Question

Three Carnot engines operate in series between a heat source at a temperature $T_1$ and a heat sink at temperature $T_4$ (see figure). There are two other reservoirs at temperature $T_2$, and $T_3$, as shown, with $T_2 > T_2 > T_3 > T_4$ . The three engines are equally efficient if:

• A. $T_{2} =\left(T_{1}^{2} T_{4}\right)^{1/3} ; T_{3} =\left(T_{1} T_{4}^{2}\right)^{1/3} $
• B. $T_{2} =\left(T_{1} T_{4}^{2} \right)^{1/3} ; T_{3} =\left(T_{1}^{2} T_{4}\right)^{1/3} $
• C. $T_{2} =\left(T_{1}^{3} T_{4}\right)^{1/4} ; T_{3} =\left(T_{1} T_{4}^{3}\right)^{1/4} $
• D. $T_{2} =\left(T_{1} T_{4}\right)^{1/2} ; T_{3} =\left(T_{1}^{2} T_{4}\right)^{1/3} $

Answer 1

The Correct Option is A

 To solve this problem, we need to check under which conditions the three Carnot engines operating in series are equally efficient. The efficiency of a Carnot engine is given by:

\(\eta = 1 - \frac{T_{\text{C}}}{T_{\text{H}}}\)

where \( T_{\text{H}} \) is the temperature of the heat source and \( T_{\text{C}} \) is the temperature of the heat sink.

For the three engines to be equally efficient:

1. The first engine operates between \( T_1 \) and \( T_2 \):
2. The second engine operates between \( T_2 \) and \( T_3 \):
3. The third engine operates between \( T_3 \) and \( T_4 \):

For the engines to have equal efficiency:

\(\frac{T_2}{T_1} = \frac{T_3}{T_2} = \frac{T_4}{T_3}\)

This leads to the ratios:

1. \(\frac{T_2}{T_1} = \frac{T_3}{T_2} = \frac{T_4}{T_3} = k\)
2. \(T_2 = kT_1\)
3. \(T_3 = kT_2 = k^2T_1\)
4. \(T_4 = kT_3 = k^3T_1\)

Solving these, we find:

\(k^3 T_1 = T_4 \implies k = \left(\frac{T_4}{T_1}\right)^{1/3}\)

Thus, the equations for \( T_2 \) and \( T_3 \) become:

1. \(T_2 = k T_1 = \left(\frac{T_4}{T_1}\right)^{1/3} T_1 = \left(T_1^2 T_4\right)^{1/3}\)
2. \(T_3 = k^2 T_1 = \left(\frac{T_4}{T_1}\right)^{2/3} T_1 = \left(T_1 T_4^2\right)^{1/3}\)

So, under these conditions, the engines are equally efficient. Therefore, the correct answer is:

\(T_2 = \left(T_1^2 T_4\right)^{1/3} ; T_3 = \left(T_1 T_4^2\right)^{1/3}\)