<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def vanderwaals(target, pressure='pore.pressure', temperature='pore.temperature', critical_pressure='pore.critical_pressure', critical_temperature='pore.critical_temperature'):
 r""" Uses Van der Waals equation of state to calculate the density of a real gas Parameters target : OpenPNM Object The object for which these values are being calculated. This controls the length of the calculated array, and also provides access to other necessary thermofluid properties. pressure : string The dictionary key containing the pressure values in Pascals (Pa) temperature : string The dictionary key containing the temperature values in Kelvin (K) critical_pressure : string The dictionary key containing the critical pressure values in Pascals (Pa) critical_temperature : string The dictionary key containing the critical temperature values in Kelvin (K) Returns ------- rho, the density in [mol/m3] """


    P = target[pressure]/100000
    T = target[temperature]
    Pc = target[critical_pressure]/100000  # convert to bars
    Tc = target[critical_temperature]
    R = 83.1447
    a = 27*(R**2)*(Tc**2)/(64*Pc)
    b = R*Tc/(8*Pc)
    a1 = -1/b
    a2 = (R*T+b*P)/(a*b)
    a3 = -P/(a*b)
    a0 = sp.ones(sp.shape(a1))
    coeffs = sp.vstack((a0, a1, a2, a3)).T
    density = sp.array([sp.roots(C) for C in coeffs])
    value = sp.real(density[:, 2])*1e6  # Convert it to mol/m3
    return value