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📐 Carnot Heat Engine — Complete Theory
🔁 The Carnot Cycle
The Carnot engine is an idealized thermodynamic cycle that provides the maximum possible efficiency between two temperature reservoirs. It consists of four reversible processes:
- 1→2 (Isothermal Expansion): Heat absorbed QH from hot reservoir at TH, volume increases.
- 2→3 (Adiabatic Expansion): No heat exchange, temperature drops from TH to TC.
- 3→4 (Isothermal Compression): Heat rejected QC to cold reservoir at TC, volume decreases.
- 4→1 (Adiabatic Compression): No heat exchange, temperature rises from TC back to TH.
⚡ Key Equations & Formulas
Ideal Gas Law: PV = nRT
Isothermal Process (ΔU=0): Q = W = nRT ln(V2/V1)
Adiabatic Process (Q=0): TVγ-1 = constant , PVγ = constant , γ = Cp/Cv = 5/3 (monatomic)
Heat Absorbed (Isothermal Expansion a→b): QH = nRTH ln(Vb/Va)
Heat Rejected (Isothermal Compression c→d): QC = nRTC ln(Vc/Vd) = nRTC ln(Vb/Va) (since Vc/Vd = Vb/Va)
Net Work Done: Wnet = QH - QC
Carnot Efficiency (Maximum): ηmax = 1 - TC/TH (Absolute temperatures)
Clausius Theorem (Reversible Cycle): ∮ dQ/T = 0
Adiabatic Relations for Volumes: Vc = Vb (TH/TC)1/(γ-1) , Vd = Va (TH/TC)1/(γ-1)
📈 P-V Diagram & Work
The area enclosed by the cycle on the P-V diagram equals the net work done. The cycle consists of two isotherms (curves at constant T) and two adiabats (steeper curves). Efficiency depends only on the temperature ratio: η = 1 - TC/TH.
🌍 Real-World Implications
No real engine can exceed Carnot efficiency due to the Second Law of Thermodynamics. Carnot's theorem establishes the upper bound for thermal engines.
🧪 Variables used in this simulation
n = 1 mol, R = 8.314 J/(mol·K), γ = 1.67. Sliders allow you to explore how TH, TC, Va, Vb affect QH, QC, work & efficiency.
