How to use the thermo.utils.log function in thermo

was available for the fluid.
    Analytical integrals are available for this expression.

    Examples
    --------
    >>> Zabransky_quasi_polynomial(330, 591.79, -3.12743, 0.0857315, 13.7282, 1.28971, 6.42297, 4.10989)
    165.4728226923247

    References
    ----------
    .. [1] Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski.
       Heat Capacity of Liquids: Critical Review and Recommended Values.
       2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.
    '''
    Tr = T/Tc
    return R*(a1*log(1-Tr) + a2/(1-Tr) + a3 + a4*Tr + a5*Tr**2 + a6*Tr**3)

dP_dT = R/x0 - da_alpha_dT/x1
            dP_dV = -x3*x4 + x5*x7
            d2P_dT2 = -d2a_alpha_dT2/x1
            d2P_dV2 = -2.*a_alpha*x5*x5*x1**-3 + 2.*x7 + 2.*x3*x0**-3
            d2P_dTdV = -R*x4 + da_alpha_dT*x5*x6
            H_dep = x12*(T*da_alpha_dT - a_alpha) - x3 + x8
            S_dep = -R*log((V*x3/(x0*x8))**2)/2. + da_alpha_dT*x12  # Consider Real part of the log only via log(x**2)/2 = Re(log(x))
            Cv_dep = -T*d2a_alpha_dT2*x13*(-log(((x14 - x15 + x16)/(x14 + x15 - x16))**2)*0.5) # Consider Real part of the log only via log(x**2)/2 = Re(log(x))
        else:
            dP_dT = R/(V - b) - da_alpha_dT/(V**2 + V*delta + epsilon)
            dP_dV = -R*T/(V - b)**2 - (-2*V - delta)*a_alpha/(V**2 + V*delta + epsilon)**2
            d2P_dT2 = -d2a_alpha_dT2/(V**2 + V*delta + epsilon)
            d2P_dV2 = 2*(R*T/(V - b)**3 - (2*V + delta)**2*a_alpha/(V**2 + V*delta + epsilon)**3 + a_alpha/(V**2 + V*delta + epsilon)**2)
            d2P_dTdV = -R/(V - b)**2 + (2*V + delta)*da_alpha_dT/(V**2 + V*delta + epsilon)**2
            H_dep = P*V - R*T + 2*(T*da_alpha_dT - a_alpha)*catanh((2*V + delta)/sqrt(delta**2 - 4*epsilon)).real/sqrt(delta**2 - 4*epsilon)
            S_dep = -R*log(V) + R*log(P*V/(R*T)) + R*log(V - b) + 2*da_alpha_dT*catanh((2*V + delta)/sqrt(delta**2 - 4*epsilon)).real/sqrt(delta**2 - 4*epsilon)
            Cv_dep = -T*(sqrt(1/(delta**2 - 4*epsilon))*log(V - delta**2*sqrt(1/(delta**2 - 4*epsilon))/2 + delta/2 + 2*epsilon*sqrt(1/(delta**2 - 4*epsilon))) - sqrt(1/(delta**2 - 4*epsilon))*log(V + delta**2*sqrt(1/(delta**2 - 4*epsilon))/2 + delta/2 - 2*epsilon*sqrt(1/(delta**2 - 4*epsilon))))*d2a_alpha_dT2
        return [dP_dT, dP_dV, d2P_dT2, d2P_dV2, d2P_dTdV, H_dep, S_dep, Cv_dep]

7-13. doi:10.1016/j.fluid.2008.03.007.
        .. [2] Walas, Stanley M. Phase Equilibria in Chemical Engineering. 
           Butterworth-Heinemann, 1985.
        '''
        if self.phase in ['l', 'l/g']:
            if xs is None:
                xs = self.zs
            self.phis_l = self.fugacity_coefficients(self.Z_l, zs=xs)
            self.fugacities_l = [phi*x*self.P for phi, x in zip(self.phis_l, xs)]
            self.lnphis_l = [log(i) for i in self.phis_l]
        if self.phase in ['g', 'l/g']:
            if ys is None:
                ys = self.zs
            self.phis_g = self.fugacity_coefficients(self.Z_g, zs=ys)
            self.fugacities_g = [phi*y*self.P for phi, y in zip(self.phis_g, ys)]
            self.lnphis_g = [log(i) for i in self.phis_g]

Ammonia, from [1]_.

