diff --git a/Chapter_8/statistical-physics.ipynb b/Chapter_8/statistical-physics.ipynb index 42609d9..41ef520 100644 --- a/Chapter_8/statistical-physics.ipynb +++ b/Chapter_8/statistical-physics.ipynb @@ -265,7 +265,7 @@ "Similarly, there is an average energy associated with the other two velocity components, which produces a net average energy per molecule of $\\frac{3}{2}kT$. In a monatomic gas (e.g., $\\rm He$ or $\\rm Ar$), virtually all of the energy is in the translational kinetic energy.\n", "\n", "```{margin}\n", - "Rigid rotator model\n", + "**Rigid rotator model**\n", "```\n", "\n", "**Consider a diatomic gas (e.g., $\\rm O_2$) as two atoms connected by a massless rod (i.e., a rigid rotator).** Then the molecule can have *rotational* kinetic energy in addtion to translational kinetic energy. The crucial questions are:\n", @@ -289,7 +289,7 @@ "Each of these rotational energies $(K_x\\ \\text{or}\\ K_y)$ is quadratic in angular velocity, so the equipartition theorem instructs us to add $2\\left(\\frac{1}{2}kT\\right) = kT$ per molecule to the translational kinetic energy for a total of $\\frac{5}{2}kT$.\n", "\n", "```{margin}\n", - "Spring model\n", + "**Spring model**\n", "```\n", "\n", "Sometimes it is a better approximation to think of atoms connected by a massless spring rather than a rigid rod. In this model, we use the potential energy of a spring but in radial coordinates:\n", @@ -377,7 +377,321 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "## Maxwell Speed Distribution" + "## Maxwell Speed Distribution\n", + "\n", + "The general form of the Maxwell velocity distribution is given as\n", + "\n", + "\\begin{align*}\n", + "f(\\vec{v})\\ d^3\\vec{v} = C e^{-\\frac{1}{2}\\beta m v^2}\\ d^3\\vec{v},\n", + "\\end{align*}\n", + "\n", + "where $C = {C^\\prime}^3 = \\left(\\frac{\\beta m}{2\\pi}\\right)^{3/2}$.\n", + "\n", + "The distribution $f(\\vec{v})$ is a function of the speed $v$ in the exponent and not the velocity $\\vec{v}$, but it is still a velocity distribution due to the phase space volume element $d^3\\vec{v}$. We can turn this velocity distribution into a speed distribution $F(v)$ using the definition\n", + "\n", + "\\begin{align*} \n", + "F(v)\\ dv =& \\text{ probability of finding a particle } \\\\ \n", + "& \\text{ with speed between } v\\ \\text{and}\\ v + dv.\n", + "\\end{align*}\n", + "\n", + "**Consider the analogous problem in 3-D position space $(x,\\ y,\\ z)$.** Some distribution of particles exist, where a particle can be located with $(x,\\ y,\\ z)$ with a distance $r = \\sqrt{x^2+ y^2 + z^2}$ from the origin and has a position vector $\\vec{r}$ measured relative to the origin. Then\n", + "\n", + "\\begin{align*}\n", + "f(x,\\ y,\\ z)\\ d^3\\vec{r} = & \\text{ probability of finding a particle} \\\\\n", + "& \\text{ between }\\vec{r}\\ \\text{and}\\ \\vec{r} + d^3\\vec{r},\n", + "\\end{align*}\n", + "\n", + "with $d^3\\vec{r} = dx\\ dy\\ dz$. To shift to the scalar *radial* distribution we introduce the definition \n", + "\n", + "\\begin{align*}\n", + "F(r)dr = & \\text{ probability of finding a particle} \\\\\n", + "& \\text{ between }r\\ \\text{and}\\ r + dr.\n", + "\\end{align*}\n", + "\n", + "The space between $r$ and $r+dr$ is a *spherical shell*, and thus, we must integrate over that volume to transform to the radial distribution. These two distributions are related by the volume of spherical shell, or $4\\pi r^2$. We may write\n", + "\n", + "\\begin{align}\n", + "F(r)dr = f(x,\\ y,\\ z) 4\\pi r^2\\ dr.\n", + "\\end{align}\n", + "\n", + "Returning to the problem of the speed distribution $F(v)$ from the velocity distribution $f(\\vec{v})$. We need to integrate over a spherical shell in the velocity phase space, where we replace $r\\rightarrow v$ from the previous example. The desired speed distribution is\n", + "\n", + "```{margin}\n", + "**Maxwell speed distribution**\n", + "```\n", + "\n", + "\\begin{align}\n", + "F(v)\\ dv = 4\\pi Ce^{-\\frac{1}{2}\\beta m v^2}v^2\\ dv.\n", + "\\end{align}\n", + "\n", + "The Maxwell speed distribution as derived from purely classical considerations, where it gives a nonzero probability of finding a particle with a speed greater than $c$. Therefore, it is only valid in the classical limit.\n", + "\n", + "The assymetry of the distribution curve leads to an interesting result:\n", + "\n", + "> the most probable speed $v_{\\rm mp}$, the mean speed $\\bar{v}$, and the root-mean-square speed $v_{\\rm rms}$ are all slightly different from each other.\n", + "\n", + "To find the most probable speed, we simply find *maximum* speed in the probabilty curve, or\n", + "\n", + "\\begin{align}\n", + "\\frac{dF}{dv} &= 4\\pi C \\frac{d}{dv}\\left[e^{-\\frac{1}{2}\\beta m v^2}v^2 \\right], \\\\\n", + "&= 2ve^{-\\frac{1}{2}\\beta m v^2} - \\left(\\beta m v\\right)v^2 e^{-\\frac{1}{2}\\beta m v^2}.\n", + "\\end{align}\n", + "\n", + "Replacing $v\\rightarrow v_{\\rm mp}$ to denote the *most probable* speed and setting the above result equal to zero, we can solve for $v_{\\rm mp}$:\n", + "\n", + "```{margin}\n", + "**Most probable speed $v_{\\rm mp}$**\n", + "```\n", + "\n", + "\\begin{align}\n", + "2v_{\\rm mp}e^{-\\frac{1}{2}\\beta m v_{\\rm mp}^2} &= \\left(\\beta m v_{\\rm mp}\\right)v_{\\rm mp}^2 e^{-\\frac{1}{2}\\beta m v_{\\rm mp}^2}, \\\\\n", + "v_{\\rm mp} &= \\sqrt{\\frac{2}{\\beta m}} = \\sqrt{\\frac{2kT}{m}}.\n", + "\\end{align}\n", + "\n", + "A particle moving at the most probable speed has a kinetic energy $K_{\\rm mp} = \\frac{1}{2}mv_{\\rm mp}^2 = kT$.\n", + "\n", + "The *mean speed* is found by integrating (summing up) the probabilities for individual speeds, or\n", + "\n", + "\\begin{align*}\n", + "\\bar{v} = \\int_0^\\infty vF(v)\\ dv = 4\\pi C \\int_0^\\infty v^3e^{-\\frac{1}{2}\\beta m v^2}dv.\n", + "\\end{align*}\n", + "\n", + "See a list of [integrals containing exponential functions](https://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions) and using $(k=1,\\ n=3,\\ a=\\beta m/2)$ to find\n", + "\n", + "```{margin}\n", + "**Mean speed $\\bar{v}$**\n", + "```\n", + "\n", + "\\begin{align*}\n", + "\\bar{v} &= 4\\pi C \\left(\\frac{1}{2(\\beta m/2)^2}\\right) = 8\\pi \\left(\\frac{\\beta m}{2\\pi}\\right)^{3/2} \\left(\\frac{1}{(\\beta m)^2}\\right), \\\\\n", + " &= \\frac{4}{\\sqrt{2\\pi}} \\left(\\frac{1}{\\sqrt{\\beta m}}\\right), \\\\ &= \\frac{2}{\\sqrt{\\pi}} \\sqrt{\\frac{2kT}{m}}, \\\\\n", + " &= \\frac{2}{\\sqrt{\\pi}} v_{\\rm mp}.\n", + "\\end{align*}\n", + "\n", + "We define the **root-mean-square** $v_{\\rm rms}$ to be $ v_{\\rm rms} \\equiv \\left(\\overline{v^2}\\right)^{1/2}$. We cannot simply use the result for the mean speed $\\bar{v}$ directly. Instead, we need to find the mean using the probability function $F(v)$, or\n", + "\n", + "\\begin{align*}\n", + "\\overline{v^2} = \\int_0^\\infty v^2F(v)\\ dv = 4\\pi C \\int_0^\\infty v^4e^{-\\frac{1}{2}\\beta m v^2}dv.\n", + "\\end{align*}\n", + "\n", + "See a list of [integrals containing exponential functions](https://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions) and using $(k=2,\\ n=4,\\ a=\\beta m/2)$ to find\n", + "\n", + "\\begin{align}\n", + "\\overline{v^2} &= 4\\pi C \\int_0^\\infty v^4e^{-\\frac{1}{2}\\beta m v^2}dv = 4\\pi C \\left(\\frac{3!!}{8a^2}\\sqrt{\\frac{\\pi}{a}}\\right), \\\\\n", + "&= 4\\pi \\left(\\frac{\\beta m}{2\\pi}\\right)^{3/2} \\frac{3}{2(\\beta m)^2} \\sqrt{\\frac{2\\pi}{\\beta m}}, \\\\\n", + "&= \\frac{3(\\beta m)^{3/2}}{(\\beta m)^{5/2} } = \\frac{3}{\\beta m}, \\\\\n", + "&= \\frac{3kT}{m}.\n", + "\\end{align}\n", + "\n", + "Then\n", + "\n", + "```{margin}\n", + "**Root-mean-square speed $v_{\\rm rms}$**\n", + "```\n", + "\n", + "\\begin{align}\n", + "v_{\\rm rms} = \\left(\\overline{v^2}\\right)^{1/2} = \\sqrt{\\frac{3kT}{m}} = \\sqrt{\\frac{3}{2}}v_{\\rm mp}.\n", + "\\end{align}\n", + "\n", + "A particle moving with the mean squared speed has a kinetic energy \n", + "\n", + "$$ \\overline{K} = \\frac{1}{2}m\\overline{v^2} = \\frac{1}{2}m \\left(\\frac{3kT}{m}\\right) = \\frac{3}{2}kT $$\n", + "\n", + "in keeping with our basic law of kinetic theory.\n", + "\n", + "The **standard deviation** of the molecular speeds $\\sigma_v$ is \n", + "\n", + "\\begin{align}\n", + "\\sigma_v &= \\left(\\overline{v^2} - \\bar{v}^2\\right)^{1/2} = \\left(\\frac{3kT}{m} - \\frac{8kT}{m\\pi}\\right)^{1/2}, \\\\\n", + "&= \\left[\\frac{(3\\pi-8)kT}{m\\pi}\\right]^{1/2},\\\\\n", + "&= \\left[3-\\left(\\frac{8}{\\pi}\\right)\\right]^{1/2} \\sqrt{\\frac{kT}{m}}, \\\\\n", + "&= \\left[\\frac{3}{2}-\\left(\\frac{8}{2\\pi}\\right)\\right]^{1/2} v_{\\rm mp}.\n", + "\\end{align}\n", + "\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "```{exercise}\n", + ":class: orange\n", + "\n", + "**Computre the mean molecular speed $\\bar{v}$ in hydrogen $({\\rm H_2})$ and radon $({\\rm Rn})$ gas, both at room temperature $293\\ {\\rm K}$. Use the longest-lived radon isotope, which has a molar mass of $222\\ {\\rm u}$. Compare the results.**\n", + "\n", + "The mean molecular speed is given by \n", + "\n", + "$$ \\bar{v} = \\frac{2}{\\sqrt{\\pi}} \\sqrt{\\frac{2kT}{m}}, $$\n", + "\n", + "which can be applied to each type of gas. The only difference between results will be due to the molar mass $m$. The molar mass of $\\rm H_2$ is $2.01568\\ {\\rm u}$, which is found by multiplying the mass of hydrogen by two. The conversion from atomic mass units ${\\rm u}$ to $\\rm kg$ is $1.660539 \\times 10^{-27}\\ {\\rm kg/u}$. Let's re-write the mean molecular speed so that all the physical constants are combined and so that we can use atomic mass units directly. This gives\n", + "\n", + "\\begin{align}\n", + "\\bar{v} = \\left(2\\sqrt{\\frac{2k}{\\pi u}}\\right)\\sqrt{\\frac{T\\ \\text{(in K)}}{m\\ \\text{(in u)}}} = 145.5081\\sqrt{\\frac{T}{m}}.\n", + "\\end{align}\n", + "\n", + "Then, we compute the mean molecular speed for molecular hydrogen $\\rm H_2$ as\n", + "\n", + "$$ \\bar{v}_{\\rm H_2} = 145.5081\\sqrt{\\frac{293}{2.01568}} = 1750\\ {\\rm m/s}.$$\n", + "\n", + "The mean molecular speed for radon ${\\rm Rn}$ can be calculated a mass ratio because the temperature is the same in both cases. From this method we find that the mean speed is:\n", + "\n", + "$$ \\bar{v}_{\\rm Rn} = \\sqrt{\\frac{2.01568}{222}}\\bar{v}_{\\rm H_2} = 166\\ {\\rm m/s}.$$\n", + "\n", + "The hydrogen molecule is ${\\sim}10\\times$ faster on average. That's to be expected because the mass ratio $(m_{\\rm Rn}/m_{\\rm H_2}) \\sim \\sqrt{100} = 10.$\n", + "\n", + "```\n" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "----For hydrogen-----\n", + "The mean molecular speed of molecular hydrogen (H_2) is 1750 m/s.\n", + "The mean molecular speed using our simplified equation is 1750 m/s.\n", + "----For radon-----\n", + "The mean molecular speed of radon gas (Rn) is 166 m/s.\n", + "The mass ratio (m_Rn/m_H2) is 110 and the ratio of speeds (v_H2/v_Rn) is 10.5.\n" + ] + } + ], + "source": [ + "import numpy as np\n", + "from scipy.constants import physical_constants \n", + "\n", + "def mean_molecular_speed(T,m):\n", + " #T = absolute temperature in K\n", + " #m = molar mass in u\n", + " return (2/np.sqrt(np.pi))*np.sqrt((2*k*T)/(m*u))\n", + "\n", + "k = physical_constants['Boltzmann constant'][0]\n", + "u = physical_constants['atomic mass constant'][0]\n", + "mean_speed_constant = 2/np.sqrt(np.pi)*np.sqrt(2*k/u)\n", + "\n", + "T = 293 #room temperature in K\n", + "m_H2 = 2.01568 #molar mass in u\n", + "m_Rn = 222 #molar mass in u\n", + "mass_ratio = m_Rn/m_H2\n", + "\n", + "v_H2 = np.round(mean_molecular_speed(T,m_H2),-1)\n", + "v_H2_constant = np.round(mean_speed_constant*np.sqrt(T/m_H2),-1)\n", + "\n", + "print(\"----For hydrogen-----\")\n", + "print(\"The mean molecular speed of molecular hydrogen (H_2) is %i m/s.\" % v_H2)\n", + "print(\"The mean molecular speed using our simplified equation is %i m/s.\" % v_H2_constant)\n", + "\n", + "print(\"----For radon-----\")\n", + "v_Rn = v_H2/np.sqrt(mass_ratio)\n", + "print(\"The mean molecular speed of radon gas (Rn) is %i m/s.