{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "48ab1797-e684-449d-b47a-5c4a688e5679",
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    "tags": []
   },
   "source": [
    "# Introduction to Thermodynamics and Statistical Physics"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "80900063-339b-4a4c-9ea6-2774c29554fa",
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   "source": [
    "In this lecture, we are going to discuss the 2nd law of thermodynamics. In particular, we'll cover:\n",
    "\n",
    "* The second law of thermoydnamics.\n",
    "* Heat Engines.\n",
    "* Clausius' Theorem.\n",
    "* Entropy.\n",
    "* Irreversibility.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d90ad1f5-0dd9-4979-8873-849d2a93e514",
   "metadata": {},
   "source": [
    "## Heat engines & and the second law"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5b3da7a1-ed69-494b-ac38-846b7d491613",
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   "source": [
    "At the end of the last lecture, we stated the second law of thermodynamics in 2 different ways:\n",
    "\n",
    "1. No process is possible whose sole result is the transfer of heat from a colder to a hotter body. (Clasius)\n",
    "2. No process is possible whose sole result is the complete conversion of heat into work. (Kelvin)\n",
    "\n",
    "So, assuming that these statements are true, let's consider the second statement. This says that no process can completely convert heat into work, meaning there must be a limit to the efficiency of converting heat into work. As such, we want to figure what this efficiency is. First, we are going to define efficiency as:\n",
    "$$\n",
    "    \\eta = \\frac{W}{Q_{\\rm H}}\n",
    "$$\n",
    "That is, we will supply a system with heat $Q_{\\rm H}$, and the system will then do some work $W$. To figure out what this efficiency is and how the various terms in that equation work, we will consider the Carnot engine."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "85a9515c-a391-42b5-bb71-0f2e7140ff12",
   "metadata": {},
   "source": [
    "## The Carnot Engine"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "06da2e44-bd69-4fdd-89e9-9a4e487db6bd",
   "metadata": {},
   "source": [
    "Consider the following engine.\n",
    "\n",
    "![Carnot_Engine](Figures/Carnot_Engine.png)\n",
    "\n",
    "It is able to draw heat $Q_{\\rm H}$ from a reservoir which is at a temperature $T_H$. It then dumps heat $Q_{\\rm L}$ to reservoir which is at a temperature $T_L$ while doing some work.\n",
    "\n",
    "We are going to let the engine perform the following steps: an isothermal expansion, followed by an adiabatic expansion, followed by an isothermal compression, and then another adiabatic compression. These 4 processes will trace out a path shown below in the P-V diagram, where the process follows a series of isotherms and adiabats. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "001cad0e-c3b5-4895-afe9-9c70fbd023b4",
   "metadata": {},
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 600x450 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import astropy.units as u\n",
    "import astropy.constants as c\n",
    "\n",
    "def adiabatic(V_i,P_i,V):\n",
