Combined Gas Law Practice Problems

Mastering the Combined Gas Law: Practice Problems and In-Depth Explanations

The combined gas law is a cornerstone of chemistry and physics, allowing us to predict how the pressure, volume, and temperature of a gas will change under different conditions. Understanding and applying this law is crucial for anyone studying gases, from high school students to advanced researchers. This comprehensive guide provides a deep dive into the combined gas law, including numerous practice problems with detailed solutions, to help you master this fundamental concept. We'll cover everything from the basic formula to more complex scenarios, ensuring you develop a strong understanding of gas behavior.

Understanding the Combined Gas Law

The combined gas law neatly combines Boyle's Law, Charles's Law, and Gay-Lussac's Law into a single, powerful equation. It states that for a fixed amount of gas, the ratio of the product of pressure and volume to the absolute temperature remains constant. Mathematically, this is expressed as:

(P₁V₁)/T₁ = (P₂V₂)/T₂

Where:

• P₁ represents the initial pressure of the gas.
• V₁ represents the initial volume of the gas.
• T₁ represents the initial absolute temperature of the gas (always in Kelvin).
• P₂ represents the final pressure of the gas.
• V₂ represents the final volume of the gas.
• T₂ represents the final absolute temperature of the gas (always in Kelvin).

Remember that temperature must be expressed in Kelvin (K). To convert from Celsius (°C) to Kelvin, use the formula: K = °C + 273.15. Ignoring this crucial step is a common source of error!

Practice Problems: A Step-by-Step Approach

Let's tackle a series of practice problems, starting with simpler scenarios and progressing to more challenging ones. Each problem will be solved step-by-step, emphasizing the importance of clearly outlining the known variables and the desired unknown.

Problem 1: Simple Application

A gas occupies a volume of 5.0 L at a pressure of 1.0 atm and a temperature of 25°C. What will be the volume of the gas if the pressure is increased to 2.0 atm and the temperature is increased to 50°C?

Solution:

1. List the known variables:

   • P₁ = 1.0 atm
   • V₁ = 5.0 L
   • T₁ = 25°C + 273.15 = 298.15 K
   • P₂ = 2.0 atm
   • T₂ = 50°C + 273.15 = 323.15 K
   • V₂ = ? (This is what we need to find)

2. Apply the combined gas law formula:

   (P₁V₁)/T₁ = (P₂V₂)/T₂

3. Rearrange the formula to solve for V₂:

   V₂ = (P₁V₁T₂)/(P₂T₁)

4. Substitute the known values and calculate:

   V₂ = (1.0 atm * 5.0 L * 323.15 K) / (2.0 atm * 298.15 K) V₂ ≈ 2.7 L

Therefore, the volume of the gas will be approximately 2.7 L under the new conditions.

Problem 2: Volume Change with Constant Pressure

A sample of gas has a volume of 2.5 L at 20°C. If the temperature is increased to 40°C while keeping the pressure constant, what is the new volume?

Solution:

Notice that this problem simplifies the combined gas law. Since pressure is constant (P₁ = P₂), the equation simplifies to Charles' Law: V₁/T₁ = V₂/T₂

1. Convert temperatures to Kelvin:

   • T₁ = 20°C + 273.15 = 293.15 K
   • T₂ = 40°C + 273.15 = 313.15 K

2. Rearrange the formula to solve for V₂:

   • V₂ = (V₁T₂)/T₁

3. Substitute and calculate:

   • V₂ = (2.5 L * 313.15 K) / 293.15 K
   • V₂ ≈ 2.67 L

The new volume is approximately 2.67 L.

Problem 3: Pressure Change with Constant Volume

A gas in a rigid container (constant volume) has a pressure of 1.2 atm at 27°C. What will be the pressure if the temperature is increased to 127°C?