\" % v_Rn)\n", + "print(\"The mass ratio (m_Rn/m_H2) is %i and the ratio of speeds (v_H2/v_Rn) is %1.1f.\" % (mass_ratio, np.sqrt(mass_ratio)))\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "```{exercise}\n", + ":class: orange\n", + "\n", + "**What fraction of the molecules in an ideal gas in equilibrium has speeds with $\\pm1\\%$ of the most probable speed $v_{\\rm mp}$?**\n", + "\n", + "The Maxwell speed distribution function provides the probability of finding a particle within an interval of speeds. The fraction of molecules within the a given speed interval is equal to the integrated probability over the interval. Mathematically, this is expressed by the number of molecules at a particular speed $N(v)$: \n", + "\n", + "\\begin{align}\n", + "P(\\pm1\\%) = \\frac{N(1.01v_{\\rm mp})-N(0.99v_{\\rm mp})}{N} = \\int_{0.99v_{\\rm mp}}^{1.01v_{\\rm mp}} F(v)dv.\n", + "\\end{align}\n", + "\n", + "The indefinite integral introduces the [error function](https://en.wikipedia.org/wiki/Error_function), which is beyond the scope of this course. However, we can obtain an approximate solution by calculating $F(v_{\\rm mp})$ and multiplying by $dv \\approx \\Delta v = 0.02v_{\\rm mp}$. This solution works for a small window, where wider intervals are easily evaluated using numerical methods (e.g., [Simpson's rule](https://saturnaxis.github.io/CompPhysics/Chapter_4/Integration.html#simpson-s-rule)).\n", + "\n", + "Recall the most probable speed $v_{\\rm mp} = \\sqrt{\\frac{2}{\\beta m}}$ and substitute to get\n", + "\n", + "\\begin{align*}\n", + "F(v_{\\rm mp}) &= 4\\pi \\left(\\frac{\\beta m}{2\\pi}\\right)^{3/2} v_{\\rm mp}^2 e^{-\\frac{1}{2}\\beta m v_{\\rm mp}^2}, \\\\\n", + "&= 4\\pi \\sqrt{\\frac{\\beta m}{2\\pi}}\\left(\\frac{\\beta m}{2\\pi}\\right) \\left(\\frac{2}{\\beta m}\\right) e^{-\\frac{1}{2}\\beta m \\frac{2}{\\beta m}}, \\\\\n", + "&= \\frac{4}{e} \\sqrt{\\frac{\\beta m}{2\\pi}}.\n", + "\\end{align*}\n", + "\n", + "Then applying our approximation\n", + "\n", + "\\begin{align*}\n", + "P(\\pm1\\%) &= F(v_{\\rm mp})(0.02v_{\\rm mp}),\\\\\n", + "&= \\frac{4}{e} \\sqrt{\\frac{\\beta m}{2\\pi}} (0.02) \\sqrt{\\frac{2}{\\beta m}}, \\\\\n", + "&= \\frac{4}{\\sqrt{\\pi}}(0.02 e^{-1}) \\approx 0.017.\n", + "\\end{align*}\n", + "\n", + "The python code plots the Maxwell speed distribution for an ideal gas using $g=v/v_{\\rm mp}$ for the $x$-axis coordinate, where the vertical lines denote the mean, root-mean-squared, and most probable speeds. It also numerically calculates (via Simpson's rule) the proability and finds good agreement with our approximation ($0.0166$).\n", + "\n", + "```" + ] + }, + { + "cell_type": "code", + "execution_count": 74, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The fraction of molecules with speeds within 1 percent of v_mp is 0.0166.\n" + ] + }, + { + "data": { + "image/png": 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vAQCZjdXPWxIJLiYiIkI3btzQzZs37WaKAAAgs/Dy8lKOHDkUGBgof39/q8NJEs9bAEBmZ+XzlkSCCwkPD9e5c+esDgMAgHSJjY1VaGioQkNDVbRoUeXMmdPqkOzwvAUAuAMrn7ckElxE0aJFzUpN9uzZlTt3bvn5+cnDg2EsAACZR3x8vCIjIxUaGqpbt27p3Llz8vb2dpmWCTxvAQDuwOrnLYkEF1GzZk1Jdys1xYoVy3T9SwEAkCQPDw9lz55dAQEBCgkJ0a1bt3Tjxg2XSSTwvAUAuAOrn7ek311EpUqVJEm5c+emUgMAyPRsNpty584tSbp586bF0fyN5y0AwJ1Y9bwlkeACPDw8zP4sfn5+FkcDAIBjJDzTYmNjXWJWBJ63AAB3ZMXzlkSCC/D09DSX6aMJAHAXiZ9prpBI4HkLAHBHVjxveYoCAAAAAIAUI5EAAAAAAABSjEQCAAAAAABIMRIJAAAAAAAgxbysDgBwpPRO5eUKg4EBAODqeN4CQNZGIgFuhYoJAADOx/MWALI2ujYAAAAAAIAUI5EAAAAAAABSjEQCAAAAAABIMRIJcDvBwcGy2Wyy2WxavXr1ffd9/fXXZbPZVLt2bfp7ZiHcI3B13KPIDLhP8SDcI3B13KNpRyIBbqdEiRLKlSuXJOnAgQPJ7nfq1Cn997//lSR9/vnn6R6BGpkH9whcHfcoMgPuUzwI9whcHfdo2pFIgFuqVq2apPv/Qhg2bJiio6PVvn17PfLIIxkVGlwE9whcHfcoMgPuUzwI9whcHfdo2pBIgFuqXr26pOR/IWzbtk0zZ86Up6enxowZk5GhwUVwj8DVcY8iM+A+xYNwj8DVcY+mjZfVASD1IiPvvlIqMFD6Z+ubGzeklHbt8fGRsmWzXxcdLd25k/IYcuSQPD3t18XESN7eKT9Hajwos/j222/LMAw9//zzqly5snOCsBI3yQNl+XvEanGRd18p5Z3EPRp9Q1IK71EPH8nrH/doXLQUl4p71CuH5PGPezQ+RvLgHnVX/Cp9sCx/n3KTPFCWv0esxvP2gbhH04ZEQib06afSv/+d8v1DQ6X/7/pjKlny7nMrJYKCpMmT7dfNmCH16ZPyGHbtkmrWtF+3ZInUpUvKz5EaCZnF69ev6+LFiypUqJC5beHChVq3bp2yZ8+uf6fmQmYm3CQPlOXvEasd+FTan4pr2yNU8sllv25BSSkmhfdo6SCp4WT7dWdmSFtScY/+a5eUu6b9unNLpOJdUn6OVOAetR6/Sh8sy9+n3CQPlOXvEavxvH0g7tG0oWsD3FK1atXMQVASZxfj4uI0ZMgQSdKQIUNUsGBBS+KD9bhH4Oq4R5EZcJ/iQbhH4Oq4R9OGRALcUvbs2VWqVClJ9r8QfvjhBx0+fFhFixbVG2+8YVF0cAWOukdiYmIUExPjrDCRhfF7DJkB9ykehOctXB2/x9LIgOVq1KhhzJo1yzh48KARFxf3wP0jIgwjNDTlr/j4e88RFpby42/fvvf4qKjUxRAbe+85oqMf+FHTpXPnzoYk48UXXzQMwzBu3rxpFCxY0JBkTJo0yblvbjVukhRJ7T0yadIkQ5KxdOlSY/jw4UaxYsUMDw8PY9euXea25cuXG++++65RpEgRIyAgwGjXrp1x7tw5wzAM44svvjBKly5t+Pr6Go0aNTIOHjx4z3vcvn3bGDZsmFGuXDnDz8/PyJMnj1GnTh3j22+/deq1yHCxEYYRFZryV1L3aFRYyo+PSeIejY1KXQxxSdyjca55j6bnPkw4x7Jly4whQ4YYhQsXNvz8/IwmTZoY27ZtS1X8cXFxxsGDB1P8fEtQuXJlo3Llyql6r5TgeescPG9DuUkegOethXjepgjP29RjjIRMyM/v7is9AgPTd7yPz91Xejhr4KcE1atX14IFC8zM4meffaZLly6pRo0aeu6555z75lbjJkmRtN4jb731lnx9ffX6669LkvLkyWNuGzJkiPz9/TV06FCFhIRo7Nixevzxx9W6dWvNnTtXgwYN0o0bNzRmzBg9/vjj2rdvn91cxAMGDNCMGTPUv39/Va9eXbdv39aBAwe0fv169e/f30lXwgKefndf6eGTznvU0+fuKz2cNPBTgrTeo+m9DyXpnXfeUXx8vN566y2Fh4frm2++0SOPPKKtW7dmmcGm+FWaMjxvuUkehOethXjepgjP29QjkQC3lXgqlwsXLuiLL76QJH3++efy8KBXD9J+j9hsNm3cuFE+SVTcfHx8tHbtWnn+/4jY0dHR+uqrr3Tx4kXt379f/v7+kiQ/Pz8NGzZMmzdvVqNGjczjFyxYoOeff15ff/21wz4nMq+03qPpvQ8lKTw8XHv27FH27NklST169FCNGjU0bNgwLViwwOGfFZkXz1s8CM9buDqet6nHb3e4rYSpXMLCwvTCCy/o9u3b+te//qVWrVpZHBlcRVrvkRdeeCHJSk3CNs9E02o9/PDDkqRnnnnGfJhIUoMGDSRJJ06csDs+V65c2r59u86ePZv6DwS3k557ND33oSS99NJLZqVGkqpWrao2bdpo6dKl9FOGHZ63eBCet3B1PG9Tj0QC3Fb58uXNH+AlS5bI09NTn332mcVRwZWk9R4pW7ZssttKlixpV871/9N8lShRIsn1169ft1v/xRdf6NChQypZsqRq1qypN998Uxs3bnxgTHBPab1H03sfSlKlSpXuWVexYkVFRUXp/PnzKQkfWQTPWzwIz1u4Op63qUciAW7Lw8NDVapUMct9+/a1KwNpvUcSZ5j/KXFWOiXrDcOwK3fr1k2nT5/WTz/9pBo1amjGjBlq0qSJXnnllQfGBfeT1ns0vfchkBo8b/EgPG/h6njeph5jJMCtbd++3eoQ4OJc8R7Jly+fevfurd69eys2NlY9e/bUf//7X7311lsqXbq01eEhg1l1jx4+fFidOnWyW3fkyBH5+vqqcOHClsQE1+WKv0vhWlzxHuF5i8R43qYOLRIAwEXExcUpLCzMbp2Xl5fZb+/atWsWRIWs6n//+59u375tlvfv36+lS5eqdevWyfZZBoDMgOctXElmfd7SIgEAXMTNmzdVpEgRdenSRTVr1lTevHl16NAhTZgwQVWqVFGtWrWsDhFZSM6cOdWoUSP17t1b4eHhGj9+vPz8/PTJJ59YHRoApAvPW7iSzPq8JZEAAC4iW7ZsevXVV7VixQr9+eefunPnjooVK6aXX35Zw4YNS7a/HeAM//nPf7Ry5UqNGTNGoaGhqlOnjsaOHauqVataHRoApAvPW7iSzPq8tRnuNOJDJlWzZk299957qlKliipWrMicywAAy0yePFl9+vTR6tWr1aJFi3SdKz4+XkeOHJGkVD3fEga4OnDgQLre/5943gIAXEVmf97yBAUAAAAAAClGIgEAAAAAAKQYiQQAAAAAAJBiJBIAAICpd+/eMgwj3f01AQBA8jL785ZEAgAAAAAASDESCQAAAAAAIMVIJAAAAAAAgBQjkQAAAAAAAFKMRAIAAAAAAEgxEgkAAAAAACDFSCQAAAAAAIAUI5HgAuLi4szl+Ph4CyMBAMBxEj/TbDabhZHcxfMWAOCOrHjekkhwAfHx8QoPD5ckRUZGWhwNAACOkfBM8/LycolEAs9bAIA7suJ5SyLBRRw+fFiSFBoaKsMwLI4GAID0MQxDoaGhkqQcOXJYHM3feN4CANyJVc9brwx7J9zX7t27JUm3bt1SSEiIcufOLT8/P3l4kOsBAGQe8fHxioyMVGhoqG7duiVJCgwMtDiqv/G8BQC4A6uftyQSXMS5c+dUtGhRnTt3Trdu3TJvBgAAMrOiRYvK39/f6jBMPG8BAO4oo5+3JBJcSM6cOeXt7a0bN27o5s2bio2NtTokAABSzcvLSzly5FBgYKBLJRES8LwFALgDK5+3JBJcjL+/v/z9/VWoUCEZhkH/TQBApmKz2VxiYMUH4XkLAMjMrH7ekkhwYVbfHAAAZAU8bwEASB1GFgIAAAAAAClGIgEAAAAAAKQYiQQAAAAAAJBiJBIAAAAAAECKkUgAAAAAAAApRiIBAAAAAACkGIkEAAAAAACQYiQSAAAAAABAipFIAAAAAAAAKUYiAQAAAAAApBiJBAAAAAAAkGJeVgcAwHEuXbqkNWvWaO/evTpx4oTCwsJkGIZy5sypUqVKqUqVKmrevLlKlSolm81mdbgAAMBFREVF6ebNm7p9+3ayrzt37igyMlJRUVGKjo5O9t/Y2FjFx8ebr7i4OLtyfHy8DMOQp6enPDw87F6J13l5ecnHx0e+vr7y8fFJctnX11cBAQEKCAhQtmzZklxOeFH3ARyHRAKQyd25c0fTpk3TlClTtHnz5hQd89BDD6lXr1564YUXVLBgQSdHCAAAMkJ0dLSuXbumq1evmq+EclhYmMLCwnTjxg3z38TLUVFRVofvVJ6ensqZM6dy5sypwMDAJF958+ZN8pU7d255enpa/REAl0IiAcikoqKi9PXXX2vMmDG6evXqPdt9vaWhHf8uf7pIioq5u3zo0CG9//77+uijj9S3b1998MEHKly4cAZFDgAAUioyMlKXLl3ShQsXdPHiRV28eNFu+eLFi2bSIDw83OpwH+h+9RNniouLU2hoqEJDQ1N9rM1mU65cuZQvXz4VKFBABQsWTPJVqFAhFSxYUAEBAU74BIBrIZEAZEKrVq3SCy+8oJMnT96zrXjx4mrSpInCrwVrZPeN5vqVIVV1KTRKx44dM9dFR0fru+++09SpUzVixAi9/vrr8vb2zpDPAABAVhcREaGQkBCdPXs2yVdISIjCwsIyPK7E3QUSv7JlyyZ/f3+ze0FS/544cUKXLl2SzWZL8pXdJ1Yjuy813+tQfGvdivaSYRj3vOLj45U/f36VLl062a4UkZGRZreLxP8mLDuCYRhmEiJxPSo5OXPmVNGiRe1exYoVsysXKFBAHh4MV4fMi0QCkIlER0dryJAh+uqrr+zWZ8uWTf369VOfPn1Us2ZN2Ww2jfrwPW0JjTX3adu+lQYOflvXrl3T9OnT9e233+rChQuS7naPGDJkiGbOnKmZM2eqTJkyGfmxAABwS9HR0QoODtapU6d08uRJnTp1ylw+ffp0ki0KHcHf31/58uUzX3nz5lWePHnMJvy5cuVKcjlHjhzKli2bvLzS/ifCokWLtHPnzmS3e9litCU0zCxXr1VPsUbyX2LUqFFDXbp0SVMshmEoIiJCt27dMrtyJPUKDw9XaGiorl+/rmvXrtm94uLiUv2+4eHhCg8P16FDh5Ldx8fHRyVKlFDJkiWTfBUrVowvd+DSSCQAmcSlS5fUvXt3bdz4dysDT09PvfLKK3r//feVL18+u/1jDW8tvfqve85TpkwZvf/++xoyZIh+/PFHjRw5UleuXJEk/fXXX6pTp46mTZumDh06OPcDAQDgBqKionTy5EkdPXrUfB0/flwnT55USEiI4uPjHfI+efPmVaFChVSoUCEVLlzYXC5UqJAKFChglzTIli2bQ97TGZKrnziDzWZTtmzZlC1bNhUoUCDVxxuGofDwcDOpcPXqVV26dCnZ19WrV2UYRorOHR0drePHj+v48eNJbvfw8FDx4sVVpkwZlS1b9p5XYGBgqj8P4EgkEoBM4OjRo2rbtq1OnTplrqtataqmTJmi2rVrp+mcPj4+GjBggJ566im9/fbbmjhxoiQpLCxMHTt21HvvvacPP/yQEY4BAFmeYRi6dOmSDhw4oMOHD9slDU6fPp2uZIGHh4cKFy6s4sWL272KFSum4sWLq0iRIipQoIB8fX0d+ImQEjabzWytkZLWmrGxsbp8+bLOnz+vkJAQnTt3LsnXzZs3H3iu+Ph4nTlzRmfOnNHq1avv2Z4nTx6VLVtWFSpUUIUKFVSxYkVzmTEakBFIJAAu7sCBA3rkkUd0+fJlc13v3r313//+V/7+/uk+f+7cufXjjz+qVatWev7553X79m1J0scff6xLly7p22+/ZaRiAECWYBiGLl68qAMHDujgwYN2/6ZlkD7p7h+jxYoVU+nSpVWmTBm7f0uUKKHChQunqysBXIeXl5eKFCmiIkWKqG7dusnuFx4ebiYJzpw5o+DgYLtyQtfT+7l+/bquX7+u7du337OtWLFidsmFhx56SFWqVFHRokX5gggO47a/tSIjIzV69Gj98ssvCg4OVp48edS2bVt9+OGHKlasWKrO9eeff2rcuHHavn27bty4ody5c6t+/fp6/fXX9eijjzrpEwB3Z1do2bKl2fVAkj777DO9+eabDn8QPPnkk6pevbq6d++uw4cPS5J++OEH3b59W5MnT6afHgDArURERGj//v3as2eP+dq/f3+aEgaenp4qU6aM+Y1w+fLlVbZsWTNZQGsCJJYzZ05Vq1ZN1apVS3J7VFSUgoODdfLkSZ04ceKeV0RExH3PHxISopCQEK1atcpufY4cOVS5cmVVqVLF7t/ixYuTYECquWUiITIyUo8++qg2bdqkwoULq3Pnzjp9+rQmTZqkxYsXa/PmzSpbtmyKzjV27Fjzj7bGjRuraNGiOnnypJYsWaIlS5bo22+/Vf/+/Z38iZAVXbx4Uf/617/MJIKHh4cmTZqk5557LkXHe9piVTfw7yz1jhv1HnhM5cqVtWXLFrVr106bNm2SJM2YMUN37tzRzJkzSSYAADKlixcvaufOnXZJg6NHj6a6S0LBggX10EMPqVKlSmbSoEKFCipVqhTPyBRKqn4SZ7jlnyRp5uvrq/Lly6t8+fL3bEtoNXPixAkdP35cx44d05EjR3TkyBEdO3ZMUVFRyZ735s2b2rp1q7Zu3Wq3Pnv27KpataqqV69uvqpVq6ZcuXI5+qPBjbjlT+0nn3yiTZs2qWHDhlq2bJmyZ88u6e+kQN++fbV27doHnufKlSsaNmyYfHx8tHLlSjVp0sTcNmfOHD3++ON688039cwzz5jvATjC7du31bFjR505c0bS3WaRU6dO1dNPP53ic3jZYtQ2/9/TK+0Or5mi4wIDA7Vs2TJ17txZK1eulCTNnz9fL774on766Scy1gAAl3bp0iX99ddf2rFjh3bs2KG//vpL58+fT9U5ChYsaPetbcJy3rx5nRR11pFU/YREQsrZbDYVLlxYhQsXtvvbRJLi4uJ09uxZHTlyREePHtWRI0d0+PBhHTp06L4/A7du3dKWLVu0ZcsWu/UlSpQwkwrVq1dXjRo1VKFCBbq8QpIbJhJiYmI0fvx4SdKECRPs/sB/4403NGXKFK1bt84cnf5+tm7dqujoaLVt2/aeH9Tu3burevXq2rNnjw4ePKj69es7/sMgS4qLi9PTTz+tHTt2mOvGjh2bqiRCegUEBGjx4sV64okntGjRIknS5MmTVbRoUX388ccZFgcAAPcTGhqqbdu2afv27WbSICQkJMXH+/n5qVq1aqpRo4Zq1Kih6tWrq0qVKiQMkCl5enqqVKlSKlWqlNq0aWO3LSwsTAcPHjRfCeN/3O/nJTg4WMHBwVq8eLG5Llu2bKpZs6Zq1aql2rVrq1atWqpSpYp8fHyc9rngmtwukbBhwwaFhYWpbNmyqlWr1j3be/Toob1792rRokUPTCSktD9bnjx50hQrkJQPPvhACxYsMMsDBw7U4MGDMzwOPz8/zZw5U23atNG6deskSaNGjVLRokX18ssvZ3g8AICsLTY2VgcOHDC/Od2yZYs5pk9KFClSRDVr1jSTBjVq1FD58uX5dhVZQq5cudSoUSM1atTIbn14eLgOHjyoffv2ae/evdq7d6/27NmjGzduJHmeO3fuaNOmTWYXWEny9vZWtWrVzORC/fr1Vb16dZILbs7tEgl79uyRpGSnxEtYn7Df/dSrV0+BgYFatWqVNmzYYNcqYe7cudq7d68aNWqkcuXKOSByQFq5cqVGjx5tljt27KivvvrKsu4Efn5+mj9/vpo2baoDBw5IupvYKF++vFq1amVJTACArOHatWvmHyxbtmzR9u3bzZmFHqRw4cKqW7eu6tSpY/5bqFAhJ0cMZD45c+ZUgwYN1KBBA3OdYRg6e/asmVhISC4kN65ITEyMdu7cqZ07d5rTifv4+KhGjRqqV6+e+apUqRKJOzfidomE4OBgSUp2ZoaE9Qn73U+uXLn0448/6umnn1azZs3MwRZPnTql7du3q23btpo8ebLDYkfWdvnyZT3zzDMyDEOSVLFiRc2YMcPyX7i5c+fWvHl/qE6dWbp5M1zx8dKTT/bWrl2bVbx4cUtjAwC4j+DgYK1fv958HTx4MEXH5c2bVw8//LDq1q1rJg2KFCni5GgB92Wz2VSiRAmVKFFCHTp0MNffvn1be/fu1a5du8zEwb59xxQb+2aioz+VFKXo6Ght377dbnrK7Nmzmy0WGjRooIYNG/Kzmom5XSLh1q1bku7230lKQECA3X4P0qNHD+XJk0c9e/bUhg0bzPUFCxbUI488kqo+dFWqVEly/YkTJ1I8iwTcU3x8vIKCgnTx4kVJd7O4v/76q8sM4pk/f3HdvPmGWb527Sv16NFD69atY0orAC6H563rMwxDBw8etEscnD179oHHeXp6qmbNmuY3qA0aNFDZsmUZCBjIAAEBAWrYsKEaNmxorrt8OVoFC/7dhaF27fXav3+DoqOj7zn+1q1bWrdundllVpKKFy+uhg0bmomFWrVqUbfMJNwukZDwbW5yD5SE7Sn1xRdf6J133lGXLl00cuRIlSlTRidPntQHH3ygt99+W1u2bNHs2bPTHTeytvHjx+vPP/80y59//rlq1qxpXUApsG3bNr322mv69ttvrQ4FAODiDMPQ0aNHtXr1aq1evVpr1qzR5cuXH3hcgQIF1KRJE/MPjdq1ayf7ZRGAjPfPcRBWrlypbNmitW/fPrNFwvbt23XgwIEku0WcPXtWZ8+e1cyZM83z1apVSw0bNlSTJk3UuHFjuiW5KLdLJOTIkUOSku1Dd+fOHUlK0Te9a9eu1VtvvaXatWtr1qxZ8vDwkCRVq1ZNs2fPVr169TRnzhwtW7ZMrVu3fuD5EvqY/1Ny35wgazh9+rSGDx9uljt16qSBAwdaGFHKfffdd3rkkUf0+OOPWx0KAJh43lrPMAydOnVKq1ev1qpVq7RmzZoUTcFYtmxZNW3aVE2bNlWTJk1Uvnx5WhsAmYyPj4/q1KmjOnXqqH///