Ideal Gas Equation Practice Problems

Mastering the Ideal Gas Equation: Practice Problems and Solutions

The Ideal Gas Law, PV = nRT, is a cornerstone of chemistry and physics. Understanding and applying this equation is crucial for tackling numerous problems involving gases. This article provides a comprehensive guide, working through various practice problems of increasing complexity, to solidify your understanding of the Ideal Gas Law and its applications. We'll cover various scenarios, focusing on problem-solving strategies and explaining the underlying scientific principles. Whether you're a high school student, an undergraduate, or simply looking to refresh your knowledge, this guide will equip you with the tools to confidently tackle ideal gas equation problems.

Understanding the Ideal Gas Law: A Quick Recap

Before diving into the problems, let's briefly review the Ideal Gas Law:

• P: Pressure (usually in atmospheres, atm, or Pascals, Pa)
• V: Volume (usually in liters, L, or cubic meters, m³)
• n: Number of moles (mol)
• R: Ideal gas constant (0.0821 L·atm/mol·K or 8.314 J/mol·K, depending on the units used)
• T: Temperature (always in Kelvin, K; K = °C + 273.15)

The Ideal Gas Law describes the behavior of an ideal gas, a theoretical gas whose particles have negligible volume and do not interact with each other. While real gases deviate from ideal behavior under certain conditions (high pressure, low temperature), the Ideal Gas Law provides a good approximation for many practical applications.

Practice Problems: From Basic to Advanced

Let's now tackle a range of practice problems. Remember to always identify the known and unknown variables before applying the Ideal Gas Law.

Problem 1: Basic Application

A sample of nitrogen gas (N₂) occupies a volume of 5.00 L at a pressure of 1.00 atm and a temperature of 25°C. How many moles of nitrogen gas are present?

Solution:

1. Identify knowns: V = 5.00 L, P = 1.00 atm, T = 25°C + 273.15 = 298.15 K. R = 0.0821 L·atm/mol·K.
2. Identify unknown: n (moles of N₂)
3. Apply the Ideal Gas Law: PV = nRT => n = PV/RT
4. Substitute and solve: n = (1.00 atm × 5.00 L) / (0.0821 L·atm/mol·K × 298.15 K) ≈ 0.204 mol

Therefore, approximately 0.204 moles of nitrogen gas are present.

Problem 2: Determining Pressure

A 2.50 mol sample of oxygen gas (O₂) is contained in a 10.0 L vessel at 27°C. What is the pressure of the gas in atmospheres?

Solution:

1. Identify knowns: n = 2.50 mol, V = 10.0 L, T = 27°C + 273.15 = 300.15 K, R = 0.0821 L·atm/mol·K.
2. Identify unknown: P (pressure)
3. Apply the Ideal Gas Law: PV = nRT => P = nRT/V
4. Substitute and solve: P = (2.50 mol × 0.0821 L·atm/mol·K × 300.15 K) / 10.0 L ≈ 6.16 atm

The pressure of the oxygen gas is approximately 6.16 atm.

Problem 3: Finding Volume at a Different Temperature

A balloon filled with helium gas has a volume of 2.50 L at 20°C and 1.00 atm. What will be its volume if the temperature is increased to 35°C while the pressure remains constant? (This problem showcases Charles's Law, a direct application of the Ideal Gas Law).

Solution:

This problem can be solved using the following ratio derived from the Ideal Gas Law, keeping pressure and moles constant:

V₁/T₁ = V₂/T₂

1. Identify knowns: V₁ = 2.50 L, T₁ = 20°C + 273.15 = 293.15 K, T₂ = 35°C + 273.15 = 308.15 K.
2. Identify unknown: V₂ (final volume)
3. Solve for V₂: V₂ = V₁T₂/T₁ = (2.50 L × 308.15 K) / 293.15 K ≈ 2.63 L

The volume of the balloon will increase to approximately 2.63 L.

Problem 4: Gas Mixtures and Partial Pressures (Dalton's Law)

A container holds a mixture of 0.50 mol of nitrogen gas and 0.75 mol of oxygen gas at a total pressure of 2.00 atm and a temperature of 25°C. What is the partial pressure of each gas?

Solution:

Dalton's Law of Partial Pressures states that the total pressure of a mixture of gases is the sum of the partial pressures of the individual gases. The partial pressure of a gas is the pressure that gas would exert if it occupied the container alone. We can use the mole fraction to determine the partial pressure.

