def a_alpha_and_derivatives(self, T, full=True, quick=True):
        r'''Method to calculate `a_alpha` and its first and second
        derivatives for this EOS. Returns `a_alpha`, `da_alpha_dT`, and 
        `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives` for more 
        documentation. Uses the set values of `Tc`, `a`, `S1`, and `S2`. 
        
        .. math::
            a\alpha(T) = a\left[1 + S_1\left(1-\sqrt{T_r}\right) + S_2\frac{1
            - \sqrt{T_r}}{\sqrt{T_r}}\right]^2
        
            \frac{d a\alpha}{dT} = a\frac{Tc}{T^{2}} \left(- S_{2} \left(\sqrt{
            \frac{T}{Tc}} - 1\right) + \sqrt{\frac{T}{Tc}} \left(S_{1} \sqrt{
            \frac{T}{Tc}} + S_{2}\right)\right) \left(S_{2} \left(\sqrt{\frac{
            T}{Tc}} - 1\right) + \sqrt{\frac{T}{Tc}} \left(S_{1} \left(\sqrt{
            \frac{T}{Tc}} - 1\right) - 1\right)\right)

            \frac{d^2 a\alpha}{dT^2} = a\frac{1}{2 T^{3}} \left(S_{1}^{2} T
            \sqrt{\frac{T}{Tc}} - S_{1} S_{2} T \sqrt{\frac{T}{Tc}} + 3 S_{1}
            S_{2} Tc \sqrt{\frac{T}{Tc}} + S_{1} T \sqrt{\frac{T}{Tc}} 
            - 3 S_{2}^{2} Tc \sqrt{\frac{T}{Tc}} + 4 S_{2}^{2} Tc + 3 S_{2} 
            Tc \sqrt{\frac{T}{Tc}}\right)
        '''
        a, Tc, S1, S2 = self.a, self.Tc, self.S1, self.S2
        if not full:
            return a*(S1*(-(T/Tc)**0.5 + 1.) + S2*(-(T/Tc)**0.5 + 1)*(T/Tc)**-0.5 + 1)**2
        else:
            if quick:
                x0 = (T/Tc)**0.5
                x1 = x0 - 1.
                x2 = x1/x0
                x3 = S2*x2
                x4 = S1*x1 + x3 - 1.
                x5 = S1*x0
                x6 = S2 - x3 + x5
                x7 = 3.*S2
                a_alpha = a*x4*x4
                da_alpha_dT = a*x4*x6/T
                d2a_alpha_dT2 = a*(-x4*(-x2*x7 + x5 + x7) + x6*x6)/(2.*T*T)
            else:
                a_alpha = a*(S1*(-sqrt(T/Tc) + 1) + S2*(-sqrt(T/Tc) + 1)/sqrt(T/Tc) + 1)**2
                da_alpha_dT = a*((S1*(-sqrt(T/Tc) + 1) + S2*(-sqrt(T/Tc) + 1)/sqrt(T/Tc) + 1)*(-S1*sqrt(T/Tc)/T - S2/T - S2*(-sqrt(T/Tc) + 1)/(T*sqrt(T/Tc))))
                d2a_alpha_dT2 = a*(((S1*sqrt(T/Tc) + S2 - S2*(sqrt(T/Tc) - 1)/sqrt(T/Tc))**2 - (S1*sqrt(T/Tc) + 3*S2 - 3*S2*(sqrt(T/Tc) - 1)/sqrt(T/Tc))*(S1*(sqrt(T/Tc) - 1) + S2*(sqrt(T/Tc) - 1)/sqrt(T/Tc) - 1))/(2*T**2))
            return a_alpha, da_alpha_dT, d2a_alpha_dT2
def a_alpha_and_derivatives(self, T, full=True, quick=True):
        r'''Method to calculate `a_alpha` and its first and second
        derivatives for this EOS. Returns `a_alpha`, `da_alpha_dT`, and 
        `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives` for more 
        documentation. Uses the set values of `Tc`, `a`, `S1`, and `S2`. 
        