pLu/m22a9cubd++XVu3btXmzZuT7GoeHR2trVu3auvWrfrqq68kSeXKlVPjxo3VpEkTNWnSRBUrVuT3ggtwu0RCiRIlJCnZqUwS1ifsdz9Tp06VJHXr1s1MIiTw9PRUt27dtGvXLq1ZsyZFiQTgnwzD0IABA8wEV758+fTjjz+6/C/HmjVraffuNZKk/v37q0mTJipcuLC1QQEALHX9+nWtXLlSy5Yt0/Lly3XmzJn77m+z2VSjRg27xAHPEsA9BQQEmImABOfPn9eWLVu0efNmbdmyRTt27FBkZOQ9xx4/flzHjx/XlClTJN0dFyXhXC1atFCtWrUsH1MsK3K7REKNGjUkSTt37kxye8L66tWrP/BcCUmHnDlzJrk9Yf3169dTHScgSTNnztQff/xhlr/88kvlz5/fwohSZuLEiWrWrLpu376t69evq2/fvvr9999dPgECAHCc6Ohobd682Uwc7Nix44FdSKtVq6aWLVuqZcuWatasGVNoA1lYkSJF1K1bN3Xr1k3S3d8pe/fu1ebNm7Vx40Zt2LBB586du+e4a9euacGCBeZ06Tlz5lSzZs3UokULtWzZUjVq1CCxkAHcLpHQuHFjBQYG6sSJE9q1a5dq1apltz1hPIPEI5AmJ6E/zo4dO5LcnjAKaalSpdIRMbKq0NBQvfrqq2b5scce09NPP21hRClXpkwZjR07Vi+99JIk6c8//9R3332nl19+2eLIAADOdOzYMf3xxx9atmyZ1qxZ88DpGCtVqqRHHnlELVu2VPPmzTNFshyANXx8fMzZVwYNGiTDMBQcHKwNGzaYrwMHDtyTsAwPD9fixYu1ePFiSXdn3mvWrJlatmypFi1aqHr16ve0Lkf6uV0iwcfHRwMHDtSoUaM0cOBALVu2zJypYezYsdq7d6+aNGmievXqmcd88803+uabb9S1a1eNHj3aXN+lSxdNnTpV06dPV48ePdSxY0dz24IFCzRjxgx5eHioa9euGfcB4TaGDx9uDjTl7++v7777zqHf6EfF++qrU4PtymmVM6d06pR9+YUXXtDChQu1ZMkSSdKbb76pxx57TOXKlUvz+wAAXEtkZKTWrVun33//XUuWLNHx48fvu3+hQoX02GOPqXXr1nr00UfpqoB7OLJ+AuslVUd0FJvNppIlS6pkyZLml22hoaHavHmz1q9fr3Xr1mnbtm2KjY21Oy4sLEwLFy7UwoULJd2dyrx58+Zma6gqVaqQWHAAt0skSNJ7772nFStWaNOmTSpfvryaNm2qM2fOaOvWrcqbN68mTZpkt//Vq1d15MgRXbhwwW59ly5d9Pjjj2vWrFnq1KmT6tatq9KlS+vUqVNmK4VRo0apYsWKGfbZ4B4OHDig77//3iwnzAjiWB66EZvbMWfykO5teGPTjz/+qGrVqunq1auKiIjQyy+/rGXLltHFAQAyseDgYP3+++/6/ffftXLlSnMcn6T4+fmpefPmZvKgatWqPAPwAI6rn8B6SdcRnSd37txq166d2rVrJ+nulJKbNm0yZ4TZsWOH4uLi7I4JDQ3V/PnzNX/+fEl3xyRr3ry52RWicuXK/N5KA7dMJPj5+Wn16tUaPXq0ZsyYofnz5yt37twKCgrSRx99pOLFi6foPDabTb/99pvatm2rKVOmaO/evdq9e7dy5cqldu3aadCgQWrbtq2TPw3c0dtvv21OgVO+fHm99tpr1gaURoUKFdKECRPUs2dPSdKKFSv0888/69lnn7U4MgBASsXHx2vbtm3mN3jJzXqRoFq1amrbtq1at26tJk2ayM/PL4MiBQB72bNnV+vWrc2B72/evKkNGzZozZo1Wr16tf766697pp28evWq5syZozlz5ki6O81sq1atzPPQkiplbMaDRsWB0yVMR/WgBzfcw7Jly9SmTRuzPG/ePHXp0sXh7/Pvf//7nnWvvfaaAgMDHfo+hmGoY8eOZheHfPny6dChQ8qXL59D3wdA1uGs5yLP279FRERo5cqVWrBggRYtWqRLly4lu2+2bNnUqlUrtW/fXv/6179S/IUMsqZFixYlO+h5WtSoUcMp9SRkDTdu3NCGDRu0evVqrVmzRjt37nzgoLDVq1dX69at1aZNG7dPlqbnueiWLRIAVxUXF6e33nrLLDdr1kydO3e2MKL0s9lsmjBhgjno1tWrV/XWW29p8uTJVocGAEjkypUrWrx4sRYuXKhly5bdt8tChQoVzObDzZo1k68v/dgBZD6BgYFq37692rdvL+nu+Anr1683u0Ls2bPnnsTC3r17tXfvXn3++efy9/dXixYtzMRCpUqV6Abx/0gkABlo8uTJ2rdvn1n+4osvnPbLyMcjUi+X+NYsfxuc9hkVwsOlxDOm7t1rP5hOyZIl9dFHH+mNN96QJE2ZMkV9+vRR8+bN0/yeAID0CwkJ0dy5czVnzhxt2LDhnia+CTw9PdW0aVN16tRJHTt2ZOBcOE1S9ZPoePf9xtfdPaiO6Gpy5cqljh07moPoX79+XatXr9bSpUu1dOlSBQcH2+0fERGhP/74w5yuvXjx4mrTpo1at26tVq1aKXfurDveB4kEIINERUXZdTd45plnVLduXae9n02GcnnfsCunVXy8dOaMffmfBg0apOnTp+uvv/6SJA0ePFh//fUX8/gCQAY7ceKEmTzYunVrsvtlz55dbdu2VefOndWuXTvlyZMnA6NEVuXI+gmsl5I6oivLkyePunfvru7du8swDB09etRMKqxZs+aelltnz57Vjz/+qB9//FEeHh6qX7++2Vqhfv368vLKOn9eZ51PCljsp59+0tmzZyVJ3t7e+vjjjy2OyLG8vLw0YcIENWjQQJK0Z88e/fjjj3rppZcsjgwA3N/BgwfNwcP27NmT7H5FixZVp06d1LlzZ7Vo0YIuCwDw/2w2mypWrKiKFSvq1VdfVVRUlDZu3KilS5dq2bJl2r17t93+8fHx2rJli7Zs2aIPP/xQgYGBatOmjTp06KB27dopb9681nyQDEIiAcgAUVFR+uSTT8xyv379VLJkSQsjco6HH35Yzz33nKZOnSpJevfdd/XEE09k6WZfAOAsR48e1W+//abffvvtvgNllS1b1vzGrV69evTvBYAU8PX11SOPPKJHHnlE//nPf3Tx4kUtX75cy5Yt07Jly3T58mW7/W/cuKGZM2dq5syZ8vDwUMOGDdWxY0d16NDBLaeYJJEAZICJEycqJCRE0t3WCMOHD7c4Iuf59NNPNXfuXN26dUvXrl3Tv//9b3311VdWhwUAbuHUqVOaOXOmfv3113u+HUusSpUq6t69u7p166bq1au7XQUWADJaoUKF9Oyzz+rZZ59VfHy89uzZo2XLlmnp0qXasGGDYmJizH3j4+O1ceNGbdy4UUOHDlXp0qXVoUMHdejQQc2bN3eL1mAkEgAn+2drhOeff96tp84qXLiw3n33XQ0bNkyS9M033+ill17SQw89ZHFkAJA5hYSEaObMmfrtt9+0bdu2ZPerXbu22fKgYsWKGRghAGQtHh4eqlWrlmrVqqUhQ4bo1q1bWrVqlRYvXqzFixfrwoULdvufOnVK48eP1/jx45U9e3a1bt1aHTp0UPv27VWgQAGLPkX6kEgAnGzixIk6d+6cJMnHx8f8A9udvfbaa/rhhx908uRJxcXFafjw4Zo3b57VYQFAphEaGqrZs2fr559/1rp165Ldr3r16nryySf1xBNPqGzZshkYIQAgQfbs2dWpUyd16tRJ8fHx2rVrlxYvXqxFixaZA5EnuHXrlubOnau5c+fKZrOpfv36ZheIzNSCjEQC4ESxsbEaM2aMWXb31ggJ/Pz8NGbMGPXo0UOSNH/+fG3atEmNGjWyODIAcF2RkZFasmSJfv75Z/3++++Kjo5Ocr+HHnpIPXv2VM+ePVWpUqUMjhIAcD8eHh6qU6eO6tSpoxEjRuj8+fNasmSJFi9erOXLlysiIsLc1zAMbd26VVu3btV7772n4sWLq0OHDurcubNatmwpHx8fCz/J/ZFIAJxo9uzZOvP/c+J4eXlp6NChFkeUcbp166b69eubzXDfeecdrV+/PtNkWQEgI8THx2vt2rWaPn26Zs+erRs3biS5X9myZfXkk0+qZ8+eqlq1Kr9LASCTKFKkiF544QW98MILioiI0OrVq80uEAkzuiU4e/asvv32W3377bcKDAxUhw4d1K1bN7Vp00YBAQEWfYKkeVgdAOCuDMPQZ599ZpaffPLJLNEaIYHNZrNrjbFx40YtWrTIwogAwHUcOnRIQ4cOVcmSJfXII49o4sSJ9yQRChQooFdffVXbtm3TsWPH9PHHH6tatWokEQAgk/L391e7du303//+V2fOnNHu3bv18ccf6+GHH77nd/uNGzc0ffp0de/eXfnz51fXrl01bdo0hYaGWhS9PVokAE6yevVq7dy50yy/9dZbFkZjjebNm6tzmzaquXSpJOlsv36KPXVKXtmzWxwZAGS80NBQ/frrr5o8eXKygyYGBASoa9euevrpp9WqVSt5eVFVAwB3ZLPZVKNGDdWoUUPvvvuuLl26pD/++EMLFizQ0qVLFRERIV9JQyUpIkKaP18vzJ+vOC8vtWzZUt26dVPnzp1VuHBhS+Ln6QQ4SeLWCI899phq1KiRoe8fb3jq8K2KduW08vaWOne2L6fURyNH6sT/JxJ09ap+nT5dz7z0UppjAYDMJDY2VsuWLdPkyZO1YMGCJMc98PT0VJs2bfT000+rc+fOLtd8FXAkR9ZPYL301BFhr2DBgurdu7d69+6t27dva+nSpVoyc6bqzZmjmNhYSZKnpKjYWC1fvlzLly/XgAED1LBhQ3Xr1k1du3ZVmTJlMixeEgmAE+zbt09//vmnWX777bczPIYYw0e/XXjKIecKCJDmz0/bsdUaNNBnzz6radOmSZLKjBmjnn37ypsnDQA3dvDgQU2ePFnTpk3TxYsXk9ynVq1aCgoK0lNPPZVpp/8CUsuR9RNYLz11RCQvICBA3bp1U7du3RQdHa3Vq1dr7ty5yj5/vu5cvmzuZxiGNm3apE2bNumtt95SzZo11bVrV3Xr1k1VqlRxalc4xkgAnGDs2LHmco0aNdSqVSsLo7HeiBEj5Ol59xuHkydPaurUqRZHBACOd+vWLf30009q1KiRqlSpos8+++yeJEKBAgX0xhtvaM+ePdq5c6cGDx5MEgEAkCwfHx+1adNG//vf/3T+/HmtX79er7/+ukqWLHnPvrt379aIESNUrVo1VahQQUOHDtWuXbtkGIbD4yKRADjY1atX9csvv5jlN998M8sPjFW2bFkFBQWZ5Y8//jjZac0AIDMxDEPbtm3Tiy++qMKFC6tfv37avHmz3T7e3t7q1q2bFi5cqJCQEH3xxReqXr26RREDADIrT09PNWnSRGPHjtWpU6e0c+dOvffee6pcufI9+x4/flz/+c9/VLt2bVWsWFHvvfee9u3b57CkAokEwMEmTpyoqKgoSXf7OvXs2dPiiFzDe++9Zw4advr0aU2ePNnagAAgHa5du6Zx48apRo0aevjhh/XDDz/o1q1bdvvUrFlT48eP14ULFzRnzhx17NiRbl0AAIew2WyqVauWPvroIx04cECHDh3SJ598onr16t2z77FjxzRq1ChVr15dVapU0b///W8dOnQoXe/PGAmAA8XFxenbb781yy+88IJ8fHwsicVDcSofcNQsH7tdIc3niomRliz5u9y+fSoG0/n/g0tLGteypQYvX65Y3W2VEBQUJF9f3zTHBQAZyTAMbdiwQd99951mz56dZMuqwMBAPf300+rXr59q165tQZSAa0uqfhIvBlzMrNJVR8SDpeICV6pUScOGDdOwYcMUHBysuXPnaubMmfe0kjt06JBGjhypkSNHytfXV2XLlk1TaCQSAAf6/fffdebMGUl3mx69ZOHsBN4e0XqyyG9m+dMTQ9J8rtu3pa5d/y6Hhkq5cqX+4AGSRnh56WpsrM6ePavJkydbeo0AICXCwsI0bdo0fffddzp48GCS+zRr1kzPP/+8unfvrmzZsmVwhEDmkVT9JCre38KIkB7pqiPiwdJ4gUuUKKHXXntNr732moKDgzVr1iz99ttv2r59u91+Ca2o04KuDYADTZgwwVzu3LmzihUrZmE0runZZ581l8eMGaPY/5/OBgBciWEY2r59u/r166ciRYro1VdfvSeJUKBAAb3zzjs6fPiw1q5dq2effZYkAgDApZQoUUJvvvmmtm3bphMnTujTTz9VrVq10n1eEgmAgxw7dkxLly41ywMHDrQwGtc1ePBguxkcZs2aZXFEAPC3W7du6YcfflDdunVVv359/fTTT4qIiLDbp1WrVpo1a5ZCQkL0n//8RxUrVrQoWgAAUq5MmTIaMmSIdu7cqSNHjih//vxpPheJBMBBEo+NULlyZbVo0cK6YFxYyZIl9eSTT5rl0aNHO2VKGgBIjaNHj+q1115T0aJF9eKLL2rnzp122/Pmzau3335bx44d0/Lly9WjRw8GTgQAZFoVKlQgkQBYLSoqSlOmTDHLAwYMyPJTPt7P0KFDzeV9+/ZpSeJBZAAgg8TFxWnx4sVq27atKlasqHHjxik8PNxun6ZNm2r69OkKCQnRmDFjVK5cOYuiBQDAdZBIABxg/vz5un79uiTJ399fzzzzjMURubaqVauqU6dOZplWCQAy0vXr1/X555+rfPny6tixo123NEnKmTOnBg0apP3792vdunXq1auX/Pz8LIoWAADXw6wNgANMnDjRXH788ccVGBhoYTSZw7Bhw7Rw4UJJ0qZNm7R+/Xo1a9bM4qgAuLvz58+raNGiioyMvGdblSpVNHDgQD3zzDPKnj27BdEBAJA50CIBSKfTp09r+fLlZvn555+3MJrMo0GDBnbjSIwePdq6YABkGWFhYXZJBE9PT3Xv3l2rV6/Wvn371L9/f5IIAAA8AIkEIJ0mTZpkLleoUEFNmjSxMJrMZdiwYebyn3/+ec/gZgDgLPnz59e7776rU6dOafbs2WrRogVj2wAAkEIkEoB0iIuLs0sk9O3bl4poKjz22GOqXbu2Wf70008tjAZAVuDn56dp06bp7Nmz+vjjj1W8eHGrQwIAINMhkQCkw/Lly3X27FlJd5vHBgUFWRxR5mKz2exaJcyZM0enTp2yMCIA7q5MmTJ65pln5Ovra3UoAABkWgy2CKTDTz/9ZC536NBBhQoVsjAae9HxPpocEmRXTqvs2aXVq+3Ljjq4a9euKleunI4fP674+Hh9/fXX+vLLL9McKwAAcF2OrJ/AeumqI+LBXPgCk0gA0ig0NFQLFiwwy/369bMwmnsZ8tSZiNIOOZeXl5RoXESHHuzp6anBgwdr0KBBku7OgDFy5EhmvgAAwA05sn4C66WrjogHc+ELTNcGII1mz56t6OhoSXcH7Wrbtq3FEWVevXv3Vq5cuSRJN2/etJtOEwAAAIBrIZEApNG0adPM5aeeekre3t4WRpO5Zc+eXS+++KJZ/vrrrxUbG2thRAAAAACSQyIBSIPTp09r/fr1ZvmZZ56xMBr3MHDgQHl6ekqSzpw5o/nz51sbEAAAAIAkMUYCkAY///yzuVyxYkXVrVvXwmiS5m2LUq8iM8zyjPO90nyuW7ekDh3+Li9enIqxXlJ4cPHixfXEE0/ol19+kSSNHTtWPXr0SHPMAADA9SRVP4kxmEUls0pXHREP5sIXmEQCkEqGYdglEp599lnZbDYLI0qahy1epbKdsSunVWystHatfdkZB7/++utmImHz5s3aunWrHn744VRGCwAAXFWS9RPDwoCQLumqI+LBXPgC07UBSKUdO3boyJEjZvnpp5+2MBr3Uq9ePTVu3NgsMw0kAAAA4HpIJACplHiQxaZNm6pUqVLWBeOG3njjDXN59uzZCg4OtjAaAAAAAP9EIgFIhdjYWP32229m+dlnn7UwGvfUuXNnlS59d37puLg4TZgwweKIAAAAACRGIgFIhbVr1+ry5cuSJG9vbwYDdAJPT08NGjTILE+cOFGRkZEWRgQAAAAgMRIJQCr8+uuv5nKbNm2UO3duC6NxX3369FG2bNkkSdeuXbNrBQIAAADAWiQSgBSKjo7W3LlzzXLPnj0tjMa95cqVy24Qy2+++cbCaAAAAAAkRiIBSKEVK1bo+vXrkiRfX1916tTJ4ojc2yuvvGIu79ixQ9u2bbMwGgAAAAAJSCQAKZS4eX379u2VM2dOC6NxfzVq1FCTJk3MMoMuAgAAAK6BRAKQApGRkZo/f75ZpltDxhg4cKC5/Ouvv+rKlSsWRgMAAABAkrysDgDIDP7880+Fh4dLkrJly6b27dtbHNGDGYaHLkYVtCunlaenVKOGfTkjDu7atasKFSqkixcvKjo6WhMnTtTQoUNT8eYAAMCVOLJ+Auulq46IB3PhC0wiAUiBxN0aOnXqpICAAAujSZlow1f/C37ZIefKkUPavTvjD/bx8dGLL76oDz/8UJL07bff6u2335anC/0SBQAAKefI+gmsl646Ih7MhS8wKUDgASIjI7V48WKzTLeGjPXSSy/Jy+tuzjM4ONju/wIAAABAxiORADzAihUrdOvWLUlSQECA2rRpY3FEWUuRIkXUtWtXs8ygiwAAAIC1SCQADzB37lxzuV27dvL397cwmqwp8aCLy5cv1/Hjxy2MBgAAAMjaGCMBuI/Y2FgtXLjQLCf+ZtzV2RSvAj6XzPLl6IL32fv+4uKkffv+LlerloqxXtJ18F1NmzZVlSpVdODAAUnSjz/+qE8//TRV5wAAANZLqn5i8N1mpuWAah7ux4UvMIkE4D7Wr1+va9euSbo78F9mmK0hgY9HlPqX/J9Z/vTEkDSf6+ZNqVatv8uhoVKuXBlx8F02m00vvviiBg8eLEmaNGmSPvzwQ/n4+KTqPAAAwFpJ1U+i4mntmVk5oJqH+3HhC0z6D7iPxN0aWrVqpZw5c1oYTdb2zDPPyM/PT5J0+fJlLViwwOKIAAAAgKyJRAKQjPj4eM2bN88sZ6ZuDe4oT548evzxx83y999/b2E0AAAAQNZFIgFIxo4dO3Tu3DlJkoeHhzp16mRxRHjxxRfN5RUrVujEiRMWRgMAAABkTSQSgGQk7tbQtGlTFShQwMJoIEmNGzfWQw89ZJZ/+OEHC6MBAAAAsiYSCUASDMOwSyTQrcE1JAy6mGDSpEmKjo62MCIAAAAg6yGRACTh4MGDOnbsmFkmkeA6nnvuOfn6+kq6O+hi4uk5AQAAADgfiQQgCYlbI9StW1clSpSwMBokxqCLAAAAgLVIJABJoFuDa0vcvWH58uU6efKkhdEAAAAAWQuJBOAfTp06pd27d5vlbt26WRcMktSkSRNVqlTJLDPoIgAAAJBxSCQA/7BgwQJzuVKlSnZ/sMI1/HPQxZ9++olBFwEAAIAM4mV1AICrWbx4sbncuXNnCyNJn5h4b82/2NmunFbZskmTJtmXM+bg5D333HMaNmyYoqKidPnyZS1atEjdu3d3yLkBAIBzOLJ+Aus5qZqHBC58gUkkAIncuHFDa9euNcsdO3a0MJr0iZeX9tys5ZBz+fhIvXtbcXDy8ubNqx49emj69OmS7g66SCIBAADX5sj6CaznpGoeErjwBaZrA5DIsmXLFBsbK+nuH6oNGjSwOCLcT+LuDcuWLdPp06etCwYAAADIIkgkAIkk7tbQrl07eXp6WhgNHqRp06aqUKGCWZ48ebJ1wQAAAABZBIkE4P/FxcXp999/N8sdOnSwMBqkhM1mU9++fc3ypEmTFB8fb2FEAAAAgPtjjATg/23dulVXr16VJHl5ealNmzYWR5Q+XrZotS+wxCwvudw+zee6c0caMODv8n//m4qxXtJ18IM999xzevfddxUXF6fg4GCtWrVKrVq1ctj5AQCA4yRVP4k1fCyMCOnh5GoeXPgCk0gA/l/ibg3NmjVTYGCghdGkn6ctTjVz7jHLf15pm+ZzRUdLU6b8Xf7qq1T8DkvXwQ9WuHBhtWvXTosWLZJ0dypIEgkAALimpOonsYaFASFdnFzNgwtfYLo2AP8vcSKBbg2ZS+LuDXPnzlVoaKiF0QAAAADujUQCIOnMmTPat2+fWSaRkLm0b99eBQoUkCRFRUVpxowZFkcEAAAAuC8SCYDsWyNUrFhR5cuXtzAapJa3t7eeffZZs/zTTz9ZGA0AAADg3kgkAKJbgzvo06ePubxz507t3r3bumAAAAAAN0YiAVnerVu3tGrVKrNMIiFzqlKlih5++GGzPGnSJAujAQAAANwXiQRkeStXrlR0dLQkKVeuXGrcuLHFESGtEg+6+PPPPysqKsrCaAAAAAD35LaJhMjISI0YMUIVKlSQn5+fihQpor59+yokJCRN5zt+/LheeOEFlSpVSn5+fsqfP78aNWqkzz77zMGRI6MlTBsoSW3btpW3t7eF0SA9nnzySfn7+0uSrl+/roULF1ocEQAAAOB+3DKREBkZqUcffVQffvihbt26pc6dO6t48eKaNGmSateurRMnTqTqfPPmzVO1atU0ceJE5c2bV127dlWtWrV06tQp/e9//3PSp0BGMAxDv//+u1mmW0PmljNnTj3++ONmmUEXAQAAAMfzsjoAZ/jkk0+0adMmNWzYUMuWLVP27NklSWPHjtWbb76pvn37au3atSk61549e/Tkk08qR44cWr58uZo0aWJui4+P186dO53yGZAx9uzZowsXLkiSPDw81LZtW4sjQnr17dtXU6dOlSQtXbpUZ8+eVfHixS2OCgAAAHAfbtciISYmRuPHj5ckTZgwwUwiSNIbb7yh6tWra926dfrrr79SdL5BgwYpOjpakydPtksiSHf/8Kxbt67jgkeG+/PPP83l+vXrK2/evBZG42g2Rcb5mi/JlvYz2aTAwL9fttScKl0Hp16zZs1UtmxZSXdbnEyZMsWp7wcAAFLDcfUTWC+Dq3lZjwtfYLdrkbBhwwaFhYWpbNmyqlWr1j3be/Toob1792rRokWqU6fOfc916NAhrV+/XhUqVKDJu5tKnEhwt9YIUfF++s/JYQ45V2CgFBZmxcGpZ7PZ1KdPH7333nuS7s7eMHz4cHl4uF3eFACATMeR9RNYL4OreVmPC19gt6tZ79mzR5JUu3btJLcnrE/Y735WrlwpSXrssccUGRmpKVOmaNCgQXr11Vf1448/Kjw83EFRwwrh4eHauHGjWXa3REJWFhQUJNv/Z2xPnjypDRs2WBwRAAAA4D7crkVCcHCwJKlYsWJJbk9Yn7Df/Rw4cECS5O/vr5o1a+rIkSN224cNG6Y5c+aoWbNm6QkZFlm1apViY2MlSXny5KGbihspVqyYWrduraVLl0qSJk+ezM8pAAAA4CBu1yLh1q1bkqRs2bIluT0gIMBuv/sJDQ2VJH311Ve6fv265s6dq7CwMB05ckS9evXS1atX1aVLF3OwvgepUqVKkq/UziIBx0jcraF169by9PS0MBo4WlBQkLk8a9Ys3b5928JoAGQknrcAADiX2yUSDMOQJLNZc3LbUyIuLk6SFBsbq59//lldu3ZVYGCgKlSooOnTp6tevXoKDQ3VhAkT0h84MpRhGG49PsJdhnw9IsyXlPJ7/54zGXe7ZyW8UvFjlM6D065Lly7KmTOnpLuJw3nz5mXI+wIAgPtxXP0E1rOompd1uPAFdruuDTly5JCkZL99vHPnjiTZzebwoHMVLVpUrVu3vmd7nz59tH37dq1ZsyZFsSV0lfinKlWqpOh4OM6RI0d05swZs5zU/29m5+sRqaFl/2OWPz0xJM3nunFDyp3773JoqJQrV0YcnHb+/v564okn9OOPP0qSpkyZomeeecbp7wvAejxvAdeVVP0kKt7fwoiQHhZV87IOF77AbtcioUSJEpKkkJCQJLcnrE/Y735KlSolSSpZsuR9t1++fDmVUcJqiVsj1KxZU4ULF7YwGjhL4u4NK1eu1NmzZy2MBgAAAHAPbpdIqFGjhiRp586dSW5PWF+9evUHnith+sjr168nuf3atWuSUta6Aa7F/bs1QJIaN26ssmXLSrrbnWXatGkWRwQAAABkfm6XSGjcuLECAwN14sQJ7dq1657ts2fPliR16NDhged69NFHFRAQoBMnTiT5TWZCl4bkppqEa4qIiNDatWvNMokE92Wz2fTcc8+Z5SlTpqRqnBQAAAAA93K7RIKPj48GDhwoSRo4cKDdWAljx47V3r171aRJE9WrV89c/80336hSpUoaNmyY3bmyZcumQYMGKSYmRi+//LLduf78809NmTJFNptNL774opM/FRxp7dq1ioyMlHR3HIyGDRtaHBGcKXEi4ejRo9q6dauF0QAAAACZn9sNtihJ7733nlasWKFNmzapfPnyatq0qc6cOaOtW7cqb968mjRpkt3+V69e1ZEjR5KcxnHEiBFav369lixZovLly+vhhx/W5cuXtWXLFsXHx2vUqFGqX79+Rn00OEDibg2PPvqofHx8LIwGzlaqVCm1aNHCbEE0ZcoUNWjQwNqgAAAAgEzMaS0SYmJitH//fq1evVrz58/X6tWrtX//fsXExDjrLU1+fn5avXq13n//fWXLlk3z58/X6dOnFRQUpF27dqlcuXKpOteqVas0atQo5cqVS3/88YcOHDigli1bavHixRo+fLgTPwmcgfERsp7Egy7++uuvZosUAAAAAKnn0BYJV65c0eTJk7VkyRJt27ZNUVFR9+zj5+en+vXrq3379goKClL+/PkdGYLJ399fH374oT788MMH7jty5EiNHDky2e0+Pj4aPnw4SQM3cOrUKR05csQst2nTxsJokFG6d++uV155RXfu3FFYWJgWLlyoJ554wuqwAAAAgEzJIS0Sjh07pqeeekrFixfXkCFDtG7dOmXPnl2NGjVShw4d1KtXL7Vv316NGjVStmzZtHbtWr3zzjsqXry4evXqpePHjzsiDOCBErdGqFSpkjmFJ9xbjhw51L17d7M8ZcoUC6MBAAAAMrd0t0gYNGiQvv/+e8XFxally5bq1auXWrRoodKlSyd7zMmTJ7V69WrNmDFDM2fO1Jw5c/Tiiy9q/Pjx6Q0HuK/ly5eby7RGyFqCgoLM6R+XLl2qixcvqlChQhZHBQAAAGQ+6W6RMHHiRL388ssKDg7W8uXL1adPn/smESSpTJky6tevn1auXKkzZ86of//++umnn9IbCnBfsbGxWrVqlVl+7LHHLIwGGa1ly5YqXry4JCkuLk7Tp0+3OCIAAAAgc0p3i4STJ0+m61u9okWLaty4cfdMvQg42o4dO3Tjxg1Jkre3t5o3b25xRM4Va3hpzbXmduW08vOTRoywL2fMwY7j4eGh5557TqNGjZIkTZ48WW+88YZsNpsl8QAAkBU5sn4C67lINc99ufAFTvdPrqOaBtPEGM62YsUKc7lhw4bKnj27hdE4X5zhrbXXWzrkXH5+0n3GI3XiwY6VOJGwf/9+7dq1S7Vr17Y4KgAAsg5H1k9gPReq5rknF77ATpv+EXA1icdHoFtD1lShQgU1bNjQLDPoIgAAAJB6liQSVq9erW+++UYTJkyw67MOOMutW7e0efNms0wiIesKCgoyl2fMmKHo6GgLowEAAAAynwztlBQcHKxu3bpp586dCgwMlCTduHFDtWrV0ty5c1WyZMmMDAdZyNq1axUTEyNJCgwMVJ06dSyOCFbp2bOnBg8erKioKF29elV//PGHOnfubHVYAAAAQKaRoYmEl156SfHx8dq3b5+qVKkiSTpw4ICCgoLUv39//fHHHxkZDrKQxN0aHnnkEXl5uf/APp62GDXJvcEsbwhtkuZzRUZKn376d3no0FSM9ZKugx0vV65c6ty5s2bOnCnpbvcGEgkAAGSMpOoncYa3hREhPVysmud+XPgCZ+hfU2vWrNGaNWvMJIIkValSRd98840eeeSRjAwFWUzigRazSrcGL1usWuRda5a3hDVI87kiI6V///vv8muvpTKRkOaDnSMoKMhMJCxevFhXr15Vvnz5LI0JAICsIKn6CYmEzMsFq3nuxYUvcIaOkVCgQAFly5btnvXZsmVT3rx5MzIUZCHnz5/XgQMHzHJWSSQgea1btzZniomJidEvv/xicUQAAABA5pGhiYQhQ4ZoyJAhCgsLM9eFhYVp+PDheueddzIyFGQhiVsjlCpVSmXLlrUwGrgCLy8vPfPMM2aZ2RsAAACAlMvQrg2zZs3Szp07VaxYMVWsWFE2m02HDx+Wl5eX7ty5o3nz5pn7MpsDHOWf0z7abDYLo4GrCAoK0ueffy5J+uuvv3TgwAG7blcAAAAAkpahiYRSpUqpVKlSduuqVauWkSEgizEMI0uOj4AHq1q1qmrXrq2dO3dKutsqYcyYMRZHBQAAALi+DE0kTJo0KSPfDtD+/ft18eJFSZLNZmNQT9gJCgoyEwk///yzRo8eLU9PT4ujAgAAAFxbho6RAGS0xK0R6tSpw6CesPPUU0+ZU4FeuHDB7n4BAAAAkLQMTSRcvnxZvXv3VtGiReXl5SVPT0+7F+Bo/xwfAUgsf/78ateunVmeOnWqhdEAAAAAmUOGdm0ICgrSkSNHNHDgQBUuXJhB7+BU0dHRWrv273mKW7VqZWE0cFVBQUFauHChJGnevHkKDw9Xzpw5LY4KAAAAcF0ZmkjYsGGD1qxZozp16mTk2yKL2rp1q+7cuSNJ8vPzU6NGjSyOCK6offv2yp07t0JDQxUREaFZs2apX79+VocFAAAAuKwM7dpQsmRJujAgwySeQrRJkyby8/OzMBq4Kl9fXz311FNmme4NAAAAwP1laIuETz75REOHDtW0adOUP3/+jHxrZEGJEwlZcbaGqHg/fXpiiF05rQIDpdBQ+3LGHJwxgoKC9N///leStG7dOp06dUqlS5e2OCoAANyPI+snsF4mqOZlbi58gTO0RULCH3OFCxdW8eLFVaZMGbsX4Ch37tzR5s2bzXJWTCRINkXF+5svKe1jkthsUq5cf79SNbxJug7OGPXq1VPFihXNMq0SAABwFsfVT2C9TFDNy9xc+AJnaIuE3r17a9u2berXrx+DLcKpNm7cqJiYGElSzpw5GZcD92Wz2RQUFKThw4dLuptI+OCDD/gdBQAAACQhQxMJf/75pxYtWqSWLVtm5NsiC0rcraF58+by8srQWx2Z0DPPPKN3331XhmHo5MmT2rhxo5o0aWJ1WAAAAIDLydCuDfnz51ehQoUy8i2RRa1cudJczprdGpBaxYsXt7tXpkyZYmE0AAAAgOvK0ETC8OHD9dFHH5lNzgFnCAsL019//WWWs2oiwdcjUkPKjDZfvh6RaT7XjRv23bNu3MiogzNWUFCQuTxz5kxFRERYGA0AAO7HkfUTWC8TVfMyJxe+wBna3nvGjBnatWuXihQpokqVKsnb29tue+Lm6EBarVu3TvHx8ZKkfPnyqWrVqhZHZBVDfp5RduU0n8mw/71lpOZU6To4Y3Xr1k0vv/yybt++rfDwcC1YsEBPPvmk1WEBAOBGHFc/gfUyUTUvc3LhC5yhiYRSpUqpVKlSGfmWyIISJ6RatmwpD48MbXiDTCwgIEA9evQwuzVMmTKFRAIAAADwDw5PJCxevFgdOnRIctukSZMc/XbAPRInErJqtwakXVBQkJlIWLZsmS5cuKDChQtbHBUAAADgOhz+VW2nTp3UvHlzbdmyxdGnBh7o8uXL2rdvn1kmkYDUat68uUqUKCFJio+P1/Tp0y2OCAAAAHAtDk8kdO3aVevXr1fjxo3VvXt3HTlyxNFvASRrzZo15nLRokVVvnx564JBpuTh4aFnn33WLE+ZMkWGC/VHAwAAAKzm8ETCnDlztGXLFjVv3lzz5s1TtWrV9NJLL+nChQuOfivgHv+c9tFms1kYDTKr5557zlzev3+/du/ebV0wAAAAgItxyih09evX16pVq/THH3+oatWq+uGHH1S+fHm9++67Cg8Pd8ZbApIYHwGOUaFCBTVs2NAsJ4yZAAAAAMBJiYQEbdq00c6dOzV9+nQVKlRIo0ePVpkyZfTll18qOjramW+NLCg4OFjHjx83yyQSkB6JWyXMmDFDMTExFkYDAAAAuI4MmRfvqaee0uHDh/XNN9/I29tbb775pipUqKBp06ZlxNsji1i9erW5XK5cOXPAPCAtevbsKV9fX0nSlStX9Oeff1ocEQAAAOAaMiSRIEleXl4aMGCATpw4oeHDh+vcuXPq06dPRr09sgC6NcCRcufOrU6dOpllujcAAAAAd3k58+S3b9/WoUOHdPDgQfPfgwcP6vTp04qLi2MgPDiMYRgkEv4hzvDU7vAaduW08vGRgoLsyxlzsLWee+45zZo1S5K0aNEiXb9+XXny5LE4KgAAMi9H1k9gvUxczcscXPgCOzyR8MYbb5iJg5CQEHN9wvRpHh4eKlOmjKpWrapq1ao5+u2RRR07dszufmvRooV1wbiIWMNHCy51dci5smWTJk+24mBrtWnTRgUKFNDly5cVHR2t3377TS+//LLVYQEAkGk5sn4C62Xial7m4MIX2OGJhK+++spcLlKkiKpWrWomDapWraoqVarIz8/P0W+LLC5xa4SqVauqYMGCFkYDd+Ht7a2nn35aX375paS73RtIJAAAACCrc3giYcKECWbiIFeuXI4+PZAkujXAWZ577jkzkbB161YdOXJEFStWtDgqAAAAwDoOH2zx5ZdfVtOmTUkiIMPEx8fbzdhAIgGOVLNmTVWvXt0sT5061cJoAAAAAOulu0VC+fLl1blzZ3Xo0EFNmzaVpycDpiBjHThwQFevXpV0dwyOZs2aWRyRa/BQrKrl2GeW991M+5gk0dHSjBl/l3v1SsVYL+k62DUEBQXpzTfflCRNmzZNH330kTw8MmzSGwAA3EZS9ZN4547/Didyg2qea3PhC+yQn9qxY8fqyy+/VGBgoNq1a6eOHTuqbdu2CgwMdMTpgftau3atuVyzZk3lzp3bwmhch7dHjLoUWmCWD9+ulOZz3bkjJZ6ttUuXVPwOS9fBrqFXr1565513FBcXp7Nnz2rNmjW0fAEAIA2Sqp9ExZNIyKzcoJrn2lz4Aqf7K7Vjx47p0KFDGj16tKpUqaLffvtNvXr1UoECBfToo49q3LhxOnHihCNiBZK0Zs0ac5nZGuAMhQoVUps2bczylClTLIwGAAAAsJZD2uZWrFhR77zzjtavX6+LFy9q0qRJ6tixo3bs2KHXX39dFSpUUOXKlTV06FBt3LjRnAoSSK/4+Hi7FgkkEuAsQYnm8J0zZ45u3bplYTQAAACAdRzeyTdv3rx67rnnNHv2bF29elV//PGHXn75Zd25c0djxoxRs2bNVKBAAfXu3ZvKONLt4MGD5vgINptNTZs2tTgiuKtOnTqZ3bVu376tuXPnWhwRAAAAYA2njhbm7e2tNm3a6JtvvtHp06e1e/dujRw5UmXKlNG0adP0+OOPK1++fPrhhx+cGQbc2D/HR2C2EDiLn5+fevbsaZbp3gAAAICsyuGJhMWLFye7rXr16nr//fe1detWnTt3Tv/73//UunVr3bhxw9FhIItgfARkpMTdG1avXq3g4GALowEAAACs4fBEQqdOndS8eXNt2bLlvvsVKlRIL7zwghYuXKi33nrL0WEgCzAMg0QCMlTDhg1Vrlw5SXfvv59//tniiAAAAICM5/BEQteuXbV+/Xo1btxY3bt315EjRxz9FoAkxkdAxrPZbHruuefM8pQpUxg8FgAAAFmOwxMJc+bM0ZYtW9S8eXPNmzdP1apV00svvaQLFy44+q2QxSUeH6FGjRrKnTu3hdEgq3j22WfN5aNHj2rbtm0WRgMAAABkPKcMtli/fn2tWrVKf/zxh6pWraoffvhB5cuX17vvvqvw8HBnvCWyILo1wAqlSpVS8+bNzTKDLgIAACCrceqsDW3atNHOnTs1ffp0FSpUSKNHj1aZMmX05ZdfKjo62plvDTfH+AiwUuJBF3/99VdFRUVZGA0AAACQsbwy4k2eeuopPf744/r+++/10Ucf6c0339S4ceP00Ucf2TUTBlLq0KFDunLliiTGR0hOdLyvvjvzkl05rXLkkHbtsi9nzMGuqUePHnrllVcUERGh0NBQLV68WN27d7c6LAAAXJ4j6yewnhtW81yLC19gp7ZISMzLy0sDBgzQiRMnNHz4cJ07d059+vTJqLeHm0k8PkL16tWVJ08eC6NxTYY8dCm6sPky0vHj7ukp1az598vTM6MOdk05cuRQt27dzDLdGwAASBlH1k9gPTes5rkWF77ATm2RcPv2bR06dEgHDx40/z148KBOnz6tuLg42Ww2Z7493BjdGmC1oKAgTZ8+XZL0xx9/6PLlyypQoIDFUQEAAADO5/BEwhtvvGEmDkJCQsz1CVOkeXh4qEyZMqpataqqVavm6LdHFsD4CHAFjzzyiIoWLapz584pNjZWv/zyiwYPHmx1WAAAAIDTOTyR8NVXX5nLRYoUUdWqVc2kQdWqVVWlShX5+fk5+m2RhRw+fFiXL1+WdHd8hGbNmlkcEbIiT09PPfPMM/rPf/4j6W73BhIJAAAAyAocnkiYMGGCmTjIlSuXo08P2I2PUK1aNcZHSIaPLUp9iv9klied7Zvmc928KSUez3L9+lSM9ZKug11bUFCQmUjYtWuX9u3bR0srAADuI6n6SbTBgIuZlRtX81yDC19ghycSXn75ZUefErBDt4aUsdniVcj3kl05reLipD177MsZc7Bre+ihh1SvXj1t375dkjR16lR99tlnFkcFAIDrSrJ+YlgYENLFjat5rsGFLzDDpCJTYXwEuJrnnnvOXP75558VGxtrYTQAAACA85FIQKZy9OhRXbr0dxab8RFgtaeeekre3t6SpIsXL2r58uUWRwQAAAA4V7oTCYcPH3ZEHA47D9xb4tYI1apVU968ea0LBpCUN29edejQwSxPnTrVwmgAAAAA50t3IqFq1ap6+umntX///jQdv3v3bj355JMMUIYUoVsDXFHi7g3z58/XjRs3LIwGAAAAcK50JxLef/99LVq0SDVq1FDt2rX1xRdfaMeOHYqJiUly/6ioKG3ZskWjR49WtWrVVKdOHf3+++/64IMP0hsK3BzjI8BVtWvXzmwdExkZqVmzZlkcEQAAAOA86Z61YcSIEXr55Zc1atQoTZ06VW+//bZsNpu8vb1VqlQp5c6dWzly5FB4eLiuX7+uM2fOKDY2VoZhKDAwUIMHD9awYcOUP39+R3weuLFjx47p4sWLZpnxEeAqfHx81KtXL40fP16SNHnyZD3//PMWRwUAAAA4h0OmfyxQoIDGjRunTz/9VDNnztTixYu1ceNGHT169J59CxUqpKZNm6p9+/Z64okn5Ofn54gQkAUkbo1QtWpV5cuXz7pggH8ICgoyEwkJv/8qVKhgcVQAAACA46U7kRAcHKwSJUpIkvz9/RUUFKSgoCBJ0pUrV3T58mXduHFDgYGBKlCgAC0PkGZ0a4Arq127tqpXr669e/dKkn766Sd9+umnFkcFAAAAOF66x0goXbq0XnnllSS35c+fX1WqVFGjRo1UpUoVkghIM8ZHgKuz2Wzq27evWZ4yZYpiY2MtjAgAAABwjnS3SDAMQxEREUluW7lyperWravAwMD0vg2yuOPHj+vChQtmmfERHize8NDpOyXtymnl5SU1b25fzpiDM5enn35ab7/9tmJiYnTx4kX9+eefdlNDAgCQ1TmyfgLrZaFqnjVc+AI7NZLHHntMffr00cSJE535NsgCErdGoHVLysQYvppyro9DzpU9u5TovyADD85c8uXLpy5dupizNkycOJFEAgAAiTiyfgLrZaFqnjVc+AI7PQVoGIaz3wJZAN0akFkk7t6wePFiXbp0ycJoAAAAAMejLRFcHuMjIDN57LHHVKxYMUlSbGysfv75Z4sjAgAAAByLRAJc3okTJ3T+/HmzzPgIcGWenp7q3bu3WZ44cSItswAAAOBWHDJGwrVr13T16lXly5fPEacD7CRujVC5cmUVKFDAumAyEZviVMI/2CwHR5RI87liY6UNG/4uN2mSirFe0nVw5tS7d299/PHHkqRDhw5p69atatCggcVRAQBgvaTqJ4Y8LYwI6ZEFq3kZy4UvsEOiWLx4sQoWLKg8efLooYceUuXKlVW5cmVHnBqgW0Ma+XhEq3exKWb50xND0nyuW7ekli3/LoeGSrlyZcTBmVPZsmXVokUL89796aefSCQAAKCk6ydR8f4WRoT0yILVvIzlwhc43V0bBgwYoMaNGytHjhy6du2aNmzYoO+//16vv/66bDabpk2bpsqVK6tXr1767LPPtGLFCl27ds0Rsd9XZGSkRowYoQoVKsjPz09FihRR3759FRISkq7zHjt2TP7+/rLZbGrbtq2DokVyGB8BmVW/fv3M5V9//VW3b9+2MBoAAADAcdLdIuGbb74xl0+dOqU9e/aYr927d+v06dM6fPiwDh8+rN9++83ct2jRoqpVq5YWLFiQ3hDuERkZqUcffVSbNm1S4cKF1blzZ50+fVqTJk3S4sWLtXnzZpUtWzZN537ppZcUFRXl4IiRnJMnT+rcuXNmuXnieVQBF9atWze98sorCg8P182bNzV79mwFBQVZHRYAAACQbg4dbLF06dLq0qWLRowYoblz5+rkyZO6ceOG1q1bp/Hjx6tv376qU6eO/Pz8FBISosWLFzvy7U2ffPKJNm3apIYNG+ro0aP67bfftHXrVn3xxRe6cuWK3fRsqTFx4kStXr1aL7zwgoMjRnISt0Z46KGHGB8BmUa2bNn01FNPmeWffvrJwmgAAAAAx3H6rA05cuRQkyZN9Morr+iHH37Qtm3bdPPmTR08eFAzZsxw+PvFxMRo/PjxkqQJEyYoe/bs5rY33nhD1atX17p16/TXX3+l6ryXL1/W22+/rVatWtn9cQDnolsDMrPE3RvWrVunY8eOWRgNAAAA4BiWTP/o4eGhSpUqqWfPng4/94YNGxQWFqayZcuqVq1a92zv0aOHJGnRokWpOu+rr76qiIgIffvttw6JEw9mGIbWrl1rlkkkILOpW7euqlatapYnTZpkYTQAAACAY1iSSHCmPXv2SJJq166d5PaE9Qn7pcTvv/+u3377TcOHD1e5cuXSHyRS5NSpUzp79qxZZnwEZDY2m82uK9XkyZMVGxtrYUQAAABA+rldIiE4+O68tMWKFUtye8L6hP0e5Pbt2xowYIAqVqyoIUPSPn0eUi9xt4ZKlSqpYMGC1gUDpNEzzzwjb29vSdKFCxe0dOlSiyMCAAAA0ifdsza4mlu3bkm6O9BZUgICAuz2e5D33ntPZ86c0apVq+Tj45Ou2KpUqZLk+hMnTqR5Fgl3xvgIcAf58+dXp06dNGfOHEl3B11s3769xVEB7o3nLQAAzuV2LRIMw5B0t0nx/banxI4dOzR+/Hg999xzatmypUPiQ8owPgLcSeLuDQsXLtTly5ctjAYAAABIH7drkZAjRw5Jd7skJOXOnTuSZDebQ1JiY2P1wgsvKDAwUJ9//rlDYjtw4ECS65P75iQrO336tF33E8ZHQGbWpk0bFS1aVOfOnVNsbKymTp2qt956y+qwALfF8xYAAOdyu0RCiRIlJEkhISFJbk9Yn7BfckJCQrR7924VKlRIjz/+uN22sLAwSdK2bdvUokULZc+eXYsXL05n5EgscbeGihUrqlChQtYFk0nFxPvo1/M97cppFRAgzZtnX86Yg92Dp6enevfurVGjRkmSfvjhB7355pvJtpwCAMBdObJ+AutRzXMyF77AbpdIqFGjhiRp586dSW5PWF+9evUUne/ixYu6ePFikttCQ0O1du1aBQYGpiFS3A/jI6RfvDx15PZDDjmXt7fUpYsVB7uPfv366ZNPPpFhGDp69KjWrVtHSxsAQJbjyPoJrEc1z8lc+AK73RgJjRs3VmBgoE6cOKFdu3bds3327NmSpA4dOtz3PKVKlZJhGEm+Vq9eLeluc2XDMMwWCnAcxkeAuyldurRat25tlr///nsLowEAAADSzu0SCT4+Pho4cKAkaeDAgXZjJYwdO1Z79+5VkyZNVK9ePXP9N998o0qVKmnYsGEZHi/udfr0aZ05c8Ys860t3MWLL75oLs+ePVvXrl2zMBoAAAAgbdwukSDdnbLx4Ycf1qZNm1S+fHn17NlTDRo00Jtvvqm8efNq0qRJdvtfvXpVR44c0YULFyyKGIkl7tZQoUIFFS5c2LpgAAfq2LGjChYsKEmKjo7W1KlTLY4IAAAASD23TCT4+flp9erVev/995UtWzbNnz9fp0+fVlBQkHbt2qVy5cpZHSLug/ERHMPbFq2ehX8xX9626DSf6/btu92zEl7JTIrihIPdi7e3t91UkN9//32qpqQFACCzc2T9BNajmudkLnyB3W6wxQT+/v768MMP9eGHHz5w35EjR2rkyJEpPneLFi2o/DsRiQTH8LDFqVL2I3bltIqJkRYssC9nzMHu5/nnn9fo0aMlSYcPH9aGDRvUtGlTi6MCACBjJFk/oVqdaVHNczIXvsBu2SIBmRfjI8DdlSlTRo899phZZtBFAAAAZDYkEuBSEs/WUKFCBRUpUsTCaADnSDzo4qxZs3T9+nULowEAAABSh0QCXArdGpAVdOrUSQUKFJAkRUVFadq0aRZHBAAAAKQciQS4lMSJBLo1wF35+PioT58+ZplBFwEAAJCZkEiAyzhz5oxOnz5tlkkkwJ09//zz5vLBgwe1ceNGC6MBAAAAUo5EAlxG4vERypcvr6JFi1oYDeBc5cqV06OPPmqWGXQRAAAAmQWJBLgMxkdAVpN40MWZM2cy6CIAAAAyBRIJcBkkEpDVdOnSRfnz55fEoIsAAADIPEgkwCWcOXNGp06dMsuMj4CswMfHR7179zbL3377LYMuAgAAwOV5WR0AINmPj1CuXDnGR3AAQzaFxQTaldPKw0MqWdK+nDEHu7/+/fvr888/l2EYOnLkiFatWmU3dgIAAO7EkfUTWI9qnpO58AUmkQCXQLcGx4uO99O406875Fw5c0qJJtTIwIPdX5kyZfSvf/1Lv//+uyTpv//9L4kEAIDbcmT9BNajmudkLnyBXSelgSwtcYsEEgnIagYMGGAuL1iwQCEhIRZGAwAAANwfiQRYLjg4WCdPnjTLjI+ArKZt27YqVaqUJCkuLk4//PCDtQEBAAAA90EiAZb75/gIxYoVszAaION5enqqf//+Zvn7779XdHS0hREBAAAAySORAMslHh+B1giOFK9Ar1DzJcWn/Uzxd7tnJbziU3OqdB2cdfTt21c+Pj6SpIsXL2r+/PnWBgQAgFM4rn4C61HNczIXvsAMtgjLMdCic/h6ROm10uPM8qcnhqT5XOHhUunSf5dDQ6VcuTLi4Kwjf/786tmzp6ZNmybp7qCLTzzxhMVRAQDgWEnVT6Li/S2MCOlBNc/JXPgC0yIBljp79izjIwD/L/Ggi2vXrtWBAwcsjAYAAABIGokEWCrx+Ahly5ZV8eLFLYwGsNbDDz+sWrVqmeVvv/3WwmgAAACApJFIgKXo1gD8zWaz2bVKmDp1qm7evGlhRAAAAMC9SCTAUiQSAHu9evVSYGCgJOnmzZv6+eefLY4IAAAAsEciAZY5e/asTpw4YZYZHwGQsmXLpj59+pjl//73vzIMw8KIAAAAAHskEmCZxOMjlClThvERgP/Xv39/c3n//v1av369hdEAAAAA9kgkwDKJEwl0awD+VrFiRbVq1cosjx8/3sJoAAAAAHskEmAZxkcAkjdw4EBzed68eQoODrYwGgAAAOBvJBJgiZCQEB0/ftwsMz4CYK9Dhw4qU6aMJCkuLk4TJkywOCIAAADgLhIJsMQ/x0coUaKEhdEArsfT01ODBg0yy99//71u375tYUQAAADAXV5WB4CsKXG3BlojOEes4a0/r7SxK6eVv7/05Zf25Yw5OGvr06eP3n//fd26dUthYWGaNm2a3UCMAABkNo6sn8B6VPOczIUvMIkEWILxEZwvzvDS1rCGDjmXr6/02mtWHJy1BQYGqm/fvvr6668lSePGjdOLL74oDw8akwEAMidH1k9gPap5TubCF5jaKDLcuXPnGB8BSKFBgwbJZrNJkg4fPqzly5dbHBEAAACyOhIJyHCJx0coXbq0SpYsaWE0gGsrV66cOnToYJbHjRtnYTQAAAAAiQRYgG4NQOoMHjzYXP7jjz90+PBhC6MBAABAVscYCchwJBIyhpctRo/mXWGWV15rleZzRURIw4b9XR49OhVjvaTrYEjSI488oqpVq2r//v2SpPHjxzMdJAAgU0qqfsKAi5kX1Twnc+ELTCIBGer8+fM6duyYWWZ8BOfxtMWqQe6tZnnN9RZpPldUlJS4Rf3Ikan4HZaugyFJNptNgwcP1gsvvCBJmjx5sj7++GPlzp3b4sgAAEidpOonJBIyL6p5TubCF5iuDchQq1evNpdLlSrF+AhACj399NPKmzevJOnOnTuaOHGixREBAAAgqyKRgAy1atUqc/mRRx6xMBIgc/H399eLL75olr/55hvFxsZaGBEAAACyKhIJyFAkEoC0GzBggLy87vZIO3PmjBYsWGBxRAAAAMiKSCQgw5w6dUqnT582yy1btrQuGCATKlasmHr06GGWv/jiCwujAQAAQFZFIgEZJnFrhEqVKqlIkSIWRgNkTm+88Ya5vHnzZm3cuNHCaAAAAJAVkUhAhqFbA5B+9erVs5vt5LPPPrMwGgAAAGRFJBKQIQzDIJEAOMhbb71lLi9cuFBHjhyxMBoAAABkNSQSkCEOHz6sixcvmuUWLVpYFwyQybVr104PPfSQpLtJurFjx1ocEQAAALISEgnIEIlbI9SsWVN58+a1MBogc/Pw8LBrlTBlyhRdunTJwogAAACQlZBIQIagWwPgWE8//bQKFSokSYqKitKECRMsjggAAABZhZfVAcD9xcfHa/Xq1WaZRELGiIr317+PjXTIuXLlkgzDioORHF9fX7366qsaPny4JGnChAkaMmSIAgICLI4MAIDkObJ+AutRzXMyF77AtEiA0+3Zs0ehoaGSJE9PTzVt2tTiiAD30L9/fzNxcP36dU2aNMniiAAAAJAVkEiA0yXu1lCvXj3lzJnTwmgA95E7d2698MILZnns2LGKjY21MCIAAABkBSQS4HSMjwA4z2uvvSZPT09J0qlTpzRr1iyLIwIAAIC7I5EAp4qJidG6devMMokEwLFKliypp556yix/8sknio+PtzAiAAAAuDsSCXCqHTt26NatW5IkHx8fNWrUyOKIsg5fjwiNKD/SfPl6RKT5XGFhks329yssLKMORkoMHTrUXN6/f7+WLFliYTQAACTPkfUTWI9qnpO58AUmkQCnStytoVGjRvL397cwGsA9ValSRV26dDHLo0aNkuGiI/wCAAAg8yORAKdifAQgYwwbNsxc3rp1q9asWWNdMAAAAHBrJBLgNBEREdq4caNZJpEAOE/9+vXVqlUrs/zJJ59YGA0AAADcGYkEOM3mzZsVFRUlSQoICFC9evUsjghwb8OHDzeXV6xYoW3btlkYDQAAANwViQQ4TeJuDU2bNpWPj4+F0QDur0WLFmrQoIFZHj16tIXRAAAAwF2RSIDTMD4CkLFsNptdq4T58+frwIEDFkYEAAAAd0QiAU5x8+ZNu2bVJBKAjNG+fXtVq1bNLI8aNcrCaAAAAOCOSCTAKdatW6e4uDhJUq5cuVSzZk1rAwKyCA8PD7sZHH799VcdPnzYwogAAADgbkgkwCmWL19uLrds2VKenp4WRgNkLU888YQqVaokSTIMQx999JHFEQEAAMCdkEiAU6xYscJcfuyxxyyMBMh6PD099cEHH5jlX375hVYJAAAAcBgvqwOA+zl//rzdAG8kEqwRZ3hpS+jDduW08vWVBg+2L2fMwUirJ554Qh9++KEOHz5stkqYPn261WEBALI4R9ZPYD2qeU7mwheYn1w4XOLWCCVLllTZsmUtjCbrijW8tfTqvxxyLn9/6auvrDgYaZXQKqFXr16S7rZKeP/9980uDwAAWMGR9RNYj2qek7nwBaZrAxwu8fgIjz32mGw2m4XRAFkXYyUAAADAGUgkwKEMw2B8BMBFMFYCAAAAnIFEAhzqwIEDunjxoiTJZrPp0UcftTgiIGujVQIAAAAcjTES4FCJuzXUrl1befPmtTCarM3TFqu6gdvN8o4b9dJ8rqgo6dtv/y6//HIqxnpJ18FIr6TGSnj33XdVuXJliyMDAGRFSdVPGHAx86Ka52QufIH5qYVD/XN8BFjHyxajtvmXmuXd4TXTfK6ICOn11/8u9+6dit9h6ToYjvDPGRzee+89zZ071+qwAABZUFL1ExIJmRfVPCdz4QtM1wY4TFRUlNauXWuWSSQArsHT09OuS8O8efO0bds2CyMCAABAZkYiAQ6zefNm3blzR5Lk5+enRo0aWRwRgATdu3dXnTp1zPK7775rYTQAAADIzEgkwGESd2to1qyZ/Pz8LIwGQGI2m02jRo0yyytWrNCqVassjAgAAACZFYkEOAzTPgKurXXr1mrevLlZHjZsmAzDsDAiAAAAZEYkEuAQoaGh2rFjh1kmkQC4HpvNptGjR5vlbdu2aeHChRZGBAAAgMyIRAIcYtWqVYqPj5ckFShQQNWqVbM4IgBJadiwoTp27GiW3333XcXFxVkYEQAAADIbEglwiMTjI7Rq1UoeHtxagKv6+OOPZbPZJEkHDhzQjBkzLI4IAAAAmQl/7SHdDMPQsmXLzHKrVq0sjAbAg1SvXl1PPfWUWX7vvfcUGRlpYUQAAADITEgkIN2OHTumU6dOmeXWrVtbGA2AlPjwww/l7e0tSQoODta4ceMsjggAAACZhdsmEiIjIzVixAhVqFBBfn5+KlKkiPr27auQkJAUnyMsLEwzZsxQr169VLlyZQUEBChHjhx6+OGHNW7cOMXExDjxE2Qef/75p7lcvXp1FS1a1MJoAKRE2bJlNXDgQLP8ySef6MqVKxZGBAAAgMzCy+oAnCEyMlKPPvqoNm3apMKFC6tz5846ffq0Jk2apMWLF2vz5s0qW7bsA8/z+eefa9SoUfLw8FCtWrXUsWNHXblyRRs3btS2bds0e/ZsLV26VNmyZcuAT+W6EicS2rZta2EkSCwq3ldfnRpsV06rnDmlRI1OlDNnRh0MZ3rvvfc0adIkhYWFKTw8XB9++KHGjx9vdVgAADfmyPoJrEc1z8lc+AK7ZYuETz75RJs2bVLDhg119OhR/fbbb9q6dau++OILXblyRX379k3RebJnz67hw4crODhYO3bs0K+//qqVK1dq3759KlGihDZs2KCPP/7YyZ/GtUVERGj16tVmmUSCK/HQjdjc5is9P+4eHlKpUn+/UjWWZroOhjPlyZNH77//vln+7rvvdOTIEQsjAgC4P8fVT2A9qnlO5sIX2HUicZCYmBjzG7UJEyYoe/bs5rY33nhD1atX17p16/TXX3898FxDhw7VqFGj7mmqX758eX366aeSpF9++cWB0Wc+69atMwdpy549uxo3bmxxRABS45VXXlGZMmUkSbGxsRoyZIjFEQEAAMDVuV0iYcOGDQoLC1PZsmVVq1ate7b36NFDkrRo0aJ0vU+NGjUkSefPn0/XeTK7xN0aHn30Ufn4+FgYDYDU8vX1NROjkrRgwQKtXbvWwogAAADg6twukbBnzx5JUu3atZPcnrA+Yb+0OnnypCSpUKFC6TpPZvfHH3+Yy3RrADKnHj16qGHDhmb5zTffVHx8vIURAQAAwJW53WCLwcHBkqRixYoluT1hfcJ+aZUwVVrnzp3TdZ7M7NSpU3b9qUkkuBYfj0i9XOJbs/xt8MtpPld4uFS9+t/lvXtTMdZLug5GRrDZbPriiy/UqFEjSdJff/2lqVOnqnfv3tYGBgBwO0nVT6Lj/SyMCOlBNc/JXPgCu10i4datW5KU7EwKAQEBdvulxXfffacVK1YoV65cGjp0aIqPq1KlSpLrT5w4kaJZJFzN0qVLzeVKlSqpVKlS1gWDe9hkKJf3DbtyWsXHS2fO2Jcz5mBklIYNG+qJJ57QzJkzJUlDhgxR165dFRgYaHFkQOq52/MWcCeOrJ/AelTznMyFL7DbdW0wjLu/jGw22323p9XatWs1ePBg2Ww2/fTTTypSpEi6zpeZ0a0BcC+fffaZ/P39JUmXL1/Whx9+aHFEAAAAcEVu1yIhR44ckqTbt28nuf3OnTuSZDebQ0rt3btXXbp0UXR0tL7++mt17do1VccfOHAgyfXJfXPiyqKjo7Vy5UqzTCIByPxKlCihYcOG6YMPPpAkff3113r++ef10EMPWRwZkDru9LwFAMAVuV2LhBIlSkiSQkJCktyesD5hv5Q6ceKE2rRpo7CwMI0cOVKDBg1KX6CZ3MaNG81kjb+/v5o3b25xRAAc4e2331bp0qUl3Z0O8tVXX013Sy4AAAC4F7dLJCRMy7hz584ktyesr5540IoHOH/+vB577DFdvHhRgwcP1ogRI9IfaCaXeNrHFi1ayM+PQXIAd+Dn56exY8ea5RUrVmj+/PnWBQQAAACX43aJhMaNGyswMFAnTpzQrl277tk+e/ZsSVKHDh1SdL7Q0FC1adNGp06dUp8+ffTll186NN7MivERAPfVuXNntW7d2iy/8cYbioiIsDAiAAAAuBK3SyT4+Pho4MCBkqSBAwfajZUwduxY7d27V02aNFG9evXM9d98840qVaqkYcOG2Z3rzp07ateunfbv368nnnhCP/zwQ7KDOGYl586d0759+8wyiQTAvdhsNo0bN05eXneH0Tl9+rT+85//WBwVAAAAXIXbDbYoSe+9955WrFihTZs2qXz58mratKnOnDmjrVu3Km/evJo0aZLd/levXtWRI0d04cIFu/XvvvuutmzZIk9PT3l5ealfv35Jvt/kyZOd9VFcUuLWCGXKlFH58uUtjAaAM1SqVEmDBw/WF198IUn69NNP1atXL1WoUMHiyAAAAGA1t0wk+Pn5afXq1Ro9erRmzJih+fPnK3fu3AoKCtJHH32k4sWLp+g8oaGhkqS4uDjNmDEj2f2yWiJh0aJF5nK7du1opQG4qQ8++EC//PKLzp8/r6ioKPXv318rV67kZx4AACCLc7uuDQn8/f314Ycf6vjx44qKitLFixc1efLkJJMII0eOlGEY9yQEJk+eLMMwHvjKSiIiIrR8+XKz3LFjRwujAeBMOXPm1Pjx483y6tWrNW3aNAsjAgAAgCtwyxYJcJ5Vq1aZg65lz56daR9dWLzhqcO3KtqV08rbW+rc2b6cMQfDal27dlXHjh3NlkhvvPGG2rVrp3z58lkcGQAgM3Jk/QTWo5rnZC58gUkkIFUSd2to3bq1fH19LYwG9xNj+Oi3C0855FwBAVKaZwBM18Gwms1m0zfffKNVq1bp9u3bunbtmt5+++17xpoBACAlHFk/gfWo5jmZC19gt+3aAMczDEOLFy82y3RrALKGEiVK6KOPPjLLkydP1po1a6wLCAAAAJYikYAU2717t86dOyfp7reU7dq1szgiABll0KBBqlWrlll+6aWXFBkZaWFEAAAAsAqJBKRY4m4NDRo0UIECBSyMBkBG8vLy0vfffy8Pj7uPjaNHj+rf//63xVEBAADACoyRgBRLnEigW4Pr81CcygccNcvHbldI87liYqQlS/4ut2+firFe0nUwXEndunU1aNAgjRs3TpI0ZswYde3aVfXr17c4MgBAZpFU/SReDLiYWVHNczIXvsAkEpAi58+f144dO8wyiQTX5+0RrSeL/GaWPz0xJM3nun1b6tr173JoqJQrV0YcDFczatQoLV68WCdOnFB8fLz69Omjv/76S35+flaHBgDIBJKqn0TF+1sYEdKDap6TufAFpmsDUmRJokxYqVKlVKVKFQujAWCVgIAA/fTTT2b54MGDdHEAAADIYkgkIEX+2a3BZrNZGA0AKzVr1kyvvvqqWR4zZoy2bdtmYUQAAADISCQS8EARERFasWKFWaZbA4BPPvlEZcuWlSSziwOzOAAAAGQNJBLwQCtXrlRERIQkKUeOHGrevLnFEQGwWkBAgCZNmmS2Tjp48KBGjhxpbVAAAADIECQS8ECJuzW0adNGPj4+FkYDwFU0bdpUgwYNMstjxozR2rVrLYwIAAAAGYFEAu4rPj5eixcvNssdOnSwMBoAruaTTz5RhQp3pxY1DEPPPPOMQkNDLY4KAAAAzkQiAfe1bds2nT9/XpLk4eGh9u3bWxwRAFcSEBCg6dOny8vr7mzCISEheumll2QYhsWRAQAAwFlIJOC+5s6day43b95c+fLlszAaAK6obt26+vjjj83yrFmzNGXKFAsjAgAAgDORSECyDMOwSyR069bNwmgAuLK3335bLVu2NMsDBw7U8ePHLYwIAAAAzkIiAcnat2+fTpw4YZa7du1qYTQAXJmHh4emTp2q3LlzS5Ju376tXr16KSYmxuLIAAAA4GheVgcA15W4NUKDBg1UtGhRC6NBakXH+2hySJBdOa2yZ5dWr7YvZ8zByEyKFSumH374QT169JAkbd++XcOHD9dnn31mcWQAAFfhyPoJrEc1z8lc+AKTSECy6NaQuRny1JmI0g45l5eX1KKFFQcjs+nevbv69euniRMnSpI+//xzNWrUiBZNAABJjq2fwHpU85zMhS8wXRuQpGPHjmnfvn1mmT8CAKTU119/rWrVqpnl3r1723WTAgAAQOZGIgFJmjdvnrlcvXp1lStXzsJoAGQm2bJl0+zZs5UjRw5JUnh4uHr06KGIiAiLIwMAAIAjkEhAkujWACA9KlSoYHZvkKTdu3fr1VdftTAiAAAAOApjJOAeZ86c0datW80y3RoyJ29blHoVmWGWZ5zvleZz3boldejwd3nx4lSM9ZKug5GZPf7443r11Vf19ddfS5J+/PFHNW7cWL1797Y2MACAZZKqn8QYvhZGhPSgmudkLnyBSSTgHjNnzjSXK1asaNfXGZmHhy1epbKdsSunVWystHatfTljDkZm99lnn2nr1q1mcrJ///6qXLmy6tevb3FkAAArJFk/MSwMCOlCNc/JXPgC07UB9/jtt9/M5Z49e8pms1kYDYDMzMfHRzNnzlTevHklSVFRUerSpYvOnz9vcWQAAABIKxIJsHP8+HH99ddfZrlnz54WRgPAHZQoUUKzZ8+Wl9fdRnAXLlxQly5dGHwRAAAgkyKRADuJuzVUrVpVlStXtjAaAO6iRYsWGj9+vFnevn27XnjhBRkG7VkBAAAyGxIJsPPPbg0A4Cj9+/fXgAEDzPL06dM1ZswYCyMCAABAWpBIgOnQoUPau3evWSaRAMDRvvrqK7Vs2dIsDxs2TPPmzbMwIgAAAKQWiQSYErdGqFWrlsqXL29hNADckbe3t2bNmqUyZcpIkgzDUK9evbRp0yaLIwMAAEBKkUiApLuV+cSJhCeeeMLCaAC4s7x582rRokXKlSuXJCkyMlIdO3bUkSNHrA0MAAAAKUIiAZKknTt36vDhw2aZRAIAZ6pcubIWLFggHx8fSdL169fVtm1bXbhwweLIAAAA8CAkEiBJmjZtmrncuHFjs9kxADhLs2bN9PPPP8tms0mSTp8+rfbt2+vmzZsWRwYAAID78bI6AFgvNjZWv/zyi1l+9tlnLYwGjmIYHroYVdCunFaenlKNGvbljDkY7u7xxx/XuXPn9Prrr0uSdu3apW7dumnRokXy8/OzODoAgKM5sn4C61HNczIXvsAkEqBly5bp8uXLkiQfHx+6NbiJaMNX/wt+2SHnypFD2r3bioORFbz22msKCQnRF198IUlasWKFnnjiCc2ZM0fe3t4WRwcAcCRH1k9gPap5TubCF5gUIDR16lRzuUOHDsqdO7eF0QDIisaMGaNevXqZ5UWLFunpp59WbGyshVEBAAAgKSQSsrgbN25owYIFZpluDQCs4OHhocmTJ6tLly7mulmzZqlfv36Kj4+3LjAAAADcg0RCFjdnzhxFRkZKkvLkyaN27dpZHBGArMrb21u//vqr2rZta66bOnWqBgwYIMMwLIwMAAAAiTFGQhaXeLaGJ5980pyKDZmfTfEq4HPJLF+OLnifve8vLk7at+/vcrVqqRjrJV0HI6vx9fXV3Llz1a5dO61Zs0aS9L///U82m00TJkyQhwf5bwDIzJKqnxh8t5lpUc1zMhe+wCQSsrATJ06YFXWJbg3uxscjSv1L/s8sf3piSJrPdfOmVKvW3+XQUClXrow4GFmRv7+/Fi5cqDZt2mjz5s2SpO+++04RERH6v/buPC6qcv8D+OcM24Aou+aSEmBaIu6Ka+p1weWikmaluebNay43NZVr3rQ0TZSrubRp6u3Gzdwwck2vgb4M0SwRrmgSLiQoqKSCDNvz+8MfkxPbAGfmzJz5vF8vXnKe55yH73leXzwP3zlzZvPmzbCzkAsoERFVX3nrE12Js4IRUW1wmWdiFjzBLP/ZsM8++0z//TPPPIMuXbooGA0R0e/q1q2L/fv3Izg4WN+2bds2jBkzBoWFhQpGRkREREQsJNiooqIibNmyRb/96quvQpIkBSMiIjLk7u6Ow4cP47nnntO3bd++HSNHjoROp1MwMiIiIiLbxkKCjdq3bx8yMjIAPHrA2bhx4xSOiIiorNI7EwYMGKBv+/rrrzF06FDcv39fwciIiIiIbBcLCTZq06ZN+u9HjBgBb29vBaMhIqqYi4sLvv76a4SGhurbjhw5gl69eukLokRERERkPiwk2KD09HTs379fvz1lyhQFoyEiqpqTkxN27tyJF198Ud/2008/ITg4GBcuXFAwMiIiIiLbw0KCDdqyZQtKSkoAAE899RT69u2rcERERFVzcHDAF198gTlz5ujbrl27hm7duuH48eMKRkZERERkW1hIsDHFxcXYvHmzfnvy5Mn8XHYishoajQarVq3CmjVr9A+IzcnJQf/+/fHvf/9b4eiIiIiIbAP/grQx33zzDa5evQoAsLOzw4QJE5QNiIioBmbNmoUdO3bAyckJAKDT6fDKK69g7ty5KCoqUjg6IiIiInVjIcHGrF27Vv99WFgYGjdurGA0REQ19/zzz+Po0aPw8vLSt61evRqDBw/GnTt3FIyMiIiISN1YSLAh58+fx7Fjx/TbM2fOVDAaIqLa6969O86cOYOgoCB927fffotOnTohKSlJwciIiIiI1Mte6QDIfNatW6f/vn379ujevbuC0ZCpFZY4IDpzmMF2Tbm4AFu2GG6b52Ciqvn6+uLkyZOYOHEiduzYAQD45Zdf0KVLF2zYsAHjx4/XP0+BiIiUJef6hJTHZZ6JWfAEs5BgI27fvm3wILKZM2dyYa1yJbDHufvtZBnL0RGo8eM0anUwkXHq1KmD7du3o127dli4cCGEEMjLy8PEiRNx9OhRbNy4EXXr1lU6TCIimyfn+oSUx2WeiVnwBPOtDTZi8+bNePjwIQDAx8cHo0ePVjgiIiJ5SZKE8PBwfPPNN/D09NS3//vf/0bHjh3x008/KRccERERkYqwkGADCgsLsX79ev32a6+9Bq1Wq2BERESmM3jwYPz0008Gb9+6dOkSgoODERkZieLiYgWjIyIiIrJ+LCTYgKioKFy/fh0AYG9vj6lTpyocERGRaT355JP47rvvsHDhQv3buHQ6HebMmYPevXsjNTVV4QiJiIiIrBefkaByxcXFWL58uX573Lhx/MhHG2EvFWBI/X367X23htR4rLw8YNq037c3bqzGs15qdTBRzdnb22Pp0qXo3bs3xo4di5s3bwIATpw4gaCgIKxcuRJ//etfodGwpk5EZC7lrU+KhKOCEVFtcJlnYhY8wSwkqFx0dDQuXrwI4NH7h+fPn69wRGQudlIx2tY7p98+mBVS47EKCoBt237fXrOmGv+H1epgotrr168fkpKSMH36dGzfvh0AkJeXh+nTp2Pnzp346KOP0KJFC4WjJCKyDeWtT4qEggFRrXCZZ2IWPMF8GUbFhBB477339NujRo3C008/rWBERETK8Pb2xpdffomvvvoKXl5e+vbvvvsOQUFBWLRokf6BtERERERUORYSVOzw4cM4e/asfjs8PFzBaIiIlDdq1CgkJydj+PDh+raCggIsXboUgYGBOHjwoHLBEREREVkJFhJUSgiBZcuW6bcHDx6Mtm3bKhcQEZGFaNCgAXbv3o1du3ahSZMm+vZffvkFgwYNQmhoKFJSUhSMkIiIiMiysZCgUocOHcLx48f12wsXLlQwGiIiyyJJEsLCwnDhwgXMmTMHdnZ2+r6YmBgEBgZi+vTpyMrKUjBKIiIiIsvEQoIKlZSUGDxUsV+/fujWrZuCERERWSZXV1esWrUKZ8+eRY8ePfTtxcXF2LBhAwICArBixQo8ePBAwSiJiIiILAsLCSoUFRWFxMRE/faKFSsUjIaIyPIFBQUhLi4OO3bsgJ+fn7793r17CA8Ph5+fH1atWoW8vDwFoyQiIiKyDCwkqIxOp8Nbb72l337xxRfRoUMHBSMiIrIOkiRh5MiR+N///ofIyEh4eHjo+7KysvDmm2/iqaeeQmRkJHJzcxWMlIiIiEhZLCSozIcffoirV68CAOzt7bF06VKFIyIisi5OTk544403cPnyZcyfPx8uj31e861btzBnzhw0bdoUixYtwq1btxSMlIiIiEgZLCSoSHZ2Nt5991399tSpU+Hv769gRERE1svT0xMrVqxAWloa5s6dC2dnZ33fnTt3sHTpUjRt2hSvvfYaLl68qGCkRERERObFQoKKzJs3D3fu3AHw6AFij7/FgWyRhPxiJ/0XINV8JAlwc/v9S6rOULU6mEh59evXR0REBNLS0jB79mzUqVNH36fT6fDJJ5+gZcuW6NevH3bt2oXCwkIFoyUisnTyrU9IeVzmmZgFT7C90gGQPGJjY7Flyxb99jvvvIMGDRooGBEpTVeixfu/hMsylpsbkJOjxMFElqNBgwZYvXo13nrrLXz88cdYu3YtMjMz9f1Hjx7F0aNH0bBhQ0yZMgUTJ06Er6+vcgETEVkgOdcnpDwu80zMgieYdySogE6nw9SpU/Xbbdu2xYwZMxSMiIhIvTw8PLBgwQJcuXIFmzdvRmBgoEF/RkYG3nnnHTz11FPo06cPtmzZgvv37ysULREREZH8WEhQgYiICKSkpAB49NTxjz/+GPb2vNmEiMiUnJycMGnSJCQmJuL48eN46aWX4ODgYLDPd999h0mTJuGJJ57Ayy+/jN27d/MjJImIiMjqsZBg5c6fP2/wgMVp06ahc+fOCkZERGRbJElCjx49EBUVhfT0dCxfvhzNmzc32CcvLw//+c9/8Pzzz8PHxwejRo3Cl19+yTsViIiIyCqxkGDF8vPz8fLLL6OgoAAA0LBhQyxbtkzhqMhyCDhpHuq/AFHzkcSjt2eVfonqDFWrg4msS/369bFgwQJcvHgR33//PaZOnQp3d3eDffLy8rBz50689NJL8PHxwdChQ7FhwwakpqYqEzQRkVnJtz4h5XGZZ2IWPMG8/92KzZ49G0lJSfrtLVu2wM3NTcGIyJI4afKxwP99/faK1Pk1Huu33wAPj9+3794F/vC3kYkOJrJOkiQhODgYwcHBWLNmDfbt24ddu3YhJibG4C4EnU6Hffv2Yd++fQAAf39/hISEYODAgejTpw9cXV2VOgUiIpMob32iK3Gu5AiyZFzmmZgFTzDvSLBSn3/+OT788EP99syZMzFw4EAFIyIiovI4OTkhLCwMX3zxBbKyshATE4MJEybA4/GFwf9LTU3Fhg0bEBoaCg8PD3Tp0gVz587F3r17cfv2bQWiJyIiIiqLdyRYofj4ePzlL3/Rb3fo0AHvv/9+JUcQEZElcHJywtChQzF06FAUFhYiNjYWBw8exMGDB5GcnGywb1FRERISEpCQkIDVq1cDAFq1aoXu3bujU6dO6NSpE5599tkyD3gkIiIiMjUWEqxMamoqQkNDkZ+fDwDw9PTErl27oNVqFY6MiIiqw8HBAf369UO/fv2watUqpKen49ChQzh06BC+/fZb5JTzudHJyclITk7GJ598AgDQarVo27YtOnXqhPbt26N169Z45pln4OLiYuazISIiIlui2kJCfn4+li9fjv/85z+4du0aPD09ERISgnfeeQdNmjSp1lg5OTlYvHgx