Units and Equations

Although Eqs. (1.1) to (1.4) are sufficient for the description of unit systems, they are but a small fraction of the equations needed in this book. Many such equations contain terms that represent properties of substances, and these are introduced as needed. All new quantities are measured in combinations of units already defined, and all are expressible as functions of the five base units for mass, length, time, temperature, and mole. 

GENERAL EQUATIONS. Except for the appearance of the proportionality factors Qc and J, the equations for all three unit systems are alike. In the SI system, neither constant appears in the cgs system, is omitted and J retained in the fps system both constants appear. In this text, to obtain equations in a general form for all systems, Qc and J are included for use with fps units then either or both and J may be equated to unity when the equations are used in the cgs or SI systems. 

DIMENSIONLESS EQUATIONS AND CONSISTENT UNITS. Equations derived directly from the basic laws of the physical sciences consist of terms that either have the same units or can be written in the same units by using the definitions of derived quantities to express complex units in terms of the five base ones. 

A dimensionally homogeneous equation can be used as it stands with any set of units provided that the same units for the five base units are used throughout. Units meeting this requirement are called consistent units. No conversion factors are needed when consistent units are used. 

For example, consider the usual equation for the vertical distance Z traversed by a freely falling body during time t when the initial velocity is Ug  

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Properties of aqueous hydrofluoric acid. Top row indicates unit and equation or reference used for calculation. Bold values are certified by experimental data. 

Other systems of units and equations in common use in electromagnetic theory, in addition to the SI, are the esu system, the emu system, the Gaussian system, and the system of atomic units. The conversion from SI to these other systems may be understood in the following steps. 

SI units and equations wBll be used throughout Most of the original work, however, was done in the c.g.s.-e.s.u. system. Table 1 lists the SI units for the electromagnetic properties referred to in this artide, as well as the conversion factors from the SI to the probably more-familiar e.s.u. stem. To convert all equations to their c.g.s.-e.s.u. equivalents replace Co> the permittivity of free space (8.8S x 10 C m by (47t)-. ... 

The action of water here involves neither oxidation nor reduction as far as can be told and is a simple hydrolysis. On the basis of the theoretical developments, the simplest view to take is that equation (6) part 1 involves a reduction of one carbon atom two units, and equation (7) part 1 an oxidation of one carbon atom two units. The hydrolysis reactions indicate this difference also. As pointed out at the end of Chapter IX, these explanations represent the facts most nearly at the present time, but the possibility exists that the oxidation-reduction changes are actually brought about by the reagents added in the attempt to determine the structures, water in the present instance. The possibility exists and must be borne in mind even though the evidence at present available is against this explanation and in favor of the one given in greater detail. 

To first order, the dispersion (a-a) interaction is independent of the structure in a condensed medium and should be approximately pairwise additive. Qualitatively, this is because the dispersion interaction results from a small perturbation of electronic motions so that many such perturbations can add without serious mutual interaction. Because of this simplification and its ubiquity in colloid and surface science, dispersion forces have received the most significant attention in the past half-century. The way dispersion forces lead to long-range interactions is discussed in Section VI-3 below. Before we present this discussion, it is useful to recast the key equations in cgs/esu units and SI units in Tables VI-2 and VI-3. 

Equation (9.44) treats the free-draining molecule as an assembly of independent hydrodynamic units and shows that in this limit [r ] (nf/r o)(rJ/n). 

Here i —> i is the convex and continuous function describing a plasticity yield condition, the dot denotes a derivative with respect to t, n = (ni,ri2) is the unit normal vector to the boundary F. The function v describes a vertical velocity of the plate, rriij are bending moments, (5.175) is the equilibrium equation, and equations (5.176) give a decomposition of the curvature velocities —Vij as a sum of elastic and plastic parts aijkiirikiy Vijy respectively. Let aijki x) = ajiki x) = akuj x), i,j,k,l = 1,2, and there exist two positive constants ci,C2 such that for all m = rriij ... 

Gj /k aPh.3.s the dimension of length or height and is thus designated the gas-phase height of one transfer unit, The integral is dimensionless and indicates how many of these transfer units it takes to make up the whole tower. Consequently, it is called the number of gas-phase transfer units, N. Equation 40 may therefore be written as... 

Hq and aie called the overall gas-phase height of a transfer unit and the number of overall gas-phase transfer units, respectively. In the case of a straight equiUbrium line, is often neady concentration-independent as explained earher. In such cases, use of equation 47 is especially convenient... 

