def Yu_Lu(self, T, full=True, quick=True):
        r'''Method to calculate `a_alpha` and its first and second
        derivatives according to Yu and Lu (1987) [1]_. Returns `a_alpha`, 
        `da_alpha_dT`, and `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives`  
        for more documentation. Four coefficients needed.
        
        .. math::
            \alpha = 10^{c_{4} \left(- \frac{T}{Tc} + 1\right) \left(
            \frac{T^{2} c_{3}}{Tc^{2}} + \frac{T c_{2}}{Tc} + c_{1}\right)}

        References
        ----------
        .. [1] Yu, Jin-Min, and Benjamin C. -Y. Lu. "A Three-Parameter Cubic 
           Equation of State for Asymmetric Mixture Density Calculations." 
           Fluid Phase Equilibria 34, no. 1 (January 1, 1987): 1-19. 
           doi:10.1016/0378-3812(87)85047-1. 
        '''
        c1, c2, c3, c4 = self.alpha_function_coeffs
        T, Tc, a = self.T, self.Tc, self.a
        a_alpha = a*10**(c4*(-T/Tc + 1)*(T**2*c3/Tc**2 + T*c2/Tc + c1))
        if not full:
            return a_alpha
        else:
            da_alpha_dT = a*(10**(c4*(-T/Tc + 1)*(T**2*c3/Tc**2 + T*c2/Tc + c1))*(c4*(-T/Tc + 1)*(2*T*c3/Tc**2 + c2/Tc) - c4*(T**2*c3/Tc**2 + T*c2/Tc + c1)/Tc)*log(10))
            d2a_alpha_dT2 = a*(10**(-c4*(T/Tc - 1)*(T**2*c3/Tc**2 + T*c2/Tc + c1))*c4*(-4*T*c3/Tc - 2*c2 - 2*c3*(T/Tc - 1) + c4*(T**2*c3/Tc**2 + T*c2/Tc + c1 + (T/Tc - 1)*(2*T*c3/Tc + c2))**2*log(10))*log(10)/Tc**2)
            return a_alpha, da_alpha_dT, d2a_alpha_dT2
def Yu_Lu(self, T, full=True, quick=True):
        r'''Method to calculate `a_alpha` and its first and second
        derivatives according to Yu and Lu (1987) [1]_. Returns `a_alpha`, 
        `da_alpha_dT`, and `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives`  
        for more documentation. Four coefficients needed.
        
        .. math::
            \alpha = 10^{c_{4} \left(- \frac{T}{Tc} + 1\right) \left(
            \frac{T^{2} c_{3}}{Tc^{2}} + \frac{T c_{2}}{Tc} + c_{1}\right)}

        References
        ----------
        .. [1] Yu, Jin-Min, and Benjamin C. -Y. Lu. "A Three-Parameter Cubic 
           Equation of State for Asymmetric Mixture Density Calculations." 
           Fluid Phase Equilibria 34, no. 1 (January 1, 1987): 1-19. 
           doi:10.1016/0378-3812(87)85047-1. 
        '''
        c1, c2, c3, c4 = self.alpha_function_coeffs
        T, Tc, a = self.T, self.Tc, self.a
        a_alpha = a*10**(c4*(-T/Tc + 1)*(T**2*c3/Tc**2 + T*c2/Tc + c1))
        if not full:
            return a_alpha
        else:
            da_alpha_dT = a*(10**(c4*(-T/Tc + 1)*(T**2*c3/Tc**2 + T*c2/Tc + c1))*(c4*(-T/Tc + 1)*(2*T*c3/Tc**2 + c2/Tc) - c4*(T**2*c3/Tc**2 + T*c2/Tc + c1)/Tc)*log(10))
            d2a_alpha_dT2 = a*(10**(-c4*(T/Tc - 1)*(T**2*c3/Tc**2 + T*c2/Tc + c1))*c4*(-4*T*c3/Tc - 2*c2 - 2*c3*(T/Tc - 1) + c4*(T**2*c3/Tc**2 + T*c2/Tc + c1 + (T/Tc - 1)*(2*T*c3/Tc + c2))**2*log(10))*log(10)/Tc**2)
            return a_alpha, da_alpha_dT, d2a_alpha_dT2
def Yu_Lu ( self , T , full = True , quick = True ) : c1 , c2 , c3 , c4 = self . alpha_function_coeffs T , Tc , a = self . T , self . Tc , self . a a_alpha = a * 10 ** ( c4 * ( - T / Tc + 1 ) * ( T ** 2 * c3 / Tc ** 2 + T * c2 / Tc + c1 ) ) if not full : return a_alpha else : da_alpha_dT = a * ( 10 ** ( c4 * ( - T / Tc + 1 ) * ( T ** 2 * c3 / Tc ** 2 + T * c2 / Tc + c1 ) ) * ( c4 * ( - T / Tc + 1 ) * ( 2 * T * c3 / Tc ** 2 + c2 / Tc ) - c4 * ( T ** 2 * c3 / Tc ** 2 + T * c2 / Tc + c1 ) / Tc ) * log ( 10 ) ) d2a_alpha_dT2 = a * ( 10 ** ( - c4 * ( T / Tc - 1 ) * ( T ** 2 * c3 / Tc ** 2 + T * c2 / Tc + c1 ) ) * c4 * ( - 4 * T * c3 / Tc - 2 * c2 - 2 * c3 * ( T / Tc - 1 ) + c4 * ( T ** 2 * c3 / Tc ** 2 + T * c2 / Tc + c1 + ( T / Tc - 1 ) * ( 2 * T * c3 / Tc + c2 ) ) ** 2 * log ( 10 ) ) * log ( 10 ) / Tc ** 2 ) return a_alpha , da_alpha_dT , d2a_alpha_dT2
r'''Method to calculate `a_alpha` and its first and second
        derivatives according to Yu and Lu (1987) [1]_. Returns `a_alpha`, 
        `da_alpha_dT`, and `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives`  
        for more documentation. Four coefficients needed.
        
        .. math::
            \alpha = 10^{c_{4} \left(- \frac{T}{Tc} + 1\right) \left(
            \frac{T^{2} c_{3}}{Tc^{2}} + \frac{T c_{2}}{Tc} + c_{1}\right)}

        References
        ----------
        .. [1] Yu, Jin-Min, and Benjamin C. -Y. Lu. "A Three-Parameter Cubic 
           Equation of State for Asymmetric Mixture Density Calculations." 
           Fluid Phase Equilibria 34, no. 1 (January 1, 1987): 1-19. 
           doi:10.1016/0378-3812(87)85047-1.
r Method to calculate a_alpha and its first and second derivatives according to Yu and Lu ( 1987 ) [ 1 ] _ . Returns a_alpha da_alpha_dT and d2a_alpha_dT2 . See GCEOS . a_alpha_and_derivatives for more documentation . Four coefficients needed . .. math :: \ alpha = 10^ { c_ { 4 } \ left ( - \ frac { T } { Tc } + 1 \ right ) \ left ( \ frac { T^ { 2 } c_ { 3 }} { Tc^ { 2 }} + \ frac { T c_ { 2 }} { Tc } + c_ { 1 } \ right ) }