Animated Solution for Chemistry - Basic Concepts in Chemistry: The minimum number of moles of $O_2$ required for complete combustion of $1$ mole of propane and $2$ moles of butane is …… .

Imagine you are tasked with fueling a massive engine. You have two different types of fuel: propane and butane. To make them burn perfectly without leaving any soot behind, you need to supply exactly the right amount of oxygen. This is the essence of complete combustion.

In chemistry, complete combustion of any hydrocarbon (a compound made of only carbon and hydrogen) always yields the same two products: carbon dioxide ($C{O}_2$) and water ($H_{2}O$). The challenge lies in balancing the chemical equation to find the exact molar ratio of the reactants.

Our problem gives us a specific mixture: $1$ mole of propane and $2$ moles of butane. We need to find the total minimum moles of oxygen gas ($O_2$) required to burn this entire mixture completely.

To solve this efficiently, we can rely on the master formula for the combustion of any hydrocarbon $C_x{H}_y$:

This elegant formula saves us from manually balancing equations every single time. Let's apply it to our two fuels.

First, let's look at propane, which has the chemical formula $C_3{H}_8$. Here, $x=3$ and $y=8$.

Plugging these into our master formula, the coefficient for oxygen becomes:

So, the balanced equation is:

This tells us that exactly $5$ moles of $O_2$ are required to completely burn $1$ mole of propane. Since our mixture contains exactly $1$ mole of propane, we bank these $5$ moles of oxygen for later.

Next, we tackle butane, $C_4{H}_{10}$. Here, $x=4$ and $y=10$.

Using the master formula again, the oxygen coefficient is:

The balanced equation is:

This means $1$ mole of butane requires $\frac{13}{2}$ (or $6.5$) moles of oxygen.

Here is where many students make a silly mistake! They simply add $5$ and $6.5$. But look closely at the problem statement: we have $2$ moles of butane, not $1$.

We must multiply the oxygen requirement by $2$:

Now, we simply bring it all together. The total oxygen required is the sum of the oxygen needed for the propane and the oxygen needed for the butane.

By systematically breaking down the mixture and applying the fundamental laws of stoichiometry, we arrive at our final answer of $18$. Always remember to double-check the quantities given in the question before doing your final addition!