1. Total moles: n_total = 0.50 mol + 0.75 mol = 1.25 mol
2. Mole fractions:
   • X_N₂ = n_N₂ / n_total = 0.50 mol / 1.25 mol = 0.40
   • X_O₂ = n_O₂ / n_total = 0.75 mol / 1.25 mol = 0.60
3. Partial pressures:
   • P_N₂ = X_N₂ × P_total = 0.40 × 2.00 atm = 0.80 atm
   • P_O₂ = X_O₂ × P_total = 0.60 × 2.00 atm = 1.20 atm

The partial pressure of nitrogen gas is 0.80 atm, and the partial pressure of oxygen gas is 1.20 atm.

Problem 5: Molar Mass Determination

A 0.250 g sample of an unknown gas occupies 150 mL at 27°C and 750 mmHg. What is the molar mass of the gas? (Remember to convert pressure to atm and volume to L).

Solution:

1. Convert units: P = 750 mmHg × (1 atm / 760 mmHg) ≈ 0.987 atm; V = 150 mL × (1 L / 1000 mL) = 0.150 L; T = 27°C + 273.15 = 300.15 K
2. Use the Ideal Gas Law to find moles: n = PV/RT = (0.987 atm × 0.150 L) / (0.0821 L·atm/mol·K × 300.15 K) ≈ 0.00599 mol
3. Calculate molar mass: Molar mass (g/mol) = mass (g) / moles (mol) = 0.250 g / 0.00599 mol ≈ 41.8 g/mol

The molar mass of the unknown gas is approximately 41.8 g/mol.

Problem 6: Stoichiometry and the Ideal Gas Law

What volume of carbon dioxide gas (CO₂) at STP (Standard Temperature and Pressure: 0°C and 1 atm) is produced when 10.0 g of calcium carbonate (CaCO₃) reacts completely with excess hydrochloric acid (HCl) according to the following balanced equation?

CaCO₃(s) + 2HCl(aq) → CaCl₂(aq) + H₂O(l) + CO₂(g)

Solution:

1. Convert grams of CaCO₃ to moles: Molar mass of CaCO₃ ≈ 100.1 g/mol; Moles of CaCO₃ = 10.0 g / 100.1 g/mol ≈ 0.0999 mol
2. Use stoichiometry: From the balanced equation, 1 mol of CaCO₃ produces 1 mol of CO₂. Therefore, 0.0999 mol of CaCO₃ produces 0.0999 mol of CO₂.
3. Apply Ideal Gas Law at STP: P = 1 atm, T = 0°C + 273.15 = 273.15 K, n = 0.0999 mol, R = 0.0821 L·atm/mol·K
4. Solve for V: V = nRT/P = (0.0999 mol × 0.0821 L·atm/mol·K × 273.15 K) / 1 atm ≈ 2.24 L

Approximately 2.24 L of carbon dioxide gas is produced at STP.

Frequently Asked Questions (FAQ)

Q1: What are the limitations of the Ideal Gas Law?

The Ideal Gas Law assumes that gas particles have negligible volume and do not interact with each other. This is not true for real gases, especially at high pressures and low temperatures where intermolecular forces become significant and the volume of gas molecules becomes a larger fraction of the total volume. Under these conditions, real gases deviate from ideal behavior, and more complex equations are needed to accurately describe their behavior.

Q2: Why is temperature always in Kelvin?

The Kelvin scale is an absolute temperature scale, meaning it starts at absolute zero (0 K), the theoretical point where all molecular motion ceases. Using Kelvin ensures consistent and accurate results when applying the Ideal Gas Law because calculations involving temperature ratios or differences are directly proportional. Using Celsius or Fahrenheit would introduce inconsistencies.

Q3: What if I have a mixture of gases with different molar masses?

When dealing with a mixture of gases, you can either use the total number of moles (as shown in Problem 4) or calculate the partial pressure of each gas individually using its mole fraction and then apply the ideal gas law to each component. Both approaches yield the same total pressure.

Q4: How can I improve my problem-solving skills with the Ideal Gas Law?

Practice is key! Work through many problems of varying difficulty, making sure you understand the underlying principles. Start with basic problems and gradually increase the complexity. Pay close attention to units and always convert to the correct units before applying the Ideal Gas Law. Also, try visualizing the scenario described in the problem to better grasp the concepts involved.

Conclusion

The Ideal Gas Law is a fundamental concept with wide-ranging applications. By mastering this equation and understanding its limitations, you can tackle a diverse set of problems related to gases. Remember to always identify the knowns and unknowns, convert units appropriately, and choose the correct value of the ideal gas constant. Consistent practice and careful attention to detail are crucial for success in solving ideal gas law problems. This comprehensive guide, replete with solved examples, provides a solid foundation for your understanding and application of this vital scientific principle. Continue practicing and you’ll master this important tool in your scientific arsenal!