A cylinder filled with $\backslash ($ n$=1\mathrm{.}5\backslash$mathrm$\{$~mol$\}\backslash )$ of air is capped with a moveable piston, which can slide frictionlessly inside the cylinder. The air, which you can model as an ideal diatomic gas, is initially at room temperature $(\backslash ($ T$\_\{$i$\}=295\backslash$mathrm$\{$~K$\}\backslash )).$ The piston is given a brief, downward whack with a hammer, causing the air inside the cylinder to compress adiabatically. The person swinging the hammer does $\backslash ($ W$=550\backslash$mathrm$\{$~J$\}\backslash )$ of work on the gas in getting it to its final state of maximum compression. What is the final temperature $\backslash ($ T$\_\{$f$\}\backslash )$ of the gas? For the limit check, investigate what happens to $\backslash ($ T$\_\{$f$\}\backslash )$ if the whack from the hammer is extremely weak, meaning that the person swinging it does very little work on the gas $\backslash (($W $\backslash$rightarrow $0)\backslash ).$ For your representation, draw this process on the $\backslash ($ P V $\backslash )$ plane. Make sure to include isothermal curves representing the initial$/$final temperatures, as well as an adiabatic curve representing the process that occurred.