Similarities Among Systems in Mathematics, Science & Technology

Instructor: Matt Gromlich

Matt has taught at the K-12 level and beyond for the last five years. He is currently getting his Ph.D. in Mathematics Education.

How do the concepts of balance and equilibrium extend throughout science and mathematics? How does solving algebraic equations parallel chemical equilibrium within a solution? See how crucial balance is in our universe as well as how mathematics is used to describe these constant equilibriums.

The Balance of the Universe

The universe is in constant balance. The gravitational pull of the sun keeps the planets in orbit, allowing life to continue on Earth. The gravity of Earth keeps its organisms grounded so that they can continue to live and thrive. Even within biological systems, cells remain in equilibrium, balancing the amount of water and other minerals they absorb.

The Law of Conservation of Mass states that matter can neither be created nor destroyed. This concept has been more broadly expanded to state that energy can neither be created nor destroyed (the First Law of Thermodynamics). While energy can be transformed, it remains a part of a system. The universe, as a system, continuously transforms energy to remain in a state of equilibrium.

So how do the concepts of balance and equilibrium extend throughout science and mathematics? Let's take a look.

Chemical Equilibrium

In the fields of science and mathematics, equilibrium is the state in which forces are balanced. From a chemical perspective, this means that a chemical reaction and its reverse reaction occur at equal rates.

For example, assume that table salt (NaCl) is dissolved in water. Some of the sodium chloride will break apart into ions, Na+ and Cl-. However, not all the NaCl will become ions; some will remain bonded.

At equilibrium, the net amount of bonded sodium chloride and ions will remain unchanged. Na+ and Cl- ions will bond together at the same rate that other NaCl molecules split apart. This is called chemical equilibrium. This process can be expressed using the chemical equation:

  • NaCl ↔ Na+ + Cl-

The double arrow shows that the chemical equation occurs in both the forward and reverse directions.

Equilibrium in Mathematics

Mathematics also uses the idea of equilibrium when solving equations. Consider the equation:

4x + 7 = 27

To be able to solve this equation, you have to know that the equals sign means that the two sides of the equation are currently in equilibrium. If you make a change to one side of the equation without changing the other, the equilibrium is destroyed. Therefore, if an operation is performed on one side of the equation, it must be performed on the other side as well.

To isolate the variable term, 4x, use the additive inverse of -7 on each side of the equation. This maintains equilibrium.

4x + 7 = 27

4x = 20

4x equals 20 now and equilibrium is still maintained. Now, we can finish isolating the variable by dividing by four (the coefficient) on each side. Again, this must be done to both sides of the equation so that equilibrium is conserved.

4x = 20

x = 5

Once the variable is isolated, we can now see that x is 5. Plugging this value in for x in the original equation, it is clear that the equilibrium has been maintained and the equation is balanced.

Balancing Chemical Equations

Again, matter can neither be created nor destroyed according to the Law of Conservation of Mass. In a chemical process, atoms and molecules break apart and re-form to make new ions, atoms, or molecules. This can be seen through the process of balancing chemical equations.

For example, propane is a flammable hydrocarbon that is used in a lot of barbecue grills. By igniting it, propane combusts and reacts with oxygen in the air. This produces carbon dioxide and water.

  • Propane + Oxygen → Carbon Dioxide + Water
  • C3 H8 + O2 → CO2 + H2 O

However, if we look at this equation, there are three carbon atoms on the left side of the equation and only one carbon atom on the right side. Where did the other two carbon atoms go? They cannot just disappear, so somehow we need to show what happened to these atoms. We need to balance the equation to show that equilibrium is still maintained.

This can be done by adding coefficients (a numerical constant that is placed in front of a variable or term that is multiplied by that term). In chemistry, the coefficient shows how many of that molecule is reacting or being produced.

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