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Introduction to the Combined Gas Law
The combined gas law is a vital concept in chemistry that illustrates the relationship between pressure, volume, and temperature of a fixed amount of gas. It is derived from Boyle’s law, Charles’s law, and Gay-Lussac’s law, providing a unified approach to understanding how these variables interact under different conditions. For students and professionals alike, mastering the combined gas law is essential for solving real-world problems involving gases in various scientific and engineering applications.
In this article, we will explore the fundamentals of the combined gas law, explain how to derive and apply the law, provide step-by-step solution strategies, and include an answer key for common practice problems. Whether you’re studying for an exam or working on a research project, this guide aims to enhance your understanding and confidence in using the combined gas law effectively.
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Understanding the Fundamentals of the Combined Gas Law
What is the Combined Gas Law?
The combined gas law expresses the relationship among pressure (P), volume (V), and temperature (T) for a fixed amount of gas when these variables change. It combines three key gas laws:
- Boyle’s Law: P₁V₁ = P₂V₂ (at constant T)
- Charles’s Law: V₁/T₁ = V₂/T₂ (at constant P)
- Gay-Lussac’s Law: P₁/T₁ = P₂/T₂ (at constant V)
By integrating these, the combined gas law can be written as:
\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]
where:
- \( P_1, V_1, T_1 \) are the initial pressure, volume, and temperature,
- \( P_2, V_2, T_2 \) are the final pressure, volume, and temperature.
Assumptions and Limitations
The combined gas law assumes:
- The gas behaves ideally.
- The amount of gas remains constant.
- Temperatures are in Kelvin.
- No chemical reactions occur that change the amount or type of gas.
Real gases may deviate from ideal behavior under high pressure or low temperature, but the combined gas law remains a valuable approximation for most practical purposes.
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Deriving the Combined Gas Law
The derivation involves combining the three individual laws:
1. Boyle’s Law: \( P_1 V_1 = P_2 V_2 \) (at constant T)
2. Charles’s Law: \( V_1/T_1 = V_2/T_2 \) (at constant P)
3. Gay-Lussac’s Law: \( P_1/T_1 = P_2/T_2 \) (at constant V)
By eliminating variables and focusing on conditions where all three change, we arrive at the combined gas law:
\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]
This formula allows us to calculate the unknown variable when the other five are known, making it a versatile tool in chemistry.
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Applying the Combined Gas Law: Step-by-Step Solution Strategy
To effectively use the combined gas law, follow these steps:
Step 1: Identify Known and Unknown Variables
- Write down all given data: initial pressure, volume, temperature, and the final conditions.
- Label the unknown variable you need to find.
Step 2: Convert Temperatures to Kelvin
Since the law uses absolute temperature, convert Celsius or Fahrenheit to Kelvin:
\[ T(K) = T(°C) + 273.15 \]
Step 3: Plug Values into the Formula
Insert the known values into:
\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]
Ensure units are consistent (e.g., atmospheres for pressure, liters for volume).
Step 4: Solve Algebraically for the Unknown
Rearranged as needed, solve for the variable:
- For pressure: \( P_2 = \frac{P_1 V_1 T_2}{V_2 T_1} \)
- For volume: \( V_2 = \frac{P_1 V_1 T_2}{P_2 T_1} \)
- For temperature: \( T_2 = \frac{P_2 V_2 T_1}{P_1 V_1} \)
Step 5: Perform Calculations and Check Units
- Carefully perform arithmetic operations.
- Confirm units are consistent and convert if necessary.
- Round off to appropriate significant figures.
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Common Practice Problems and Answer Key
Below are sample problems with step-by-step solutions to help reinforce understanding.
