Exercicio-Fis-Qui3-Entropia.ipynb

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   "cell_type": "markdown",
   "metadata": {
    "id": "4w-viVb_1eiU"
   },
   "source": [
    "## Exercício 1\n",
    "\n",
    "A tabela abaixo apresenta os valores de $Cº_{p,m} (J mol^{-1} K^{-1})$ para  $O_2(g)$ no intervalo de tempertaura de 298K até 1500K. a) Ajuste um polinômio de terceiro grau ao dados de capacidade calorífica em função da temperatura e construa o gráfico de $Cº_{p,m}$ versus temperatura. b) Uma segunda alternativa é ajustar uma função do tipo $Cº_{p,m} = A + BT + CT^2 + D/T^2 $ aos dados fornecidos. Ajuste esta função aos dados fornecidos e construa o gráfico de $Cº_{p,m}$ versus temperatura. c) Compare os resultados obtidos.\n",
    "\n",
    "|   |   |   |   |   |   |   |   |   |   |   |   |   |   |\n",
    "|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n",
    "|T (K)|298.15|400.|500.|600.|700.|800.|900.|1000.|1100.|1200.|1300.|1400.|1500.\n",
    "|Cp (J/K mol)|29.376| 30.106| 31.091| 32.090| 32.981| 33.733| 34.355| 34.870| 35.300| 35.667| 35.988| 36.277| 36.544\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "x8m_Xf281eiV"
   },
   "outputs": [],
   "source": [
    "# Tudo começa com a importação dos módulos necessários para fazer as contas\n",
    "import numpy as np\n",
    "from scipy.optimize import curve_fit\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.integrate import quad\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "oCanuCJP1eiY",
    "outputId": "49f6532e-d177-4594-d934-be1c6d0a8ed0"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cºp,m =  25.66508 + 0.01304T-3.81147e-06T²-5.57560e-11T³\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([-5.57559849e-11, -3.81147086e-06,  1.30449460e-02,  2.56650822e+01])"
      ]
     },
     "execution_count": 2,
     "metadata": {
      "tags": []
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T = np.array([298.15,400.,500.,600.,700.,800.,900.,1000.,1100.,1200.,1300.,1400.,1500.])\n",
    "Cp = np.array([29.376, 30.106, 31.091, 32.090, 32.981, 33.733, 34.355, 34.870, 35.300, 35.667,35.988, 36.277, 36.544])\n",
    "\n",
    "deg = 3\n",
    "z = np.polyfit(T, Cp, deg)\n",
    "y = np.poly1d(z)\n",
    "\n",
    "\n",
    "print('Cºp,m = ' + format(z[3] , ' 6.5f') + ' +' + format(z[2] , ' 6.5f') + 'T' + format(z[1] , ' 6.5e') + 'T²' + format(z[0] , ' 6.5e') + 'T³')   \n",
    "\n",
    "#imprimir só os coeficientes na forma de vetor\n",
    "z\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "Tv6hqjy81eie",
    "outputId": "461672bf-595b-4431-e5ff-010f9e6572f4"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function matplotlib.pyplot.show(*args, **kw)>"
      ]
     },
     "execution_count": 3,
     "metadata": {
      "tags": []
     },
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light",
      "tags": []
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(T,Cp,'*')\n",
    "plt.plot(T,y(T), color='green', linewidth=1)\n",
    "plt.xlabel('Temperatura (K)')\n",
    "plt.ylabel('Cºp,m (J/mol K)')\n",
    "plt.show"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "BJx5ZncO1eig",
    "outputId": "7f1e6cbf-7f2e-41eb-ff62-a175082b4d14"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cºp,m =  2.483e+01 + 1.456e-02T -4.560e-06T²  + 4.90797e+04/T²\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ 2.48294824e+01,  1.45554447e-02, -4.55956846e-06,  4.90796782e+04])"
      ]
     },
