Combined Gas Law Sample Problems

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

The combined gas law is a cornerstone of chemistry and physics, elegantly combining the principles of Boyle's, Charles's, and Gay-Lussac's laws to describe the relationship between pressure, volume, and temperature of a fixed amount of gas. Understanding this law is crucial for various applications, from predicting weather patterns to designing efficient engines. This comprehensive guide will walk you through the combined gas law, explaining its derivation, providing numerous sample problems with detailed solutions, and addressing frequently asked questions.

Understanding the Combined Gas Law

The combined gas law 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 represented as:

(P₁V₁)/T₁ = (P₂V₂)/T₂

Where:

• P₁ and P₂ represent the initial and final pressures, respectively.
• V₁ and V₂ represent the initial and final volumes, respectively.
• T₁ and T₂ represent the initial and final absolute temperatures (always in Kelvin).

It's crucial to remember that temperature must always be expressed in Kelvin. To convert Celsius to Kelvin, add 273.15: K = °C + 273.15

This law assumes the amount of gas remains constant throughout the process and that the gas behaves ideally (obeying the ideal gas law). Real gases may deviate from this law at high pressures and low temperatures.

Derivation of the Combined Gas Law

The combined gas law is a direct consequence of Boyle's, Charles's, and Gay-Lussac's laws:

• Boyle's Law: At constant temperature, the volume of a gas is inversely proportional to its pressure (V ∝ 1/P).
• Charles's Law: At constant pressure, the volume of a gas is directly proportional to its absolute temperature (V ∝ T).
• Gay-Lussac's Law: At constant volume, the pressure of a gas is directly proportional to its absolute temperature (P ∝ T).

Combining these laws, we get: V ∝ T/P. Introducing a constant of proportionality (k), we arrive at the equation PV/T = k. For two different states of the same gas, we have: (P₁V₁)/T₁ = (P₂V₂)/T₂. This is the combined gas law.

Sample Problems and Solutions

Let's delve into several sample problems to solidify your understanding of the combined gas law.

Problem 1: Simple Pressure Change

A gas sample occupies 5.00 L at a pressure of 1.00 atm and a temperature of 25°C. If the pressure is increased to 2.50 atm while the temperature remains constant, what will be the new volume?

Solution:

1. Convert temperature to Kelvin: T₁ = 25°C + 273.15 = 298.15 K (T₂ remains the same since the temperature is constant).
2. Apply the combined gas law: (P₁V₁)/T₁ = (P₂V₂)/T₂
3. Plug in the values: (1.00 atm * 5.00 L) / 298.15 K = (2.50 atm * V₂) / 298.15 K
4. Solve for V₂: V₂ = (1.00 atm * 5.00 L * 298.15 K) / (2.50 atm * 298.15 K) = 2.00 L

Therefore, the new volume is 2.00 L.

Problem 2: Changes in Pressure, Volume, and Temperature

A gas occupies a volume of 2.00 L at a pressure of 1.50 atm and a temperature of 27°C. If the pressure is changed to 2.00 atm and the temperature is increased to 127°C, what will be the new volume?

Solution:

1. Convert temperatures to Kelvin: T₁ = 27°C + 273.15 = 300.15 K; T₂ = 127°C + 273.15 = 400.15 K
2. Apply the combined gas law: (P₁V₁)/T₁ = (P₂V₂)/T₂
3. Plug in the values: (1.50 atm * 2.00 L) / 300.15 K = (2.00 atm * V₂) / 400.15 K
4. Solve for V₂: V₂ = (1.50 atm * 2.00 L * 400.15 K) / (2.00 atm * 300.15 K) = 2.00 L

Therefore, the new volume will be approximately 2.00 L.

Problem 3: Finding the Initial Pressure

A gas sample has a volume of 3.00 L at a temperature of 300 K and a pressure of 1.20 atm. The volume is decreased to 2.00 L and the temperature is increased to 360 K. What is the new pressure?

Solution:

1. Apply the combined gas law: (P₁V₁)/T₁ = (P₂V₂)/T₂
2. Plug in the values: (1.20 atm * 3.00 L) / 300 K = (P₂ * 2.00 L) / 360 K
3. Solve for P₂: P₂ = (1.20 atm * 3.00 L * 360 K) / (2.00 L * 300 K) = 2.16 atm

The new pressure is 2.16 atm.

Problem 4: A more complex scenario involving all three variables

A balloon filled with helium has a volume of 1.5 L at 25°C and 1 atm. It is then heated to 50°C and taken to a mountain where the pressure is 0.8 atm. What is the new volume of the balloon?

Solution:

1. Convert temperatures to Kelvin: T₁ = 25°C + 273.15 = 298.15 K; T₂ = 50°C + 273.15 = 323.15 K
2. Apply the combined gas law: (P₁V₁)/T₁ = (P₂V₂)/T₂
3. Plug in the values: (1 atm * 1.5 L) / 298.15 K = (0.8 atm * V₂) / 323.15 K
4. Solve for V₂: V₂ = (1 atm * 1.5 L * 323.15 K) / (0.8 atm * 298.15 K) ≈ 2.04 L

The new volume of the balloon is approximately 2.04 L.

Problem 5: A Real-World Application

A scuba diver's tank contains 10 L of air at 200 atm and 20°C. When the diver is underwater at a depth where the pressure is 3 atm and the temperature is 10°C, what is the volume of the air?

Solution:

1. Convert temperatures to Kelvin: T₁ = 20°C + 273.15 = 293.15 K; T₂ = 10°C + 273.15 = 283.15 K
2. Apply the combined gas law: (P₁V₁)/T₁ = (P₂V₂)/T₂
3. Plug in the values: (200 atm * 10 L) / 293.15 K = (3 atm * V₂) / 283.15 K
4. Solve for V₂: V₂ = (200 atm * 10 L * 283.15 K) / (3 atm * 293.15 K) ≈ 642 L

The volume of air expands to approximately 642 L at depth, which is why scuba divers need appropriately sized tanks.

Explaining the Science Behind the Combined Gas Law

The combined gas law is a direct consequence of the kinetic molecular theory of gases. This theory postulates that gases consist of tiny particles in constant random motion. Pressure is a result of these particles colliding with the container walls. Volume is the space occupied by these particles, and temperature is a measure of the average kinetic energy of these particles.

The combined gas law reflects how changes in pressure, volume, and temperature affect the kinetic energy of gas particles and their interactions. For example, increasing the temperature increases the kinetic energy, leading to more frequent and forceful collisions with the container walls, thus increasing pressure or volume (or both, depending on the constraints). Increasing pressure compresses the gas, reducing its volume.

Frequently Asked Questions (FAQ)

• Q: What happens if one of the variables remains constant?

  A: 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 the pressure is constant, it becomes Charles's Law; and if the volume is constant, it becomes Gay-Lussac's Law.

• Q: Why must temperature be in Kelvin?

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

• Q: Does the combined gas law apply to all gases?

  A: The combined gas law is most accurate for ideal gases. Real gases deviate from ideal behavior at high pressures and low temperatures due to intermolecular forces and the finite volume of gas molecules.

Conclusion

The combined gas law is a powerful tool for understanding and predicting the behavior of gases. By mastering this law and its application, you gain a deeper understanding of the fundamental principles governing the physical world. This comprehensive guide, with its detailed examples and explanations, equips you to confidently solve various problems involving pressure, volume, and temperature changes in gases. Remember to always practice and apply the principles to solidify your understanding and develop problem-solving skills. Consistent practice is key to mastering this crucial concept in chemistry and physics.