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    "# General cubics\n",
    "\n",
    "The reduced residual Helmholtz energy for the main cubic EOS (van der Waals, Peng-Robinson, and Soave-Redlich-Kwong) can be written in a common form:\n",
    "\n",
    "$$ \\begin{equation}\n",
    "\\label{eq:alphar_from_psi}\n",
    "\\alpha^r = \\psi^{(-)} - \\dfrac{\\tau a_m}{RT_r } \\psi^{(+)}.\n",
    "\\end{equation}$$\n",
    "\n",
    "$$ \\begin{eqnarray}\n",
    "\\psi^{(-)} &=& \\int_0^\\delta\\dfrac{b_m\\rho_r }{1-b_m\\delta\\rho_r }{\\rm d}\\delta \\label{eq:psiminusintegral}\\\\\n",
    "           &=&-\\ln(1-b_m\\rho ). \\label{eq:psiminusresult}\n",
    "\\end{eqnarray} $$\n",
    "\n",
    "$$ \\begin{eqnarray}\n",
    "\\psi^{(+)} &=& \\int_0^\\delta \\dfrac{\\rho_r}{\\left(1+\\Delta_1 b_m\\delta\\rho_r \\right)\\left(1+\\Delta_2 b_m\\delta\\rho_r \\right)} {\\rm d}\\delta \\label{eq:psiplusintegral}\\\\\n",
    "           &=& \\dfrac{\\ln\\left(\\dfrac{\\Delta_1 b_m\\rho+1}{\\Delta_2b_m\\rho+1}\\right)}{b_m(\\Delta_1-\\Delta_2)}\\label{eq:psiplusresult}\n",
    "\\end{eqnarray} $$\n",
    "\n",
    "with the constants given by:\n",
    "\n",
    "* vdW: $\\Delta_1=0$, $\\Delta_2=0$\n",
    "* SRK: $\\Delta_1=1$, $\\Delta_2=0$\n",
    "* PR: $\\Delta_1=1+\\sqrt{2}$, $\\Delta_2=1-\\sqrt{2}$\n",
    "\n",
    "The quantities $a_m$ and $b_m$ are described (with exact solutions for the numerical coefficients) for each of these EOS in https://pubs.acs.org/doi/abs/10.1021/acs.iecr.1c00847.\n",
    "\n",
    "The models in teqp are instantiated based on knowledge of the critical temperature, pressure, and acentric factor.  Thereafter all quantities are obtained from derivatives of $\\alpha^r$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "9a884dab",
   "metadata": {},
   "outputs": [],
   "source": [
    "import teqp\n",
    "\n",
    "# Values taken from http://dx.doi.org/10.6028/jres.121.011\n",
    "Tc_K = [ 190.564, 154.581, 150.687 ]\n",
    "pc_Pa = [ 4599200, 5042800, 4863000 ]\n",
    "acentric = [ 0.011, 0.022, -0.002 ]\n",
    "\n",
    "# Instantiate Peng-Robinson model\n",
    "modelPR = teqp.canonical_PR(Tc_K, pc_Pa, acentric)\n",
    "\n",
    "# Instantiate Soave-Redlich-Kwong model\n",
    "modelSRK = teqp.canonical_SRK(Tc_K, pc_Pa, acentric)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "615c7830",
   "metadata": {},
   "source": [
    "## Adjusting k_ij\n",
    "\n",
    "Fine-tuned values of $k_{ij}$ can be provided when instantiating the model, for Peng-Robinson and SRK.  A complete matrix of all the $k_{ij}$ values must be provided. This allows for asymmetric mixing models in which $k_{ij}\\neq k_{ji}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "a8c87890",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<teqp.teqp.GenericCubic at 0x20b222799b0>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k_12 = 0.01\n",
    "kmat = [[0,k_12,0],[k_12,0,0],[0,0,0]]\n",
    "teqp.canonical_PR(Tc_K, pc_Pa, acentric, kmat)\n",
    "teqp.canonical_SRK(Tc_K, pc_Pa, acentric, kmat)"
   ]
  }
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