Question

Use the Peng-Robinson equation of state to compute the specific volume, V (L/mol), for methane gas at 298 K and 40 atm. Report your answer with 2 significant figures.

Answer 1

For Methane

Critical temperature Tc = 190.4 K

Critical pressure Pc = 45.398 atm

Accentric factor w = 0.011

Temperature T = 298 K

Pressure P = 40 atm

Tr = T/Tc = 298/190.4 = 1.56513

Gas constant R = 0.0821 L-atm/mol·K

Calculate from the last equation

= [ 1 + (0.37464 + 1.54226*0.011 - 0.26992*0.011*0.011)*(1 - 1.56513^0.5)]²

= [1 + (0.391572) * (-0.25105)]²

= 0.813055

b = 0.0777961*0.0821*190.4/45.398

= 0.0267874

a = 0.457236*0.813055*0.0821*0.0821*190.4*190.4/45.398

= 2.000985

From the first equation

40 = [(0.0821*298)/(V - 0.0267874)] - [(2.000985)/(V² + 2*V*0.0267874 - 0.0267874*0.0267874)]

40 = [(24.4658)/(V - 0.0267874)] - [(2.000985)/(V² + V*0.0535748 - 0.0007175648)]

40*(V - 0.0267874)*(V² + V*0.0535748 - 0.0007175648) = 24.4658*(V² + V*0.0535748 - 0.0007175648) - 2.000985*(V - 0.0267874)

(40V - 1.071496)*(V² + V*0.0535748 - 0.0007175648) = 24.4658V² + 1.31075V - 0.0175558 - 2.000985V + 0.0536

40V³ + 2.142992V² - 0.0007175648V - 1.071496V² - 0.057405V + 0.00076887 = 24.4658V² - 0.690235V + 0.0360442

40V³ - 23.394304V² + 0.632112V - 0.03527533 = 0

This is a cubic equation so there will be three roots of this equation

V1 = 0.5594273

V2 = 0.01271 + i* 0.037612

V3 = 0.01271 - i* 0.037612

V2 and V3 are imaginary roots so it can't be the specific volumes

V1 = 0.56 L/mol is the correct answer