Problem 46 - Chemistry An Atoms First Approach

Question

Consider the following chemical equation. $$2 \mathrm{NO}_{2}(g) \longrightarrow \mathrm{N}_{2} \mathrm{O}_{4}(g)$$ If \(25.0 \mathrm{mL} \mathrm{NO}_{2}\) gas is completely converted to \(\mathrm{N}_{2} \mathrm{O}_{4}\) gas under the same conditions, what volume will the \(\mathrm{N}_{2} \mathrm{O}_{4}\) occupy?

Step-by-Step Solution

Answer

The N₂O₄ gas will occupy a volume of 12.5 mL after the complete conversion of 25.0 mL NO₂ gas, based on the given balanced chemical equation and applying Avogadro's law.

Step 1: Understand the balanced equation

The given balanced chemical equation is: \(2 \mathrm{NO}_{2}(g) \longrightarrow \mathrm{N}_{2} \mathrm{O}_{4}(g)\) This equation tells us that 2 moles of NO₂ react with each other to form 1 mole of N₂O₄.

Step 2: Identify the given information

We are given the following information: - Volume of NO₂ gas = 25.0 mL

Step 3: Apply the relation between volumes of gases in a chemical reaction

According to Avogadro's law, equal volumes of all gases, at the same temperature and pressure, have the same number of molecules. We can apply this relation as follows for the given chemical reaction: Volume of N₂O₄ gas / Volume of NO₂ gas = moles of N₂O₄ / moles of NO₂ Since the balanced equation tells us that 2 moles of NO₂ react to produce 1 mole of N₂O₄, we can write: Volume of N₂O₄ gas / Volume of NO₂ gas = 1/2

Step 4: Calculate the volume of N₂O₄ gas

Now, we can rearrange the equation we found in Step 3 to solve for the volume of N₂O₄ gas: Volume of N₂O₄ gas = (1/2) * Volume of NO₂ gas Plugging in the given volume of NO₂ gas: Volume of N₂O₄ gas = (1/2) * 25.0 mL Volume of N₂O₄ gas = 12.5 mL Therefore, the N₂O₄ gas will occupy a volume of 12.5 mL after the complete conversion of NO₂ gas.

Key Concepts

Understanding Chemical Equations

A chemical equation is a symbolic representation of a chemical reaction. It shows the reactants and products involved in the reaction along with the proportions in which they react. For example, the chemical equation \(2 \ \text{NO}_2(g) \longrightarrow \text{N}_2\text{O}_4(g)\) indicates that two molecules of nitrogen dioxide (\(\text{NO}_2\)) gas react to form one molecule of dinitrogen tetroxide (\(\text{N}_2\text{O}_4\)) gas.
This balanced equation is crucial as it maintains the Law of Conservation of Mass, meaning the quantity of each element is the same before and after the reaction.
When interpreting chemical equations, remember these points:

• Coefficients (the numbers before the formulas) tell you the number of moles of each substance involved.
• Physical states of reactants and products are indicated by letters in parentheses (like (g) for gas).
• The arrow in the equation points from reactants to products, showing the direction of the reaction.

Using this knowledge helps in predicting the amounts of products formed and reactants required in a chemical reaction.

Exploring Avogadro's Law

Avogadro's Law is a fundamental principle in chemistry that relates the volume of a gas to the amount of substance present. It states that under the same conditions of temperature and pressure, equal volumes of different gases contain an equal number of molecules.
This law is invaluable when studying gaseous reactions, as it allows for a straightforward relationship between moles and volume. It can be expressed mathematically as:

• \( V \propto n \) or \( \frac{V}{n} = k \)

where:

• \(V\) is the volume of the gas.
• \(n\) is the number of moles of gas.
• \(k\) is a constant.

In the given exercise, Avogadro's law is applied to relate the volume of \(\text{NO}_2\) gas to \(\text{N}_2\text{O}_4\) based on their mole ratio from the balanced chemical equation. Since 2 moles of \(\text{NO}_2\) form 1 mole of \(\text{N}_2\text{O}_4\), the volume of \(\text{N}_2\text{O}_4\) is half that of \(\text{NO}_2\) gas, assuming the same conditions.

The Concept of Gas Volume Conversion

Gas volume conversion is essential to interpret the results of reactions involving gases. Often, we start off knowing the volume of a reactant gas and need to find the volume of a product gas, like in the given problem. This is made simpler by using the stoichiometric relationships provided by the balanced chemical equation and Avogadro's law.
Here's how you can approach gas volume conversions with ease:

• Identify the volumes of gases given in the problem.
• Determine the mole ratio from the balanced chemical equation (reactants to products).
• Apply Avogadro's law to relate the initial and final volumes using the mole ratio.

In the exercise, we used these principles to find that the volume of \(\text{N}_2\text{O}_4\) was half of that of \(\text{NO}_2\) given the 2:1 mole ratio.
This conversion is an excellent example of how chemical equations and Avogadro's law come together to predict the behavior of gases in chemical reactions efficiently.