A Carnot engine has the same efficiency between (i) $100 \mathrm{~K}$ and $600 \mathrm{~K}$ and (ii)…
A Carnot engine has the same efficiency between (i) $100 \mathrm{~K}$ and $600 \mathrm{~K}$ and (ii) $\mathrm{T} \mathrm{K}$ and $960 \mathrm{~K}$. The temperature $T$ in kelvin of the sink is
120
160
240
320
Solution
$\begin{array}{ll}
& \text { Efficiency of a carnot engine is } \eta=1-\frac{T_C}{T_H} \\
& \text { For case (i), } \\
& T_C=100 \mathrm{~K} \text { and } T_H=600 \mathrm{~K} \\
\therefore \quad & \eta_1=1-\frac{100}{600}=\frac{5}{6} \\
& \text { For case (ii), } \\
& \mathrm{T}_{\mathrm{C}}=\mathrm{T} \mathrm{K} \text { and } \mathrm{T}_{\mathrm{H}}=960 \mathrm{~K} \\
\therefore \quad & \eta_2=1-\frac{\mathrm{T}}{960} \\
& \text { Given } \eta_1=\eta_2 \\
\therefore \quad & \frac{5}{6}=1-\frac{\mathrm{T}}{960} \\
& \frac{5}{6}=\frac{960-\mathrm{T}}{960}
\end{array}$
Solving for $\mathrm{T}$, we get $\mathrm{T}=160 \mathrm{~K}$
.