18 lntroductlon

carbon-12 atom is a fixed quantity, changing the unit of mass on the reference mass of carbon-12 will change the numbcr of atoms that are contained in onemole of carbon-12. The number of atoms in one gram mole of carbon-12 is known as Avogadro's number, A, and its most accurately known value3 is 6.0220943 x 1023

() 6.3 x 1017

Since

12_g carb( Jl-12/gmol = A atorns/gmol

Mass of one carhon- 12 atom

the number of elementary entities in 1 kgmol will be equal to 103A, in one gold brick mole will be equal to l000/3A, ami so on. Fortunately, the cakulations of the mole-mass equivalences use the rclative atomic/molecular weights and do not explicitly require Avogadro's numher or its multiples. Thus, knowledgc of its pre cise numerical value is not essential for most calculations.

The key points to rctain from the preceding discussion are the following. First, although the standard mole is the gram mole, mole units can be defincd in terms of any desired unit of mass. Second, the mole-to-mass conversion factor for any compound in any mass unit will always be numerically equal to the molecular weight of the compound. For instancc, for water thcre are 18 g per gmol, 18 kg/ kgmol, 18 gold bricks/gold-brick mole, and so forth. For this reason, thc molecular weight is oftcn dircctly written as thc mas:-;-mole convcrsion factor wilh an assigned set of units, although strictly :.;peaking it is dimensionless. We employ this loose practice in subsequent chapters because of its widespread usagc in the field.

1.3.3Equation-Solvlng Concepts

The application of the conservation laws to steady-statc systemsultimately requires the solution of sets of balance cquations for the values of unknown process flow rates. In general, the equations will be algebraic, may be linear or nonlinear, and their number muy be quite largc. Since thc solution of such cquation scts can be a formidable task, it is appropriate to devote attention to the study of efficient solution strategies and methods, as is done in subsequent chapters. For the present , we briefly recall some elementary equation-solving concepts which ought to be familiar to the reader.

Linear Equations A linear equation of the variables x 1 through xN is a function of the form

where the coefficients a; and the right-hand side b are known constants. Each term in the function contains a single variable, and each variable occurs to thc first power. If the linear equation contains but a single unknown, say, x 1, then the

unique solution to the quatio11; can be obt ined by mere division, that is, x1 = b/

3 Anon., "Metrology: A More Accurat Value of Avogadro's Number," Sciet ce,183, lll37-1038 (Sept.

20, 1974).

1 .

'

Review of Baslc Concepts19

1

a1 1f the linear equation contains more than one variable, then it will ave an infinite number of solutions since we can always set all variables but one to arbitrary values and solve the resulting single-variable equation for the remaining un nown.

The process of s2tting N - 1 variables to fixed values is equivalent (o aug- !

menting the multivariable linear equation with N - 1 trivial equations of th'e form X; = e;. This is but a special instance of the well-known fact that a system olinear equations has a unique solution if and only if the system contains as many inde

pendent equations as there are unknowns. A set of linear equations is independent if and only if no one equation in the system can be obtained by adding tdgether multiples of any of the remaining equations. 1

i

Example 1.1OThe system of equations \

X1 + X2 = 11

Ox1 + lx2 = 2: \

i

is independent beca use neither equation can be expressed as a multiple of the pther.

The system of equations 1

x1 + Ox2 = 1l

!

Ox1 + lx2 = 2l

lx1 + lx2 = 3

is dependent because the third is equal to the sum of the firs.t two.

\

:

' 1

(\)\ \ .

The solution of systems of linear equations can be accomplished in two ays:

by Cramer's rule or by variable elimination. \

Cramer's Rule Cramer's rule is a classical construction which expresses te so

lution of a system of linear equations in terms of ratios of determinants of the

aay of coefficients of the cquations. In the case of a system of two equatiqns in

two unknowns, \

a 11x 1 + a12x2 = b 1\

'

The solution wil1 be given by

det 1 b1

21

b2"22

b1a22hzllz

X= ---- =

detlll

a21

at2at,a22al2az,

llz2

b1

(= -----)b2 b2a11 - b,a2t

Xz =

llz llnll22 - llzll21

ll z z

20 lnlroductlon

The form of the solution cxtends in an obvious way to larger:systems. In each case, the denominator of the ratios will consist of the determinant of the array of

variable coefficients, while the numerator for the ith variable will consist of the

determinant of the array formed when the ith column of the array of variable

coefficients is replaced by a column of right-hand side O, the new esti ate z is used to replace the point with positive function value. The calculations continue

in this fashion until either f(z) becomes sufficiently close to zero or the dif(erence between the points xL and xR becomes sufficiently small. Observe that si ce the

interval xL x x 1within which the solution of f(x) must Iie is reduced .bY one

half with each tria! point, the size of the interval after N tria) points will be: (XR -

x 1-)f2N, Where XR and XL are the OtiaJ estimateS Wth Which the bisectOO method

is started. As a result, the desired accuracy of the estmate of the solution:x* can

also be controlled by' choosing N appropriately. The complete interval halving ;

(=)algorithm is summarized below:

1

Given N, ::1, and the bounds xn and x 1 such that f(xL) f(xR)