    >>> Campbell_Thodos(T=405.45, Tb=239.82, Tc=405.45, Pc=111.7*101325, M=17.03, dipole=1.47)
    7.347363635885525e-05

    References
    ----------
    .. [1] Campbell, Scott W., and George Thodos. "Prediction of Saturated
       Liquid Densities and Critical Volumes for Polar and Nonpolar
       Substances." Journal of Chemical & Engineering Data 30, no. 1
       (January 1, 1985): 102-11. doi:10.1021/je00039a032.
    '''
    Tr = T/Tc
    Tbr = Tb/Tc
    Pc = Pc/101325.
    s = Tbr * log(Pc)/(1-Tbr)
    Lambda = Pc**(1/3.)/(M**0.5*Tc**(5/6.))
    alpha = 0.3883 - 0.0179*s
    beta = 0.00318*s - 0.0211 + 0.625*Lambda**(1.35)
    if dipole:
        theta = Pc*dipole**2/Tc**2
        alpha -= 130540 * theta**2.41
        beta += 9.74E6 * theta**3.38
    if hydroxyl:
        beta = 0.00318*s - 0.0211 + 0.625*Lambda**(1.35) + 5.90*theta**0.835
        alpha = (0.69*Tbr - 0.3342 + 5.79E-10/Tbr**32.75)*Pc**0.145
    Zra = alpha + beta*(1-Tr)
    Vs = R*Tc/(Pc*101325)*Zra**(1+(1-Tr)**(2/7.))
    return Vs

Chemical Thermodynamics for Process Simulation. 1st edition. Weinheim:
       Wiley-VCH, 2012.
    .. [3] Maurer, G., and J. M. Prausnitz. "On the Derivation and Extension of
       the Uniquac Equation." Fluid Phase Equilibria 2, no. 2 (January 1,
       1978): 91-99. doi:10.1016/0378-3812(78)85002-X.
    '''
    cmps = range(len(xs))
    rsxs = sum([rs[i]*xs[i] for i in cmps])
    phis = [rs[i]*xs[i]/rsxs for i in cmps]
    qsxs = sum([qs[i]*xs[i] for i in cmps])
    vs = [qs[i]*xs[i]/qsxs for i in cmps]

    Ss = [sum([vs[j]*taus[j][i] for j in cmps]) for i in cmps]

    loggammacs = [log(phis[i]/xs[i]) + 1 - phis[i]/xs[i]
    - 5*qs[i]*(log(phis[i]/vs[i]) + 1 - phis[i]/vs[i]) for i in cmps]

    loggammars = [qs[i]*(1 - log(Ss[i]) - sum([taus[i][j]*vs[j]/Ss[j]
                  for j in cmps])) for i in cmps]

    return [exp(loggammacs[i] + loggammars[i]) for i in cmps]

Notes
        -----
        The object must be flashed before this routine can be used. It 
        depends on the properties T, P, zs, V_over_F, HeatCapacityGases, 
        EnthalpyVaporizations, VaporPressures, and xs.
        
        References
        ----------
        .. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
           New York: McGraw-Hill Professional, 2000.
        '''
        S = 0
        T = self.T
        P = self.P
        S -= R*sum([zi*log(zi) for zi in self.zs if zi > 0]) # ideal composition entropy composition; chemsep checked
        # Both of the mixing and vapor pressure terms have negative signs
        # Equation 6-4.4b in Poling for the vapor pressure component
        # For liquids above their critical temperatures, Psat is equal to the system P (COCO).
        if self.phase == 'g':
            S -= R*log(P/101325.) # Gas-phase ideal pressure contribution (checked repeatedly)
            for i in self.cmps:
                S += self.HeatCapacityGases[i].T_dependent_property_integral_over_T(298.15, T)
        elif self.phase == 'l':
            Psats = self._Psats(T)
            for i in self.cmps:
                Sg298_to_T = self.HeatCapacityGases[i].T_dependent_property_integral_over_T(298.15, T)
                Hvap = self.EnthalpyVaporizations[i](T)
                if Hvap is None:
                    Hvap = 0 # Handle the case of a package predicting a transition past the Tc
                Svap = -Hvap/T # Do the transition at the temperature of the liquid
                S_P = -R*log(Psats[i]/101325.)

    interpolation_property = lambda self, P: log(P)
    '''log(P) interpolation transformation by default.'''

