Page 49 - Materials Science and Engineering An Introduction

2.2 Fundamental Concepts • 21

all atoms of a given element, the number of neutrons (N) may be variable. Thus atoms of
isotope some elements have two or more different atomic masses, which are called isotopes. The
atomic weight of an element corresponds to the weighted average of the atomic masses
atomic weight 3
of the atom’s naturally occurring isotopes. The atomic mass unit (amu) may be used to
1
atomic mass unit compute atomic weight. A scale has been established whereby 1 amu is defined as of
12
12
(amu) the atomic mass of the most common isotope of carbon, carbon 12 ( C) (A 12.00000).
Within this scheme, the masses of protons and neutrons are slightly greater than unity, and
A Z + N (2.1)
The atomic weight of an element or the molecular weight of a compound may be speci-
mole fied on the basis of amu per atom (molecule) or mass per mole of material. In one mole
23
of a substance, there are 6.022 10 (Avogadro’s number) atoms or molecules. These
two atomic weight schemes are related through the following equation:
1 amu/atom (or molecule) = 1 g/mol
For example, the atomic weight of iron is 55.85 amu/atom, or 55.85 g/mol. Sometimes
use of amu per atom or molecule is convenient; on other occasions, grams (or kilograms)
per mole is preferred. The latter is used in this book.

EXAMPLE PROBLEM 2.1
Average Atomic Weight Computation for Cerium

Cerium has four naturally occurring isotopes: 0.185% of 136 Ce, with an atomic weight of
135.907 amu; 0.251% of 138 Ce, with an atomic weight of 137.906 amu; 88.450% of 140 Ce, with
142
an atomic weight of 139.905 amu; and 11.114% of Ce, with an atomic weight of 141.909 amu.
Calculate the average atomic weight of Ce.

Solution
The average atomic weight of a hypothetical element M, A M , is computed by adding fraction-
of-occurrence—atomic weight products for all its isotopes; that is,
A M = (2.2)
a f i M A i M
i
is the fraction-of-occurrence of isotope i for element M (i.e., the percentage-
In this expression, f i M
is the atomic weight of the isotope.
of-occurrence divided by 100), and A i M
For cerium, Equation 2.2 takes the form
A Ce = f 136 A 136 Ce + f 138 A 138 Ce + f 140 A 140 Ce + f 142 A 142 Ce
Ce
Ce
Ce
Ce
Incorporating values provided in the problem statement for the several parameters leads to
0.185% 0.251% 88.450%
A Ce = a b(135.907 amu) + a b(137.906 amu) + a b(139.905 amu)
100 100 100
11.114%
+ a b(141.909 amu)
100
= (0.00185)(135.907 amu) + (0.00251)(137.906 amu) + (0.8845)(139.905 amu)
+ (0.11114)(141.909 amu)
= 140.115 amu

3 The term atomic mass is really more accurate than atomic weight inasmuch as, in this context, we are dealing with
masses and not weights. However, atomic weight is, by convention, the preferred terminology and is used throughout
this book. The reader should note that it is not necessary to divide molecular weight by the gravitational constant.