Ideal Gas Law Practice Problems

Mastering the Ideal Gas Law: A Comprehensive Guide with Practice Problems

The Ideal Gas Law is a cornerstone of chemistry and physics, providing a simple yet powerful model for understanding the behavior of gases. Understanding this law is crucial for anyone studying science, engineering, or related fields. This comprehensive guide will not only explain the Ideal Gas Law itself but also walk you through a variety of practice problems, ranging from simple applications to more complex scenarios. We'll tackle different types of problems, focusing on problem-solving strategies and providing detailed explanations for each step. By the end, you'll be confident in your ability to apply the Ideal Gas Law to various situations.

Understanding the Ideal Gas Law: PV = nRT

The Ideal Gas Law is mathematically represented as:

PV = nRT

Where:

• P represents pressure (typically in atmospheres, atm, or Pascals, Pa).
• V represents volume (typically in liters, L, or cubic meters, m³).
• n represents the number of moles of gas (mol).
• R is the ideal gas constant (its value depends on the units used for P, V, and T). Common values include:
  • 0.0821 L·atm/mol·K (when using atm, L, and K)
  • 8.314 J/mol·K (when using Pa, m³, and K)
• T represents temperature (always in Kelvin, K). Remember to convert Celsius to Kelvin using the formula: K = °C + 273.15

Problem-Solving Strategies

Before diving into the problems, let's outline a general strategy for solving Ideal Gas Law problems:

1. Identify the knowns and unknowns: Carefully read the problem and determine what values are given and what needs to be calculated.
2. Choose the appropriate value of R: Select the ideal gas constant (R) that matches the units used in the problem.
3. Convert units: Ensure all values are in consistent units that match the chosen R value. This is a crucial step to avoid errors.
4. Apply the Ideal Gas Law: Substitute the known values into the equation PV = nRT and solve for the unknown variable.
5. Check your answer: Does the answer make sense in the context of the problem? Consider the units and the magnitude of the values.

Practice Problems: From Basic to Advanced

Let's work through a series of practice problems, starting with simpler examples and progressing to more challenging ones.

Problem 1: Basic Application

A sample of gas occupies 5.00 L at a pressure of 1.20 atm and a temperature of 25°C. How many moles of gas are present?

Solution:

1. Knowns: V = 5.00 L, P = 1.20 atm, T = 25°C + 273.15 = 298.15 K. Unknown: n.
2. R: We'll use R = 0.0821 L·atm/mol·K since our units are L, atm, and K.
3. Units: All units are consistent.
4. Ideal Gas Law: n = PV/RT = (1.20 atm * 5.00 L) / (0.0821 L·atm/mol·K * 298.15 K) ≈ 0.245 mol
5. Check: The number of moles is reasonable for a sample of gas at these conditions.

Problem 2: Calculating Pressure

A 2.50 mol sample of nitrogen gas (N₂) is contained in a 5.00 L vessel at 27°C. What is the pressure of the gas?

Solution:

1. Knowns: n = 2.50 mol, V = 5.00 L, T = 27°C + 273.15 = 300.15 K. Unknown: P.
2. R: R = 0.0821 L·atm/mol·K.
3. Units: All units are consistent.
4. Ideal Gas Law: P = nRT/V = (2.50 mol * 0.0821 L·atm/mol·K * 300.15 K) / 5.00 L ≈ 12.3 atm
5. Check: This pressure is relatively high, but it's plausible for a significant amount of gas in a relatively small volume.

Problem 3: Calculating Volume

0.75 moles of oxygen gas (O₂) are at a pressure of 1.10 atm and a temperature of 15°C. What is the volume of the gas?

Solution:

1. Knowns: n = 0.75 mol, P = 1.10 atm, T = 15°C + 273.15 = 288.15 K. Unknown: V.
2. R: R = 0.0821 L·atm/mol·K.
3. Units: All units are consistent.
4. Ideal Gas Law: V = nRT/P = (0.75 mol * 0.0821 L·atm/mol·K * 288.15 K) / 1.10 atm ≈ 16.1 L
5. Check: The volume is reasonable for the given amount of gas.