B0 = 0.1445 - 0.33/Tr - 0.1385/Tr**2 - 0.0121/Tr**3 - 0.000607/Tr**8
        B1 = 0.0637 + 0.331/Tr**2 - 0.423/Tr**3 - 0.008/Tr**8
    elif order == 1:
        B0 = 33*Tc/(100*T**2) + 277*Tc**2/(1000*T**3) + 363*Tc**3/(10000*T**4) + 607*Tc**8/(125000*T**9)
        B1 = -331*Tc**2/(500*T**3) + 1269*Tc**3/(1000*T**4) + 8*Tc**8/(125*T**9)
    elif order == 2:
        B0 = -3*Tc*(27500 + 34625*Tc/T + 6050*Tc**2/T**2 + 1821*Tc**7/T**7)/(125000*T**3)
        B1 = 3*Tc**2*(331 - 846*Tc/T - 96*Tc**6/T**6)/(500*T**4)
    elif order == 3:
        B0 = 3*Tc*(8250 + 13850*Tc/T + 3025*Tc**2/T**2 + 1821*Tc**7/T**7)/(12500*T**4)
        B1 = 3*Tc**2*(-662 + 2115*Tc/T + 480*Tc**6/T**6)/(250*T**5)
    elif order == -1:
        B0 = 289*T/2000. - 33*Tc*log(T)/100. + (969500*T**6*Tc**2 + 42350*T**5*Tc**3 + 607*Tc**8)/(7000000.*T**7)
        B1 = 637*T/10000. - (23170*T**6*Tc**2 - 14805*T**5*Tc**3 - 80*Tc**8)/(70000.*T**7)
    elif order == -2:
        B0 = 289*T**2/4000. - 33*T*Tc*log(T)/100. + 33*T*Tc/100. + 277*Tc**2*log(T)/2000. - (254100*T**5*Tc**3 + 607*Tc**8)/(42000000.*T**6)
        B1 = 637*T**2/20000. - 331*Tc**2*log(T)/1000. - (44415*T**5*Tc**3 + 40*Tc**8)/(210000.*T**6)
    else: 
        raise Exception('Only orders -2, -1, 0, 1, 2 and 3 are supported.')
    Br = (B0+omega*B1)
    return Br*R*Tc/Pc

Industrial & Engineering Chemistry Fundamentals 22, no. 1 (February 1,
       1983): 90-97. doi:10.1021/i100009a016.
    .. [2] Reid, Robert C.; Prausnitz, John M.; Poling, Bruce E.
       Properties of Gases and Liquids. McGraw-Hill Companies, 1987.
    '''
    Cs = [2.907741307E6, -3.312874033E6, 1.608101838E6, -4.331904871E5,
          7.062481330E4, -7.116620750E3, 4.325174400E2, -1.445911210E1,
          2.037119479E-1]

    Tr = T/Tc
    if Tr > 2:
        Tr = 2
    Vr = Vm/Vc
    if Vr > 2:
        Vr = 2
    theta = 1 + (omega - 0.011)*(0.09057 - 0.86276*log(Tr) + (0.31664 - 0.46568/Tr)*(Vr-0.5))
    psi = (1 + (omega-0.011)*(0.39490*(Vr-1.02355) - 0.93281*(Vr-0.75464)*log(Tr)))*0.288/Zc
    f = Tc/190.4*theta
    h = Vc/9.92E-5*psi
    T0 = T/f
    rho0 = 16.04/(Vm*1E6)*h  # Vm must be in cm^3/mol here.
    eta0 = 1E-7*sum([Cs[i]*T0**((i+1-4)/3.) for i in range(len(Cs))])
    k1 = 1944*eta0
    b1 = -0.25276920E0
    b2 = 0.334328590E0
    b3 = 1.12
    b4 = 0.1680E3
    k2 = (b1 + b2*(b3 - log(T0/b4))**2)/1000.*rho0

    a1 = -7.19771
    a2 = 85.67822
    a3 = 12.47183