Question 10

For a single Carnot engine, the efficiency is given by $\eta = 1 - \frac{Q_{out}}{Q_{in}}$, which implies $\frac{Q_{out}}{Q_{in}} = 1 - \eta$.

In a series of five engines where the heat rejected by one is absorbed by the next:

Heat rejected by engine 1: $Q_1 = Q_0(1 - \eta)$

Heat rejected by engine 2: $Q_2 = Q_1(1 - \eta) = Q_0(1 - \eta)^2$

Continuing this for five engines, the heat rejected by the final engine is $Q_5 = Q_0(1 - \eta)^5$.

The total work done by the system of engines is the difference between the heat absorbed by the first engine and the heat rejected by the last engine:

$W = Q_0 - Q_5 = Q_0 [1 - (1 - \eta)^5]$

The net efficiency of the system is:

$\eta_{net} = \frac{W}{Q_0} = 1 - (1 - \eta)^5$

Given $\eta_{net} = \frac{211}{243}$:

$1 - (1 - \eta)^5 = \frac{211}{243}$

$(1 - \eta)^5 = 1 - \frac{211}{243} = \frac{32}{243}$

$(1 - \eta)^5 = \left(\frac{2}{3}\right)^5$

$1 - \eta = \frac{2}{3}$

$\eta = 1 - \frac{2}{3} = \frac{1}{3} \approx 0.333$