9uzZg8zMTDzxxBMYPnw4lixZUuYhWqZ05coV9O3bF1lZWQAeLUJ37dqFZs2amS0GIiIyjSZNmmDy5MmYPHkyiouLcf78ecTFxeH48eM4fvw4bt68WeaY/Px8xMfHIz4+Xt8mSRL8/PwQGBiIVq1aITAwEIGBgQgICICzM9+HTERERLWnykJCfn4+/vSnP+HkyZNo2LAhhg0bhitXrmDLli345ptv8P3338Pf39+osW7fvo2uXbvi559/hp+fH4YPH47k5GR88MEH2L9/P+Lj4+Hl5WXiMwJSUlIQEhKCa9eu6ds2b96M3r17m/xnExGRednZ2aFt27Zo27YtZs6cCSEELl++jLi4OCQkJODMmTNITExEUVFRmWOFEEhNTUVqair27t1r0Ne4cWP4+/sjICAA/v7++i8/Pz94eHhAkiRznSIRERFZMVUWEt577z2cPHkSXbt2xeHDh/VPvY6MjMScOXMwadIkxMbGGjXWG2+8gZ9//hlhYWHYvn077O0fTdnMmTOxbt06zJ49G9u2bTPZuQDA0aNH8cILL+DOnTv6tvXr1+OVV14x6c8lIiLLIEkSmjdvjubNm2Py5MkAHhXNExMTcfr0aZw+fRrnz5/H//73P/1b38rz66+/4tdff0VcXFyZPhcXFzRp0qTCr0aNGsHb2xt2dnYmO08iIiKyDqorJBQWFmLdunUAgA0bNhh8dFbpH/1xcXH44Ycf0KFDh0rHyszMxBdffAEHBwds3LhRX0QAgIiICHz55Zf44osvsHLlSjRo0ED2c8nPz8fSpUvx3nvvQfz/Z4ZKkoS1a9fi9ddfl/3nERGR9dBqtejcuTM6d+6sbysuLkZaWhqSkpKQnJys/zclJQWFhYWVjpeXl4dLly7h0qVLFe4jSRI8PT1Rv359+Pj4wMfHx+B7Hx8feHh4wN3dHW5ubnB3d4e7uzucnJxkO28iIiJSnuoKCSdOnEBOTg78/f3Rrl27Mv0jR45EYmIiYmJiqiwkHDhwACUlJejTp0+ZQoGTkxP+/Oc/47PPPsOBAwcwYcIE2c6hqKgIX331Fd5++21cvnzZ4Gd+/vnnGDVqlGw/i4iI1MPOzg4BAQEICAjA8OHD9e3FxcW4fv06Ll++jNTUVP2/pd/n5eUZNb4QArdv38bt27dx4cIFo+PSarVligv16tWDq6srXF1dUbduXf33rq6uePjwIZ/nQEREZMFUV0g4d+4cAKB9+/bl9pe2l+5X27E+++wzo8aqSl5eHqKionDy5Ens3r0bGRkZBv0BAQH46quvyi2OEBERVcbOzg6+vr7w9fVFv379DPqEEMjOzsavv/6K9PT0cr+uX79udLGhPPn5+cjMzERmZqZR+3t4eLCQQEREZMFUV0gofRhhRZ/MUNr++EMLzTEW8Ojzv8uTkpKCkpISjBkzptx+T09P2NvbY+zYsUb9HCIAyM66ic8dH9su2IRtUXuh0WiqPVZJieF2166A0cPU6mAiUlppEaKkpARFRUUoLi4u99/S70tKSlBcXIzi4uIa/8x79+5V+VaMylR2vXVwcKiwn4hq7/79+5U+q0VCSZn1iUDF6wKtVouFCxfKGSLJiMs8EzPxBKempsLBwaFGx6qukPDgwQMAqPAztOvUqWOwn7nGqkxJSQk0Gg1atmxZq3FsQWpqKgAY/akbtkxAg4uZjx6K5unpWauxNBrg2WeVONi8mF/G41xVjxrmS6PRwNHRseodZZCSkoKHDx/KPq4kSTVeMClJDfljDTjP5iHn+oSqZuq8tqJlnkmZbJ5NPMEODg76v2mrS3WFhMcfSlhZv7nHAoDk5ORy20tfGamon37Huaoezlf1cL6Mx7mqHs5X9dT2jgG1zTPzxzw4z+bDuTYfzrV52OI8q+7Gk7p16wIAcnNzy+0vfY/n45/mYI6xiIiIiIiIiNRAdYWEpk2bAgDS09PL7S9tL93PXGMRERERERERqYHqCglt2rQBAJw9e7bc/tL2oKAgs45FREREREREpAaqKyR0794dbm5uSE1NxY8//limf+fOnQCAoUOHVjlWSEgINBoNjh8/jlu3bhn06XQ6xMTEQKPRYNCgQfIET0RERERERGThVFdIcHR0xPTp0wEA06dPN3i+QWRkJBITE9GjRw906tRJ375+/Xq0bNkS4eHhBmM1bNgQL730EgoKCjBt2jQUFRXp++bNm4esrCy8/PLLeOKJJ0x8VkRERERERESWQRLV/egBK5Cfn4/evXvj1KlTaNiwIXr27ImrV6/i1KlT8PLyQnx8PAICAvT7L168GEuWLMH48eOxdetWg7Gys7MRHByM1NRU+Pv7o2PHjkhOTkZSUhL8/f0RHx8Pb29vM58hERERERERkTJUd0cCAGi1Whw7dgyLFi2Ci4sLoqOjceXKFYwfPx4//vijQRGhKt7e3jh9+jRmzJiBgoIC7NmzB7/99humT5+OhIQEFhGIiIiIiIjIpqjyjgQiIiIiIiIiMg1V3pFARERERERERKbBQgIRERERERERGY2FBCIiIiIiIiIyGgsJRERERERERGQ0FhKIiIiIiIiIyGgsJJhAfn4+3n77bTz99NPQarVo1KgRJk2ahPT09GqPlZOTg7/97W9o1qwZnJyc0KxZM8yaNQs5OTnyB64AuebK19cXkiRV+JWSkmKiMzCfH374AStWrEBYWBgaN24MSZKg1WprPJ6ac0vOuVJ7buXl5SE6OhqTJ09GUFAQ6tWrhzp16qBNmzZ455138ODBg2qPqebcknu+1J5fABAZGYmwsDA0b94cbm5u+pwYP348kpOTqz2eteeXnGuEgwcPYtCgQfD29oaDgwPq16+PoUOH4ujRoyaI3PrIOdcHDhxA//794e7uDhcXF7Ru3RoREREoKioyQeTWhesT85BznmNjY7FkyRIMGTIEPj4+kCQJLVu2lDli6yXXXOfk5CAqKgovv/wynn32WdSpUwd169ZFly5dsHbtWhQWFpogejMSJKuHDx+Kbt26CQCiYcOG4oUXXhCdO3cWAISPj4+4fPmy0WNlZ2eL5s2bCwDCz89PvPDCC6JVq1YCgAgICBDZ2dkmPBPTk3OumjVrJgCI8ePHl/t148YNE56JeQwbNkwAMPhycnKq0Vhqzy0550rtufXpp5/q56hVq1Zi1KhRYuDAgaJu3boCgGjZsqW4efOm0eOpPbfkni+155cQQnh5eQmtVis6d+4sRowYIUaMGCGefvppAUA4OjqK/fv3Gz2WteeXnNe91atXCwBCkiTRo0cPMXr0aNGpUyd9fn744YcmPBPLJ+dcr1ixQgAQGo1GdO3aVYSGhor69esLAGLAgAGisLDQhGdi+bg+MQ8557lNmzZlxmrRooXMEVsvueZ64cKF+v87OnToIEaPHi369u0rnJycBADRo0cPkZuba4IzMA8WEmS2aNEiAUB07dpV3L9/X99eesHv1auX0WO98sorAoAICwszuEjNmDFDABDjxo2TNXZzk3OuShfjarZixQrxj3/8Q8TExIjMzMxaXUDUnltyzpXac2vbtm3ir3/9q7h06ZJB+40bN0S7du0EAPHSSy8ZPZ7ac0vu+VJ7fgkhxIkTJ8TDhw/LtG/cuFEAEI0aNRJFRUVGjWXt+SXXde/WrVvC0dFRODo6iuPHjxv07dy5U0iSJFxcXAx+hq2Ra64TEhKEJEnCwcFBHDp0SN+ek5MjevXqJQCIFStWyB6/NeH6xDzknOc333xTLFu2TBw+fFicPXuWhYQ/kGuuly9fLv7+97+L9PR0g/ZLly6Jpk2bCgAiPDxcrrDNTt2rFzMrKCgQ7u7uAoA4e/Zsmf6goCABQJw5c6bKsTIyMoRGoxEODg4iMzPToC8/P1/4+PgIOzu7Mn3WQs65EsI2FuN/VNP/1NSeW+VhIaFmTp48qZ87nU5X5f62mFuPq+58CWHb+SWEEAEBAQKASE5OrnJfa88vOa97MTExAoAICQkpt7/01cZTp07VOm5rJOdcT548WQAQU6ZMKdOXlJSkv8PB2GKYLeD6xDxqs7Z5XFpaGgsJVZBrrh8XFRUlAAhfX19ZxzUnPiNBRidOnEBOTg78/f3Rrl27Mv0jR44EAMTExFQ51oEDB1BSUoJevXqhQYMGBn1OTk7485//jOLiYhw4cECe4M1Mzrmi6lF7bpF82rRpAwDQ6XS4fft2lfvbem5Vd74IsLOzAwA4OjpWua+155ec1z0nJyejfqanp2f1glQJOef6hx9+AAD07t27TF+rVq3g7e2NrKwsnDx5snZBk9X/jhNVR+ma4caNGwpHUnMsJMjo3LlzAID27duX21/aXrqfucayRKY6v4iICEydOhWzZs3CJ598gqysrNoFqkJqzy1TscXc+uWXXwAADg4ORv1BYuu5Vd35epwt5te//vUvXLx4EU8//TT8/Pyq3N/a80vO+Dt16gQ3Nzf897//xYkTJwz6du/ejcTERHTr1g0BAQG1jNo6yTnXubm5AAAPD49y+0t/1y0176yJtf+OE1VH6ZrhiSeeUDiSmrNXOgA1uXbtGgCgSZMm5faXtpfuZ66xLJGpzm/evHkG22+88QY++OADTJ48uQZRqpPac8tUbDG31q5dCwAICQkx6hVQW8+t6s7X42whvyIiIpCcnIzc3FxcuHABycnJaNSoEaKioqDRVP26hrXnl5zxu7u7Y9OmTRgzZgx69eqF7t27o3HjxkhLS8Pp06cREhKCrVu3yha7tZFzrn18fPDzzz/j6tWrZfpKSkpw/fp1AMCVK1dqGC2VsvbfcaLqKF0zDBs2TOFIao53JMio9GO/XFxcyu2vU6eOwX7mGssSyX1+oaGh2L17N65evYq8vDwkJSVh9uzZ0Ol0ePXVVxEdHS1L3Gqg9tySm63m1v79+7F582Y4ODjg3XffNeoYW86tmswXYFv5dejQIWzbtg07d+5EcnIynnzySURFRaFDhw5GHW/t+SV3/CNHjsSBAwfg5eWFEydOYPv27UhISED9+vXRt29feHl5yRO4FZJzrp977jkAwLZt28r0bd++HQ8fPgQA3L9/v0ax0u+s/XecyFgfffQRjhw5And3dyxYsEDpcGqMhQQZCSEAAJIkVdpv7rEskdzn98EHH2DEiBFo2rQpnJ2d0apVK6xevRobN24EAMyfP792AauI2nNLbraYWxcuXMDYsWMhhEBERIT+fXxVsdXcqul8AbaVX0eOHIEQAnfv3kVcXBxatGiB3r17Y9myZUYdb+35JXf8q1evRv/+/dGrVy8kJibiwYMHSExMRNeuXfHmm29i9OjRtY7ZWsk516+//jrc3NwQHx+PCRMm4PLly8jJycH27dvx+uuvw97+0c29xtxVQ5Wz9t9xImPExsZi1qxZkCQJn332GRo1aqR0SDXG//VkVLduXQC/v5/uj/Ly8gAArq6uZh3LEpnr/F599VXUr18fly5dQlpaWq3GUgu155a5qDW30tPTERISgrt372L27NmYNWuW0cfaYm7VZr4qo9b8Ah7dlt+zZ0/s378fHTp0wKJFi3D69Okqj7P2/JIz/tjYWMydOxdt27bFjh070Lp1a9SpUwetW7fGzp070a5dO+zatQuHDx+W7wSsiJxz3bhxY+zZsweenp7Ytm0bmjdvDg8PD7z44ot48sknMWnSJAAVP0OBjGftv+NEVUlMTMTw4cNRUFCAtWvXYsSIEUqHVCssJMioadOmAB4tLMtT2l66n7nGskTmOj+NRgN/f38AQEZGRq3GUgu155a5qDG3srOz0b9/f1y7dg0TJ07EqlWrqnW8reVWbeerMmrMrz9ycHDA6NGjIYQw6un51p5fcsb/r3/9CwAQFhZW5pVwOzs7hIWFAQC+++67moZr1eTOlT59+iA1NRUfffQRpk2bhunTp+Pzzz9HQkICcnJyADz6BAeqHWv/HSeqTGpqKgYOHIicnBwsXrwYM2bMUDqkWuPDFmVUejvr2bNny+0vbQ8KCjLrWJbInOd39+5dAKxgl1J7bpmTmnLr/v37GDRoEFJSUhAWFoZPP/20wttLK2JLuSXHfFVFTflVEW9vbwAw6lMqrD2/5Iy/9A+qevXqldtf2n7nzp1qx6kGpsgVd3d3vPbaawZtRUVFiI2NhUajQa9evWoYLZWy9t9xoorcuHED/fv3R2ZmJmbNmoW3335b6ZDkIUg2Op1OuLm5CQDi7NmzZfqDgoIEAJGQkFDlWDdu3BAajUY4OjqKmzdvGvTl5+cLHx8fodFoREZGhmzxm5Occ1WZpKQkIUmScHFxETqdrlZjWRoAwsnJqdrHqT23ylPTuaqMmnIrPz9f9OnTRwAQAwcOrPH52EpuyTVflVFTflVm/PjxAoCIiIiocl9rzy85r3vjxo0TAMS4cePK7R87dqwAIJYvX17ruK2RudYYW7duFQDE4MGDazWO2nB9Yh5yrW3S0tIEANGiRQsZolKn2sz1nTt3RGBgoAAgJk6cKEpKSmSOTjksJMhs4cKFAoDo1q2bePDggb599erVAoDo0aOHwf7r1q0TLVq0EAsWLCgz1pgxYwQA8fzzz4vCwkJ9+8yZMwUAMXbsWNOdiBnINVcHDx4UZ86cKTP+uXPnxDPPPCMAiJkzZ5rmJBRU1X9qtpxbf1TTubKF3CoqKhIjRowQAETPnj1Fbm5ulcfYcm7JOV+2kF9xcXHiyy+/NMgFIYQoKCgQH3zwgdBoNMLZ2Vlcu3ZN36fm/JLrurd7924BQNjZ2Ymvv/7aoC86OlpoNBqh0WhESkqK6U7Gwsm5Hjtz5kyZxf/hw4dF3bp1hVartel5Lg/XJ+ZRm3l+HAsJVavpXOfm5org4GABQLzwwguiqKjI1KGaFd/aILO33noLR44cwcmTJ9G8eXP07NkTV69exalTp+Dl5YUtW7YY7J+dnY2LFy+W+x7YNWvWID4+Hrt27ULLli3RsWNHJCcnIykpCf7+/vjnP/9prtMyCbnm6vvvv8eSJUvQrFkz+Pv7w8fHB2lpaTh79iyKiorw3HPPYfny5eY8NZPYt29fmY+VKygoQHBwsH570aJFGDJkCADbzi255soWcmv9+vXYs2cPgEe3mU+bNq3c/VatWqW/Dd2Wc0vO+bKF/EpNTcXEiRPh7e2NDh06wMvLC9nZ2Th//jwyMjKg1WqxdetWPPnkk/pj1Jxfcl33hg8fjlGjRmHHjh0IDQ1Fx44d8dRTTyEtLQ1nzpwBACxbtgwtWrQw27lZGjnXY88//zyKi4vRunVruLm54eLFi/jxxx/h7OyMnTt32vQ8A1yfmIuc87xp0yZs2rQJAKDT6QAAV69eNRhr48aNaN++veznYQ3kmuuFCxciPj4ednZ2sLe3x+TJk8v9eVu3bpX3BMyEhQSZabVaHDt2DMuXL0dUVBSio6Ph4eGB8ePH49133zVYLFXF29sbp0+fxttvv43o6Gjs2bMHDRo0wPTp07FkyRJ4enqa8ExMT665GjhwIK5fv47Tp0/j3Llz+O2331CvXj306NEDY8aMwcSJE2FnZ2fiszG9rKwsnDp1yqBNCGHQZsz7jAH155Zcc2ULuVX6PnwA+j+Qy7N48WL9H8aVUXtuyTlftpBfzz33HP7+978jNjYWiYmJyM7OhqOjI3x9fTFy5EjMnDkTAQEBRo9n7fkl13VPkiRs374dISEh2LZtGxITE/HTTz/B3d0dgwcPxowZMxASEmLis7Fscq7Hpk6diujoaJw6dQoPHjxAw4YN8Ze//AXz58+Hn5+fCc/COnB9Yh5yznN6enqZsfLz8w3a7t27V4torZtcc126ZiguLkZUVFSF+1lrIUESgh/KSkRERERERETG4cc/EhEREREREZHRWEggIiIiIiIiIqOxkEBERERERERERmMhgYiIiIiIiIiMxkICERERERERERmNhQQiIiIiIiIiMhoLCURERERERERkNBYSiIiIiIiIiMhoLCQQERERERERkdFYSCAiIiIiIiIio7GQQERERERERERGYyGBiIiIiIiIiIzGQgIRERERERERGY2FBCJSrbZt20KSJBw9elTpUIiIiFSL11si2yMJIYTSQRARye3q1avw9fWFu7s7bt26BQcHB6VDIiIiUh1eb4lsE+9IICJV2rt3LwBg8ODBXNQQERGZCK+3RLaJhQQiUqXShc2wYcMUjoSIiEi9eL0lsk0sJBCRRbl27RokSYIkSTh27Fil+77xxhuQJAnt27fH4+/Sunv3LuLi4uDo6IiQkBBZxyYiIlIDXm+JqDZYSCAii9K0aVO4u7sDAJKTkyvcLy0tDRs3bgQArFq1CpIk6fv27duHoqIi9O3bF/Xq1ZN1bCIiIjXg9ZaIaoOFBCKyOK1btwZQ+eIjPDwcBQUFGDJkCPr27WvQV9ltlrUdm4iISC14vSWimmIhgYgsTlBQEICKFx8JCQn46quvYGdnh5UrVxr06XQ6HDx4EJIkITQ0VNaxiYiI1ITXWyKqKRYSiMjiVPUqxptvvgkhBF599VU8++yzBn3//e9/8eDBA3Ts2BGNGjWSdWwiIiI14fWWiGqKhQQisjilr2LcuXMHmZmZBn1ff/014uLi4OrqiiVLlpQ5Njo6GkDFT4+uzdhERERqwustEdUUCwlEZHFat26tf+DS469kFBcXY/78+QCA+fPno0GDBgbHCSEQExMDABg+fLisYxMREakNr7dEVFMsJBCRxXF1dYWvry8Aw8XHp59+ipSUFDRu3BizZ88uc1xCQgIyMjLg7++PVq1ayTo2ERGR2vB6S0Q1xUICEVmkPz6k6cGDB1i8eDEAYOnSpXBxcSlzTGVPj67t2Fu3boUkSThy5AjeeustNG7cGK6urhgyZAhu3LgBAIiMjISfnx+0Wi26d++OCxculDvGt99+iwULFqBRo0ZwdnZGz549cfr0aWOmhYiISFa83hJRTdgrHQARUXmCgoKwd+9e/eIjIiICN2/eRJs2bTBu3Lhyj6nq/Zq1GbvU/Pnz4ezsjAULFiA9PR2RkZEYNWoUBgwYgN27d2PGjBn47bffsHLlSowaNQrnz58v87nY8+bNQ0lJCebOnYt79+5h/fr16Nu3L06dOsUHThERkVnxektENcFCAhFZpMdfxcjIyMDq1asBAKtWrYJGU/ZmqsuXL+PChQvw9vZG9+7dZR37cY6OjoiNjYWdnR0AoKCgAGvWrEFmZiaSkpLg7OwMANBqtQgPD8f333+Pbt26GYxx7949nDt3Dq6urgCAkSNHok2bNggPD9e/ykNERGQOvN4SUU3wrQ1EZJFKPzYqJycHU6ZMQW5uLgYNGoR+/fqVu3/pqyNDhw7VLzrkGvtxU6ZMMRi/S5cuAICxY8fqFzUAEBwcDABITU0tM8Zrr72mX9QAQGBgIAYOHIhDhw6hsLCwyhiIiIjkwustEdUECwlEZJGaN2+uXyjs27cPdnZ2iIiIqHB/Y9+vWZOxH9esWTODbXd3dwBA06ZNy22/c+dOmTFatmxZpq1FixbQ6XT6938SERGZA6+3RFQTLCQQkUXSaDQGT4KeNGlShU+GzsrKwsmTJ+Hs7IwBAwbIOvYfVfTqS0XtQgijxiUiIlICr7dEVBN8RgIRWSxjn6wcExODkpIS9OvXr9wnQNdmbFNISUlBaGioQdvFixfh5OSEhg0bKhQVERHZKl5viai6eEcCEVm90tsshw8frmwgRvr444+Rm5ur305KSsKhQ4cwYMAAODo6KhgZERFRxXi9JaJSvCOBiKxe9+7d0a5dO6Per2kJ6tWrh27dumHChAm4d+8e1q1bB61Wi/fee0/p0IiIiCrE6y0RlWIhgYis3rx585QOoVref/99HD16FCtXrsTdu3fRoUMHREZGIjAwUOnQiIiIKsTrLRGVkgSfTEJEZBZbt27FxIkTcezYMfTu3VvpcIiIiFSJ11si0+MzEoiIiIiIiIjIaCwkEBEREREREZHRWEggIiIiIiIiIqPxGQlEREREREREZDTekUBERERERERERmMhgYiIiIiIiIiMxkICERERERERERmNhQQiIiIiIiIiMhoLCURERERERERkNBYSiIiIiIiIiMhoLCQQERERERERkdFYSCAiIiIiIiIio7GQQERERERERERGYyGBiIiIiIiIiIzGQgIRERERERERGY2FBCIiIiIiIiIyGgsJRERERERERGQ0FhKIiIiIiIiIyGgsJBARERERERGR0VhIICIiIiIiIiKjsZBAREREREREREb7P+7yTeUGjYmPAAAAAElFTkSuQmCC", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "import numpy as np \n", + "import matplotlib.pyplot as plt \n", + "from scipy.integrate import simpson\n", + "\n", + "def Maxwell_speed(g):\n", + " #g = v/v_mp or v = g*v_mp\n", + " return 4/np.sqrt(np.pi)*g**2*np.exp(-g**2)\n", + "\n", + "v_rng = np.arange(0.99,1.01,0.0001)\n", + "prob = simpson(Maxwell_speed(v_rng),v_rng)\n", + "print(\"The fraction of molecules with speeds within 1 percent of v_mp is %1.4f.\" % prob)\n", + "\n", + "fig = plt.figure(figsize=(8,4),dpi=150)\n", + "ax1 = fig.add_subplot(121)\n", + "ax2 = fig.add_subplot(122)\n", + "ax_list = [ax1,ax2]\n", + "\n", + "g = np.arange(0, 3,0.01)\n", + "for i in range(0,2):\n", + " ax = ax_list[i]\n", + " ax.plot(g,Maxwell_speed(g),'k-',lw=1.5)\n", + " ax.axvline(2/np.sqrt(np.pi),0,Maxwell_speed(2/np.sqrt(np.pi)),color='b',linestyle='--',label='$\\\\overline{v}$')\n", + " ax.axvline(np.sqrt(1.5),0,Maxwell_speed(np.sqrt(1.5)),color='r',linestyle='--',label='$v_{\\\\rm rms}$')\n", + " ax.axvline(1,0,Maxwell_speed(1),color='orange',linestyle='--',label='$v_{\\\\rm mp}$')\n", + " ax.fill_between(v_rng,0,Maxwell_speed(v_rng),color='gray')\n", + "\n", + " ax.legend(loc='best',ncols=3,fontsize='large')\n", + " \n", + " ax.set_xlabel(\"$v/v_{\\\\rm mp}$\",fontsize='large')\n", + " ax.set_ylim(0.,1)\n", + "\n", + "ax1.set_ylabel(\"$F(v/v_{\\\\rm mp})$\")\n", + "ax1.set_xlim(0,3)\n", + "ax2.set_xlim(0.75,1.25)\n", + "ax2.set_title(\"Zoomed $x$\")\n", + "ax2.set_yticklabels([])\n", + "\n", + "fig.subplots_adjust(wspace=0.05);\n" ] }, { diff --git a/docs/Chapter_8/statistical-physics.html b/docs/Chapter_8/statistical-physics.html index f9a0e65..dbcc7b2 100644 --- a/docs/Chapter_8/statistical-physics.html +++ b/docs/Chapter_8/statistical-physics.html @@ -537,18 +537,18 @@

