EEE460-Handout K.E. Holbert

ATOMIC NUMBER DENSITY Number of Atoms (n) and Number Density (N) The number of atoms or molecules (n) in a mass (m) of a pure material having atomic or molecular weight (M) is easily computed from the following equation using Avogadro's number (NAv = 6.022×1023 atoms or molecules per gram-mole):

MNm

n Av= (1)

In some situations, the atomic number density (N), which is the concentration of atoms or molecules per unit volume (V), is an easier quantity to find when the density (ρ) is given

MN

VnN Avρ

== (2)

Number Density for Compounds For a chemical compound (mixture) Z, which is composed of elements X and Y, the number (atom) density of the compound is calculated from

mix

AvmixmixZ M

NNN

ρ== (3)

In some cases, the desired quantity is the number density of the compound constituents. Specifically, if qp YXZ = , then there are p atoms of X and q atoms of Y for every molecule of Z; hence

ZY

ZX

NqNNpN

==

(4)

Example: Calculate the number density of natural uranium in UO2 with 3g/cm5.102

=UOρ .

322233

UO

UOUOU atoms/cm1034.2

g/mole)]9994.15(20289.238[)atoms/mole10022.6)(g/cm5.10(

2

22

×=+

×===

MN

NN Avρ

Number Density Given Atom Fraction (Abundance) Oftentimes, it is necessary to compute the concentration of an individual isotope j given its fractional presence (abundance) γj in the element

elementtheofatomsofnumberTotal

jisotopeofatomsofNumberj =γ (5)

Many times, the fraction γj is stated as an atom percent, which is abbreviated a/o. The atomic number density of isotope j is then

elem

Avelemjelemjj M

NNN

ργγ == (6)

If the element has a non-natural abundance of its isotopes (that is, the elemental material is either enriched or depleted), then it is necessary to compute the atomic weight of the element (Melem) from the sum of all the atomic weights of the isotopes (Mj) rather than use the tabulated Melem value found in a reference

∑= jjelem MM γ (7)

EEE460-Handout K.E. Holbert

Example: Find the U-235 concentration for 3 a/o in UO2. Solution: To solve this example, Equations 4, 3 and 7 are progressively substituted into Eq. 6.

320

235233

O235235238238

UO235U

UO

UO235U

UO235-UU235U235U

atoms/cm1003.7

UatomsUatoms03.0

g/mole)]16(2)03.0)(235()97.0)(238[()atoms/mole10022.6)(g/cm5.10(

2

2

2

2

2

×=

−−

++×=

++==

==

−−

−−

MMMN

MN

NNN

AvAv

γγργρ

γ

γγ

Number Density Given Weight Fraction (Enrichment) Other times, when working with nuclear fuels such as uranium, the enrichment may be specified in terms of weight percent or weight fraction, ωi, of isotope i:

elementtheofmassTotal

iisotopeofMassi =ω (8)

The atomic number density of isotope i is

i

Avelemi

i

Avii M

NMN

Nρωρ

== (9)

Clearly, if the material is enriched, then the atomic weight of the material differs from its natural reference value, and the enriched atomic weight, if needed, should be computed from

∑=i i

i

elem MMω1 (10)

Example: Find the U-235 concentration for 4% enriched UO2. Solution: First compute the molecular weight of the enriched uranium, which is basically 4% U-235 and 96% U-238 since the U-234 component is negligible.

moleg/g9.237

23896.0

23504.01

U

238U

238U

235U

235U

U

⋅=

+=+=−

−

−

−

MMMMωω

Next, use Equation 9 and the fact that 2

2UO

UUOU M

Mρρ =

3202

235

32

23235

UO

UUO

235U

235U

235U

U235U235U

atoms/cm1049.9

UOg)]16(29.237[Ug9.237

/moleUg235)/cmUOg5.10)(atoms/mole10022.6(

UgUg04.0

22

×=

⋅+⋅

⋅⋅×

⋅⋅=

==−

−

−

−− M

MM

NM

NN AvAv ρωρω