        .. math::
            a\alpha(T) = a\left[1 + S_1\left(1-\sqrt{T_r}\right) + S_2\frac{1
            - \sqrt{T_r}}{\sqrt{T_r}}\right]^2
        
            \frac{d a\alpha}{dT} = a\frac{Tc}{T^{2}} \left(- S_{2} \left(\sqrt{
            \frac{T}{Tc}} - 1\right) + \sqrt{\frac{T}{Tc}} \left(S_{1} \sqrt{
            \frac{T}{Tc}} + S_{2}\right)\right) \left(S_{2} \left(\sqrt{\frac{
            T}{Tc}} - 1\right) + \sqrt{\frac{T}{Tc}} \left(S_{1} \left(\sqrt{
            \frac{T}{Tc}} - 1\right) - 1\right)\right)

            \frac{d^2 a\alpha}{dT^2} = a\frac{1}{2 T^{3}} \left(S_{1}^{2} T
            \sqrt{\frac{T}{Tc}} - S_{1} S_{2} T \sqrt{\frac{T}{Tc}} + 3 S_{1}
            S_{2} Tc \sqrt{\frac{T}{Tc}} + S_{1} T \sqrt{\frac{T}{Tc}} 
            - 3 S_{2}^{2} Tc \sqrt{\frac{T}{Tc}} + 4 S_{2}^{2} Tc + 3 S_{2} 
            Tc \sqrt{\frac{T}{Tc}}\right)
        '''
        a, Tc, S1, S2 = self.a, self.Tc, self.S1, self.S2
        if not full:
            return a*(S1*(-(T/Tc)**0.5 + 1.) + S2*(-(T/Tc)**0.5 + 1)*(T/Tc)**-0.5 + 1)**2
        else:
            if quick:
                x0 = (T/Tc)**0.5
                x1 = x0 - 1.
                x2 = x1/x0
                x3 = S2*x2
                x4 = S1*x1 + x3 - 1.
                x5 = S1*x0
                x6 = S2 - x3 + x5
                x7 = 3.*S2
                a_alpha = a*x4*x4
                da_alpha_dT = a*x4*x6/T
                d2a_alpha_dT2 = a*(-x4*(-x2*x7 + x5 + x7) + x6*x6)/(2.*T*T)
            else:
                a_alpha = a*(S1*(-sqrt(T/Tc) + 1) + S2*(-sqrt(T/Tc) + 1)/sqrt(T/Tc) + 1)**2
                da_alpha_dT = a*((S1*(-sqrt(T/Tc) + 1) + S2*(-sqrt(T/Tc) + 1)/sqrt(T/Tc) + 1)*(-S1*sqrt(T/Tc)/T - S2/T - S2*(-sqrt(T/Tc) + 1)/(T*sqrt(T/Tc))))
                d2a_alpha_dT2 = a*(((S1*sqrt(T/Tc) + S2 - S2*(sqrt(T/Tc) - 1)/sqrt(T/Tc))**2 - (S1*sqrt(T/Tc) + 3*S2 - 3*S2*(sqrt(T/Tc) - 1)/sqrt(T/Tc))*(S1*(sqrt(T/Tc) - 1) + S2*(sqrt(T/Tc) - 1)/sqrt(T/Tc) - 1))/(2*T**2))
            return a_alpha, da_alpha_dT, d2a_alpha_dT2
def a_alpha_and_derivatives ( self , T , full = True , quick = True ) : a , Tc , S1 , S2 = self . a , self . Tc , self . S1 , self . S2 if not full : return a * ( S1 * ( - ( T / Tc ) ** 0.5 + 1. ) + S2 * ( - ( T / Tc ) ** 0.5 + 1 ) * ( T / Tc ) ** - 0.5 + 1 ) ** 2 else : if quick : x0 = ( T / Tc ) ** 0.5 x1 = x0 - 1. x2 = x1 / x0 x3 = S2 * x2 x4 = S1 * x1 + x3 - 1. x5 = S1 * x0 x6 = S2 - x3 + x5 x7 = 3. * S2 a_alpha = a * x4 * x4 da_alpha_dT = a * x4 * x6 / T d2a_alpha_dT2 = a * ( - x4 * ( - x2 * x7 + x5 + x7 ) + x6 * x6 ) / ( 2. * T * T ) else : a_alpha = a * ( S1 * ( - sqrt ( T / Tc ) + 1 ) + S2 * ( - sqrt ( T / Tc ) + 1 ) / sqrt ( T / Tc ) + 1 ) ** 2 da_alpha_dT = a * ( ( S1 * ( - sqrt ( T / Tc ) + 1 ) + S2 * ( - sqrt ( T / Tc ) + 1 ) / sqrt ( T / Tc ) + 1 ) * ( - S1 * sqrt ( T / Tc ) / T - S2 / T - S2 * ( - sqrt ( T / Tc ) + 1 ) / ( T * sqrt ( T / Tc ) ) ) ) d2a_alpha_dT2 = a * ( ( ( S1 * sqrt ( T / Tc ) + S2 - S2 * ( sqrt ( T / Tc ) - 1 ) / sqrt ( T / Tc ) ) ** 2 - ( S1 * sqrt ( T / Tc ) + 3 * S2 - 3 * S2 * ( sqrt ( T / Tc ) - 1 ) / sqrt ( T / Tc ) ) * ( S1 * ( sqrt ( T / Tc ) - 1 ) + S2 * ( sqrt ( T / Tc ) - 1 ) / sqrt ( T / Tc ) - 1 ) ) / ( 2 * T ** 2 ) ) return a_alpha , da_alpha_dT , d2a_alpha_dT2
r'''Method to calculate `a_alpha` and its first and second
        derivatives for this EOS. Returns `a_alpha`, `da_alpha_dT`, and 
        `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives` for more 
        documentation. Uses the set values of `Tc`, `a`, `S1`, and `S2`. 
        
