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Anish Jain

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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

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Question:

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/cm3combinedwith the solution containing B at 3.6 g/cm3 combined with the

solution containing C at 5.3 g/cm3makes 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 thechemical 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?

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Simplified Version:

Three Ingredients A,B,C

Defined by fixed volume

1.5 g/cm

3

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

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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

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Rewrite In Matrix Form:

Solve:

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Converting back to equation form:

a=1.5cm3, b=3.1cm3, c=2.2cm3

Demonstrates use of linear algebra for simple chemistry

problem

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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).

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Question:

Balance the chemical equation xC2H 6+ yO2

zCO2+ tH2O

by finding out how much of each molecule is needed tosatisfy the Law of Conservation of Matter. The amount of

each molecule needed is represented by x, y, z, and t.

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The amount of each type of atom is written in parentheses:

(2x)C+(6x)H+(2y)O=(z)C+(2z)O+(2t)H+(t)O

We can break this down into three equations by matching

them up by the atom: 2x=z

6x=2t

2y=2z+t

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First rewrite equations:

2x-z=0

6x-2t=0

2y-2z-t=0Write in Matrix Form:

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Can simplify matrix to:

Writing back in equation form:

x=2/6t

y=7/6t

z=2/3t

t=1t

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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