8.2. Maxwell Velocity Distribution

where \(d^3\vec{v} = dv_x\ dv_y\ dv_z\), or a volume element in the phase space. The product of a velocity distribution funciton with a phase space volume plays a role analogous to the probability density \(\Psi^*\Psi\) in quantum theory.

Maxwell proved that the probability distribution funciton is proportional to \(e^{-mv^2/(2kT)}\), where \(m\) is the molecular mass, \(v\) is the molecular speed, \(k\) is Boltzmann’s constant, and \(T\) is the absolute temperature. Then, we may write

-
-(8.1)#\[\begin{align} +
+(8.1)#\[\begin{align} f(\vec{v})\ d^3\vec{v} = Ce^{-\frac{1}{2}\beta mv^2}\ d^3\vec{v}, \end{align}\]

where \(C\) is aproportionality factor and \(\beta \equiv (kT)^{-1}\); not to be confused with \(\beta = v/c\) from relativity. We can expand the above expression as

-
-(8.2)#\[\begin{align} +
+(8.2)#\[\begin{align} f(\vec{v})\ d^3\vec{v} = Ce^{-\frac{1}{2}\beta m(v_x^2+v_y^2+v_z^2)}\ d^3\vec{v}, \end{align}\]

and use the properties of exponents to rewrite as three factors that each contain one of the three velocity components. They are defined as

-
-(8.3)#\[\begin{align} +
+(8.3)#\[\begin{align} g(v_x)\ dv_x &\equiv C^\prime e^{-\frac{1}{2}\beta mv_x^2}\ dv_x, \\ g(v_y)\ dv_y &\equiv C^\prime e^{-\frac{1}{2}\beta mv_y^2}\ dv_y, \\ g(v_z)\ dv_z &\equiv C^\prime e^{-\frac{1}{2}\beta mv_z^2}\ dv_z, @@ -595,24 +595,24 @@

8.2. Maxwell Velocity Distribution

Recalling that \(a=\beta m/2\), we have

-
-(8.6)#\[\begin{align} +
+(8.6)#\[\begin{align} C^\prime &= \frac{1}{2}\sqrt{\frac{4a}{\pi}}, \\ &= \frac{1}{2}\sqrt{\frac{2\beta m}{\pi}} = \sqrt{\frac{\beta m}{2\pi}}, \end{align}\]

and

-
-(8.7)#\[\begin{align} +
+(8.7)#\[\begin{align} g(v_x)\ dv_x = \sqrt{\frac{\beta m}{2\pi}} e^{-\frac{1}{2}\beta mv_x^2}\ dv_x. \end{align}\]

With this distribution we can calculate \(\bar{v}_x\) (mean value of \(v_x\)) as

-
-(8.8)#\[\begin{align} +
+(8.8)#\[\begin{align} \bar{v}_x = \int_{-\infty}^\infty v_x\ g(v_x)\ dv_x = C^\prime \int_{-\infty}^\infty v_x e^{-\frac{1}{2}\beta mv_x^2}\ dv_x = 0 \end{align}\]

because \(v_x\) is an odd function. The result makes sense because in a random distribution of velocities one woul expect the velocity components to be evenly distributed about \(v_x = 0\). The mean value of \(v_x^2\) is

-
-(8.9)#\[\begin{align} +
+(8.9)#\[\begin{align} \bar{v}_x^2 &= \int_{-\infty}^\infty v_x^2\ g(v_x)\ dv_x, \\ &= 2C^\prime \int_{0}^\infty v_x^2 e^{-\frac{1}{2}\beta mv_x^2}\ dv_x. \end{align}\]
@@ -627,14 +627,14 @@

8.2. Maxwell Velocity Distribution

Alternatively, one could use an integral table with exponential functions. The final result via substitution is

-
-(8.10)#\[\begin{align} +
+(8.10)#\[\begin{align} \bar{v}_x^2 &= 2C^\prime I_2 = 2 \left(\frac{\beta m}{2\pi}\right)^{1/2} \frac{\sqrt{\pi}}{4}\left(\frac{2}{\beta m}\right)^{3/2}, \\ &= \frac{1}{\beta m} = \frac{kT}{m}. \end{align}\]

There’s nothing special about the \(x\)-direction, so the results for the \(x\)-, \(y\)-, and \(z\)-velocity components are identical. The three components can be summed to find the mean translational kinetic energy \(\overline{K}\) of a molecule:

-
-(8.11)#\[\begin{align} +
+(8.11)#\[\begin{align} \overline{K} = \frac{1}{2}m \left(\bar{v}_x^2 + \bar{v}_y^2 + \bar{v}_z^2\right) = \frac{1}{2}m \left(\frac{3kT}{m}\right) = \frac{3}{2}kT. \end{align}\]

This is one of the principal results of kinetic theory.

@@ -692,7 +692,7 @@

8.3. Equipartition Theorem\(\frac{3}{2}kT\). In a monatomic gas (e.g., \(\rm He\) or \(\rm Ar\)), virtually all of the energy is in the translational kinetic energy.

Consider a diatomic gas (e.g., \(\rm O_2\)) as two atoms connected by a massless rod (i.e., a rigid rotator). Then the molecule can have rotational kinetic energy in addtion to translational kinetic energy. The crucial questions are:

    @@ -713,7 +713,7 @@

    8.3. Equipartition Theorem\((K_x\ \text{or}\ K_y)\) is quadratic in angular velocity, so the equipartition theorem instructs us to add \(2\left(\frac{1}{2}kT\right) = kT\) per molecule to the translational kinetic energy for a total of \(\frac{5}{2}kT\).

    Sometimes it is a better approximation to think of atoms connected by a massless spring rather than a rigid rod. In this model, we use the potential energy of a spring but in radial coordinates:

    @@ -828,6 +828,260 @@

    8.3. Equipartition Theorem

    8.4. Maxwell Speed Distribution#

    +

    The general form of the Maxwell velocity distribution is given as

    +
    +\[\begin{align*} +f(\vec{v})\ d^3\vec{v} = C e^{-\frac{1}{2}\beta m v^2}\ d^3\vec{v}, +\end{align*}\]
    +

    where \(C = {C^\prime}^3 = \left(\frac{\beta m}{2\pi}\right)^{3/2}\).

    +

    The distribution \(f(\vec{v})\) is a function of the speed \(v\) in the exponent and not the velocity \(\vec{v}\), but it is still a velocity distribution due to the phase space volume element \(d^3\vec{v}\). We can turn this velocity distribution into a speed distribution \(F(v)\) using the definition

    +
    +\[\begin{align*} +F(v)\ dv =& \text{ probability of finding a particle } \\ +& \text{ with speed between } v\ \text{and}\ v + dv. +\end{align*}\]
    +

    Consider the analogous problem in 3-D position space \((x,\ y,\ z)\). Some distribution of particles exist, where a particle can be located with \((x,\ y,\ z)\) with a distance \(r = \sqrt{x^2+ y^2 + z^2}\) from the origin and has a position vector \(\vec{r}\) measured relative to the origin. Then

    +
    +\[\begin{align*} +f(x,\ y,\ z)\ d^3\vec{r} = & \text{ probability of finding a particle} \\ +& \text{ between }\vec{r}\ \text{and}\ \vec{r} + d^3\vec{r}, +\end{align*}\]
    +

    with \(d^3\vec{r} = dx\ dy\ dz\). To shift to the scalar radial distribution we introduce the definition

    +
    +\[\begin{align*} +F(r)dr = & \text{ probability of finding a particle} \\ +& \text{ between }r\ \text{and}\ r + dr. +\end{align*}\]
    +

    The space between \(r\) and \(r+dr\) is a spherical shell, and thus, we must integrate over that volume to transform to the radial distribution. These two distributions are related by the volume of spherical shell, or \(4\pi r^2\). We may write

    +
    +(8.12)#\[\begin{align} +F(r)dr = f(x,\ y,\ z) 4\pi r^2\ dr. +\end{align}\]
    +

    Returning to the problem of the speed distribution \(F(v)\) from the velocity distribution \(f(\vec{v})\). We need to integrate over a spherical shell in the velocity phase space, where we replace \(r\rightarrow v\) from the previous example. The desired speed distribution is

    + +
    +(8.13)#\[\begin{align} +F(v)\ dv = 4\pi Ce^{-\frac{1}{2}\beta m v^2}v^2\ dv. +\end{align}\]
    +

    The Maxwell speed distribution as derived from purely classical considerations, where it gives a nonzero probability of finding a particle with a speed greater than \(c\). Therefore, it is only valid in the classical limit.

    +

    The assymetry of the distribution curve leads to an interesting result:

    +
    +

    the most probable speed \(v_{\rm mp}\), the mean speed \(\bar{v}\), and the root-mean-square speed \(v_{\rm rms}\) are all slightly different from each other.

    +
    +

    To find the most probable speed, we simply find maximum speed in the probabilty curve, or

    +
    +(8.14)#\[\begin{align} +\frac{dF}{dv} &= 4\pi C \frac{d}{dv}\left[e^{-\frac{1}{2}\beta m v^2}v^2 \right], \\ +&= 2ve^{-\frac{1}{2}\beta m v^2} - \left(\beta m v\right)v^2 e^{-\frac{1}{2}\beta m v^2}. +\end{align}\]
    +

    Replacing \(v\rightarrow v_{\rm mp}\) to denote the most probable speed and setting the above result equal to zero, we can solve for \(v_{\rm mp}\):

    + +
    +(8.15)#\[\begin{align} +2v_{\rm mp}e^{-\frac{1}{2}\beta m v_{\rm mp}^2} &= \left(\beta m v_{\rm mp}\right)v_{\rm mp}^2 e^{-\frac{1}{2}\beta m v_{\rm mp}^2}, \\ +v_{\rm mp} &= \sqrt{\frac{2}{\beta m}} = \sqrt{\frac{2kT}{m}}. +\end{align}\]
    +

    A particle moving at the most probable speed has a kinetic energy \(K_{\rm mp} = \frac{1}{2}mv_{\rm mp}^2 = kT\).

    +

    The mean speed is found by integrating (summing up) the probabilities for individual speeds, or

    +
    +\[\begin{align*} +\bar{v} = \int_0^\infty vF(v)\ dv = 4\pi C \int_0^\infty v^3e^{-\frac{1}{2}\beta m v^2}dv. +\end{align*}\]
    +

    See a list of integrals containing exponential functions and using \((k=1,\ n=3,\ a=\beta m/2)\) to find

    + +
    +\[\begin{align*} +\bar{v} &= 4\pi C \left(\frac{1}{2(\beta m/2)^2}\right) = 8\pi \left(\frac{\beta m}{2\pi}\right)^{3/2} \left(\frac{1}{(\beta m)^2}\right), \\ + &= \frac{4}{\sqrt{2\pi}} \left(\frac{1}{\sqrt{\beta m}}\right), \\ &= \frac{2}{\sqrt{\pi}} \sqrt{\frac{2kT}{m}}, \\ + &= \frac{2}{\sqrt{\pi}} v_{\rm mp}. +\end{align*}\]
    +

    We define the root-mean-square \(v_{\rm rms}\) to be \( v_{\rm rms} \equiv \left(\overline{v^2}\right)^{1/2}\). We cannot simply use the result for the mean speed \(\bar{v}\) directly. Instead, we need to find the mean using the probability function \(F(v)\), or

    +
    +\[\begin{align*} +\overline{v^2} = \int_0^\infty v^2F(v)\ dv = 4\pi C \int_0^\infty v^4e^{-\frac{1}{2}\beta m v^2}dv. +\end{align*}\]
    +

    See a list of integrals containing exponential functions and using \((k=2,\ n=4,\ a=\beta m/2)\) to find

    +
    +(8.16)#\[\begin{align} +\overline{v^2} &= 4\pi C \int_0^\infty v^4e^{-\frac{1}{2}\beta m v^2}dv = 4\pi C \left(\frac{3!!}{8a^2}\sqrt{\frac{\pi}{a}}\right), \\ +&= 4\pi \left(\frac{\beta m}{2\pi}\right)^{3/2} \frac{3}{2(\beta m)^2} \sqrt{\frac{2\pi}{\beta m}}, \\ +&= \frac{3(\beta m)^{3/2}}{(\beta m)^{5/2} } = \frac{3}{\beta m}, \\ +&= \frac{3kT}{m}. +\end{align}\]
    +

    Then

    + +
    +(8.17)#\[\begin{align} +v_{\rm rms} = \left(\overline{v^2}\right)^{1/2} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3}{2}}v_{\rm mp}. +\end{align}\]
    +

    A particle moving with the mean squared speed has a kinetic energy

    +
    +\[ \overline{K} = \frac{1}{2}m\overline{v^2} = \frac{1}{2}m \left(\frac{3kT}{m}\right) = \frac{3}{2}kT \]
    +

    in keeping with our basic law of kinetic theory.

    +

    The standard deviation of the molecular speeds \(\sigma_v\) is

    +
    +(8.18)#\[\begin{align} +\sigma_v &= \left(\overline{v^2} - \bar{v}^2\right)^{1/2} = \left(\frac{3kT}{m} - \frac{8kT}{m\pi}\right)^{1/2}, \\ +&= \left[\frac{(3\pi-8)kT}{m\pi}\right]^{1/2},\\ +&= \left[3-\left(\frac{8}{\pi}\right)\right]^{1/2} \sqrt{\frac{kT}{m}}, \\ +&= \left[\frac{3}{2}-\left(\frac{8}{2\pi}\right)\right]^{1/2} v_{\rm mp}. +\end{align}\]
    +
    + +

    Exercise 8.3

    +
    +

    Computre the mean molecular speed \(\bar{v}\) in hydrogen \(({\rm H_2})\) and radon \(({\rm Rn})\) gas, both at room temperature \(293\ {\rm K}\). Use the longest-lived radon isotope, which has a molar mass of \(222\ {\rm u}\). Compare the results.

    +

    The mean molecular speed is given by

    +
    +\[ \bar{v} = \frac{2}{\sqrt{\pi}} \sqrt{\frac{2kT}{m}}, \]
    +

    which can be applied to each type of gas. The only difference between results will be due to the molar mass \(m\). The molar mass of \(\rm H_2\) is \(2.01568\ {\rm u}\), which is found by multiplying the mass of hydrogen by two. The conversion from atomic mass units \({\rm u}\) to \(\rm kg\) is \(1.660539 \times 10^{-27}\ {\rm kg/u}\). Let’s re-write the mean molecular speed so that all the physical constants are combined and so that we can use atomic mass units directly. This gives

    +
    +(8.19)#\[\begin{align} +\bar{v} = \left(2\sqrt{\frac{2k}{\pi u}}\right)\sqrt{\frac{T\ \text{(in K)}}{m\ \text{(in u)}}} = 145.5081\sqrt{\frac{T}{m}}. +\end{align}\]
    +

    Then, we compute the mean molecular speed for molecular hydrogen \(\rm H_2\) as

    +
    +\[ \bar{v}_{\rm H_2} = 145.5081\sqrt{\frac{293}{2.01568}} = 1750\ {\rm m/s}.\]
    +

    The mean molecular speed for radon \({\rm Rn}\) can be calculated a mass ratio because the temperature is the same in both cases. From this method we find that the mean speed is:

    +
    +\[ \bar{v}_{\rm Rn} = \sqrt{\frac{2.01568}{222}}\bar{v}_{\rm H_2} = 166\ {\rm m/s}.\]
    +

    The hydrogen molecule is \({\sim}10\times\) faster on average. That’s to be expected because the mass ratio \((m_{\rm Rn}/m_{\rm H_2}) \sim \sqrt{100} = 10.\)

    +
    +
    +
    +
    +
    import numpy as np
+from scipy.constants import physical_constants 
+
+def mean_molecular_speed(T,m):
+    #T = absolute temperature in K
+    #m = molar mass in u
+    return (2/np.sqrt(np.pi))*np.sqrt((2*k*T)/(m*u))
+
+k = physical_constants['Boltzmann constant'][0]
+u = physical_constants['atomic mass constant'][0]
+mean_speed_constant = 2/np.sqrt(np.pi)*np.sqrt(2*k/u)
+
+T = 293 #room temperature in K
+m_H2 = 2.01568 #molar mass in u
+m_Rn = 222 #molar mass in u
+mass_ratio = m_Rn/m_H2
+
+v_H2 = np.round(mean_molecular_speed(T,m_H2),-1)
+v_H2_constant = np.round(mean_speed_constant*np.sqrt(T/m_H2),-1)
+
+print("----For hydrogen-----")
+print("The mean molecular speed of molecular hydrogen (H_2) is %i m/s." % v_H2)
+print("The mean molecular speed using our simplified equation is %i m/s." % v_H2_constant)
+
+print("----For radon-----")
+v_Rn = v_H2/np.sqrt(mass_ratio)
+print("The mean molecular speed of radon gas (Rn) is %i m/s." % v_Rn)
+print("The mass ratio (m_Rn/m_H2) is %i and the ratio of speeds (v_H2/v_Rn) is %1.1f." % (mass_ratio, np.sqrt(mass_ratio)))
+
    +
    +
    +
    +
    ----For hydrogen-----
+The mean molecular speed of molecular hydrogen (H_2) is 1750 m/s.
+The mean molecular speed using our simplified equation is 1750 m/s.
+----For radon-----
+The mean molecular speed of radon gas (Rn) is 166 m/s.
+The mass ratio (m_Rn/m_H2) is 110 and the ratio of speeds (v_H2/v_Rn) is 10.5.
+
    +
    +
    +
    +
    + +

    Exercise 8.4

    +
    +

    What fraction of the molecules in an ideal gas in equilibrium has speeds with \(\pm1\%\) of the most probable speed \(v_{\rm mp}\)?

    +

    The Maxwell speed distribution function provides the probability of finding a particle within an interval of speeds. The fraction of molecules within the a given speed interval is equal to the integrated probability over the interval. Mathematically, this is expressed by the number of molecules at a particular speed \(N(v)\):

    +
    +(8.20)#\[\begin{align} +P(\pm1\%) = \frac{N(1.01v_{\rm mp})-N(0.99v_{\rm mp})}{N} = \int_{0.99v_{\rm mp}}^{1.01v_{\rm mp}} F(v)dv. +\end{align}\]
    +

    The indefinite integral introduces the error function, which is beyond the scope of this course. However, we can obtain an approximate solution by calculating \(F(v_{\rm mp})\) and multiplying by \(dv \approx \Delta v = 0.02v_{\rm mp}\). This solution works for a small window, where wider intervals are easily evaluated using numerical methods (e.g., Simpson’s rule).

    +

    Recall the most probable speed \(v_{\rm mp} = \sqrt{\frac{2}{\beta m}}\) and substitute to get

    +
    +\[\begin{align*} +F(v_{\rm mp}) &= 4\pi \left(\frac{\beta m}{2\pi}\right)^{3/2} v_{\rm mp}^2 e^{-\frac{1}{2}\beta m v_{\rm mp}^2}, \\ +&= 4\pi \sqrt{\frac{\beta m}{2\pi}}\left(\frac{\beta m}{2\pi}\right) \left(\frac{2}{\beta m}\right) e^{-\frac{1}{2}\beta m \frac{2}{\beta m}}, \\ +&= \frac{4}{e} \sqrt{\frac{\beta m}{2\pi}}. +\end{align*}\]
    +

    Then applying our approximation

    +
    +\[\begin{align*} +P(\pm1\%) &= F(v_{\rm mp})(0.02v_{\rm mp}),\\ +&= \frac{4}{e} \sqrt{\frac{\beta m}{2\pi}} (0.02) \sqrt{\frac{2}{\beta m}}, \\ +&= \frac{4}{\sqrt{\pi}}(0.02 e^{-1}) \approx 0.017. +\end{align*}\]
    +

    The python code plots the Maxwell speed distribution for an ideal gas using \(g=v/v_{\rm mp}\) for the \(x\)-axis coordinate, where the vertical lines denote the mean, root-mean-squared, and most probable speeds. It also numerically calculates (via Simpson’s rule) the proability and finds good agreement with our approximation (\(0.0166\)).

