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1. A heat engine is run with a large block of metal as a reservoir with initial temperature T_i and constant heat capacitance C. The ocean is used as a cold reservoir, with constant temperature T_0. What is the maximum work that could be done by the engine in terms of T_i T_0 and C
2. C = \frac{dQ}{dT} For one cycle of the engine: Efficiency E = \frac{dW}{dQ}Here's My attempt: (Can someone please verify if my answer is correct?)
If E = \frac{dW}{dQ} for one cycle
Then:
E = \frac{dW}{CdT}
dW = CE(T)dT where E(T) = formula for Carnot Efficiency = \frac{T - T_0}{T}
Adding up the work done in every cycle for every infinitessimal change in the metal block's temp gives:
W = C\int_{T_0}^{T_i} E(T)dT
W = C\int_{T_0}^{T_i} \frac{T - T_0}{T}dT
W = C\int_{T_0}^{T_i} 1 - \frac{T_0}{T} dT
W = C((T_i- T_0) - T_0\ln(\frac{T_i}{T_0}))
Does this formula look correct?
2. C = \frac{dQ}{dT} For one cycle of the engine: Efficiency E = \frac{dW}{dQ}Here's My attempt: (Can someone please verify if my answer is correct?)
If E = \frac{dW}{dQ} for one cycle
Then:
E = \frac{dW}{CdT}
dW = CE(T)dT where E(T) = formula for Carnot Efficiency = \frac{T - T_0}{T}
Adding up the work done in every cycle for every infinitessimal change in the metal block's temp gives:
W = C\int_{T_0}^{T_i} E(T)dT
W = C\int_{T_0}^{T_i} \frac{T - T_0}{T}dT
W = C\int_{T_0}^{T_i} 1 - \frac{T_0}{T} dT
W = C((T_i- T_0) - T_0\ln(\frac{T_i}{T_0}))
Does this formula look correct?