Thermodynamics maximum and minimum temperatures

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SUMMARY

The discussion centers on calculating the maximum and minimum temperatures of an ideal gas during a reversible process around a circular path defined by the equation (V-10)² + (P-10)² = 25. The key conclusion is that the maximum temperature occurs where the product PV is maximized, while the minimum temperature occurs where PV is minimized. The use of the Lagrange Multiplier method is recommended for finding these extrema, as it allows for optimization under constraints. The graph of the circular path in the PV-plane is crucial for visualizing these temperature extremes.

PREREQUISITES
  • Understanding of ideal gas laws, specifically PV = NRT
  • Familiarity with Lagrange Multipliers for optimization
  • Basic knowledge of thermodynamics and temperature concepts
  • Graphical interpretation of equations in the PV-plane
NEXT STEPS
  • Study the application of Lagrange Multipliers in thermodynamic problems
  • Explore graphical methods for visualizing thermodynamic processes
  • Learn about the implications of the ideal gas law in different thermodynamic cycles
  • Investigate the relationship between pressure, volume, and temperature in real gases
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Students and professionals in thermodynamics, physicists, and engineers involved in gas behavior analysis and optimization of thermodynamic processes.

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In the following question,

"two moles of an ideal gas, in an initial state P=10 atm, V=5 liters, are taken reversibly in a clockwise direction around a circular path given by (V-10)^2 + (P-10)^2 =25. Calculate the amount of work done by the gas as a result of the process, and calculate the maximum and minimum temperatures attained by the gas during the cycle."

how do you find the max/min temperatures? at first, i thought of trying to take the first derivative of the path, but the results are weird...
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The graph of (V-10)2+ (P-10)2= 25 is a circle, in the PV-plane, with center at (10,10) and radius 5. Since PV= NRT, T= PV/NR. Maximum temperature occurs on that circle where PV is a maximum, Minimum Temperature where PV is a minimum. It should be clear where that occurs from the graph. If not, maximize (and minimize) PV with the constraint (V-10)2+ (P-10)2= 25 (Lagrange Multiplier method probably is best).
 
during a discussion with a classmate, he suggested to draw a 45 degree angle line through the origin... but why?
 

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