Solution:

Since the volume is constant (V₁ = V₂), this problem simplifies to Gay-Lussac's Law: P₁/T₁ = P₂/T₂

1. Convert temperatures to Kelvin:

   • T₁ = 27°C + 273.15 = 300.15 K
   • T₂ = 127°C + 273.15 = 400.15 K

2. Rearrange the formula to solve for P₂:

   • P₂ = (P₁T₂)/T₁

3. Substitute and calculate:

   • P₂ = (1.2 atm * 400.15 K) / 300.15 K
   • P₂ ≈ 1.6 atm

The new pressure is approximately 1.6 atm.

Problem 4: More Complex Scenario

A gas sample with a volume of 10.0 L at 20°C and 1.5 atm pressure is compressed to a volume of 5.0 L and heated to 40°C. What is the final pressure?

Solution:

This problem requires the full combined gas law equation.

1. Convert temperatures to Kelvin:

   • T₁ = 20°C + 273.15 = 293.15 K
   • T₂ = 40°C + 273.15 = 313.15 K

2. Rearrange the combined gas law formula to solve for P₂:

   • P₂ = (P₁V₁T₂)/(V₂T₁)

3. Substitute and calculate:

   • P₂ = (1.5 atm * 10.0 L * 313.15 K) / (5.0 L * 293.15 K)
   • P₂ ≈ 3.2 atm

The final pressure is approximately 3.2 atm.

Explanation of Scientific Principles

The combined gas law is a direct consequence of the kinetic molecular theory of gases. This theory describes gases as collections of tiny particles (atoms or molecules) in constant, random motion. The pressure exerted by a gas is due to the collisions of these particles with the walls of the container. Temperature is a measure of the average kinetic energy of these particles—higher temperature means faster-moving particles. Volume represents the space available for these particles to move around in.

• Boyle's Law (constant temperature): At constant temperature, the volume of a gas is inversely proportional to its pressure. If you decrease the volume, the particles collide more frequently with the walls, increasing the pressure.

• Charles's Law (constant pressure): At constant pressure, the volume of a gas is directly proportional to its absolute temperature. As temperature increases, particles move faster and require more space, resulting in increased volume.

• Gay-Lussac's Law (constant volume): At constant volume, the pressure of a gas is directly proportional to its absolute temperature. As temperature increases, faster-moving particles collide more forcefully with the walls, increasing the pressure.

The combined gas law elegantly combines these relationships, allowing us to predict changes in all three variables simultaneously, provided the amount of gas remains constant. It's important to remember that the combined gas law, like other ideal gas laws, is most accurate at relatively low pressures and high temperatures where intermolecular forces are minimal. At high pressures or low temperatures, deviations from the ideal gas behavior can occur, requiring more complex equations of state.

Frequently Asked Questions (FAQ)

Q1: What happens if I forget to convert Celsius to Kelvin?

A1: Your calculations will be significantly incorrect. The combined gas law relies on the absolute temperature scale (Kelvin), where zero Kelvin represents the absolute absence of thermal energy. Using Celsius will lead to erroneous results.

Q2: Can I use the combined gas law for mixtures of gases?

A2: The combined gas law applies to a single gas or a mixture of gases behaving ideally. For complex mixtures with non-ideal behavior, more advanced techniques might be needed.

Q3: What if one of the variables remains constant?

A3: If one of the variables (pressure, volume, or temperature) remains constant, the combined gas law simplifies to one of the individual gas laws (Boyle's, Charles's, or Gay-Lussac's Law).

Q4: Are there limitations to the combined gas law?

A4: Yes, the combined gas law is based on the ideal gas assumption. Deviations from ideal behavior occur at high pressures and low temperatures where intermolecular forces become significant.

Conclusion

The combined gas law is a powerful tool for understanding and predicting gas behavior under changing conditions. By mastering the formula and practicing various problem types, you'll build a strong foundation in gas laws and develop crucial problem-solving skills applicable to many scientific fields. Remember to always pay attention to units, particularly ensuring temperatures are in Kelvin, to ensure accurate calculations. Through consistent practice and a solid understanding of the underlying principles, you can confidently tackle even the most complex gas law problems. Continue practicing and exploring different scenarios to solidify your understanding and expertise in this fundamental area of physics and chemistry.