def Androulakis(self, T, full=True, quick=True):
        r'''Method to calculate `a_alpha` and its first and second
        derivatives according to Androulakis et al. (1989) [1]_. Returns `a_alpha`, 
        `da_alpha_dT`, and `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives`  
        for more documentation. Three coefficients needed.
        
        .. math::
            \alpha = c_{1} \left(- \left(\frac{T}{Tc}\right)^{\frac{2}{3}}
            + 1\right) + c_{2} \left(- \left(\frac{T}{Tc}\right)^{\frac{2}{3}} 
            + 1\right)^{2} + c_{3} \left(- \left(\frac{T}{Tc}\right)^{
            \frac{2}{3}} + 1\right)^{3} + 1

        References
        ----------
        .. [1] Androulakis, I. P., N. S. Kalospiros, and D. P. Tassios. 
           "Thermophysical Properties of Pure Polar and Nonpolar Compounds with
           a Modified VdW-711 Equation of State." Fluid Phase Equilibria 45, 
           no. 2 (April 1, 1989): 135-63. doi:10.1016/0378-3812(89)80254-7. 
        '''
        c1, c2, c3 = self.alpha_function_coeffs
        T, Tc, a = self.T, self.Tc, self.a
        a_alpha = a*(c1*(-(T/Tc)**(2/3) + 1) + c2*(-(T/Tc)**(2/3) + 1)**2 + c3*(-(T/Tc)**(2/3) + 1)**3 + 1)
        if not full:
            return a_alpha
        else:
            da_alpha_dT = a*(-2*c1*(T/Tc)**(2/3)/(3*T) - 4*c2*(T/Tc)**(2/3)*(-(T/Tc)**(2/3) + 1)/(3*T) - 2*c3*(T/Tc)**(2/3)*(-(T/Tc)**(2/3) + 1)**2/T)
            d2a_alpha_dT2 = a*(2*(T/Tc)**(2/3)*(c1 + 4*c2*(T/Tc)**(2/3) - 2*c2*((T/Tc)**(2/3) - 1) - 12*c3*(T/Tc)**(2/3)*((T/Tc)**(2/3) - 1) + 3*c3*((T/Tc)**(2/3) - 1)**2)/(9*T**2))
            return a_alpha, da_alpha_dT, d2a_alpha_dT2
def Androulakis(self, T, full=True, quick=True):
        r'''Method to calculate `a_alpha` and its first and second
        derivatives according to Androulakis et al. (1989) [1]_. Returns `a_alpha`, 
        `da_alpha_dT`, and `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives`  
        for more documentation. Three coefficients needed.
        
        .. math::
            \alpha = c_{1} \left(- \left(\frac{T}{Tc}\right)^{\frac{2}{3}}
            + 1\right) + c_{2} \left(- \left(\frac{T}{Tc}\right)^{\frac{2}{3}} 
            + 1\right)^{2} + c_{3} \left(- \left(\frac{T}{Tc}\right)^{
            \frac{2}{3}} + 1\right)^{3} + 1