    "    gamma = 5/3\n",
    "    return (P_i*V_i**gamma)/(V**gamma)\n",
    "\n",
    "def isothermal(V_i,P_i,V):\n",
    "    T = P_i*V_i/c.R\n",
    "    return (c.R*T/V)\n",
    "\n",
    "P_is = np.arange(3,5,1)*u.Pa\n",
    "V_i = 1*u.m**3\n",
    "V = np.arange(0.5,2.02,0.01)*u.m**3\n",
    "\n",
    "plt.figure(figsize=[4,3],dpi=150)\n",
    "#Plotting the full isotherms & Adiabats\n",
    "for P_i in P_is:\n",
    "    P_a = adiabatic(V_i,P_i,V)\n",
    "    P_isot = isothermal(V_i,P_i,V)\n",
    "    if P_i == 3.0*u.Pa:\n",
    "        plt.plot(V,P_a,'C0-',label='Adiabat, $PV^\\gamma=const$')\n",
    "        plt.plot(V,P_isot,'C1--',label='Isotherm, $PV=const$')\n",
    "    else:\n",
    "        plt.plot(V,P_a,'C0-',label='')\n",
    "        plt.plot(V,P_isot,'C1--',label='')\n",
    "\n",
    "gamma = 5/3\n",
    "V_A = (P_is[0]/P_is[1])**(1/(gamma-1))\n",
    "P_A = P_is[1]*V_i/V_A\n",
    "plt.text(V_A.value,P_A.value,'A')\n",
    "\n",
    "V_B = V_i\n",
    "P_B = P_is[1]\n",
    "plt.text(V_B.value,P_B.value,'B')\n",
    "\n",
    "V_C = (P_is[1]/P_is[0])**(1/(gamma-1))\n",
    "P_C = P_is[0]*V_i/V_C\n",
    "plt.text(V_C.value,P_C.value,'C')\n",
    "\n",
    "V_D = V_i\n",
    "P_D = P_is[0]\n",
    "plt.text(V_D.value,P_D.value-0.8,'D')\n",
    "\n",
    "# Process A-B\n",
    "V_temp = np.arange(V_A.value,V_B.value,0.001)*u.m**3\n",
    "P_temp = isothermal(V_A,P_A,V_temp)\n",
    "plt.plot(V_temp,P_temp,'k-',label='')\n",
    "\n",
    "# Process B-C\n",
    "V_temp = np.arange(V_B.value,V_C.value,0.001)*u.m**3\n",
    "P_temp = adiabatic(V_B,P_B,V_temp)\n",
    "plt.plot(V_temp,P_temp,'k-',label='')\n",
    "\n",
    "# Process C-D\n",
    "V_temp = np.arange(V_C.value,V_D.value,-0.001)*u.m**3\n",
    "P_temp = isothermal(V_C,P_C,V_temp)\n",
    "plt.plot(V_temp,P_temp,'k-',label='')\n",
    "\n",
    "# Process D-A\n",
    "V_temp = np.arange(V_D.value,V_A.value,-0.001)*u.m**3\n",
    "P_temp = adiabatic(V_D,P_D,V_temp)\n",
    "plt.plot(V_temp,P_temp,'k-',label='')\n",
    "\n",
    "frame1 = plt.gca()\n",
    "frame1.axes.get_xaxis().set_ticks([])\n",
    "frame1.axes.get_yaxis().set_ticks([])\n",
    "plt.xlim(0.5,2)\n",
    "plt.ylim(0.0,8)\n",
    "plt.ylabel(\"Pressure\")\n",
    "plt.xlabel(\"Volume\")\n",
    "plt.title(\"Adiabats and Isotherms for\\n different initial conditions.\")\n",
    "plt.legend()\n",
    "plt.savefig(\"Figures/Carnot_Cycle.jpg\")\n",
    "plt.show()"
   ]
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    "Because the processes B-C and D-A are adiabatic, and A-B and C-D are isothermal, we can make the following statement. Heat $Q_{\\rm H}$ enters the engine during process A-B, and heat $Q_{\\rm L}$ leaves during C-D, and no heat is lost duing B-C and D-A. Finally, the process is cyclic, meaning there is no change in the internal energy of our engine. As such, the work done by the engine is given by\n",
    "$$\n",
    "    W = Q_{\\rm H}-Q_{\\rm L}\n",
    "$$\n",
    "In order to understand this expression, we need to figure out what $Q_{\\rm H}$ and $Q_{\\rm H}$ might be. As such, let's consider an ideal gas and follow it during each of the 4 stages.\n",
    "\n",
    "1. A-B, Isothermal expansion. $Q_{\\rm H} = R T_{\\rm H} \\ln \\frac{V_B}{V_A}$.\n",
