QUESTION $5$: Carnot cycles $(25$ Marks$)$

$($a$)$

$($i$)$ List and describe, with the help of a T$-$S diagram, the four processes that constitute the Carnot heat engine cycle.

$($ii$)$ For a Carnot Heat Engine Cycle operating between hot and cold reservoirs at $T_H$ and $T_C,$ derive an equation for its efficiency.

$($iii$)$ For a Reversed Carnot Cycle operating between hot and cold reservoirs at $T_H$ and $T_C,$ prove the relationship between the Heat Pump and Refrigeration Coefficients of performance: COP ${?}_{hp}=$ COP ${?}_{\text{r}\text{e}\text{f}}+1$

$(11$ Marks$)$

$($b$)$ An inventor claims the devices below operate thermodynamic cycles between a cold reservoir at $-23C$ and a hot reservoir at and $Q_C$ are the heat gains from the Hot and Cold reservoirs, respectively.$)$

$($i$)$ Calculate the maximum $($Carnot$)$ Heat Engine efficiency and Coefficient of Performance $($refrigeration$)$ possible between these temperatures.

$(5$ Marks$)$

$($ii$)$ For each device below decide whether it is a Heat Engine or refrigeration cycle and whether the claimed data violate the First or Second laws of thermodynamics. $($Heat and Work are positive when system gains energy.$)$

$\backslash$table$[[$Device$,\backslash$table$[[Q_H$