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

Answer: (A)

$T_{2} =\left(T_{1}^{2} T_{4}\right)^{1/3} ; T_{3} =\left(T_{1} T_{4}^{2}\right)^{1/3}$

Answer 2

$t_{1} = 1- \frac{T_{2}}{T_{1}} = 1- \frac{T_{2}}{T_{2}} = 1 - \frac{T_{4}}{T_{3}}$ $\Rightarrow \frac{T_{2} }{T_{1}} = \frac{T_{3}}{T_{4}}= \frac{T_{4}}{T_{3}}$ $\Rightarrow T_{2} = \sqrt{T_{1}T_{2}} = \sqrt{T_{1} \sqrt{T_{2}T_{4}}}$ $T_{3} = \sqrt{T_{2}T_{4}}$ $T_{2}^{3/4} = \sqrt{T_{1}^{1/2}} T^{1/4}_{4}$ $T_{2} =T_{1}^{2/3} T_{4}^{1/3}$