    "2. B-C, Adiabatic expansion. $\\frac{T_{\\rm H}}{T_{\\rm L}} = \\left(\\frac{V_{\\rm C}}{V_{\\rm B}}\\right)^{\\gamma - 1}$.\n",
    "3. C-D, Isothermal compression. $Q_{\\rm L} = R T_{\\rm L} \\ln \\frac{V_C}{V_D}$.\n",
    "4. B-C, Adiabatic compression. $\\frac{T_{\\rm L}}{T_{\\rm H}} = \\left(\\frac{V_{\\rm A}}{V_{\\rm D}}\\right)^{\\gamma - 1}$.\n",
    "\n",
    "Combining all of these together thus gives:\n",
    "$$\n",
    "    \\frac{Q_H}{Q_L} = \\frac{T_H}{T_L}\n",
    "$$\n",
    "It's also important to note that all of these processes are being done reversible, so we are free to change the directions of the arrows in our schematic (we'll come across this later in this lecture).\n",
    "\n",
    "For the Carnot engine with an ideal gas, we can thus express the efficiency in terms of the temperatures of the isoterms by:\n",
    "\\begin{align}\n",
    "    \\eta &= \\frac{Q_{\\rm H}-Q_{\\rm L}}{Q_{\\rm H}}=1-\\frac{Q_{\\rm L}}{Q_{\\rm H}}\\\\\n",
    "    \\eta &= 1-\\frac{T_{\\rm L}}{T_{\\rm H}}\n",
    "\\end{align}\n",
    "\n",
    "It turns out that this expression for efficiency is the highest efficiency any engine can achieve for a given $T_{\\rm H}$ and $T_{\\rm L}$, assuming Clausius's statement of the second law."
   ]
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   "source": [
    "## The equivalence of Clausius's and Kelvin's statement of the 2nd law"
   ]
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   "id": "c7f58cfb-d210-4945-8fd9-63d16ac0b102",
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    "We now want to ask 2 questions.\n",
    "1. If a system violates Kelvin's statement, it also violates Clausius' statement.\n",
    "2. If a system violates Clausius' statement, it also violates Kelvin's statement.\n",
    "\n",
    "To answer these questions, we're going to take the Carnot engine and hook up some special equipment, and see what happens.\n",
    "\n",
    "### 1. If a system violates Kelvin's statement, it also violates Clausius' statement.\n",
    "Let's assume we have the following system. This system violates Kelvin's statement, as the Kelvin violator is converting heat solely into work.\n",
    "\n",
    "![Kelvin Violator](Figures/Kelvin_Violator.png)\n",
    "\n",
    "In this case, we would have $Q_{\\rm  H}' = W$ from the Kelvin violator, and $Q_{\\rm H} = Q_{\\rm L} + W$ from the Carnot engine. The heat dumped into the reservor at $T_{rm H}$ is then $Q_{H}-Q_{H}'=Q_{L}$. Thus, the net process of the entire system is a flow of energy of $Q_{\\rm L}$ from the cold reservoir to the hot reservoir, which violates Clausius' statement.\n",
    "\n",
    "### 2. If a system violates Clausius' statement, it also violates Kelvin's statement.\n",
    "Let's assume we have the following system. This system violates Clausius' statement, as the Clausius violator is transferring heat from a colder body to a hotter body.\n",
    "\n",
    "![Clausius Violator](Figures/Clausius_Violator.png)\n",
    "\n",
    "The first law implies that for the Carnot engine, $Q_{\\rm H}-Q_{\\rm L} = W$. The sole effect of the whole system is thus to convert the heat $Q_{\\rm H}-Q_{\\rm L}$ into work. This violates Kelvin's law!"
