{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Aliatges binaris: Llei de la palanca"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Quan tenim un aliatge a la zona de coexistència de fases, podem fàcilment conèixer la composició de la fase sòlida  $C_{\\alpha}$ i de la fase líquida $C_{L}$ buscant la intersecció de la isoterma amb les línies *liquidus* i *solidus* :"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](img/LleiPalanca.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Una mica més complicat és calcular quin percentatge de l'aliatge el tindrem en fase sòlida i en fase líquida. Fent una mica d'àlgebra arribem a la llei de la palanca, on $W_{\\alpha}$ és el percentatge en fase $\\alpha$ (sòlid) i $W_{L}$ el percentatge en fase líquida:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$W_{\\alpha} \\cdot C_{\\alpha} + (1-W_{\\alpha}) \\cdot C_{L} = C_{0}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$W_{\\alpha} \\cdot C_{\\alpha} + C_{L}-W_{\\alpha} \\cdot C_{L} = C_{0}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$W_{\\alpha} \\cdot C_{\\alpha} -W_{\\alpha} \\cdot C_{L} = C_{0} - C_{L}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$W_{\\alpha} (C_{\\alpha} -C_{L}) = C_{0} - C_{L}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$W_{\\alpha} = \\frac{C_{0}-C_{L}}{C_{\\alpha}-C_{L}}$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fent un càlcul anàlog per $W_{L}$:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$W_{L} = \\frac{C_{\\alpha}-C_{0}}{C_{\\alpha}-C_{L}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exemple: aliatge Ni-Cu"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si tenim un aliatge amb un 53% de Ni, a $1300$ $^{0}C$ tindrem coexistència de dos fases: una fase líquida amb un 45% de Ni i una sòlida amb un 58% de Ni "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](img/CuNi.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculem quin percentatge tindrem a cada fase:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "C0=0.53\n",
    "CL=0.45\n",
    "Calfa=0.58"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6153846153846158"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Walfa=(C0-CL)/(Calfa-CL)\n",
    "Walfa"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.38461538461538425"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "WL=(Calfa-C0)/(Calfa-CL)\n",
    "WL"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Es a dir, tindrem un 61,5 % en fase sòlida i un 38,5 % en fase líquida"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Crèdits"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Imatges estretes de [https://sites.google.com/site/federubiotecindbachillerato/home/2o-bachillerato/materiales/4-diagramas-de-equilibrio](https://sites.google.com/site/federubiotecindbachillerato/home/2o-bachillerato/materiales/4-diagramas-de-equilibrio)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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