Math Problem Statement
If two separate systems, one containing a monatomic gas, and the other a diatomic gas undergo an adiabatic process from V1 to V2, which would have the lower final pressure, p2? What would be the ratio of the two final pressures? Give your answer in terms of V1 and V2. Assume the gases are at approximately room temperature.
Solution
To compare the final pressures of a monatomic gas and a diatomic gas undergoing an adiabatic process, we can use the following relationship for adiabatic processes:
where . For an ideal monatomic gas, , and for an ideal diatomic gas, .
Final Pressures
For each gas, we can express the initial and final states in terms of pressure, volume, and :
-
Monatomic Gas: Rearranging gives:
-
Diatomic Gas: Rearranging gives:
Comparison of Final Pressures
To find which gas has a lower final pressure, we need to analyze the exponents:
- For the monatomic gas:
- For the diatomic gas:
Since (because and ), the diatomic gas's final pressure will be lower if the same initial pressure is applied to both. Thus:
Ratio of Final Pressures
The ratio of the final pressures can be expressed as:
Calculating the exponent:
Thus, the ratio of the final pressures is:
Conclusion
- The diatomic gas has the lower final pressure .
- The ratio of the final pressures is .
Would you like to explore this further or have any questions? Here are some related topics to consider:
- How does temperature affect the pressure in an adiabatic process?
- What is the significance of the adiabatic process in thermodynamics?
- How do real gases differ from ideal gases in adiabatic processes?
- What are the implications of this analysis in practical applications like engines?
- How does the value of change with temperature or gas type?
Tip: Always ensure to identify the type of gas you're working with as it significantly affects the calculations in thermodynamics.
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Math Problem Analysis
Mathematical Concepts
Thermodynamics
Adiabatic Processes
Gas Laws
Formulas
PV^γ = constant
P2_mono = P1 * (V1/V2)^(5/3)
P2_di = P1 * (V1/V2)^(7/5)
Theorems
Adiabatic Process Theorem
Ideal Gas Law
Suitable Grade Level
Grades 11-12
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