Equations of balance

The possible number of valence stmctures for a given boron hydride has been defined exacdy using three general equations of balance. For a borane... 

Further checks, uhich can readily be verified from the equations of balance, are (a) the number of atoms in a neutral borane molecule = 2(s + I + y + x), and (b) there are as many framework electrons as (here are atoms in a neutral borane B H , since each BH group supplies 2 electrons and each of the m — n) "extra H atoms supplies I electron, making n + m in all. 

Three-center Bonding in Boranes Lipscomb s Equations of Balance... 

Consider the closo-BnHn2 (6 < n < 12) boranes (Figure 1-2). Such deltahedral boranes cannot have any terminal BH2 groups or three-center two-electron B-H-B bonds but acquire two extra electrons from the —2 charge on the ion. Therefore s = x = 0 in the equations of balance (la) and (lb) these reduce to (2a) and (2b) in which n is the number of boron atoms in the deltahedron corresponding to p in (la) and (lb) ... 

Such equations vaguely resemble Lipscomb s equations of balance for the polyhedral boranes [Eqs. (1) and (2)]. 

However, used qualitatively, localized bond schemes are less satisfactory for rationalizing the shapes of such relatively symmetrical species as the closo anions B H , for most of which it is not possible to find a single localized bond arrangement that appears appropriate for the symmetry. For example, the equations of balance for the anion require its... 

For a general formulation of the Zintl-Klemm concept, consider an intermetallic AmX phase, where A is the more electropositive element, t3 pically an alkali or an alkaline earth metal. Both A and X, viewed as individual atoms, are assumed to follow the octet rule leading to transfer of electrons from A to X, i.e., A AF, X —> X , so that mp = nq. The anionic unit X arising from this electron transfer is considered to be a pseudoatom, which exhibits a structural chemistry closely related to that of the isoelectronic elements [11]. Since bonding also is possible in the cationic units, the numbers of electrons involved in A-A and X-X bonds of various types (caa and exx> respectively) as well as the number of electrons e not involved in localized bonds can be generated from the numbers of valence electrons on A and X, namely and ex, respectively, by the following equations of balance ... 

From equation (14)i we could obtain the Darcy s law, if we neglect the inertial terms and the mass exchange and make suitable constitutive hypotheses on fields m,bf and Tf. The equation of balance (15)i for the volume fraction generalize the classical Langmuir s evolution equation, while the balance (15)2 for the microstretch Us includes the Wilmanski s porosity balance as well as the equation which rules the changes of internal surfaces area of the pores (see [8, 11, 1], respectively). The energy balance equations do not appear at all because the process is assumed to be isothermal. 

The fundamental theory of fluid mechanics is expressed in the mathematical language of continuum tensor field calculus. An exhaustive treatment of this subject is found in the treatise by Truesdell and Toupin (1960). Two fundamental classes of equations are required (1) the generic equations of balance and (2) the constitutive relations. 

The generic equations of balance are statements of truth, which is a priori self-evident and which must apply to all continuum materials regardless of their individual characteristics. Constitutive relations relate diffusive flux vectors to concentration gradients through phenomenological parameters called transport coefficients. They describe the detailed response characteristics of specific materials. There are seven generic principles (1) conservation of mass, (2) balance of linear momentum, (3) balance of ro-... 

The four balance or conservation principles can all be represented in terms of a general equation of balance written in integral form as... 

The equations of balance for open systems from Clraps. 2 and 5 are summarized here hr Table 7.1 for easy reference. Included are Eqs. (7.1) and (7.2), restricted forms of tire mass balance. These equations are the basis for the themrodynanricanalysis ofprocesses in tlris and the next two clrapters. When combined with themrodynanric property statements, they allow calculation of process rates and system states. 

General Equations of Balance Balance Equations for Steady-Flow Processes Balance Equations for Siirgle-SEeam Steady-Flow Processes... 

General equations of balance Balance equations for steady-flow processes Balance equations for single-stream steady-flow processes ... 

Summary of Equations of Balance for Open Systems Only the most general equations of mass, energy, and entropy balance appear in the preceding sections. In each case important applications require less general versions. The most common restrictedTcase is for steady flow processes, wherein the mass and thermodynamic properties of the fluid within the control volume are not time-dependent. A further simplification results when there is but one entrance and one exit to the control volume. In this event, m is the same for both streams, and the equations may be divided through by this rate to put them on the basis of a unit amount of fluid flowing through the control volume. Summarized in Table 4-3 are the basic equations of balance and their important restricted forms. 

Let us now consider balances of micromomentum from the expression (11) of the kinetic energy, we assume local equations of balance of micromomentum in accordance with Truesdell s second principles ... 

Our first experimental system consisted of a single-stage chemostat to which nutrient solution for the bacterial prey was fed at a constant rate and in which the interaction between the prey and the predator took place. The equation of balance for the two populations can be written as ... 

Our second experimental system consisted of two chemostats linked in series. In the first vessel the bacterial prey was allowed to come to steady state and then fed into the second stage vessel which contained the amoebae. As the limiting nutrient source for the bacteria is virtually exhausted under steady state conditions, it was assumed that no further growth of prey occurred in the second vessel. In the second vessel the equation of balance for the predator is ... 

---