Problem 1: Calculating Final Pressure
Given:
- Initial pressure, \( P_1 = 1.00\, \text{atm} \)
- Initial volume, \( V_1 = 10.0\, \text{L} \)
- Initial temperature, \( T_1 = 273\, \text{K} \)
- Final volume, \( V_2 = 20.0\, \text{L} \)
- Final temperature, \( T_2 = 546\, \text{K} \)
- Find: \( P_2 \)
Solution:
Using the combined gas law:
\[ P_2 = \frac{P_1 V_1 T_2}{V_2 T_1} \]
Plug in the values:
\[ P_2 = \frac{(1.00\, \text{atm})(10.0\, \text{L})(546\, \text{K})}{20.0\, \text{L} \times 273\, \text{K}} \]
Calculate numerator:
\[ 1.00 \times 10.0 \times 546 = 5460 \]
Calculate denominator:
\[ 20.0 \times 273 = 5460 \]
Thus,
\[ P_2 = \frac{5460}{5460} = 1.00\, \text{atm} \]
Answer: The final pressure \( P_2 \) is 1.00 atm.
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Problem 2: Determining Final Volume
Given:
- \( P_1 = 2.00\, \text{atm} \)
- \( V_1 = 5.00\, \text{L} \)
- \( T_1 = 300\, \text{K} \)
- \( P_2 = 1.00\, \text{atm} \)
- \( T_2 = 600\, \text{K} \)
- Find: \( V_2 \)
Solution:
Rearranged formula:
\[ V_2 = \frac{P_1 V_1 T_2}{P_2 T_1} \]
Insert values:
\[ V_2 = \frac{2.00\, \text{atm} \times 5.00\, \text{L} \times 600\, \text{K}}{1.00\, \text{atm} \times 300\, \text{K}} \]
Calculate numerator:
\[ 2.00 \times 5.00 \times 600 = 6000 \]
Calculate denominator:
\[ 1.00 \times 300 = 300 \]
Compute:
\[ V_2 = \frac{6000}{300} = 20.0\, \text{L} \]
Answer: The final volume \( V_2 \) is 20.0 L.
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Problem 3: Finding Final Temperature
Given:
- \( P_1 = 1.00\, \text{atm} \)
- \( V_1 = 10.0\, \text{L} \)
- \( T_1 = 273\, \text{K} \)
- \( P_2 = 1.00\, \text{atm} \)
- \( V_2 = 20.0\, \text{L} \)
- Find: \( T_2 \)
Solution:
Rearranged formula:
\[ T_2 = \frac{P_2 V_2 T_1}{P_1 V_1} \]
Substitute values:
\[ T_2 = \frac{1.00\, \text{atm} \times 20.0\, \text{L} \times 273\, \text{K}}{1.00\, \text{atm} \times 10.0\, \text{L}} \]
Calculate numerator:
\[ 1.00 \times 20.0 \times 273 = 5460 \]
Calculate denominator:
\[ 1.00 \times 10.0 = 10.0 \]
Compute:
\[ T_2 = \frac{5460}{10} = 546\, \text{K} \]
Answer: The final temperature
Frequently Asked Questions
What is the combined gas law and how is it used?
The combined gas law relates pressure, volume, and temperature of a gas, showing how they change together when the amount of gas remains constant. It is used to solve problems involving changes in these variables simultaneously.
What is the formula for the combined gas law?
The formula is (P1 × V1) / T1 = (P2 × V2) / T2, where P is pressure, V is volume, T is temperature in Kelvin, and the subscripts 1 and 2 refer to initial and final states.
How do you solve a problem using the combined gas law?
Identify the known values for initial and final states, convert temperatures to Kelvin, then substitute into the formula and solve for the unknown variable.
Why must temperatures be in Kelvin when using the combined gas law?
Because Kelvin is an absolute temperature scale, ensuring proportionality and correct calculations when applying gas laws; using Celsius or Fahrenheit can lead to incorrect results.
Can the combined gas law be used when the amount of gas changes?
No, the combined gas law assumes the amount of gas remains constant. For changing amounts, other laws like the ideal gas law with moles are applicable.
What are common mistakes to avoid when solving combined gas law problems?
Common mistakes include not converting temperatures to Kelvin, mixing units of pressure or volume, and forgetting to use initial and final values correctly. Always double-check units and conversions.
Where can I find answer keys for combined gas law practice problems?
Answer keys can be found in chemistry textbooks, online educational resources, and instructor-provided materials to help verify your solutions and understand problem-solving steps.
