As shown in the figure, five Carnot engines, each with efficiency \(\eta\) and same number of cycles per unit time, are operating between six heat reservoirs. The amount of heat released per cycle by one engine is completely absorbed by the next engine. Consider \(Q_0\) to be the amount of heat absorbed per cycle by the first engine and \(W\) as the amount of total work done by all the engines per cycle, then the net efficiency of the system is found to be
\[
\eta_{\mathrm{net}}
=
\frac{W}{Q_0}
=
\frac{211}{243}
\]
The value of \(\eta\) is _______.
Show Hint
Rather than adding up each engine's work one by one, think about the chain as a single combined system: heat rejected by one engine is fully picked up by the next, so those transfers cancel out inside the system. Focus only on what heat enters the whole chain and what heat finally leaves it, then connect that difference to the total work done.
Step 1: Understanding the Concept:
A Carnot engine working between two temperatures has an efficiency \( \eta = 1 - \frac{Q_{out}}{Q_{in}} \). When engines are arranged in series such that the heat rejected by one is the input for the next, the overall efficiency of the chain can be expressed as a product of terms related to individual efficiencies. The total work done is the sum of the work from each engine. Step 2: Key Formula or Approach:
For each engine \( i \): \( Q_{i, out} = Q_{i, in} (1 - \eta) \).
For \( N \) engines in series: \( Q_{N, out} = Q_{1, in} (1 - \eta)^N \).
Net efficiency \( \eta_{net} = 1 - \frac{Q_{N, out}}{Q_{1, in}} = 1 - (1 - \eta)^N \). Step 3: Detailed Explanation:
Given \( N = 5 \) engines and \( \eta_{net} = \frac{211}{243} \).
\[ 1 - (1 - \eta)^5 = \frac{211}{243} \]
\[ (1 - \eta)^5 = 1 - \frac{211}{243} = \frac{243 - 211}{243} = \frac{32}{243} \]
We recognize that \( 32 = 2^5 \) and \( 243 = 3^5 \).
\[ (1 - \eta)^5 = \left( \frac{2}{3} \right)^5 \]
Taking the fifth root of both sides:
\[ 1 - \eta = \frac{2}{3} \]
\[ \eta = 1 - \frac{2}{3} = \frac{1}{3} \approx 0.333... \] Step 4: Final Answer:
The value of the individual engine efficiency is \( 1/3 \). Expressed as a numerical value, this is approximately 0.33.
