quantity such as includes a subscript following the A s5unbol to indicate the kind of chemical process. The subscript r denotes a reaction or process in general. The meanings of vap, sub fus, and trs were described in Sec. 8.3.1. Subscripts for specific kinds of reactions and processes are listed in Sec. D.2 of Appendix D and are illustrated in sections to follow. 

This relation is analogous to Eq. 11.3.9, using molar differential reaction quantities in place... 

Applying the general definition of a molar differential reaction quantity (Eq. 11.2.15) to the Gibbs energy, we obtain the definition of the molar reaction Gibbs energy or molar Gibbs... 

We recognize the partial derivative on the right side of each of these relations as a molar differential reaction quantity ... 

Since the value of a standard molar reaction quantity is independent of the standard molar integral and differential quantities are identical (page 317) ... 

Unlike Qp, Na is not a conserved quantity and varies down the length of the tube. Consider a differential element of length and volume h.zAc. The molar flow entering the element is IV (z) and that leaving the element is lV (z+Az), the difference being due to reaction within the volume element. A balance on component A gives... 

The differential reactor uses a thin catalyst bed in which only small changes of concentration and temperature occur (Fig. 3.3-3). The rate of reaction, r, can be obtained from the difference in concentration, Ac, over the catalyst bed or its thickness, Ax, the volumetric throughput, v, or the molar throughput, nges, and the quantity of catalyst, WK, using the material balance ... 

Eqs. 1 to 3 relate the rate of production Rj of the balanced reaction component y to the molar amounts or their derivatives with respect to the time variable (reaction time or space time, see above). From the algebraic eq. 2 for the CSTR reactor the rate of production, Rj, may be calculated very simply by introducing the molar flow rates at the inlet and outlet of the reactor these quantities are easily derived from the known flow rate and the analytically determined composition of the reaction mixture. With a plug-flow or with a batch reactor we either have to limit the changes of conversion X or mole amount n7 to very low values so that the derivatives or dAy/d( //y,0) or dn7/d/ could be approximated by differences AXj/ (Q/Fj,0) or An7/A, (differential mode of operation), or to measure experimentally the dependence of Xj or nj on the space or reaction time in a broader region this dependence is then differentiated graphically or numerically. 

At relatively low pressures, what dimensionless differential equations must be solved to generate basic information for the effectiveness factor vs. the intrapellet Damkohler number when an isothermal irreversible chemical reaction occurs within the internal pores of flat slab catalysts. Single-site adsorption is reasonable for each component, and dual-site reaction on the catalytic surface is the rate-limiting step for A -h B C -h D. Use the molar density of reactant A near the external surface of the catalytic particles as a characteristic quantity to make all of the molar densities dimensionless. Be sure to define the intrapellet Damkohler number. Include all the boundary conditions required to obtain a unique solution to these ordinary differential equations. 

This relation can be used to define various other isochoric heat quantities such as integral and differential, molar and specific heats of reaction and the corresponding heat capacities. The most well known of these quantities is the (global or integral) heat capacity at constant volume or isochoric heat capacity, which we got to know briefly in Sect. 9.1 ... 

As we have seen in the case above of internal energy [Eq. (24.12)], this relation can be useful for defining various isobaric heat quantities such as integral and differential, molar and specific heats of reaction, transition, solution, mixing, etc. These are all produced similarly at constant p and T and, depending upon the process in question, each one can have various symbols and names. We will be content with only two examples, one integral quantity and one differential quantity ... 

We must emphasize two points. The first one has already been mentioned. The molar reaction Gibbs function change is endowed with an instantaneous value. It depends on the reaction extent and, of course, on the initial reactants and products concentrations. As a result, the potential difference E is also instantaneous. This is the reason why the above reasoning involved differential quantities. 

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