Animated Solution for Chemistry - Basic Concepts in Chemistry: At $300\text{ K}$ and $1$ atmospheric pressure, $10\text{ mL}$ of a hydrocarbon required $55\text{ mL}$ of $O_2$ for complete combustion and $40\text{ mL}$ of $C{O}_2$ is formed. The formula of the hydrocarbon is

Imagine you are in a laboratory, and you have a sealed container—a eudiometer tube—filled with a mystery gas. You know it's a hydrocarbon, meaning it's made entirely of carbon and hydrogen. Let's call it $C_x{H}_y$. Your mission is to find its exact chemical formula.

To do this, you decide to burn it completely. You measure out exactly $10\text{ mL}$ of this hydrocarbon and ignite it with a spark in the presence of excess oxygen. After the reaction, you measure the gases left behind. You find that it took exactly $55\text{ mL}$ of oxygen ($O_2$) to burn it completely, and the reaction produced $40\text{ mL}$ of carbon dioxide ($C{O}_2$).

How can we use these volumes to unlock the formula $C_x{H}_y$? Let's dive into the beautiful stoichiometry of combustion.

The key to solving any hydrocarbon combustion problem is the general balanced chemical equation. When a hydrocarbon $C_x{H}_y$ burns in oxygen, it always produces carbon dioxide and water. By balancing the carbon and hydrogen atoms on both sides, we get this elegant master equation:

This equation tells us the exact molar ratio of the reactants and products. For every $1\text{ mole}$ of hydrocarbon, we need $\left(x+\frac{y}{4}\right)\text{ moles}$ of oxygen, and we will produce $x\text{ moles}$ of carbon dioxide.

Now, here is where physics makes our chemistry easier. The problem states that the reaction happens at $300\text{ K}$ and $1\text{ atmospheric pressure}$. Because the temperature and pressure are constant, we can invoke Avogadro's Law, which states that the volume of a gas is directly proportional to its number of moles ($V\propto n$).

This means we can read our molar equation directly as a volume equation!

A Crucial Catch: Notice that we are ignoring the water ($H_{2}O$). Why? Because at $300\text{ K}$ (which is roughly room temperature, $27^{\circ }\text{C}$), water is a liquid. The volume occupied by a few drops of liquid water is practically zero compared to the large volumes of the gases. Therefore, in eudiometry problems at room temperature, we completely ignore the volume of water formed.

Finding $x$ and $y$

We started with $10\text{ mL}$ of our hydrocarbon. Let's scale our volume equation by a factor of $10$:

- Volume of $C_x{H}_y$ reacted $=10\text{ mL}$ - Volume of $O_2$ required $=10\left(x+\frac{y}{4}\right)\text{ mL}$ - Volume of $C{O}_2$ produced $=10x\text{ mL}$

Finding $x$ (The Carbon Atoms): We are given that $40\text{ mL}$ of $C{O}_2$ was formed. We can set up a simple equation:

Solving for $x$, we get:

Our hydrocarbon has $4$ carbon atoms!

Finding $y$ (The Hydrogen Atoms): We are given that $55\text{ mL}$ of $O_2$ was required for complete combustion. Let's set up the equation for oxygen:

We already know that $x=4$. Let's substitute that in:

Now, let's carefully expand the bracket to avoid any silly algebraic mistakes:

Subtracting $40$ from both sides:

Dividing by $2.5$:

Our hydrocarbon has $6$ hydrogen atoms!

We have successfully decoded the mystery. With $x=4$ and $y=6$, the chemical formula of our hydrocarbon is $C_4{H}_6$.

This could be a molecule like butyne or butadiene. By simply measuring the volumes of gases before and after a fiery reaction, we were able to count the invisible atoms inside the molecule. That is the true power of stoichiometry!