If 22.5 L Of Nitrogen At 748 Mm Hg And 273 K Are Compressed To 725 Mm Hg And 50.0 Degree C At Constant pressure, it involves understanding the fundamental principles of gas laws, particularly Boyle's Law and Charles's Law, and how they apply to the behavior of gases during compression and temperature changes. This scenario provides an excellent opportunity to explore the relationships between pressure, volume, temperature, and amount of gas, as well as to perform calculations based on the combined gas law. Whether you're a student studying chemistry or a professional working in gas-related industries, understanding these principles is key to predicting and controlling gas behavior in various applications.
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Understanding the Fundamentals of Gas Laws
Before delving into the specific problem, it's essential to grasp the core concepts of the gas laws that govern the behavior of gases under different conditions.
Boyle's Law
- States that the pressure of a fixed amount of gas is inversely proportional to its volume at constant temperature.
- Mathematically: \( P1 V1 = P2 V2 \) (at constant temperature and amount of gas).
Charles's Law
- Describes the direct relationship between temperature and volume at constant pressure.
- Expressed as: \( \frac{V1}{T1} = \frac{V2}{T2} \) (at constant pressure).
Gay-Lussac's Law
- Indicates that pressure is directly proportional to temperature when volume is held constant.
- Expressed as: \( \frac{P1}{T1} = \frac{P2}{T2} \).
The Combined Gas Law
- Integrates Boyle's, Charles's, and Gay-Lussac's laws to relate pressure, volume, and temperature simultaneously.
- Mathematically: \( \frac{P1 V1}{T1} = \frac{P2 V2}{T2} \).
Analyzing the Given Scenario
The problem provides the initial conditions of nitrogen gas and asks to understand how the gas's volume and temperature change when compressed under differing pressure conditions.
Initial Conditions
- Volume (\( V_1 \)): 22.5 liters
- Pressure (\( P_1 \)): 748 mm Hg
- Temperature (\( T_1 \)): 273 K
Final Conditions
- Pressure (\( P_2 \)): 725 mm Hg
- Temperature (\( T_2 \)): 50.0°C (which is 323 K when converted to Kelvin)
- Volume (\( V_2 \)): To be determined
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Step-by-Step Solution Using the Combined Gas Law
The core of solving this problem involves applying the combined gas law:
\[
\frac{P1 V1}{T1} = \frac{P2 V2}{T2}
\]
Rearranged to solve for the unknown final volume:
\[
V2 = V1 \times \frac{P1}{P2} \times \frac{T2}{T1}
\]
Let's proceed with detailed steps:
Step 1: Convert all temperatures to Kelvin
- \( T_1 = 273\, \text{K} \)
- \( T_2 = 50.0^\circ C + 273 = 323\, \text{K} \)
Step 2: Substitute known values into the equation
\[ V_2 = 22.5\, \text{L} \times \frac{748\, \text{mm Hg}}{725\, \text{mm Hg}} \times \frac{323\, \text{K}}{273\, \text{K}} \]Step 3: Calculate the ratios
- Pressure ratio: \( \frac{748}{725} \approx 1.031 \)
- Temperature ratio: \( \frac{323}{273} \approx 1.183 \)
Step 4: Final calculation
\[ V_2 \approx 22.5\, \text{L} \times 1.031 \times 1.183 \] \[ V_2 \approx 22.5\, \text{L} \times 1.219 \approx 27.44\, \text{L} \]Result: The final volume of nitrogen after compression and heating is approximately 27.44 liters.
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Implications and Practical Applications
Understanding how gases behave during compression and temperature changes is vital in many real-world applications.
Industrial Gas Storage and Transportation
- Accurate calculations ensure safe and efficient storage of gases.
- Compressed gases like nitrogen are used in food preservation, electronics manufacturing, and medical applications.
Refrigeration and Air Conditioning
- Knowledge of gas laws helps design systems that efficiently manage pressure and temperature changes.
Chemical Reaction Engineering
- Precise control over gas conditions improves reaction outcomes and yields.
Environmental Monitoring
- Predicting gas behavior aids in modeling atmospheric phenomena and pollution dispersion.
Additional Considerations in Gas Law Calculations
While the calculations above provide a straightforward application of the combined gas law, several factors can influence the outcome:
Real Gas Behavior
- Under high pressures or low temperatures, gases deviate from ideal behavior.
- Van der Waals Equation accounts for molecular size and intermolecular forces for more accurate predictions.
Measurement Accuracy
- Precise measurement of pressure, temperature, and volume is crucial.
- Calibration of instruments ensures reliable data.
Effect of Gas Purity
- Impurities can alter the gas's behavior, especially in sensitive applications.
Conclusion
In summary, the process of compressing nitrogen from 22.5 liters at 748 mm Hg and 273 K to a new state at 725 mm Hg and 50.0°C involves understanding and applying the combined gas law. By converting temperatures to Kelvin, calculating the relevant ratios, and substituting into the formula, we find that the final volume of nitrogen is approximately 27.44 liters. This example illustrates the practical application of fundamental gas laws in scientific and industrial contexts, emphasizing the importance of precise calculations and understanding the underlying principles governing gas behavior. Whether for designing industrial processes, ensuring safety standards, or conducting scientific research, mastering these concepts is essential for predicting and controlling the behavior of gases under varying conditions.
