## SectionA.3Geology: Phases and Components

### SubsectionA.3.1Activities

#### DefinitionA.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.

#### ObservationA.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*}

#### ActivityA.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.

#### ActivityA.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*}
##### (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.

#### ActivityA.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 $$\vec{p}_2 = {\rm CaMgSiO_4}\text{.}$$

### SubsectionA.3.2Slideshow

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