$12-127$ A natural gas liquefaction plant produces liquid natural gas at $1$ atm and $-161\mathrm{.}5\backslash$deg C when natural gas enters at $1$ atm and $20\backslash$deg C$.$ Treating the natural gas as methane, determine the minimum work input and maximum $($or reversible$)$ COP. If the liquefaction is done by a Carnot refrigerator between temperature limits T$\_\{$H$\}=20$ deg $*$ Cand$*$T$\_\{$L$\}$ with the same reversible COP, determine the temperature TL$.$ Various properties ofmethane before and after liquefaction process are given as follows