TBIL-LA Geology: Phases and Components

Section A.3 Geology: Phases and Components

Subsection A.3.1 Activities

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 ( $C a O$ ) or Silicon Dioxide ( $S i O_{2}$ ).

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.

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Observation A.3.2.

Consider the 3 components

{\vec{c}}_{1} = C a O {\vec{c}}_{2} = M g O and {\vec{c}}_{3} = S i O_{2}

and the 5 phases:

\begin{aligned} {\vec{p}}_{1} & = C a_{3} M g S i_{2} O_{8} & {\vec{p}}_{2} & = C a M g S i O_{4} & {\vec{p}}_{3} & = C a S i O_{3} \\ {\vec{p}}_{4} & = C a M g S i_{2} O_{6} & {\vec{p}}_{5} & = C a_{2} M g S i_{2} O_{7} \end{aligned}

Geologists already know (or can easily deduce) that

\begin{aligned} {\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{aligned}

since, for example:

{\vec{c}}_{1} + {\vec{c}}_{3} = CaO + {SiO}_{2} = {CaSiO}_{3} = {\vec{p}}_{3}

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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.

{\vec{p}}_{1} = [\begin{matrix} 3 \\ 1 \\ 2 \end{matrix}], {\vec{p}}_{2} = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}], {\vec{p}}_{3} = [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}], {\vec{p}}_{4} = [\begin{matrix} 1 \\ 1 \\ 2 \end{matrix}], {\vec{p}}_{5} = [\begin{matrix} 2 \\ 1 \\ 2 \end{matrix}] .

Determine if the set of phases is linearly dependent or linearly independent.

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Activity A.3.4.

Geologists are interested in knowing all the possible chemical reactions among the 5 phases:

{\vec{p}}_{1} = {Ca}_{3} {MgSi}_{2} O_{8} = [\begin{matrix} 3 \\ 1 \\ 2 \end{matrix}] {\vec{p}}_{2} = {CaMgSiO}_{4} = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] {\vec{p}}_{3} = {CaSiO}_{3} = [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}]

{\vec{p}}_{4} = {CaMgSi}_{2} O_{6} = [\begin{matrix} 1 \\ 1 \\ 2 \end{matrix}] {\vec{p}}_{5} = {Ca}_{2} {MgSi}_{2} O_{7} = [\begin{matrix} 2 \\ 1 \\ 2 \end{matrix}] .

That is, they want to find numbers $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ such that

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.

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(a)

Set up a system of equations equivalent to this vector equation.

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(b)

Find a basis for its solution space.

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(c)

Interpret each basis vector as a vector equation and a chemical equation.

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Activity A.3.5.

We found two basis vectors $[\begin{matrix} 1 \\ - 2 \\ - 2 \\ 1 \\ 0 \end{matrix}]$ and $[\begin{matrix} 0 \\ - 1 \\ - 1 \\ 0 \\ 1 \end{matrix}],$ corresponding to the vector and chemical equations

\begin{aligned} 2 {\vec{p}}_{2} + 2 {\vec{p}}_{3} & = {\vec{p}}_{1} + {\vec{p}}_{4} & 2 C a M g S i O_{4} + 2 C a S i O_{3} & = C a_{3} M g S i_{2} O_{8} + C a M g S i_{2} O_{6} \\ {\vec{p}}_{2} + {\vec{p}}_{3} & = {\vec{p}}_{5} & C a M g S i O_{4} + C a S i O_{3} & = C a_{2} M g S i_{2} O_{7} \end{aligned}

Combine the basis vectors to produce a chemical equation among the five phases that does not involve ${\vec{p}}_{2} = C a M g S i O_{4} .$

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Subsection A.3.2 Slideshow

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Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/geology.slides.html.

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