        .. math::
            a\alpha(T) = a\left[1 + S_1\left(1-\sqrt{T_r}\right) + S_2\frac{1
            - \sqrt{T_r}}{\sqrt{T_r}}\right]^2
        
            \frac{d a\alpha}{dT} = a\frac{Tc}{T^{2}} \left(- S_{2} \left(\sqrt{
            \frac{T}{Tc}} - 1\right) + \sqrt{\frac{T}{Tc}} \left(S_{1} \sqrt{
            \frac{T}{Tc}} + S_{2}\right)\right) \left(S_{2} \left(\sqrt{\frac{
            T}{Tc}} - 1\right) + \sqrt{\frac{T}{Tc}} \left(S_{1} \left(\sqrt{
            \frac{T}{Tc}} - 1\right) - 1\right)\right)

            \frac{d^2 a\alpha}{dT^2} = a\frac{1}{2 T^{3}} \left(S_{1}^{2} T
            \sqrt{\frac{T}{Tc}} - S_{1} S_{2} T \sqrt{\frac{T}{Tc}} + 3 S_{1}
            S_{2} Tc \sqrt{\frac{T}{Tc}} + S_{1} T \sqrt{\frac{T}{Tc}} 
            - 3 S_{2}^{2} Tc \sqrt{\frac{T}{Tc}} + 4 S_{2}^{2} Tc + 3 S_{2} 
            Tc \sqrt{\frac{T}{Tc}}\right)
r Method to calculate a_alpha and its first and second derivatives for this EOS . Returns a_alpha da_alpha_dT and d2a_alpha_dT2 . See GCEOS . a_alpha_and_derivatives for more documentation . Uses the set values of Tc a S1 and S2 . .. math :: a \ alpha ( T ) = a \ left [ 1 + S_1 \ left ( 1 - \ sqrt { T_r } \ right ) + S_2 \ frac { 1 - \ sqrt { T_r }} { \ sqrt { T_r }} \ right ] ^2 \ frac { d a \ alpha } { dT } = a \ frac { Tc } { T^ { 2 }} \ left ( - S_ { 2 } \ left ( \ sqrt { \ frac { T } { Tc }} - 1 \ right ) + \ sqrt { \ frac { T } { Tc }} \ left ( S_ { 1 } \ sqrt { \ frac { T } { Tc }} + S_ { 2 } \ right ) \ right ) \ left ( S_ { 2 } \ left ( \ sqrt { \ frac { T } { Tc }} - 1 \ right ) + \ sqrt { \ frac { T } { Tc }} \ left ( S_ { 1 } \ left ( \ sqrt { \ frac { T } { Tc }} - 1 \ right ) - 1 \ right ) \ right )