How to Calculate Volume Changes Using the Combined Gas Law

📚 Understanding the Combined Gas Law

The Combined Gas Law is a powerful tool in chemistry that allows us to analyze how the volume of a gas changes when temperature and pressure also change. It's essentially a combination of Boyle's Law, Charles's Law, and Gay-Lussac's Law.

📜 History and Background

The Combined Gas Law wasn't discovered by a single scientist but evolved from the individual observations of Boyle, Charles, and Gay-Lussac. They each studied the relationship between two of the variables (pressure, volume, and temperature) while keeping the third constant. Combining their findings led to the Combined Gas Law.

⚗️ Key Principles

The Combined Gas Law is expressed mathematically as:

$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

Where:

• 📊 $P_1$ is the initial pressure.
• 📦 $V_1$ is the initial volume.
• 🌡️ $T_1$ is the initial absolute temperature (in Kelvin).
• 🧭 $P_2$ is the final pressure.
• 🧱 $V_2$ is the final volume.
• 🔥 $T_2$ is the final absolute temperature (in Kelvin).

Important Notes:

• 🌡️ Temperature must always be in Kelvin. To convert from Celsius to Kelvin, use the formula: $K = °C + 273.15$.
• 📏 Pressure and volume units must be consistent on both sides of the equation (e.g., both pressures in atm, both volumes in liters).

🧪 How to Calculate Volume Changes

To calculate volume changes, rearrange the Combined Gas Law to solve for $V_2$:

$V_2 = \frac{P_1V_1T_2}{P_2T_1}$

Steps:

• 📝 Identify: Identify the knowns ($P_1, V_1, T_1, P_2, T_2$) from the problem statement.
• 🔄 Convert: Convert temperatures to Kelvin and ensure consistent pressure and volume units.
• Plug: Plug the values into the rearranged formula.
• 🧮 Solve: Solve for $V_2$.

🌍 Real-world Examples

Example 1: Inflating a Balloon

A balloon has a volume of 1.0 L at 25°C and 1.0 atm. If the temperature is increased to 50°C and the pressure is decreased to 0.5 atm, what is the new volume of the balloon?

Solution:

• 🏷️ $P_1 = 1.0 \text{ atm}$
• 🎈 $V_1 = 1.0 \text{ L}$
• 🌡️ $T_1 = 25 + 273.15 = 298.15 \text{ K}$
• 📉 $P_2 = 0.5 \text{ atm}$
• 🔥 $T_2 = 50 + 273.15 = 323.15 \text{ K}$

Using the formula:

$V_2 = \frac{(1.0 \text{ atm})(1.0 \text{ L})(323.15 \text{ K})}{(0.5 \text{ atm})(298.15 \text{ K})} = 2.16 \text{ L}$

Example 2: Syringe Volume

A gas in a syringe has a volume of 50.0 mL at standard temperature and pressure (STP: 0°C and 1 atm). If the pressure is increased to 2.0 atm and the temperature is increased to 100°C, what is the new volume?

Solution:

• 📌 $P_1 = 1.0 \text{ atm}$
• 💉 $V_1 = 50.0 \text{ mL}$
• 🧊 $T_1 = 0 + 273.15 = 273.15 \text{ K}$
• 📈 $P_2 = 2.0 \text{ atm}$
• ♨️ $T_2 = 100 + 273.15 = 373.15 \text{ K}$

Using the formula:

$V_2 = \frac{(1.0 \text{ atm})(50.0 \text{ mL})(373.15 \text{ K})}{(2.0 \text{ atm})(273.15 \text{ K})} = 34.1 \text{ mL}$

📝 Conclusion

The Combined Gas Law is essential for predicting how volume changes with pressure and temperature. By understanding the formula and practicing with examples, you can confidently solve a wide range of gas-related problems! 🎉

🤔 Practice Quiz

Test your understanding with these practice problems:

1. 🎈 A gas occupies 10.0 L at STP. What volume will it occupy at 300 K and 1.5 atm?
2. 🔥 A container of gas has a volume of 5.0 L at 20°C and 2.0 atm. If the pressure is changed to 1.0 atm and the temperature is increased to 40°C, what is the new volume?
3. 🧊 A gas occupies a volume of 25.0 mL at 25°C and 760 torr. What volume will it occupy at 50°C and 800 torr?

Answers:

1. $V_2 = 7.39 \text{ L}$
2. $V_2 = 10.51 \text{ L}$
3. $V_2 = 26.46 \text{ mL}$