     "execution_count": 4,
     "metadata": {
      "tags": []
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Definir a função que será ajustada\n",
    "def func(T, a, b, c, d):\n",
    "\n",
    "    return a + b*T + c*T**2 + d/T**2\n",
    "\n",
    "# ajuste da função usando mínimos quadrados\n",
    "# popt são os parâmetros do modelo obtidos após ajuste\n",
    "# pcov é a matriz de covariância de popt. A partir dela pode ser calculado o desvio padrão de cada parâmetro do ajuste\n",
    "popt, pcov = curve_fit(func, T, Cp)\n",
    "\n",
    "\n",
    "print('Cºp,m = ' + format(popt[0] , ' 6.3e') + ' +' + format(popt[1] , ' 6.3e') + 'T ' + format(popt[2] , ' 6.3e') + 'T²  +' + format(popt[3] , ' 6.5e') + '/T²')   \n",
    "\n",
    "#imprimir só os coeficientes na forma de vetor\n",
    "popt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "9WmF5tXN1eij",
    "outputId": "c33af5cb-dc2e-4425-f4af-3abeb7c887f8"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function matplotlib.pyplot.show(*args, **kw)>"
      ]
     },
     "execution_count": 5,
     "metadata": {
      "tags": []
     },
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light",
      "tags": []
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(T,Cp,'*')\n",
    "plt.plot(T,func(T, *popt), color='green', linewidth=1)\n",
    "plt.xlabel('Temperatura (K)')\n",
    "plt.ylabel('Cºp,m (J/mol K)')\n",
    "plt.show"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "CtnAunUN1eil"
   },
   "source": [
    "## Exercício 2\n",
    "\n",
    "Gráficos de Cp/T versus T são usados para determinar a entropia de um material, uma vez que o valor de entropia é igual a área sob a curva. A tabela abaixo apresenta a variação de Cp do $Na_2SO_4$ em função da temperatura. a) Ajuste um polinômio de terceira ordem que passa pela origem aos dados fornecidos e calcule a entropia do $Na_2SO_4$ a 90 K. b) Em baixas temperaturas é possível ajustar uma função do tipo aT³ aos valores de Cp/T (“a” é uma constante). Investigue em qual intervalo de temperaturas esta aproximação é válida. c) Ajuste uma função do tipo $ Cº_{p,m} = A + BT + CT^2 + \\frac{D}{T^2} $ e calcule a entropia a 90 K. d) Ajuste uma função do tipo $ Cº_{p,m} = A + BT + CT^2 + DT^3 $ e calcule a entropia a 90 K. Compare os ajustes dos itens (c) e (d).\n",
    "\n",
    "|   |   |   |   |   |   |   |   |   |  \n",
    "|---|---|---|---|---|---|---|---|---|\n",
    "|Cp($cal K^{-1}$ )|0.171|0.286|0.626|1.615|4.346|7.032|10.48|13.28|15.33\n",
    "|T(K)|13.74|16.25|20.43|27.73|41.11|52.72|68.15|82.96|95.71\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "BfTySfc91eil"
   },
   "source": [
    "### Solução (a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "B0nEK58q1eim",
    "outputId": "2ab77df0-165b-470d-ff76-7c298a2c34b5"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cºp,m = -5.746e-02T  + 5.167e-03T² -3.03363e-05T³\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([-5.74633196e-02,  5.16679456e-03, -3.03363216e-05])"
      ]
     },
     "execution_count": 6,
     "metadata": {
      "tags": []
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T = np.array([13.74,16.25,20.43,27.73,41.11,52.72,68.15,82.96,95.71])\n",
    "Cp = np.array([0.171,0.286,0.626,1.615,4.346,7.032,10.48,13.28,15.33])\n",
    "\n",
    "# Definir a função que será ajustada\n",
    "def func(T, b, c, d):\n",
    "\n",
    "    return b*T + c*T**2 + d*T**3\n",
    "\n",
    "# ajuste da função usando mínimos quadrados\n",
