Mastering the Combined Gas Law: A Comprehensive Guide with Worked Examples
The combined gas law is a crucial concept in chemistry, bringing together Boyle's, Charles's, and Gay-Lussac's laws to describe the relationship between pressure, volume, and temperature of a gas. Understanding this law is essential for solving a wide range of gas-related problems. This guide will walk you through the combined gas law, answer common questions, and provide worked examples to solidify your understanding.
The Combined Gas Law Equation:
The combined gas law is expressed mathematically as:
(P₁V₁)/T₁ = (P₂V₂)/T₂
Where:
- P₁ = initial pressure
- V₁ = initial volume
- T₁ = initial temperature (in Kelvin!)
- P₂ = final pressure
- V₂ = final volume
- T₂ = final temperature (in Kelvin!)
Remember: Temperature must always be in Kelvin (K). To convert from Celsius (°C) to Kelvin, add 273.15: K = °C + 273.15
Understanding the Relationship Between Pressure, Volume, and Temperature
The combined gas law highlights the interconnectedness of a gas's pressure, volume, and temperature. If one variable changes, the others will adjust accordingly, assuming the amount of gas remains constant.
-
Direct Relationship: Temperature and volume have a direct relationship (Charles's Law). As temperature increases, volume increases, and vice-versa, keeping pressure constant.
-
Inverse Relationship: Pressure and volume have an inverse relationship (Boyle's Law). As pressure increases, volume decreases, and vice-versa, keeping temperature constant.
-
Direct Relationship: Pressure and temperature have a direct relationship (Gay-Lussac's Law). As temperature increases, pressure increases, and vice-versa, keeping volume constant.
The combined gas law allows us to handle scenarios where all three variables change simultaneously.
How to Solve Combined Gas Law Problems: A Step-by-Step Approach
-
Identify the knowns and unknowns: Carefully read the problem and list the given values (P₁, V₁, T₁, P₂, V₂, T₂) and the variable you need to solve for.
-
Convert units: Ensure all temperature values are in Kelvin. Pressure and volume units must be consistent throughout the calculation (e.g., all in atm and L).
-
Rearrange the equation: Solve the combined gas law equation for the unknown variable.
-
Plug in values and solve: Substitute the known values into the rearranged equation and calculate the answer.
-
Check your answer: Does the answer make sense in the context of the problem? For example, if the temperature increases and the pressure is constant, the volume should also increase.
Worked Examples
Let's tackle some examples to illustrate the application of the combined gas law.
Example 1: A gas sample has a volume of 2.00 L at 25°C and 1.00 atm. What will be its volume if the temperature is increased to 50°C and the pressure is increased to 2.00 atm?
-
Knowns: V₁ = 2.00 L, T₁ = 25°C + 273.15 = 298.15 K, P₁ = 1.00 atm, T₂ = 50°C + 273.15 = 323.15 K, P₂ = 2.00 atm. Unknown: V₂
-
Rearrange: V₂ = (P₁V₁T₂)/(P₂T₁)
-
Solve: V₂ = (1.00 atm * 2.00 L * 323.15 K) / (2.00 atm * 298.15 K) = 1.08 L
Example 2: A gas occupies 5.0 L at 20°C and 760 mmHg. If the volume is decreased to 2.5 L and the pressure increased to 1520 mmHg, what will the new temperature be in °C?
-
Knowns: V₁ = 5.0 L, T₁ = 20°C + 273.15 = 293.15 K, P₁ = 760 mmHg, V₂ = 2.5 L, P₂ = 1520 mmHg. Unknown: T₂
-
Rearrange: T₂ = (P₂V₂T₁) / (P₁V₁)
-
Solve: T₂ = (1520 mmHg * 2.5 L * 293.15 K) / (760 mmHg * 5.0 L) = 293.15 K. Converting back to Celsius: 293.15 K - 273.15 = 20°C (Note: This indicates an error may be present in the problem's parameters given the direct relationship between pressure and temperature).
Frequently Asked Questions (FAQs)
What happens if one of the variables remains constant?
If a variable remains constant, you can simplify the combined gas law equation. For example, if the temperature is constant, the equation becomes Boyle's Law: P₁V₁ = P₂V₂. Similarly, if pressure is constant it simplifies to Charles' Law and if volume is constant it simplifies to Gay-Lussac's Law.
Why is it important to use Kelvin for temperature?
Kelvin is an absolute temperature scale, meaning it starts at absolute zero (0 K), where all molecular motion ceases. Using Kelvin ensures accurate calculations because it accounts for the relationship between temperature and molecular kinetic energy. Using Celsius or Fahrenheit would lead to incorrect results.
What if I have a different unit for pressure or volume?
It is crucial to maintain consistency in units throughout your calculations. You will need to convert all units to the same system (e.g., atm for pressure and Liters for volume) before applying the combined gas law.
This guide provides a solid foundation for understanding and applying the combined gas law. Remember to practice with various problems to master this important concept in chemistry. Consistent practice and attention to detail, especially regarding unit conversions and temperature in Kelvin, are key to success.
