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23. The Second Law of Thermodynamics
The Carnot Cycle
9:11 minutes
Textbook Question
Textbook Question(II) (a) Show that the work done by a Carnot engine is equal to the area enclosed by the Carnot cycle on a PV diagram, Fig. 20–5. (See Section 19–7.)
(b) Generalize this to any reversible cycle.
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Verified step by step guidance
1
Identify the four processes of the Carnot cycle on the PV diagram: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression.
Understand that the work done in each process is represented by the area under the curve for that process on the PV diagram. For isothermal processes, use the formula $W = nRT imes ext{ln}(rac{V_f}{V_i})$, and for adiabatic processes, use $W = rac{P_fV_f - P_iV_i}{ ext{gamma} - 1}$, where gamma is the heat capacity ratio.
Calculate the total work done by the engine by summing the work done in each of the four processes. Note that the work done during the expansion phases will be positive and the work done during the compression phases will be negative.
Visualize the total work done by the Carnot engine as the area enclosed by the cycle on the PV diagram. This area can be geometrically interpreted as the net area enclosed by the paths of the four processes.
Generalize this concept to any reversible cycle by understanding that the work done in any reversible cycle is also represented by the area enclosed by the cycle on the PV diagram. This is because the integral of $PdV$ over the cycle path gives the net work done, which geometrically corresponds to the enclosed area.
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Textbook Question
A Carnot heat engine uses a hot reservoir consisting of a large amount of boiling water and a cold reservoir consisting of a large tub of ice and water. In 5 minutes of operation, the heat rejected by the engine melts 0.0400 kg of ice. During this time, how much work W is performed by the engine?
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