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    "# Cubic Plus Association (CPA)\n",
    "\n",
    "The combination of a cubic EOS with association with the association term. The sum of the terms goes like:\n",
    "$$\n",
    "\\alpha^{\\rm r} = \\alpha^{\\rm r}_{\\rm cub} + \\alpha^{\\rm r}_{\\rm assoc}\n",
    "$$\n",
    "\n",
    "## Cubic part\n",
    "\n",
    "The residual contribution to $\\alpha$ is expressed as the sum :\n",
    "$$\n",
    "\\alpha^{\\rm r}_{\\rm cub,rep} +\\alpha^{\\rm r}_{\\rm cub,att}\n",
    "$$ where the cubic parts come from\n",
    "\n",
    "The repulsive part of the cubic EOS contribution:\n",
    "$$\n",
    "\\alpha^{\\rm r}_{\\rm cub,rep} = -\\ln(1 - b_{\\rm mix}\\rho) \n",
    "$$\n",
    "The attractive part of the cubic EOS contribution:\n",
    "$$\n",
    "\\alpha^{\\rm r}_{\\rm cub,att} = -\\frac{a_{\\rm mix}}{RT}\\dfrac{\\ln\\left(\\frac{\\Delta_1 b_{\\rm mix}\\rho + 1}{\\Delta_2b_{\\rm mix}\\rho + 1}\\right)}{b_{\\rm mix}\\cdot(\\Delta_1 - \\Delta_2)}\n",
    "$$\n",
    "with the coefficients depending on the cubic type:\n",
    "\n",
    "SRK: $\\Delta_1=1$, $\\Delta_2=0$\n",
    "\n",
    "PR: $\\Delta_1=1+\\sqrt{2}$, $\\Delta_2=1-\\sqrt{2}$\n",
    "\n",
    "The mixture models used for the $a_{\\rm mix}$ and $b_{\\rm mix}$ are the classical ones:\n",
    "\n",
    "$$\n",
    "a_{\\rm mix} = \\sum_i\\sum_jx_ix_j(1-k_{ij})a_{ij}(T)\n",
    "$$\n",
    "with x the mole fraction, $k_{ij}$ a weighting parameter\n",
    "$$\n",
    "a_{ij}(T) = \\sqrt{a_ia_j}\n",
    "$$\n",
    "and\n",
    "$$\n",
    "a_{i}(T) = a_{0i}\\left[1+c_{1i}(1-\\sqrt{T/T_{{\\rm crit},i}})\\right]^2\n",
    "$$\n",
    "and for $b$:\n",
    "$$\n",
    "b_{\\rm mix} = \\sum_ix_ib_i\n",
    "$$\n",
    "so there are three cubic parameters per fluid that need to be obtained though fitting: $b_{i}$, $a_{0i}$, $c_{1i}$. The value of $a_{\\rm ij}$ depends on temperature while $b_{\\rm mix}$ does not.\n",
    "\n",
    "## Association part\n",
    "\n",
    "For the association, one must have a solid understanding of the association approach that is being applied. To this end, a short discussion of the general approach is required. \n",
    "\n"
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