Linear Algebra: Application to Chemistry

Linear Algebra Application: ChemistryAnish Jain

Chemistry• Linear Algebra required for chem. majors• Useful for balancing chemical equations• Can solve basic math problems in chemistry• Martin Cockett, Graham Doggett• Discuss and teach these uses

First ApplicationQuestion:It takes three different ingredients A, B, and C, to produce a certain chemical substance. A, B, and C have to be dissolved in water separately before they interact to form the chemical. Suppose that the solution containing A at 1.5 g/cm3 combined with the solution containing B at 3.6 g/cm3 combined with the solution containing C at 5.3 g/cm3 makes 25.07 g of the chemical. If the proportion for A, B, C in these solutions are changed to 2.5 g/cm3, 4.3 g/cm3, and 2.4 g/cm3, respectively (while the volumes remain the same), then 22.36 g of the chemical is produced. Finally, if the proportions are 2.7 g/cm3, 5.5 g/cm3, and 3.2 g/cm3, respectively, then 28.14 g of the chemical is produced. What are the volumes (in cubic centimeters) of the solutions containing A, B, and C?

Too Many Words..

Simplified Version:• Three Ingredients A,B,C• Defined by fixed volume• 1.5 g/cm3 of A + 3.6 g/cm3 of B+ 5.3 g/cm3 of C=

25.07 g• 2.5 g/cm3 of A + 4.3 g/cm3 of B+ 2.4 g/cm3 of C=

22.36 g • 2.7 g/cm3 of A + 5.5 g/cm3 of B+ 3.2 g/cm3 of C=

28.14 g

Write As An Equation• Represent volumes with a, b, and c respectively:

• 1.5 a+ 3.6 b+ 5.3 c= 25.07 • 2.5 a+ 4.3 b+ 2.4 c= 22.36 • 2.7 a+ 5.5 b+ 3.2 c= 28.14

Write In Matrix Form And Solve

Rewrite In Matrix Form:

Solve:

Final Answer• Converting back to equation form:• a=1.5cm3, b=3.1cm3, c=2.2cm3

• Demonstrates use of linear algebra for simple chemistry problem

Second Application• Linear Algebra can be used to balance chemical

equations• Law of Conservation of Matter:• Mass is neither created nor destroyed in any chemical

reaction. Therefore balancing of equations requires the same number of atoms on both sides of a chemical reaction. The mass of all the reactants (the substances going into a reaction) must equal the mass of the products (the substances produced by the reaction).

Example

Question: Balance the chemical equation xC2H  6 +  yO2 → zCO2 +  tH2O

by finding out how much of each molecule is needed to satisfy the Law of Conservation of Matter. The amount of each molecule needed is represented by x, y, z, and t.

Rewrite As Linear Equations

The amount of each type of atom is written in parentheses:(2x)C+(6x)H+(2y)O=(z)C+(2z)O+(2t)H+(t)OWe can break this down into three equations by matching them up by the atom:• 2x=z• 6x=2t• 2y=2z+t

Rewrite as Matrix

First rewrite equations:• 2x-z=0• 6x-2t=0• 2y-2z-t=0Write in Matrix Form:

Solve

Can simplify matrix to:

Writing back in equation form:• x=2/6t• y=7/6t• z=2/3t• t=1t

Final Answer• t can be any real number and equation would be

balanced• However, small integer numbers are preferred• Set t=6:2C2H  6 +  7O2 → 4CO2 +  6H2O