   ]
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   "source": [
    "## Refrigerators"
   ]
  },
  {
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   "id": "139374ad-bf9e-4188-b216-dab969fa478e",
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   "source": [
    "Let's now consider a engine running in the reverse direction to that discussed above.\n",
    "\n",
    "![Clausius Violator](Figures/Reverse_Carnot_Engine.png)\n",
    "\n",
    "In this case, work is put into the engine, and heat flows from the cold temperature reservoir to hot reservoir. The relevant efficiency in this case is\n",
    "$$\n",
    "    \\eta = \\frac{Q_{\\rm L}}{W}\n",
    "$$\n",
    "as we want to know what flow of heat we can achieve for a given amount of work. Using the above arguements that we used for the Carnot engine, it's then easy to show that\n",
    "$$\n",
    "    \\eta = \\frac{T_{\\rm L}}{T_{\\rm H}-T_{\\rm L}}\n",
    "$$\n",
    "which is  greater than 100%. So, when we design an engine to do work, the efficiency will be less than 1. However, when we design an engine to exchange heat between bodies, the efficiency will be greater than 1."
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "## Clausius' Theorem"
   ]
  },
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   "id": "42298f4c-ff14-4e29-a746-8a67a7260b2f",
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   "source": [
    "Consider one cycle of a Carnot engine. Over this cycle, $Q_{\\rm H}$ enters the engine and $Q_{\\rm L}$ leaves the engine. We know from earlier discussions that\n",
    "$$\n",
    "    \\frac{Q_{\\rm H}}{Q_{\\rm L}} = \\frac{T_{\\rm H}}{T_{\\rm L}}\n",
    "$$\n",
    "We will now define $\\Delta Q_{\\rm rev}$ as the heat entering the system at each point of the process, such that\n",
    "$$\n",
    "    \\sum \\frac{\\Delta Q_{\\rm rev}}{T} = \\frac{Q_{\\rm H}}{T_{\\rm H}} + \\left(-\\frac{Q_{\\rm L}}{T_{\\rm L}}\\right) = 0\n",
    "$$\n",
    "Rather than just summing up all the contributions, we could instead integrate, in which case we get\n",
    "$$\n",
    "    \\oint \\frac{{\\rm d} Q_{\\rm rev}}{T} = 0\n",
    "$$\n",
    "While this is a useful result, we must now remember that the Carnot engine does everything reversible. So now we want to consider what this inequality looks like for real world systems.\n",
    "\n",
    "Consider now the following setup. We have two engines which are connected to two reservoirs, one at temperature $T_{\\rm H}$ and one at $T_{\\rm L}$. One engine is a Carnot engine, and so everything is performed reversibly, while the other one is not.\n",
    "\n",
    "![Clausius_Inequality](Figures/Clausius_Inequality.png)\n",
    "\n",
    "Now let's consider the efficincies of both engines. For the Carnot engine, I'll use subscript R for reversible, and for the other engine I'll use I for irreversible.\n",
    "\\begin{align}\n",
    "    \\eta_R &= 1-\\frac{Q_{L,R}}{Q_{H,R}}\\\\\n",
    "    \\eta_I &= 1-\\frac{Q_{L,I}}{Q_{H,I}}\n",
    "\\end{align}\n",
    "So we stated earlier that a Carnot engine is the most efficient type of engine we can have. As such\n",
    "$$\n",
    "    \\eta_{R} \\geq \\eta_{I}\n",
    "$$\n",
    "These leads to\n",
    "\\begin{align}\n",
    "    1-\\frac{Q_{L,R}}{Q_{H,R}} &\\geq 1-\\frac{Q_{L,I}}{Q_{H,I}}\\\\\n",
    "    \\frac{Q_{L,R}}{Q_{H,R}} &\\leq \\frac{Q_{L,I}}{Q_{H,I}}\\\\\n",
    "\\end{align}\n",
    "Now recalling that for a Carnot engine we have\n",
    "$$\n",
    "    \\frac{Q_H}{Q_L} = \\frac{T_H}{T_L}\n",
    "$$\n",
    "we then get\n",
    "\\begin{align}\n",
    "    \\frac{T_{L}}{T_{H}} &\\leq \\frac{Q_{L,I}}{Q_{H,I}}\\\\\n",
    "    \\frac{Q_{H,I}}{T_{H}} - \\frac{Q_{L,I}}{T_{L}} &\\leq 0\\\\\n",
    "\\end{align}\n",
    "which is the same as \n",
    "$$\n",
    "    \\oint \\frac{{\\rm d} Q}{T} \\leq 0\n",
    "$$\n",
    "There are many other ways of arriving at this inequality, which I recommend you look up. The identity will become important in the next lecture."
