Thermology is a branch of physics that studies the properties and laws of matter in a hot state, which originated from human exploration of hot and cold phenomena. Human beings live in the natural world of alternating seasons and changing climate, and the phenomenon of heat and cold was one of the first natural phenomena they observed and understood.
Solving for $T_{3}$ gives
$$
T_{3}=\left(\frac{p_{4}}{p_{3}}\right)^{-0.40} T_{4}=\left(\frac{1}{5}\right)^{-0.40}(250 \mathrm{~K})=476 \mathrm{~K}=203^{\circ} \mathrm{C}
$$
The same analysis applied to the $2 \rightarrow 1$ adiabatic expansion gives
$$T_{2}=\left(\frac{p_{1}}{p_{2}}\right)^{-0.40} T_{1}=\left(\frac{1}{5}\right)^{-0.40}(200 \mathrm{~K})=381 \mathrm{~K}=108^{\circ} \mathrm{C}$$
SOLVE Example $19.3$ found that the reservoir temperatures had to be $T_{\mathrm{C}} \geq 250 \mathrm{~K}$ and $T_{\mathrm{H}} \leq 381 \mathrm{~K}$. A Carnot refrigerator operating between $250 \mathrm{~K}$ and $381 \mathrm{~K}$ has
$$
K_{\text {Carnot }}=\frac{T_{\mathrm{C}}}{T_{\mathrm{H}}-T_{\mathrm{C}}}=\frac{250 \mathrm{~K}}{381 \mathrm{~K}-250 \mathrm{~K}}=1.9
$$
ASsess This is the minimum value of $K_{\text {Carmot }}$. It will be even higher if $T_{\mathrm{C}}>250 \mathrm{~K}$ or $T_{\mathrm{H}}<381 \mathrm{~K}$. The coefficient of performance of the reasonably realistic refrigerator of Example $19.3$ is less than $60 \%$ of the limiting value.
PHAS0006 COURSE NOTES ：
Work is done in all four processes of the Carnot cycle, but heat is transferred only during the two isothermal processes.
The thermal efficiency of any heat engine is
$$
\eta=\frac{W_{\text {out }}}{Q_{\mathrm{H}}}=1-\frac{Q_{\mathrm{C}}}{Q_{\mathrm{H}}}
$$
We can determine $\eta_{\text {Carmot }}$ by finding the heat transfer in the two isothermal processes.
Process $1 \rightarrow 2:$ Table $19.1$ gives us the heat transfer in an isothermal process at temperature $T_{\mathrm{C}}$