        References
        ----------
        .. [1] Androulakis, I. P., N. S. Kalospiros, and D. P. Tassios. 
           "Thermophysical Properties of Pure Polar and Nonpolar Compounds with
           a Modified VdW-711 Equation of State." Fluid Phase Equilibria 45, 
           no. 2 (April 1, 1989): 135-63. doi:10.1016/0378-3812(89)80254-7. 
        '''
        c1, c2, c3 = self.alpha_function_coeffs
        T, Tc, a = self.T, self.Tc, self.a
        a_alpha = a*(c1*(-(T/Tc)**(2/3) + 1) + c2*(-(T/Tc)**(2/3) + 1)**2 + c3*(-(T/Tc)**(2/3) + 1)**3 + 1)
        if not full:
            return a_alpha
        else:
            da_alpha_dT = a*(-2*c1*(T/Tc)**(2/3)/(3*T) - 4*c2*(T/Tc)**(2/3)*(-(T/Tc)**(2/3) + 1)/(3*T) - 2*c3*(T/Tc)**(2/3)*(-(T/Tc)**(2/3) + 1)**2/T)
            d2a_alpha_dT2 = a*(2*(T/Tc)**(2/3)*(c1 + 4*c2*(T/Tc)**(2/3) - 2*c2*((T/Tc)**(2/3) - 1) - 12*c3*(T/Tc)**(2/3)*((T/Tc)**(2/3) - 1) + 3*c3*((T/Tc)**(2/3) - 1)**2)/(9*T**2))
            return a_alpha, da_alpha_dT, d2a_alpha_dT2
def Androulakis ( self , T , full = True , quick = True ) : c1 , c2 , c3 = self . alpha_function_coeffs T , Tc , a = self . T , self . Tc , self . a a_alpha = a * ( c1 * ( - ( T / Tc ) ** ( 2 / 3 ) + 1 ) + c2 * ( - ( T / Tc ) ** ( 2 / 3 ) + 1 ) ** 2 + c3 * ( - ( T / Tc ) ** ( 2 / 3 ) + 1 ) ** 3 + 1 ) if not full : return a_alpha else : da_alpha_dT = a * ( - 2 * c1 * ( T / Tc ) ** ( 2 / 3 ) / ( 3 * T ) - 4 * c2 * ( T / Tc ) ** ( 2 / 3 ) * ( - ( T / Tc ) ** ( 2 / 3 ) + 1 ) / ( 3 * T ) - 2 * c3 * ( T / Tc ) ** ( 2 / 3 ) * ( - ( T / Tc ) ** ( 2 / 3 ) + 1 ) ** 2 / T ) d2a_alpha_dT2 = a * ( 2 * ( T / Tc ) ** ( 2 / 3 ) * ( c1 + 4 * c2 * ( T / Tc ) ** ( 2 / 3 ) - 2 * c2 * ( ( T / Tc ) ** ( 2 / 3 ) - 1 ) - 12 * c3 * ( T / Tc ) ** ( 2 / 3 ) * ( ( T / Tc ) ** ( 2 / 3 ) - 1 ) + 3 * c3 * ( ( T / Tc ) ** ( 2 / 3 ) - 1 ) ** 2 ) / ( 9 * T ** 2 ) ) return a_alpha , da_alpha_dT , d2a_alpha_dT2
r'''Method to calculate `a_alpha` and its first and second
        derivatives according to Androulakis et al. (1989) [1]_. Returns `a_alpha`, 
        `da_alpha_dT`, and `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives`  
        for more documentation. Three coefficients needed.
        
        .. math::
            \alpha = c_{1} \left(- \left(\frac{T}{Tc}\right)^{\frac{2}{3}}
            + 1\right) + c_{2} \left(- \left(\frac{T}{Tc}\right)^{\frac{2}{3}} 
            + 1\right)^{2} + c_{3} \left(- \left(\frac{T}{Tc}\right)^{
            \frac{2}{3}} + 1\right)^{3} + 1

        References
        ----------
        .. [1] Androulakis, I. P., N. S. Kalospiros, and D. P. Tassios. 
           "Thermophysical Properties of Pure Polar and Nonpolar Compounds with
           a Modified VdW-711 Equation of State." Fluid Phase Equilibria 45, 
           no. 2 (April 1, 1989): 135-63. doi:10.1016/0378-3812(89)80254-7.
r Method to calculate a_alpha and its first and second derivatives according to Androulakis et al . ( 1989 ) [ 1 ] _ . Returns a_alpha da_alpha_dT and d2a_alpha_dT2 . See GCEOS . a_alpha_and_derivatives for more documentation . Three coefficients needed . .. math :: \ alpha = c_ { 1 } \ left ( - \ left ( \ frac { T } { Tc } \ right ) ^ { \ frac { 2 } { 3 }} + 1 \ right ) + c_ { 2 } \ left ( - \ left ( \ frac { T } { Tc } \ right ) ^ { \ frac { 2 } { 3 }} + 1 \ right ) ^ { 2 } + c_ { 3 } \ left ( - \ left ( \ frac { T } { Tc } \ right ) ^ { \ frac { 2 } { 3 }} + 1 \ right ) ^ { 3 } + 1