   ]
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    "Clausius' theorem states that\n",
    "$$\n",
    "    \\oint \\frac{{\\rm d} Q_{\\rm rev}}{T} = 0\n",
    "$$\n",
    "This means that the quantity\n",
    "$$\n",
    "    \\int ^A_B \\frac{{\\rm d} Q_{\\rm rev}}{T}\n",
    "$$\n",
    "is a function of state (that is, it has an exact value for a given equilibrium state of the system). Another way of saying this is that this quantity is an exact differential, and is path independant. We are going to now define this function of state as the **entropy** S of the system such that\n",
    "$$\n",
    "    {\\rm d} S = \\frac{{\\rm d} Q_{\\rm rev}}{T}\n",
    "$$\n",
    "and \n",
    "$$\n",
    "    S(B)-S(A) = \\int ^A_B \\frac{{\\rm d} Q_{\\rm rev}}{T}\n",
    "$$\n",
    "Recalling that for an adiabatic process, ${\\rm d} Q_{\\rm rev} = 0$, this would also mean that there is no change in entropy during this process. As such, an adiabatic process can also be called an isentropic process."
   ]
  },
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   "source": [
    "## Irreversibility"
   ]
  },
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   "id": "5138b0d2-574e-41b4-8ce0-7293a4f335c7",
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    "S is defined in relation to a reversible change of heat. Taking Clausius' theorem for reversible processes again, we have that\n",
    "$$\n",
    "    \\oint \\frac{{\\rm d} Q_{\\rm rev}}{T} = 0\n",
    "$$\n",
    "We are now going to consider a cycle which starts with some irreversible process, followed by a reversible process. Such a process is given in the P-V diagram below.\n",
    "\n",
    "![Irreversible](Figures/Entropy_Irreversible_Process.png)\n",
    "\n",
    "The Clausius' inequality tell us that \n",
    "$$\n",
    "    \\oint \\frac{{\\rm d} Q_{\\rm rev}}{T} \\leq 0\n",
    "$$\n",
    "We can then split this integral into two segments\n",
    "$$\n",
    "    \\int ^B_A \\frac{{\\rm d} Q}{T} + \\int ^A_B \\frac{{\\rm d} Q_{\\rm rev}}{T} \\leq 0\n",
    "$$\n",
    "which simplifies to \n",
    "$$\n",
    "    \\int ^B_A \\frac{{\\rm d} Q}{T} \\leq \\int ^B_A \\frac{{\\rm d} Q_{\\rm rev}}{T}\n",
    "$$\n",
    "This is true no matter how close the curves joining A to B and B to A get. As such, we can also write the change in entropy of a system to be\n",
    "$$\n",
    "    {\\rm d} S = \\frac{{\\rm d} Q_{\\rm rev}}{T} \\geq \\frac{{\\rm d} Q}{T} \n",
    "$$\n",
    "Now consider a thermally isolated system. This means that ${\\rm d} Q$ is always 0 for this system. Then, for any process within the system, the inequality becomes\n",
    "$$\n",
    "    {\\rm d} S \\geq 0.\n",
    "$$\n",
    "It's worth considering exactly what this condition is now saying. For any system that is thermally isolated, the entropy will stay the same for reversible processes, or will increase for irreversible processes. This is another way of stating the second law of thermodynamics: **\"The entropy of an isolated system tends to a maximum\"**. \n",
    "\n",
    "This has some very fundamental consequences. For example, if the Universe is a isolated system, then it means that entropy of the Universe can only increase."
   ]
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