Applying Boyle's Law to Gas Stoichiometry Problems

🧪 Boyle's Law and Stoichiometry: An Introduction

Boyle's Law describes the relationship between the pressure and volume of a gas at constant temperature and number of moles. In gas stoichiometry, we often need to correct gas volumes measured at one set of conditions to another. Boyle's Law provides a crucial tool for this correction, allowing us to accurately relate volumes and amounts of gases in chemical reactions.

📜 Historical Context

Boyle's Law is named after Robert Boyle, who formulated it in 1662. Through meticulous experimentation, Boyle observed that for a fixed amount of gas kept at a constant temperature, pressure and volume are inversely proportional. This discovery laid the foundation for understanding gas behavior and its role in chemical calculations.

🔑 Key Principles of Boyle's Law

• 📏 Inverse Proportionality: Boyle's Law states that the pressure ($P$) of a gas is inversely proportional to its volume ($V$) when the temperature ($T$) and the number of moles ($n$) are kept constant. Mathematically, this is expressed as $P \propto \frac{1}{V}$.
• ➗ Constant Value: The product of pressure and volume ($PV$) remains constant if the temperature and number of moles are unchanged. This can be written as $P_1V_1 = P_2V_2$, where $P_1$ and $V_1$ are the initial pressure and volume, and $P_2$ and $V_2$ are the final pressure and volume.
• 🌡️ Constant Temperature: Boyle's Law is only valid when the temperature is constant. Changes in temperature will affect the relationship between pressure and volume, requiring the use of the Combined Gas Law or the Ideal Gas Law.
• ⚖️ Constant Moles: The amount of gas (number of moles) must remain constant. If gas is added or removed from the system, Boyle's Law cannot be directly applied without accounting for the change in the number of moles.

🧮 Applying Boyle's Law in Stoichiometry: Worked Examples

Let's explore how Boyle's Law is used to solve gas stoichiometry problems. Here are some examples:

1. Example 1: Volume Change Calculation

   A gas occupies a volume of 10.0 L at a pressure of 2.0 atm. If the pressure is increased to 4.0 atm while keeping the temperature constant, what is the new volume?

   Solution:

   • 📝 Identify the knowns: $P_1 = 2.0 \text{ atm}$, $V_1 = 10.0 \text{ L}$, $P_2 = 4.0 \text{ atm}$
   • 🧮 Apply Boyle's Law: $P_1V_1 = P_2V_2$
   • ➗ Solve for $V_2$: $V_2 = \frac{P_1V_1}{P_2} = \frac{(2.0 \text{ atm})(10.0 \text{ L})}{4.0 \text{ atm}} = 5.0 \text{ L}$
2. Example 2: Pressure Change Calculation

   A gas has a pressure of 700 mmHg in a 5.0 L container. If the volume is changed to 2.5 L at constant temperature, what is the new pressure?

   Solution:

   • 📝 Identify the knowns: $P_1 = 700 \text{ mmHg}$, $V_1 = 5.0 \text{ L}$, $V_2 = 2.5 \text{ L}$
   • 🧮 Apply Boyle's Law: $P_1V_1 = P_2V_2$
   • ➗ Solve for $P_2$: $P_2 = \frac{P_1V_1}{V_2} = \frac{(700 \text{ mmHg})(5.0 \text{ L})}{2.5 \text{ L}} = 1400 \text{ mmHg}$
3. Example 3: Stoichiometry with Boyle's Law

   Consider the reaction: $N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)$. If 5.0 L of $N_2$ reacts completely at 2.0 atm, what volume of $H_2$ is required at 2.0 atm, assuming constant temperature?

   Solution:

   • 🧪 According to the balanced equation, 1 mole of $N_2$ reacts with 3 moles of $H_2$.
   • ⚖️ Since the pressure and temperature are constant, the volume ratio is the same as the mole ratio.
   • ➗ Therefore, the volume of $H_2$ required is $3 \times 5.0 \text{ L} = 15.0 \text{ L}$.

🌍 Real-World Applications

• 🤿 Scuba Diving: Divers need to understand Boyle's Law to predict how the volume of air in their lungs changes with depth due to pressure variations.
• 🎈 Weather Balloons: Meteorologists use Boyle's Law to predict how the volume of weather balloons will change as they rise into the atmosphere where the pressure decreases.
• 🚗 Internal Combustion Engines: The compression of air-fuel mixtures in internal combustion engines follows Boyle's Law, influencing engine efficiency and performance.

💡 Conclusion

Boyle's Law is a fundamental principle in chemistry and physics, particularly useful in gas stoichiometry. By understanding the inverse relationship between pressure and volume, we can accurately predict and calculate gas behavior in various scenarios. Mastering Boyle's Law enhances our ability to solve complex stoichiometric problems and appreciate its practical applications in the real world.