# Heat Engine with Finite Heat Capacity- Is my answer correct?

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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?

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