Explain the liquefaction of $CO_2$ gas using the Thomas Andrews plot,including the effects of temperature and pressure,and define the critical constants.

(N/A) Thomas Andrews obtained the first complete data on the pressure-volume-temperature relationships of a substance in both gaseous and liquid states using $CO_2$.
Key observations from the Andrews isotherms:
$(i)$ At high temperatures (e.g.,$50^{\circ}C$),the isotherms resemble those of an ideal gas,and the gas cannot be liquefied regardless of the pressure applied.
$(ii)$ As the temperature is lowered,the shape of the curve changes,showing significant deviation from ideal behavior. $A$ horizontal portion appears,representing the coexistence of liquid and gas phases.
$(iii)$ Critical Constants $(T_C, P_C, V_C)$: Point $E$ represents the critical state. At $30.98^{\circ}C$ $(T_C)$,$CO_2$ remains a gas up to $73 \ atm$ $(P_C)$. At this pressure,liquid $CO_2$ appears for the first time. The volume at this point is the critical volume $(V_C)$.