Problem $5\mathrm{.}33.$ An inventor proposes to make a heat engine using water$/$ice as the working substance, taking advantage of the fact that water expands as it freezes. A weight to be lifted is placed on top of a piston over a cylinder of water at $\backslash (1^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The system is then placed in thermal contact with a low$-$temperature reservoir at $\backslash (-1^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ until the water freezes into ice, lifting the weight. The weight is then removed and the ice is melted by putting it in contact with a high$-$temperature reservoir at $\backslash (1^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The inventor is pleased with this device because it can seemingly perform an unlimited amount of work while absorbing only a finite amount of heat. Explain the flaw in the inventor's reasoning, and use the Clausius$-$Clapeyron relation to prove that the maximum efficiency of this engine is still given by the Carnot formula, $\backslash (1-$T$\_\{$c$\}/$ T$\_\{$h$\}\backslash ).$