    "# popt são os parâmetros do modelo obtidos após ajuste\n",
    "# pcov é a matriz de covariância de popt. A partir dela pode ser calculado o desvio padrão de cada parâmetro do ajuste\n",
    "popt, pcov = curve_fit(func, T, Cp)\n",
    "\n",
    "\n",
    "print('Cºp,m = ' + format(popt[0] , ' 6.3e') + 'T  +' + format(popt[1] , ' 6.3e') + 'T² ' + format(popt[2] , ' 6.5e') + 'T³')   \n",
    "\n",
    "#imprimir só os coeficientes na forma de vetor\n",
    "popt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "faLoRz2a1eio",
    "outputId": "4a054856-27e6-40c7-95fa-b87e7ac836bd"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function matplotlib.pyplot.show(*args, **kw)>"
      ]
     },
     "execution_count": 7,
     "metadata": {
      "tags": []
     },
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light",
      "tags": []
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(T,Cp,'*')\n",
    "plt.plot(T,func(T, *popt), color='green', linewidth=1)\n",
    "plt.xlabel('Temperatura (K)')\n",
    "plt.ylabel('Cºp,m (J/mol K)')\n",
    "plt.show"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "KJFVadQj1eir",
    "outputId": "e61fb873-b487-49b7-a9ef-0bb21df78abd"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Delta S (J/K mol)=  8.382\n",
      "erro da integração = 1.006e-13\n"
     ]
    }
   ],
   "source": [
    "# Fazendo a integração no intervalo desejado\n",
    "\n",
    "def func(T):\n",
    "\n",
    "    return popt[0] + popt[1]*T + popt[2]*T**2\n",
    "\n",
    "ans, err = quad(func, 0, 90) #calcula a integral da função da acima e o erro da integração\n",
    "print('Delta S (J/K mol)= ' + format(ans , '6.3f'))\n",
    "print('erro da integração = ' + format(err, '6.3e'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "qrCXZFn41eiu"
   },
   "source": [
    "### Solução (b)\n",
    "É preciso analisar o intervalo no qual o ajusto pedido é válido."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "eNbxXVsf1eiv",
    "outputId": "e2105af7-ac4f-4bc0-9579-923b5c38b530"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a= 6.383e-05\n",
      "desvio padrão do ajuste = 1.919e-06\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<function matplotlib.pyplot.show(*args, **kw)>"
      ]
     },
     "execution_count": 26,
     "metadata": {
      "tags": []
     },
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light",
      "tags": []
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "Ti = T[:5] \n",
    "Cpi = Cp[:5]\n",
    "# Definir a função que será ajustada\n",
    "def func(Ti, a):\n",
    "\n",
    "    return a*Ti**3\n",
    "\n",
    "a, pcov = curve_fit(func, Ti, Cpi) # faz o ajuste do modelo\n",
    "\n",
    "perr = np.sqrt(np.diag(pcov)) # calcula o desvio padrão do modelo\n",
    "\n",
    "#imprimir só os coeficientes na forma de vetor\n",
    "print('a= ' + format(a[0] , '6.3e'))\n",
    "print('desvio padrão do ajuste = ' + format(perr[0], '6.3e'))\n",
    "\n",
    "plt.plot(Ti,Cpi,'*')\n",
    "plt.plot(Ti,func(Ti, *a), color='green', linewidth=1)\n",
    "plt.xlabel('Temperatura (K)')\n",
    "plt.ylabel('Cºp,m (J/mol K)')\n",
    "plt.show"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "k57fhhQK1eix"
   },
   "source": [
    "### Solução conjunta dos itens (c) e (d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "-skiuH1z1eiy",
