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Linear Algebra for Team-Based Inquiry Learning
2022 Edition
| Steven Clontz |
Drew Lewis |
| University of South Alabama |
University of South Alabama |
|
|
September 14, 2022
Section A.3: Geology: Phases and Components
Definition A.3.1
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 Silicon 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.
Observation A.3.2
Consider the 3 components
\begin{equation*}
\vec{c}_1={\rm CaO} \hspace{1em} \vec{c}_2={\rm MgO} \hspace{1em} \text{and } \vec{c}_3={\rm SiO_2}
\end{equation*}
and the 5 phases:
\begin{align*}
\vec{p}_1 &= {\rm Ca_3MgSi_2O_8} & \vec{p}_2 &= {\rm CaMgSiO_4} & \vec{p}_3 &= {\rm CaSiO_3}\\
\vec{p}_4 &= {\rm CaMgSi_2O_6} & \vec{p}_5 &= {\rm Ca_2MgSi_2O_7}
\end{align*}
Geologists already know (or can easily deduce) that
\begin{align*}
\vec{p}_1 &= 3\vec{c}_1 + \vec{c}_2 + 2 \vec{c}_3 & \vec{p}_2 &= \vec{c}_1 +\vec{c}_2 + \vec{c}_3 &
\vec{p}_3 &= \vec{c}_1 + 0\vec{c}_2 + \vec{c}_3\\
\vec{p}_4 &= \vec{c}_1 +\vec{c}_2 + 2\vec{c}_3 & \vec{p}_5 &= 2\vec{c}_1 + \vec{c}_2 + 2 \vec{c}_3
\end{align*}
since, for example:
\begin{equation*}
\vec c_1+\vec c_3 = \mathrm{CaO} + \mathrm{SiO_2} = \mathrm{CaSiO_3} = \vec p_3
\end{equation*}
Activity A.3.3
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.3.4
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*}
Part 1.
Set up a system of equations equivalent to this vector equation.
Activity A.3.4
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*}
Part 2.
Find a basis for its solution space.
Activity A.3.4
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*}
Part 3.
Interpret each basis vector as a vector equation and a chemical equation.
Activity A.3.5
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 \(\vec{p}_2 = {\rm CaMgSiO_4}\text{.}\)