The basic form of the equation is normally modified so that the differential is expressed in pressure units and the flow coefficient is divided into the product of an experimentally deterrnined discharge coefficient, iC, and a series of calculated coefficients. In this form, for concentric restrictions ... 

Where T)is flame temperature in K MC is moisture content of the waste, expressed on a total weight basis SR is defined as stoichiometric ratio or moles O2 avadable/moles O2 required for complete oxidation of the carbon, hydrogen, and sulfur in the fuel, ie, 1/SR = equivalence ratio and is temperature of the combustion air, expressed in K. In Fnglish units, this equation is as follows ... 

The thermal glass-transition temperatures of poly(vinyl acetal)s can be determined by dynamic mechanical analysis, differential scanning calorimetry, and nmr techniques (31). The thermal glass-transition temperature of poly(vinyl acetal) resins prepared from aliphatic aldehydes can be estimated from empirical relationships such as equation 1 where OH and OAc are the weight percent of vinyl alcohol and vinyl acetate units and C is the number of carbons in the chain derived from the aldehyde. The symbols with subscripts are the corresponding values for a standard (s) resin with known parameters (32). The formula accurately predicts that resin T increases as vinyl alcohol content increases, and decreases as vinyl acetate content and aldehyde carbon chain length increases. 

The essential differences between sequential-modular and equation-oriented simulators are ia the stmcture of the computer programs (5) and ia the computer time that is required ia getting the solution to a problem. In sequential-modular simulators, at the top level, the executive program accepts iaput data, determines the dow-sheet topology, and derives and controls the calculation sequence for the unit operations ia the dow sheet. The executive then passes control to the unit operations level for the execution of each module. Here, specialized procedures for the unit operations Hbrary calculate mass and energy balances for a particular unit. FiaaHy, the executive and the unit operations level make frequent calls to the physical properties Hbrary level for the routine tasks, enthalpy calculations, and calculations of phase equiHbria and other stream properties. The bottom layer is usually transparent to the user, although it may take 60 to 80% of the calculation efforts. 

For consistency, clearance here is expressed in cm /s although the more common clinical units, and those used later in this chapter, are ml,/min. Combination and rearrangement of equations 6—8 allows clearance to be estimated from mass-transfer coefficient and vice versa the conditions of countercurrent flow with no dialysate recycling are shown below. 

The dimensional equations are usually expansions of the dimensionless expressions in which the terms are in more convenient units and in which all numerical factors are grouped together into a single numerical constant. In some instances, the combined physical properties are represented as a linear function of temperature, and the dimension equation resolves into an equation containing only one or two variables. 

Where ij. is defined as the kinematic viscosity (centistokes), and is a constant with a value of 2,213.8 in USCS units and 353.68 in SI units. An empirical relation for the Fanning friction factor is the Colebrook-White equation ... 

United States (equation 5.1-4), where /, is the site intensity in MM units and / is the ep ntral intensity in MM units. The site intensity is converted to the instmmental peak ground acceleration using an equation like equation 5.1-5. 

This discussion of sources of curvature in Br insted-type plots should suggest caution in the interpretation of observed curvature. There is a related matter, concerning particularly item 5 in this list, namely, the effect of a change in transition state structure. Br nsted-type plots are sometimes linear over quite remarkable ranges, of the order 10 pK units, and this linearity has evoked interest because it seems to be incompatible with Marcus theory, which we reviewed in Section 5.3. The Marcus equation (Eq. 5-69) for the plot of log k against log K of the same reaction series requires curvature, the slope of the plot being the coefficient a. given by Eq. (5-67). A Brjinsted plot, however, is not a Marcus plot, because it correlates rates and equilibria of different reactions. The slope p of a Br nsted plot is defined p = d log kobs/d pK, which we can expand as... 

The rate constants (in absolute solvents unless otherwise specified) are measured at a temperature giving a convenient reaction rate and calculated for a reference temperature used for comparison. These constants have all been converted to the same units and tabulated as 10 A . Where comparisons could otherwise not be made, pseudo-unimolecular constants (Tables IX and XIII, and as footnoted in Tables X to XIV) are used. The reader is referred to the original articles for the specific limits of error and the rate equations used in the calculations. The usual limits of error were for k, 1-2% or or 2-5% and logio A, 5%, with errors up to double these figures for some of the high-temperature reactions. 

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