Combined Gas Law Sample Problem

Mastering the Combined Gas Law: A Comprehensive Guide with Sample Problems

The combined gas law is a fundamental concept in chemistry and physics, describing the relationship between pressure, volume, and temperature of a fixed amount of gas. Understanding this law is crucial for solving various problems related to gas behavior. This article provides a comprehensive guide to the combined gas law, including its derivation, applications, and detailed solutions to sample problems of varying complexity. We'll delve into the theory, offer practical examples, and address frequently asked questions, equipping you with a solid understanding of this important scientific principle.

Understanding the Combined Gas Law

The combined gas law elegantly combines Boyle's Law, Charles's Law, and Gay-Lussac's Law into a single 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, it's 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 (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 (in Kelvin)

It's crucial to remember that temperature must always be expressed in Kelvin. To convert Celsius to Kelvin, add 273.15 to the Celsius temperature (K = °C + 273.15). Failing to do this will lead to inaccurate results.

Derivation of the Combined Gas Law

The combined gas law isn't arbitrarily created; it's a direct consequence of the individual gas laws:

• Boyle's Law: At constant temperature, the volume of a gas is inversely proportional to its pressure (P₁V₁ = P₂V₂).
• Charles's Law: At constant pressure, the volume of a gas is directly proportional to its absolute temperature (V₁/T₁ = V₂/T₂).
• Gay-Lussac's Law: At constant volume, the pressure of a gas is directly proportional to its absolute temperature (P₁/T₁ = P₂/T₂).

By combining these laws, we arrive at the combined gas law equation. For instance, starting with Boyle's law and incorporating the temperature proportionality from Charles's law, we get the combined gas law equation. This demonstrates the interconnectedness of these fundamental gas laws.

Solving Sample Problems: A Step-by-Step Approach

Let's work through several sample problems to illustrate the application of the combined gas law. We'll break down each problem into a series of clear steps, making the solution process easy to follow.

Problem 1: Simple Pressure-Volume-Temperature Change

A sample of gas has a volume of 2.50 L at 25°C and 1.00 atm pressure. What will be its volume if the temperature is increased to 50°C and the pressure is increased to 1.50 atm?

Step 1: Convert Celsius to Kelvin

T₁ = 25°C + 273.15 = 298.15 K T₂ = 50°C + 273.15 = 323.15 K

Step 2: Apply the Combined Gas Law

(P₁V₁)/T₁ = (P₂V₂)/T₂ (1.00 atm * 2.50 L) / 298.15 K = (1.50 atm * V₂) / 323.15 K

Step 3: Solve for V₂

V₂ = (1.00 atm * 2.50 L * 323.15 K) / (298.15 K * 1.50 atm) V₂ ≈ 1.81 L

Therefore, the final volume of the gas will be approximately 1.81 L.

Problem 2: Finding the Final Pressure

A gas occupies 5.00 L at 20°C and 760 mmHg. If the volume is reduced to 2.00 L and the temperature is increased to 100°C, what is the new pressure?

Step 1: Convert Celsius to Kelvin and Units to atm

T₁ = 20°C + 273.15 = 293.15 K T₂ = 100°C + 273.15 = 373.15 K P₁ = 760 mmHg * (1 atm / 760 mmHg) = 1 atm

Step 2: Apply the Combined Gas Law

(P₁V₁)/T₁ = (P₂V₂)/T₂ (1 atm * 5.00 L) / 293.15 K = (P₂ * 2.00 L) / 373.15 K

Step 3: Solve for P₂

P₂ = (1 atm * 5.00 L * 373.15 K) / (293.15 K * 2.00 L) P₂ ≈ 3.18 atm

The new pressure of the gas will be approximately 3.18 atm.

Problem 3: A More Complex Scenario

A balloon filled with helium gas has a volume of 10.0 L at 25°C and 1 atm. If the balloon rises to an altitude where the temperature is -20°C and the pressure is 0.5 atm, what is the new volume of the balloon?

Step 1: Convert Celsius to Kelvin

T₁ = 25°C + 273.15 = 298.15 K T₂ = -20°C + 273.15 = 253.15 K

Step 2: Apply the Combined Gas Law

(P₁V₁)/T₁ = (P₂V₂)/T₂ (1 atm * 10.0 L) / 298.15 K = (0.5 atm * V₂) / 253.15 K

Step 3: Solve for V₂

V₂ = (1 atm * 10.0 L * 253.15 K) / (298.15 K * 0.5 atm) V₂ ≈ 17.0 L

The new volume of the balloon at higher altitude will be approximately 17.0 L. This demonstrates how changes in pressure and temperature significantly affect gas volume.

Limitations of the Combined Gas Law

While the combined gas law is incredibly useful, it's important to acknowledge its limitations:

• Ideal Gas Behavior: The combined gas law assumes the gas behaves ideally. At high pressures or low temperatures, real gases deviate from ideal behavior, and the combined gas law may not provide accurate results.
• Constant Amount of Gas: The law applies only when the amount of gas remains constant. If gas is added or removed, the equation is no longer valid.
• No Chemical Reactions: The law doesn't account for chemical reactions that might occur within the gas system.

For situations involving non-ideal gases or significant changes in the amount of gas, more sophisticated equations of state, such as the van der Waals equation, may be necessary.

Frequently Asked Questions (FAQ)

Q1: Why must temperature be in Kelvin?

A1: Kelvin is an absolute temperature scale, meaning it starts at absolute zero (0 K), the theoretical point where all molecular motion ceases. Using Celsius or Fahrenheit would lead to inaccurate results because these scales have arbitrary zero points.

Q2: What happens if one of the variables remains constant?

A2: If one variable remains constant, the combined gas law simplifies to one of the individual gas laws. For example, if the temperature is constant, it becomes Boyle's Law; if pressure is constant, it becomes Charles's Law; and if volume is constant, it becomes Gay-Lussac's Law.

Q3: Can the combined gas law be used for mixtures of gases?

A3: The combined gas law can be applied to mixtures of gases, provided the gases behave ideally and don't react with each other. The total pressure of the mixture should be used in the calculation.

Q4: How do I handle units in the combined gas law calculations?

A4: Consistency is key. Make sure all pressure units are the same (e.g., atm, mmHg, Pa), all volume units are the same (e.g., L, mL, m³), and temperature is always in Kelvin.

Conclusion

The combined gas law provides a powerful tool for understanding and predicting the behavior of gases under various conditions. By mastering this law and its applications, you gain valuable insights into the fundamental principles of thermodynamics. Remember to always convert temperatures to Kelvin and maintain unit consistency for accurate results. Through careful application and understanding of its limitations, the combined gas law can solve a wide array of problems in chemistry and physics. Practice solving various problems using the steps outlined above to strengthen your understanding and problem-solving skills. The more you practice, the more confident and proficient you will become in applying this essential scientific principle.