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Section A.1 Geology: Phases and Components
Definition A.1.1.
In geology, a phase is any physically separable material in the system, such as various minerals or liquids.
A component is a chemical compound necessary to make up the phases; these are usually oxides such as Calcium Oxide (\({\rm CaO}\)) or Silicone Dioxide (\({\rm SiO_2}\)).
In a typical application, a geologist knows how to build each phase from the components, and is interested in determining reactions among the different phases.
Activity A.1.3.
To study this vector space, each of the three components \(\vec c_1,\vec c_2,\vec c_3\) may be considered as the three components of a Euclidean vector.
\begin{equation*}
\vec{p}_1 = \left[\begin{array}{c} 3 \\ 1 \\ 2 \end{array}\right],
\vec{p}_2 = \left[\begin{array}{c} 1 \\ 1 \\ 1 \end{array}\right],
\vec{p}_3 = \left[\begin{array}{c} 1 \\ 0 \\ 1 \end{array}\right],
\vec{p}_4 = \left[\begin{array}{c} 1 \\ 1 \\ 2 \end{array}\right],
\vec{p}_5 = \left[\begin{array}{c} 2 \\ 1 \\ 2 \end{array}\right].
\end{equation*}
Determine if the set of phases is linearly dependent or linearly independent.
Activity A.1.4.
Geologists are interested in knowing all the possible chemical reactions among the 5 phases:
\begin{equation*}
\vec{p}_1 = \mathrm{Ca_3MgSi_2O_8} = \left[\begin{array}{c} 3 \\ 1 \\ 2 \end{array}\right] \hspace{1em}
\vec{p}_2 = \mathrm{CaMgSiO_4} = \left[\begin{array}{c} 1 \\ 1 \\ 1 \end{array}\right] \hspace{1em}
\vec{p}_3 = \mathrm{CaSiO_3} = \left[\begin{array}{c} 1 \\ 0 \\ 1 \end{array}\right]
\end{equation*}
\begin{equation*}
\vec{p}_4 = \mathrm{CaMgSi_2O_6} = \left[\begin{array}{c} 1 \\ 1 \\ 2 \end{array}\right] \hspace{1em}
\vec{p}_5 = \mathrm{Ca_2MgSi_2O_7} = \left[\begin{array}{c} 2 \\ 1 \\ 2 \end{array}\right].
\end{equation*}
That is, they want to find numbers \(x_1,x_2,x_3,x_4,x_5\) such that
\begin{equation*}
x_1\vec{p}_1+x_2\vec{p}_2+x_3\vec{p}_3+x_4\vec{p}_4+x_5\vec{p}_5 = 0.
\end{equation*}
(a)
Set up a system of equations equivalent to this vector equation.
(b)
Find a basis for its solution space.
(c)
Interpret each basis vector as a vector equation and a chemical equation.
Activity A.1.5.
We found two basis vectors \(\left[\begin{array}{c} 1 \\ -2 \\ -2 \\ 1 \\ 0 \end{array}\right]\) and \(\left[\begin{array}{c} 0 \\ -1 \\ -1 \\ 0 \\ 1 \end{array}\right]\text{,}\) corresponding to the vector and chemical equations
\begin{align*}
2\vec{p}_2 + 2 \vec{p}_3 &= \vec{p}_1 + \vec{p}_4 & 2{\rm CaMgSiO_4}+2{\rm CaSiO_3}&={\rm Ca_3MgSi_2O_8}+{\rm CaMgSi_2O_6}\\
\vec{p}_2 +\vec{p}_3 &= \vec{p}_5 & {\rm CaMgSiO_4} + {\rm CaSiO_3} &= {\rm Ca_2MgSi_2O_7}
\end{align*}
Combine the basis vectors to produce a chemical equation among the five phases that does not involve \(\vec{p}_2 = {\rm CaMgSiO_4}\text{.}\)