Geoscience Reference
In-Depth Information
Clay
1
2
3
Quartz
C
Si
Figure 2.1
Graphic rules for conservation of mass and for mixing (arbitrary units). A mineral or rock
composition can be represented by a vector in the chemical concentration space (here Al and Si).
Points 1, 2, and 3 represent three compositions of mixtures of clay and quartz. Notice the linear
character of the mixing relationships.
Equation 2.5
is known as the closure condition and indicates that the list of constituents
is exhaustive.
Equation 2.6
indicates that the concentration of the system as a whole is
the weighted mean concentration of the composition of its components. The fraction by
mass of each component in the mixture is the “weight” by which the concentration of each
component must be multiplied to represent its contribution to the total inventory of the
element in the entire system. An example of this principle, taken from everyday life, is the
simple concept of rock. A rock is a sample, typically a piece of boulder or cliff, which has
been just knocked off by the geologist's hammer. There is no such ideal thing as a rock
in nature: it is a man-made sample, an artefact, whose composition results from a chance
combination of all crystals collected by a particular hammer blow.
Vectors provide a useful means of describing the relationships of conservation. A min-
eral, a solution, or a rock in which the concentrations of
n
elements (
i
,
n
)have
been measured are represented by a point (or a vector) in an
n
-dimensional space. In the
example above, a vector equation could be written formally in the two-dimensional space
of Si and Al concentrations:
C
Si
sed
C
Al
sed
=
1,
...
C
Si
clay
C
Al
clay
C
Si
qz
C
Al
qz
=
f
clay
+
f
qz
(2.7)
supplemented, of course, by the closure condition
f
clay
+
f
qz
=
1.
Equation 2.7
shows
the sediment vector as a linear combination of the vectors representing the components.
If the sediment is composed of three components (let us add, for the sake of illustration,
a carbonate component), then a vector formed by the concentrations of three elements,
such as Si, Al, and Ca, in the rock will lie within the triangle formed by the equivalent
vectors representing the clay, quartz, and carbonate components. Generally, when trying to
interpret the chemical composition of a mixture in terms of constituents that are sometimes
too numerous to be shown graphically, statistical methods are used, the simplest and most
effective of which is principal component analysis (PCA).
