Adiabatic Expansion - proof of PV^(gamma) = constant

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SUMMARY

The discussion focuses on the derivation of the condition for adiabatic processes, specifically the equation PV^(gamma) = constant. The proof utilizes the ideal gas law, expressed as PV = nRT, and explores the differential form PdV + VdP = nRdT. This relationship arises from the application of the product rule in calculus, where the derivative of the product of variables is expressed as d(PV) = VdP + PdV. The participants clarify the mathematical foundation behind this derivation, emphasizing its significance in understanding adiabatic processes.

PREREQUISITES
  • Understanding of the ideal gas law (PV = nRT)
  • Familiarity with calculus, specifically the product rule for derivatives
  • Knowledge of thermodynamic processes, particularly adiabatic processes
  • Basic physics concepts related to pressure, volume, and temperature
NEXT STEPS
  • Study the derivation of the adiabatic condition in thermodynamics
  • Explore the implications of the ideal gas law in different thermodynamic processes
  • Learn about the concept of gamma (γ) in relation to specific heat capacities
  • Investigate the applications of adiabatic processes in real-world systems
USEFUL FOR

Students of physics, particularly those studying thermodynamics, educators teaching these concepts, and professionals in engineering fields focusing on heat transfer and energy systems.

CAF123
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Hi,

I was looking at the proof for the derivation of the condition satisfied by adiabatic processes. (The proof can be found in many introductory physics textbooks, I am using Principles of Physics HRW 9th ed.) At some point , they say 'For an ideal gas PV=nRT and if P,V T are allowed to take on small variations we have that PdV + VdP = nRdT'. Where does the part in bold come from, specifically the PdV +VdP?

Sorry if I have overlooked something obvious.
 
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The derivative of product of variables. d(PV)=VdP +PdV
 

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