Environmental Modeling - Mathematics and Statistics for the Environment - page 81

Because moles of material = mass/molecular weight (MW), mass concentrations ( p i , w ) are related
by the following:

p

iw
i

,

C = (3.6)

iw ,

MW

For molarity M , [ X ] is the molar concentration of X .
The mole fraction ( X ) of a single chemical in water can be expressed as follows:

Molesofcomponent/chemical

Mole fraction X = (3.7)

Totalmoles of solution (moles of chemical + molesofwater)

For dilute solutions, the moles of chemical in the denominator of the above equation can be
ignored in comparison to the moles of water ( n w ) and can be approximated by

X = Molesofchemical
Molesofwater

(3.8)

If X is less than 0.02, an aqueous solution can be considered dilute. On a mass basis, similar expres-
sions can be formulated to yield mass fractions. Mass fractions can also be expressed as a percent-
age or as other ratios such as parts per million (ppm) or parts per billion (ppb).
The mole fraction of a component in a solution is simply the number of moles of that component
divided by the total moles of all of the components. We use the mole fraction because the sum of
the individual fractions should equal 1. This constraint can reduce the number of variables when
modeling mixtures of chemicals. Mole fractions are strictly additive. The sum of the mole frac-
tions of all components is equal to 1. Mole fraction X i of component i in an n -component mixture
is defined as follows:

Molesof i

X = (3.9)

i

n

+

∑

nn

i w

1

Thesum of allmole fractions =

n

=

∑ x w

1 (3.10)

1

For dilute solutions of multiple chemicals (as in the case of single-chemical systems), mole frac-
tion X i of component i in an n -component mixture can be approximated by the following:

= Molesof

i

X (3.11)

n w

Note that the preceding ratio is known as an intensive property because it is independent of the
system and the mass of the sample. An intensive property is any property that can exist at a point in
space. Temperature, pressure, and density are good examples. On the other hand, an extensive prop-
erty is any property that depends on the size (or extent) of the system under consideration. Volume
is an example. If we double the length of all edges of a solid cube, the volume increases by a factor
of eight. Mass is another. The same cube will undergo an eightfold increase mass when the length
of the edges is doubled.