Problem 4: Calculating Temperature

A balloon filled with helium gas has a volume of 2.00 L at 25°C and 1.00 atm pressure. What is the temperature of the gas if the balloon is compressed to a volume of 1.50 L at the same pressure?

Solution:

1. Knowns: V₁ = 2.00 L, T₁ = 25°C + 273.15 = 298.15 K, P₁ = 1.00 atm, V₂ = 1.50 L, P₂ = 1.00 atm. Unknown: T₂.
2. R: The value of R is not explicitly needed since we can use the combined gas law (a derived form of the Ideal Gas Law): (P₁V₁)/T₁ = (P₂V₂)/T₂. Since pressure is constant, this simplifies to V₁/T₁ = V₂/T₂.
3. Units: All units are consistent.
4. Combined Gas Law: T₂ = (V₂/V₁) * T₁ = (1.50 L / 2.00 L) * 298.15 K ≈ 223.6 K (Convert back to Celsius: 223.6 K - 273.15 = -49.6°C)
5. Check: The temperature decrease makes sense given the volume decrease.

Problem 5: Gas Mixtures

A container holds 1.00 mol of He and 2.00 mol of Ne at a total pressure of 6.00 atm and a temperature of 25°C. What is the partial pressure of He?

Solution:

This problem introduces Dalton's Law of Partial Pressures, which states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the individual gases. First, we find the total number of moles: n_total = 1.00 mol + 2.00 mol = 3.00 mol. Then, we use the Ideal Gas Law to find the total volume: V = n_totalRT/P_total = (3.00 mol * 0.0821 L·atm/mol·K * 298.15 K) / 6.00 atm ≈ 12.2 L. Finally, we use the Ideal Gas Law again to find the partial pressure of He: P_He = n_HeRT/V = (1.00 mol * 0.0821 L·atm/mol·K * 298.15 K) / 12.2 L ≈ 2.00 atm. The partial pressure of He is 2.00 atm.

Problem 6: Ideal Gas Law with Density

Calculate the density of carbon dioxide (CO₂) at 27°C and 1.00 atm pressure. The molar mass of CO₂ is 44.01 g/mol.

Solution:

Density (ρ) is mass (m) per unit volume (V): ρ = m/V. We can rearrange the Ideal Gas Law to solve for n/V: n/V = P/RT. Since n = m/M (where M is molar mass), we can substitute this into the equation to get: (m/M)/V = P/RT. This simplifies to ρ = (PM)/RT. Plugging in the values: ρ = (1.00 atm * 44.01 g/mol) / (0.0821 L·atm/mol·K * 300.15 K) ≈ 1.79 g/L.

Frequently Asked Questions (FAQ)

Q: What are the limitations of the Ideal Gas Law?

A: The Ideal Gas Law assumes that gas particles have negligible volume and that there are no intermolecular forces between them. These assumptions are not entirely accurate for real gases, especially at high pressures or low temperatures. Real gases deviate from ideal behavior under these conditions.

Q: How do I choose the right value of R?

A: Always choose the value of R that is consistent with the units of pressure, volume, and temperature used in the problem. If you use units other than those listed, you'll need to look up the corresponding R value.

Q: What if I have a mixture of gases?

A: For gas mixtures, use Dalton's Law of Partial Pressures and consider the total number of moles and the partial pressure of each gas.

Q: Can I use the Ideal Gas Law for liquids or solids?

A: No, the Ideal Gas Law applies only to gases.

Conclusion

The Ideal Gas Law is a fundamental concept in chemistry and physics. By understanding its principles and practicing problem-solving techniques, you can confidently approach a wide range of gas-related calculations. Remember to always carefully consider the units, choose the appropriate value of R, and check your answer for reasonableness. With practice, you'll master the Ideal Gas Law and its applications. This guide has provided a thorough introduction and a series of practice problems designed to help you build your skills. Keep practicing, and you'll become proficient in solving even the most complex Ideal Gas Law problems. Remember to consult your textbook or other learning resources for additional examples and explanations. Good luck!