    "outputId": "723e361f-9f24-4bf3-ebf8-8bbaf88eb325"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cºp,m = -6.436e+00 + 2.829e-01T -5.728e-04T²  + 5.61810e+02/T²\n",
      "Delta S (J/K mol)= -36.947\n",
      "erro da integração = 2.677e+01\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/lib/python3.6/dist-packages/ipykernel_launcher.py:25: IntegrationWarning: The integral is probably divergent, or slowly convergent.\n"
     ]
    }
   ],
   "source": [
    "# Considerando a primeira função\n",
    "\n",
    "\n",
    "# Definir a função que será ajustada\n",
    "def func(T, a, b, c, d):\n",
    "\n",
    "    return a + b*T + c*T**2 + d/T**2\n",
    "\n",
    "# ajuste da função usando mínimos quadrados\n",
    "# popt são os parâmetros do modelo obtidos após ajuste\n",
    "# pcov é a matriz de covariância de popt. A partir dela pode ser calculado o desvio padrão de cada parâmetro do ajuste\n",
    "popt, pcov = curve_fit(func, T, Cp)\n",
    "\n",
    "\n",
    "print('Cºp,m = ' + format(popt[0] , ' 6.3e') + ' +' + format(popt[1] , ' 6.3e') + 'T ' + format(popt[2] , ' 6.3e') + 'T²  +' + format(popt[3] , ' 6.5e') + '/T²')   \n",
    "\n",
    "#imprimir só os coeficientes na forma de vetor\n",
    "popt\n",
    "\n",
    "\n",
    "def func(T):\n",
    "\n",
    "    return popt[0]/T + popt[1] + popt[2]*T + popt[3]/T**3\n",
    "\n",
    "ans, err = quad(func, 0, 90)\n",
    "\n",
    "print('Delta S (J/K mol)= ' + format(ans , '6.3f'))\n",
    "print('erro da integração = ' + format(err, '6.3e'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "6llZALVb1ei0",
    "outputId": "c80cb920-3c0c-450f-c712-bfa6419c6787"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cºp,m = -0.20558-0.04189T +  4.85438e-03T² -2.85176e-05T³\n",
      "[-2.85176338e-05  4.85438172e-03 -4.18860690e-02 -2.05576078e-01]\n",
      "\n",
      "Delta S (J/K mol)=  0.393\n",
      "erro da integração = 1.922e+00\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/lib/python3.6/dist-packages/ipykernel_launcher.py:20: IntegrationWarning: The maximum number of subdivisions (50) has been achieved.\n",
      "  If increasing the limit yields no improvement it is advised to analyze \n",
      "  the integrand in order to determine the difficulties.  If the position of a \n",
      "  local difficulty can be determined (singularity, discontinuity) one will \n",
      "  probably gain from splitting up the interval and calling the integrator \n",
      "  on the subranges.  Perhaps a special-purpose integrator should be used.\n"
     ]
    }
   ],
   "source": [
    "# Considerando o polinômio do terceiro grau\n",
    "\n",
    "\n",
    "deg = 3\n",
    "z = np.polyfit(T, Cp, deg)\n",
    "y = np.poly1d(z)\n",
    "\n",
    "\n",
    "print('Cºp,m = ' + format(z[3] , ' 6.5f') + format(z[2] , ' 6.5f') + 'T + ' + format(z[1] , ' 6.5e') + 'T² ' + format(z[0] , ' 6.5e') + 'T³')   \n",
    "\n",
    "#imprimir só os coeficientes na forma de vetor\n",
    "print(z)\n",
    "print()\n",
    "\n",
    "\n",
    "def func(T):\n",
    "\n",
    "    return z[3]/T + z[2] + z[1]*T + z[0]*T**2\n",
    "\n",
    "ans, err = quad(func, 0, 90)\n",
    "\n",
    "print('Delta S (J/K mol)= ' + format(ans , '6.3f'))\n",
    "print('erro da integração = ' + format(err, '6.3e'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "ALYvg45p1ei3"
   },
   "source": [
