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Balancing equations

First, let's try to balance the combustion equations for these simple hydrocarbons:

Example: CH₄ + O₂ --> CO₂ + H₂O

              CH₄ + 2 O₂ --> CO₂ + 2 H₂O

C₅H₁₂ + O₂ --> CO₂ + H₂O


C₂H₆ + O₂ --> CO₂ + H₂O


Now, try with sugars:

Example: C₃H₆O₃ + O₂ --> CO₂ + H₂O
              C₃H₆O₃ + 3 O₂ --> 3 CO₂ + 3 H₂O

C₅H₁₀O₅ + O₂ --> CO₂ + H₂O

C₈H₁₆O₈ + O₂ --> CO₂ + H₂O

How about equilibrating the combustion reactions of lipids?

The molecular formula in the first reaction is stearic acid. Stearic acid is used in candles. When you burn a candle, stearic acid undergoes combustion and turns into CO₂ and H₂O.

The molecular formula in the second reaction is oleic acid. Oleic acid is the main component of olive oil. When you add olive oil to your salad, and eat it, your cells catabolize the oleic acid into CO₂ and H₂O. 


You probably noticed that the only difference between stearic acid and oleic acid is that the first one has more hydrogens. This indicates that stearic acid is a saturated fatty acid (solid at room temperature), whereas oleic acid is an unsaturated fatty acid (liquid at room temperature). 

Arachidonic acid is the fatty acid in peanut oil, the source of all phospholipids in our membranes, as well as the basic unit that our body uses to synthesize the intermediaries of inflamation (prostaglandins and thromboxanes). The equilibrated equation for the combustion of arachidonic acid looks like this:
Example: C₂₀H₃₂O_{2 + O2 --> CO2 + H2O}
              C₂₀H₃₂O_{2 + 27 O2 --> 20 CO2 + 16 H2O}


Try to equilibrate the following combustion reactions for stearic acid and oleic acid.

C₁₈H₃₆O₂ + O₂ --> CO₂ + H₂O

C₁₈H₃₄O₂ + O₂ --> CO₂ + H₂O




A note about equilibrating equations for long unsaturated fatty acid chains ... 


When you keep trying small coefficients and nothing seems to work, try using the rule of the least common multiple (back to 3rd grade algebra), and if that takes too long, just multiply the highest numbers of your main reactant to get the coefficient needed for your CO₂. Then finding the other coefficients will be just a matter of adding and subtracting. 


For example, if you try to equilibrate the combustion of palmitoleic acid, an unsaturated fatty acid common in macadamia nuts (yeah, Haagen Datz's macadamia brittle!), it will end up looking like this:

C₁₈H₃₄O₂ + O₂ --> CO₂ + H₂O (before equilibrating)

34 C₁₈H₃₄O₂ + 867 O₂ --> 612 CO₂ + 578 H₂O

This was my reasoning:

• I need a minimum of 18 carbons, 34 hydrogens and 2 oxygens.
• But when I try using 18 as the coefficient for the CO₂ that gives me 36 hydrogens, which is 2 more than what I have in one molecule of palmitoleic acid (34 hydrogens). 
• I quickly realize that it will take some number juggling to get the right coefficient.
• So, instead of losing my time trying one coefficient after the next, I find the l.c.m. or basically on the case of 18 and 34, its multiplication (search in wikipedia if you don't remember how to calculate l.c.m.).
• How to find the l.c.m:
• 18 can be decomposed as 2 * 3 * 3 = 2 * 3^2
• 34 can be decomposed as 2 * 17
• Choose the decomposition of the factor with the highest exponent
• Multiply all the chosen factors: 2 * 3^2 * 17 = 612
• In our case the number obtained is the maximum possible of 18 x 34 = 612. 
• 612 is the coefficient you will need for your CO₂.
• 
  
  34 C₁₈H₃₄O₂ + X O₂ --> 612 CO₂ + X H₂O
  
  • Now, we have the same number of carbons on each side of the equation
  
  
• Next, is equating the number of hydrogens on both sides of the equation. To do that, I multiply 34 (the coefficient of palmitoleic acids) x 34 (the number of hydrogens in one molecule of palmitoleic acid).
• 34 x 34 = 1,156
• This is the number of hydrogens I need on the side of the products, basically coming only from the molecules of water.
• 1,156 / 2 = 578
• 578 is the coefficient you will need for your H₂O
• 
  
  34 C₁₈H₃₄O₂ + X O₂ --> 612 CO₂ + 578 H₂O
  
  
  
• Finally, we need to equate the number of oxygens on each side of the equation. To do that, I need to know how many oxygens I have on each side by now:
• On the reactants side, we have 34 * 2 + X * 2 = 68 + X * 2
• On the products side, we have 612 *2 + 578 * 1 = 1,224 + 578 = 1,802 
• Ergo:
• 68 + X * 2 = 1,802
• X = (1,802 - 68) / 2
• X = 1,734 / 2 = 867
• 867 is the coefficient you will need for your O₂.
• Voila!
• 
  
  34 C₁₈H₃₄O₂ + 867 O₂ --> 612 CO₂ + 578 H₂O
  Simplify by dividing by 17
  
  

  2 C₁₈H₃₄O₂ + 51 O₂ --> 36 CO₂ + 34 H₂O