An article in Chemical and Engineering News (Sept. 28, 1987) describes a hydrothermal autoclave. This device is of constant volume, is evacuated, and then water is added so that a fraction x of the total volume is filled with liquid water and the remainder is filled with water vapor. The autoclave is then heated so that the temperature and pressure in the sealed vessel increase. It is observed that if x is greater than a “critical fill” value, x_c, the liquid volume fraction increases as the temperature increases, and the vessel becomes completely filled with liquid at temperatures below the critical temperature. On the other hand, if x < x_c the liquid evaporates as temperature is increased, and the autoclave becomes completely filled with vapor below the water critical temperature. If, however, x = x_c, the volume fraction of liquid in the autoclave remains constant as the temperature increases, and the temperature-pressure trajectory passes through the water critical point. Assuming the hydrothermal autoclave is to be loaded at 25°C, calculate the critical fill x_c

a. Using the steam tables 

b. Assuming the water obeys the Peng-Robinson equation of state