    "Os três valores de entropia calculados nos itens (a), (c) e (d) diferem quanto ao valor e sinal. Não havendo acordo entre os métodos empregados. Era de se esperar que os três resultados apresentassem uma pequena diferença e não o desacordo observado. O erro ocorre porque as funções utilizadas nos itens (c) e (d) apresentam uma idneterminação na origem, o que levará a problemas de integração. Nestes casos, a integral deve ser separada em dois intervalos:\n",
    "\n",
    "$$ \\Delta Sº = \\int_{0}^{90} \\frac{C_p}{T} dT = \\int_{0}^{T_{baixa}} \\frac{C_p}{T} dT + \\int_{T_{baixa}}^{90} \\frac{C_p}{T} dT $$\n",
    "\n",
    "Como visto no item (b), Cp pode ser ajustado a uma função do tipo $ aT^3$, de forma que a integração se torna:  \n",
    "\n",
    "$$ \\Delta Sº = \\int_{0}^{T_{baixa}} \\frac{aT^3}{T} dT + \\int_{T_{baixa}}^{90} \\frac{C_p}{T} dT $$ \n",
    "\n",
    "ou \n",
    "\n",
    "$$ \\Delta Sº = \\frac{aT^3}{3} + \\int_{T_{baixa}}^{90} \\frac{C_p}{T} dT = \\frac{C_p}{3} + \\int_{T_{baixa}}^{90} \\frac{C_p}{T} dT$$ \n",
    "\n",
    "Onde $C_p$ da primeira integral da direita se refere ao valor da capacidade calorífica em $T_{baixa}$. Em nosso caso, é maior valor T para o qual é válido o ajuste da função aT³ usada no item (c).\n",
    "Vamos resolver os problemas com as modificações necessárias. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "Cax0z-_m1ei3",
    "outputId": "77af2564-7016-433d-fdf8-c0d189db2e17"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Delta S (J/K mol)=  8.565\n",
      "erro da integração = 9.509e-14\n"
     ]
    }
   ],
   "source": [
    "# Considerando a primeira função\n",
    "# Não é preciso refazer o ajuste, pois os coeficientes já estão salvos.\n",
    "\n",
    "def func1(T):\n",
    "\n",
    "    return a[0]*T**2\n",
    "\n",
    "ans1, err1 = quad(func1, 0, 40)\n",
    "\n",
    "\n",
    "def func2(T):\n",
    "\n",
    "    return popt[0]/T + popt[1] + popt[2]*T + popt[3]/T**3\n",
    "\n",
    "ans2, err2 = quad(func2, 40, 90)\n",
    "\n",
    "DS = ans1 + ans2\n",
    "erro = err1 + err2\n",
    "\n",
    "print('Delta S (J/K mol)= ' + format(DS , '6.3f'))\n",
    "print('erro da integração = ' + format(erro, '6.3e'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "GEBrAdm91ei6",
    "outputId": "f16e3c57-b6e5-4ba7-9579-b3e5aa818ba3"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Delta S (J/K mol)=  8.556\n",
      "erro da integração = 9.499e-14\n"
     ]
    }
   ],
   "source": [
    "# Considerando a primeira função\n",
    "# Não é preciso refazer o ajuste, pois os coeficientes já estão salvos.\n",
    "# Esta primeira integração não precisa ser repetida, pois os resultado será o mesmo. Mas calcularemos de novo a integral\n",
    "# para fixar a necessidade da separação dos cálculos\n",
    "\n",
    "def func1(T):\n",
    "\n",
    "    return a[0]*T**2\n",
    "\n",
    "ans1, err1 = quad(func1, 0, 40)\n",
    "\n",
    "\n",
    "def func2(T):\n",
    "\n",
    "    return z[3]/T + z[2] + z[1]*T + z[0]*T**2\n",
    "\n",
    "ans2, err2 = quad(func2, 40, 90)\n",
    "\n",
    "\n",
    "DS = ans1 + ans2\n",
    "erro = err1 + err2\n",
    "\n",
    "print('Delta S (J/K mol)= ' + format(DS , '6.3f'))\n",
    "print('erro da integração = ' + format(erro, '6.3e'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "vCGsgonx1ei8"
   },
   "source": [
    "Verifiquem que o resultado dos itens (a), (c) e (d) são similares. A escolha do modelo matemático implica nos cálculos que pretende realizar e as propriedades que pretende calcular."
   ]
  }
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