Introduction

Gas laws are a set of principles that describe the behavior of gases under different conditions of temperature, pressure, and volume. These laws help us understand how gases interact and how their properties change in response to external factors. There are several key gas laws that are fundamental to the study of chemistry, including Boyle's Law, Charles's Law, Gay-Lussac's Law, and the Combined Gas Law.

Boyle's Law

Definition: Boyle's Law states that the pressure of a gas is inversely proportional to its volume, as long as the temperature remains constant.

Formula: $P_1V_1 = P_2V_2$

Example: If a gas at a pressure of 2 atm occupies a volume of 4 L, what will be the volume if the pressure is increased to 4 atm while keeping the temperature constant?

Solution: Using Boyle's Law formula, $P_1V_1 = P_2V_2$, we can calculate the new volume: $$(2 \text{ atm}) \times (4 \text{ L}) = (4 \text{ atm}) \times V_2$$ $$8 = 4V_2$$ $$V_2 = 2 \text{ L}$$

Charles's Law

Definition: Charles's Law states that the volume of a gas is directly proportional to its temperature, as long as the pressure remains constant.

Formula: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

Example: If a gas occupies a volume of 10 L at a temperature of 300 K, what will be the volume at 350 K if the pressure remains constant?

Solution: Applying Charles's Law formula, $\frac{V_1}{T_1} = \frac{V_2}{T_2}$, we can find the new volume: $$\frac{10}{300} = \frac{V_2}{350}$$ $$\frac{1}{30} = \frac{V_2}{350}$$ $$V_2 = \frac{1}{30} \times 350 = 11.67 \text{ L}$$

Gay-Lussac's Law

Definition: Gay-Lussac's Law states that the pressure of a gas is directly proportional to its temperature, provided that the volume remains constant.

Formula: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$

Example: If a gas at a pressure of 2 atm has a temperature of 300 K, what will be the pressure at 350 K if the volume remains constant?

Solution: Using Gay-Lussac's Law formula, $\frac{P_1}{T_1} = \frac{P_2}{T_2}$, we can calculate the new pressure: $$\frac{2}{300} = \frac{P_2}{350}$$ $$\frac{1}{150} = \frac{P_2}{350}$$ $$P_2 = \frac{1}{150} \times 350 = 2.33 \text{ atm}$$

Combined Gas Law

Definition: The Combined Gas Law combines Boyle's Law, Charles's Law, and Gay-Lussac's Law into a single expression that relates pressure, volume, and temperature of a gas.

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

Example: If a gas at 3 atm, 4 L, and 200 K is heated to 250 K while maintaining constant pressure, what will be the new volume?

Solution: Using the Combined Gas Law formula, $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$, we can solve for the new volume: $$\frac{3 \times 4}{200} = \frac{P_2 \times V_2}{250}$$ $$\frac{12}{200} = \frac{P_2 \times V_2}{250}$$ $$\frac{3}{50} = \frac{P_2 \times V_2}{250}$$ $$P_2 \times V_2 = \frac{3}{50} \times 250 = 15$$ Since pressure is constant, $P_2 = 3$ atm Therefore, $V_2 = \frac{15}{3} = 5 \text{ L}$

Common Mistakes

• Confusing the different gas laws and mixing up their formulas.
• Failing to convert units to consistent values before applying the gas laws.
• Forgetting to keep other variables constant when applying a specific gas law.

Key Points

• Boyle's Law: $P_1V_1 = P_2V_2$
• Charles's Law: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$
• Gay-Lussac's Law: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$
• Combined Gas Law: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

Practice Questions

1. A gas at a pressure of 2 atm occupies a volume of 6 L. If the pressure is doubled while keeping the volume constant, what will be the new pressure?

   Answer: $4$ atm

2. If a gas at 25°C occupies a volume of 8 L, what will be the volume at 50°C if the pressure remains constant?

   Answer: $8.5$ L

3. A gas at 3 atm and 300 K is compressed to a pressure of 6 atm while maintaining constant temperature. If the initial volume is 10 L, what will be the final volume?

   Answer: $5$ L

4. If a gas at 1 atm and 200 K occupies a volume of 5 L, what will be the temperature if the volume is decreased to 3 L while keeping the pressure constant?

   Answer: $120$ K

5. A gas at 4 atm, 8 L, and 400 K is cooled to 300 K while maintaining constant pressure. What will be the new volume?

   Answer: $6$ L