    +
    +
    +
    +
    +
    import numpy as np 
+import matplotlib.pyplot as plt 
+from scipy.integrate import simpson
+
+def Maxwell_speed(g):
+    #g = v/v_mp or v = g*v_mp
+    return 4/np.sqrt(np.pi)*g**2*np.exp(-g**2)
+
+v_rng = np.arange(0.99,1.01,0.0001)
+prob = simpson(Maxwell_speed(v_rng),v_rng)
+print("The fraction of molecules with speeds within 1 percent of v_mp is %1.4f." % prob)
+
+fig = plt.figure(figsize=(8,4),dpi=150)
+ax1 = fig.add_subplot(121)
+ax2 = fig.add_subplot(122)
+ax_list = [ax1,ax2]
+
+g = np.arange(0, 3,0.01)
+for i in range(0,2):
+    ax = ax_list[i]
+    ax.plot(g,Maxwell_speed(g),'k-',lw=1.5)
+    ax.axvline(2/np.sqrt(np.pi),0,Maxwell_speed(2/np.sqrt(np.pi)),color='b',linestyle='--',label='$\\overline{v}$')
+    ax.axvline(np.sqrt(1.5),0,Maxwell_speed(np.sqrt(1.5)),color='r',linestyle='--',label='$v_{\\rm rms}$')
+    ax.axvline(1,0,Maxwell_speed(1),color='orange',linestyle='--',label='$v_{\\rm mp}$')
+    ax.fill_between(v_rng,0,Maxwell_speed(v_rng),color='gray')
+
+    ax.legend(loc='best',ncols=3,fontsize='large')
+    
+    ax.set_xlabel("$v/v_{\\rm mp}$",fontsize='large')
+    ax.set_ylim(0.,1)
+
+ax1.set_ylabel("$F(v/v_{\\rm mp})$")
+ax1.set_xlim(0,3)
+ax2.set_xlim(0.75,1.25)
+ax2.set_title("Zoomed $x$")
+ax2.set_yticklabels([])
+
+fig.subplots_adjust(wspace=0.05);
+
    +
    +
    +
    +
    The fraction of molecules with speeds within 1 percent of v_mp is 0.0166.
+
    +
    +../_images/6acf80a5a688a4dcf9eb28c4fa3d9c1bef4f9bfe57f9c3560e061b385bd84126.png +
    +

    8.5. Classical and Quantum Statistics#

    diff --git a/docs/_images/57e9d47a126527075e8ffb05fa67e6c5d88de367cc7c80c7c5f2c58856f3fd52.png b/docs/_images/57e9d47a126527075e8ffb05fa67e6c5d88de367cc7c80c7c5f2c58856f3fd52.png new file mode 100644 index 0000000..3abb6a6 Binary files /dev/null and b/docs/_images/57e9d47a126527075e8ffb05fa67e6c5d88de367cc7c80c7c5f2c58856f3fd52.png differ diff --git a/docs/_images/6acf80a5a688a4dcf9eb28c4fa3d9c1bef4f9bfe57f9c3560e061b385bd84126.png b/docs/_images/6acf80a5a688a4dcf9eb28c4fa3d9c1bef4f9bfe57f9c3560e061b385bd84126.png new file mode 100644 index 0000000..2d13150 Binary files /dev/null and b/docs/_images/6acf80a5a688a4dcf9eb28c4fa3d9c1bef4f9bfe57f9c3560e061b385bd84126.png differ diff --git a/docs/_sources/Chapter_8/statistical-physics.ipynb b/docs/_sources/Chapter_8/statistical-physics.ipynb index 42609d9..41ef520 100644 --- a/docs/_sources/Chapter_8/statistical-physics.ipynb +++ b/docs/_sources/Chapter_8/statistical-physics.ipynb @@ -265,7 +265,7 @@ "Similarly, there is an average energy associated with the other two velocity components, which produces a net average energy per molecule of $\\frac{3}{2}kT$. In a monatomic gas (e.g., $\\rm He$ or $\\rm Ar$), virtually all of the energy is in the translational kinetic energy.\n", "\n", "```{margin}\n", - "Rigid rotator model\n", + "**Rigid rotator model**\n", "```\n", "\n", "**Consider a diatomic gas (e.g., $\\rm O_2$) as two atoms connected by a massless rod (i.e., a rigid rotator).** Then the molecule can have *rotational* kinetic energy in addtion to translational kinetic energy. The crucial questions are:\n", @@ -289,7 +289,7 @@ "Each of these rotational energies $(K_x\\ \\text{or}\\ K_y)$ is quadratic in angular velocity, so the equipartition theorem instructs us to add $2\\left(\\frac{1}{2}kT\\right) = kT$ per molecule to the translational kinetic energy for a total of $\\frac{5}{2}kT$.\n", "\n", "```{margin}\n", - "Spring model\n", + "**Spring model**\n", "```\n", "\n", "Sometimes it is a better approximation to think of atoms connected by a massless spring rather than a rigid rod. In this model, we use the potential energy of a spring but in radial coordinates:\n", @@ -377,7 +377,321 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "## Maxwell Speed Distribution" + "## Maxwell Speed Distribution\n", + "\n", + "The general form of the Maxwell velocity distribution is given as\n", + "\n", + "\\begin{align*}\n", + "f(\\vec{v})\\ d^3\\vec{v} = C e^{-\\frac{1}{2}\\beta m v^2}\\ d^3\\vec{v},\n", + "\\end{align*}\n", + "\n", + "where $C = {C^\\prime}^3 = \\left(\\frac{\\beta m}{2\\pi}\\right)^{3/2}$.\n", + "\n", + "The distribution $f(\\vec{v})$ is a function of the speed $v$ in the exponent and not the velocity $\\vec{v}$, but it is still a velocity distribution due to the phase space volume element $d^3\\vec{v}$. We can turn this velocity distribution into a speed distribution $F(v)$ using the definition\n", + "\n", + "\\begin{align*} \n", + "F(v)\\ dv =& \\text{ probability of finding a particle } \\\\ \n", + "& \\text{ with speed between } v\\ \\text{and}\\ v + dv.\n", + "\\end{align*}\n", + "\n", + "**Consider the analogous problem in 3-D position space $(x,\\ y,\\ z)$.** Some distribution of particles exist, where a particle can be located with $(x,\\ y,\\ z)$ with a distance $r = \\sqrt{x^2+ y^2 + z^2}$ from the origin and has a position vector $\\vec{r}$ measured relative to the origin. Then\n", + "\n", + "\\begin{align*}\n", + "f(x,\\ y,\\ z)\\ d^3\\vec{r} = & \\text{ probability of finding a particle} \\\\\n", + "& \\text{ between }\\vec{r}\\ \\text{and}\\ \\vec{r} + d^3\\vec{r},\n", + "\\end{align*}\n", + "\n", + "with $d^3\\vec{r} = dx\\ dy\\ dz$. To shift to the scalar *radial* distribution we introduce the definition \n", + "\n", + "\\begin{align*}\n", + "F(r)dr = & \\text{ probability of finding a particle} \\\\\n", + "& \\text{ between }r\\ \\text{and}\\ r + dr.\n", + "\\end{align*}\n", + "\n", + "The space between $r$ and $r+dr$ is a *spherical shell*, and thus, we must integrate over that volume to transform to the radial distribution. These two distributions are related by the volume of spherical shell, or $4\\pi r^2$. We may write\n", + "\n", + "\\begin{align}\n", + "F(r)dr = f(x,\\ y,\\ z) 4\\pi r^2\\ dr.\n", + "\\end{align}\n", + "\n", + "Returning to the problem of the speed distribution $F(v)$ from the velocity distribution $f(\\vec{v})$. We need to integrate over a spherical shell in the velocity phase space, where we replace $r\\rightarrow v$ from the previous example. The desired speed distribution is\n", + "\n", + "```{margin}\n", + "**Maxwell speed distribution**\n", + "```\n", + "\n", + "\\begin{align}\n", + "F(v)\\ dv = 4\\pi Ce^{-\\frac{1}{2}\\beta m v^2}v^2\\ dv.\n", + "\\end{align}\n", + "\n", + "The Maxwell speed distribution as derived from purely classical considerations, where it gives a nonzero probability of finding a particle with a speed greater than $c$. Therefore, it is only valid in the classical limit.\n", + "\n", + "The assymetry of the distribution curve leads to an interesting result:\n", + "\n", + "> the most probable speed $v_{\\rm mp}$, the mean speed $\\bar{v}$, and the root-mean-square speed $v_{\\rm rms}$ are all slightly different from each other.\n", + "\n", + "To find the most probable speed, we simply find *maximum* speed in the probabilty curve, or\n", + "\n", + "\\begin{align}\n", + "\\frac{dF}{dv} &= 4\\pi C \\frac{d}{dv}\\left[e^{-\\frac{1}{2}\\beta m v^2}v^2 \\right], \\\\\n", + "&= 2ve^{-\\frac{1}{2}\\beta m v^2} - \\left(\\beta m v\\right)v^2 e^{-\\frac{1}{2}\\beta m v^2}.\n", + "\\end{align}\n", + "\n", + "Replacing $v\\rightarrow v_{\\rm mp}$ to denote the *most probable* speed and setting the above result equal to zero, we can solve for $v_{\\rm mp}$:\n", + "\n", + "```{margin}\n", + "**Most probable speed $v_{\\rm mp}$**\n", + "```\n", + "\n", + "\\begin{align}\n", + "2v_{\\rm mp}e^{-\\frac{1}{2}\\beta m v_{\\rm mp}^2} &= \\left(\\beta m v_{\\rm mp}\\right)v_{\\rm mp}^2 e^{-\\frac{1}{2}\\beta m v_{\\rm mp}^2}, \\\\\n", + "v_{\\rm mp} &= \\sqrt{\\frac{2}{\\beta m}} = \\sqrt{\\frac{2kT}{m}}.\n", + "\\end{align}\n", + "\n", + "A particle moving at the most probable speed has a kinetic energy $K_{\\rm mp} = \\frac{1}{2}mv_{\\rm mp}^2 = kT$.\n", + "\n", + "The *mean speed* is found by integrating (summing up) the probabilities for individual speeds, or\n", + "\n", + "\\begin{align*}\n", + "\\bar{v} = \\int_0^\\infty vF(v)\\ dv = 4\\pi C \\int_0^\\infty v^3e^{-\\frac{1}{2}\\beta m v^2}dv.\n", + "\\end{align*}\n", + "\n", + "See a list of [integrals containing exponential functions](https://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions) and using $(k=1,\\ n=3,\\ a=\\beta m/2)$ to find\n", + "\n", + "```{margin}\n", + "**Mean speed $\\bar{v}$**\n", + "```\n", + "\n", + "\\begin{align*}\n", + "\\bar{v} &= 4\\pi C \\left(\\frac{1}{2(\\beta m/2)^2}\\right) = 8\\pi \\left(\\frac{\\beta m}{2\\pi}\\right)^{3/2} \\left(\\frac{1}{(\\beta m)^2}\\right), \\\\\n", + " &= \\frac{4}{\\sqrt{2\\pi}} \\left(\\frac{1}{\\sqrt{\\beta m}}\\right), \\\\ &= \\frac{2}{\\sqrt{\\pi}} \\sqrt{\\frac{2kT}{m}}, \\\\\n", + " &= \\frac{2}{\\sqrt{\\pi}} v_{\\rm mp}.\n", + "\\end{align*}\n", + "\n", + "We define the **root-mean-square** $v_{\\rm rms}$ to be $ v_{\\rm rms} \\equiv \\left(\\overline{v^2}\\right)^{1/2}$. We cannot simply use the result for the mean speed $\\bar{v}$ directly. Instead, we need to find the mean using the probability function $F(v)$, or\n", + "\n", + "\\begin{align*}\n", + "\\overline{v^2} = \\int_0^\\infty v^2F(v)\\ dv = 4\\pi C \\int_0^\\infty v^4e^{-\\frac{1}{2}\\beta m v^2}dv.\n", + "\\end{align*}\n", + "\n", + "See a list of [integrals containing exponential functions](https://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions) and using $(k=2,\\ n=4,\\ a=\\beta m/2)$ to find\n", + "\n", + "\\begin{align}\n", + "\\overline{v^2} &= 4\\pi C \\int_0^\\infty v^4e^{-\\frac{1}{2}\\beta m v^2}dv = 4\\pi C \\left(\\frac{3!!}{8a^2}\\sqrt{\\frac{\\pi}{a}}\\right), \\\\\n", + "&= 4\\pi \\left(\\frac{\\beta m}{2\\pi}\\right)^{3/2} \\frac{3}{2(\\beta m)^2} \\sqrt{\\frac{2\\pi}{\\beta m}}, \\\\\n", + "&= \\frac{3(\\beta m)^{3/2}}{(\\beta m)^{5/2} } = \\frac{3}{\\beta m}, \\\\\n", + "&= \\frac{3kT}{m}.\n", + "\\end{align}\n", + "\n", + "Then\n", + "\n", + "```{margin}\n", + "**Root-mean-square speed $v_{\\rm rms}$**\n", + "```\n", + "\n", + "\\begin{align}\n", + "v_{\\rm rms} = \\left(\\overline{v^2}\\right)^{1/2} = \\sqrt{\\frac{3kT}{m}} = \\sqrt{\\frac{3}{2}}v_{\\rm mp}.\n", + "\\end{align}\n", + "\n", + "A particle moving with the mean squared speed has a kinetic energy \n", + "\n", + "$$ \\overline{K} = \\frac{1}{2}m\\overline{v^2} = \\frac{1}{2}m \\left(\\frac{3kT}{m}\\right) = \\frac{3}{2}kT $$\n", + "\n", + "in keeping with our basic law of kinetic theory.\n", + "\n", + "The **standard deviation** of the molecular speeds $\\sigma_v$ is \n", + "\n", + "\\begin{align}\n", + "\\sigma_v &= \\left(\\overline{v^2} - \\bar{v}^2\\right)^{1/2} = \\left(\\frac{3kT}{m} - \\frac{8kT}{m\\pi}\\right)^{1/2}, \\\\\n", + "&= \\left[\\frac{(3\\pi-8)kT}{m\\pi}\\right]^{1/2},\\\\\n", + "&= \\left[3-\\left(\\frac{8}{\\pi}\\right)\\right]^{1/2} \\sqrt{\\frac{kT}{m}}, \\\\\n", + "&= \\left[\\frac{3}{2}-\\left(\\frac{8}{2\\pi}\\right)\\right]^{1/2} v_{\\rm mp}.\n", + "\\end{align}\n", + "\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "```{exercise}\n", + ":class: orange\n", + "\n", + "**Computre the mean molecular speed $\\bar{v}$ in hydrogen $({\\rm H_2})$ and radon $({\\rm Rn})$ gas, both at room temperature $293\\ {\\rm K}$. Use the longest-lived radon isotope, which has a molar mass of $222\\ {\\rm u}$. Compare the results.**\n", + "\n", + "The mean molecular speed is given by \n", + "\n", + "$$ \\bar{v} = \\frac{2}{\\sqrt{\\pi}} \\sqrt{\\frac{2kT}{m}}, $$\n", + "\n", + "which can be applied to each type of gas. The only difference between results will be due to the molar mass $m$. The molar mass of $\\rm H_2$ is $2.01568\\ {\\rm u}$, which is found by multiplying the mass of hydrogen by two. The conversion from atomic mass units ${\\rm u}$ to $\\rm kg$ is $1.660539 \\times 10^{-27}\\ {\\rm kg/u}$. Let's re-write the mean molecular speed so that all the physical constants are combined and so that we can use atomic mass units directly. This gives\n", + "\n", + "\\begin{align}\n", + "\\bar{v} = \\left(2\\sqrt{\\frac{2k}{\\pi u}}\\right)\\sqrt{\\frac{T\\ \\text{(in K)}}{m\\ \\text{(in u)}}} = 145.5081\\sqrt{\\frac{T}{m}}.\n", + "\\end{align}\n", + "\n", + "Then, we compute the mean molecular speed for molecular hydrogen $\\rm H_2$ as\n", + "\n", + "$$ \\bar{v}_{\\rm H_2} = 145.5081\\sqrt{\\frac{293}{2.01568}} = 1750\\ {\\rm m/s}.$$\n", + "\n", + "The mean molecular speed for radon ${\\rm Rn}$ can be calculated a mass ratio because the temperature is the same in both cases. From this method we find that the mean speed is:\n", + "\n", + "$$ \\bar{v}_{\\rm Rn} = \\sqrt{\\frac{2.01568}{222}}\\bar{v}_{\\rm H_2} = 166\\ {\\rm m/s}.$$\n", + "\n", + "The hydrogen molecule is ${\\sim}10\\times$ faster on average. That's to be expected because the mass ratio $(m_{\\rm Rn}/m_{\\rm H_2}) \\sim \\sqrt{100} = 10.$\n", + "\n", + "```\n" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "----For hydrogen-----\n", + "The mean molecular speed of molecular hydrogen (H_2) is 1750 m/s.\n", + "The mean molecular speed using our simplified equation is 1750 m/s.\n", + "----For radon-----\n", + "The mean molecular speed of radon gas (Rn) is 166 m/s.\n", + "The mass ratio (m_Rn/m_H2) is 110 and the ratio of speeds (v_H2/v_Rn) is 10.5.\n" + ] + } + ], + "source": [ + "import numpy as np\n", + "from scipy.constants import physical_constants \n", + "\n", + "def mean_molecular_speed(T,m):\n", + " #T = absolute temperature in K\n", + " #m = molar mass in u\n", + " return (2/np.sqrt(np.pi))*np.sqrt((2*k*T)/(m*u))\n", + "\n", + "k = physical_constants['Boltzmann constant'][0]\n", + "u = physical_constants['atomic mass constant'][0]\n", + "mean_speed_constant = 2/np.sqrt(np.pi)*np.sqrt(2*k/u)\n", + "\n", + "T = 293 #room temperature in K\n", + "m_H2 = 2.01568 #molar mass in u\n", + "m_Rn = 222 #molar mass in u\n", + "mass_ratio = m_Rn/m_H2\n", + "\n", + "v_H2 = np.round(mean_molecular_speed(T,m_H2),-1)\n", + "v_H2_constant = np.round(mean_speed_constant*np.sqrt(T/m_H2),-1)\n", + "\n", + "print(\"----For hydrogen-----\")\n", + "print(\"The mean molecular speed of molecular hydrogen (H_2) is %i m/s.\" % v_H2)\n", + "print(\"The mean molecular speed using our simplified equation is %i m/s.\" % v_H2_constant)\n", + "\n", + "print(\"----For radon-----\")\n", + "v_Rn = v_H2/np.sqrt(mass_ratio)\n", + "print(\"The mean molecular speed of radon gas (Rn) is %i m/s.\" % v_Rn)\n", + "print(\"The mass ratio (m_Rn/m_H2) is %i and the ratio of speeds (v_H2/v_Rn) is %1.1f.\" % (mass_ratio, np.sqrt(mass_ratio)))\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "```{exercise}\n", + ":class: orange\n", + "\n", + "**What fraction of the molecules in an ideal gas in equilibrium has speeds with $\\pm1\\%$ of the most probable speed $v_{\\rm mp}$?**\n", + "\n", + "The Maxwell speed distribution function provides the probability of finding a particle within an interval of speeds. The fraction of molecules within the a given speed interval is equal to the integrated probability over the interval. Mathematically, this is expressed by the number of molecules at a particular speed $N(v)$: \n", + "\n", + "\\begin{align}\n", + "P(\\pm1\\%) = \\frac{N(1.01v_{\\rm mp})-N(0.99v_{\\rm mp})}{N} = \\int_{0.99v_{\\rm mp}}^{1.01v_{\\rm mp}} F(v)dv.\n", + "\\end{align}\n", + "\n", + "The indefinite integral introduces the [error function](https://en.wikipedia.org/wiki/Error_function), which is beyond the scope of this course. However, we can obtain an approximate solution by calculating $F(v_{\\rm mp})$ and multiplying by $dv \\approx \\Delta v = 0.02v_{\\rm mp}$. This solution works for a small window, where wider intervals are easily evaluated using numerical methods (e.g., [Simpson's rule](https://saturnaxis.github.io/CompPhysics/Chapter_4/Integration.html#simpson-s-rule)).\n", + "\n", + "Recall the most probable speed $v_{\\rm mp} = \\sqrt{\\frac{2}{\\beta m}}$ and substitute to get\n", + "\n", + "\\begin{align*}\n", + "F(v_{\\rm mp}) &= 4\\pi \\left(\\frac{\\beta m}{2\\pi}\\right)^{3/2} v_{\\rm mp}^2 e^{-\\frac{1}{2}\\beta m v_{\\rm mp}^2}, \\\\\n", + "&= 4\\pi \\sqrt{\\frac{\\beta m}{2\\pi}}\\left(\\frac{\\beta m}{2\\pi}\\right) \\left(\\frac{2}{\\beta m}\\right) e^{-\\frac{1}{2}\\beta m \\frac{2}{\\beta m}}, \\\\\n", + "&= \\frac{4}{e} \\sqrt{\\frac{\\beta m}{2\\pi}}.\n", + "\\end{align*}\n", + "\n", + "Then applying our approximation\n", + "\n", + "\\begin{align*}\n", + "P(\\pm1\\%) &= F(v_{\\rm mp})(0.02v_{\\rm mp}),\\\\\n", + "&= \\frac{4}{e} \\sqrt{\\frac{\\beta m}{2\\pi}} (0.02) \\sqrt{\\frac{2}{\\beta m}}, \\\\\n", + "&= \\frac{4}{\\sqrt{\\pi}}(0.02 e^{-1}) \\approx 0.017.\n", + "\\end{align*}\n", + "\n", + "The python code plots the Maxwell speed distribution for an ideal gas using $g=v/v_{\\rm mp}$ for the $x$-axis coordinate, where the vertical lines denote the mean, root-mean-squared, and most probable speeds. It also numerically calculates (via Simpson's rule) the proability and finds good agreement with our approximation ($0.0166$).\n", + "\n", + "```" + ] + }, + { + "cell_type": "code", + "execution_count": 74, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The fraction of molecules with speeds within 1 percent of v_mp is 0.0166.\n" + ] + }, + { + "data": { + "image/png": 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    " + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "import numpy as np \n", + "import matplotlib.pyplot as plt \n", + "from scipy.integrate import simpson\n", + "\n", + "def Maxwell_speed(g):\n", + " #g = v/v_mp or v = g*v_mp\n", + " return 4/np.sqrt(np.pi)*g**2*np.exp(-g**2)\n", + "\n", + "v_rng = np.arange(0.99,1.01,0.0001)\n", + "prob = simpson(Maxwell_speed(v_rng),v_rng)\n", + "print(\"The fraction of molecules with speeds within 1 percent of v_mp is %1.4f.\" % prob)\n", + "\n", + "fig = plt.figure(figsize=(8,4),dpi=150)\n", + "ax1 = fig.add_subplot(121)\n", + "ax2 = fig.add_subplot(122)\n", + "ax_list = [ax1,ax2]\n", + "\n", + "g = np.arange(0, 3,0.01)\n", + "for i in range(0,2):\n", + " ax = ax_list[i]\n", + " ax.plot(g,Maxwell_speed(g),'k-',lw=1.5)\n", + " ax.axvline(2/np.sqrt(np.pi),0,Maxwell_speed(2/np.sqrt(np.pi)),color='b',linestyle='--',label='$\\\\overline{v}$')\n", + " ax.axvline(np.sqrt(1.5),0,Maxwell_speed(np.sqrt(1.5)),color='r',linestyle='--',label='$v_{\\\\rm rms}$')\n", + " ax.axvline(1,0,Maxwell_speed(1),color='orange',linestyle='--',label='$v_{\\\\rm mp}$')\n", + " ax.fill_between(v_rng,0,Maxwell_speed(v_rng),color='gray')\n", + "\n", + " ax.legend(loc='best',ncols=3,fontsize='large')\n", + " \n", + " ax.set_xlabel(\"$v/v_{\\\\rm mp}$\",fontsize='large')\n", + " ax.set_ylim(0.,1)\n", + "\n", + "ax1.set_ylabel(\"$F(v/v_{\\\\rm mp})$\")\n", + "ax1.set_xlim(0,3)\n", + "ax2.set_xlim(0.75,1.25)\n", + "ax2.set_title(\"Zoomed $x$\")\n", + "ax2.set_yticklabels([])\n", + "\n", + "fig.subplots_adjust(wspace=0.05);\n" ] }, { diff --git a/docs/objects.inv b/docs/objects.inv index db0176c..6098214 100644 Binary files a/docs/objects.inv and b/docs/objects.inv differ diff --git a/docs/searchindex.js b/docs/searchindex.js index 34224a1..3275897 100644 --- a/docs/searchindex.js +++ b/docs/searchindex.js @@ -1 +1 @@ -Search.setIndex({"docnames": ["Chapter_1/birth-of-modern-physics", "Chapter_10/semiconductors", "Chapter_11/nuclear-physics", "Chapter_12/particle-physics", "Chapter_2/special-theory-of-relativity", "Chapter_3/experimental-quantum-physics", "Chapter_4/structure-of-the-atom", "Chapter_5/quantum-mechanics-part1", "Chapter_6/quantum-mechanics-part2", "Chapter_7/hydrogen-atom", "Chapter_8/statistical-physics", "Chapter_9/molecules-and-lasers", "LICENSE", "Preamble/HW_template", "Preamble/Markdown-basics", "Preamble/Python-basics", "Preamble/who-for", "README", "home"], "filenames": ["Chapter_1/birth-of-modern-physics.ipynb", "Chapter_10/semiconductors.ipynb", "Chapter_11/nuclear-physics.ipynb", "Chapter_12/particle-physics.ipynb", "Chapter_2/special-theory-of-relativity.ipynb", "Chapter_3/experimental-quantum-physics.ipynb", "Chapter_4/structure-of-the-atom.ipynb", "Chapter_5/quantum-mechanics-part1.ipynb", "Chapter_6/quantum-mechanics-part2.ipynb", "Chapter_7/hydrogen-atom.ipynb", "Chapter_8/statistical-physics.ipynb", "Chapter_9/molecules-and-lasers.ipynb", "LICENSE.md", "Preamble/HW_template.ipynb", "Preamble/Markdown-basics.ipynb", "Preamble/Python-basics.ipynb", "Preamble/who-for.ipynb", "README.md", "home.md"], "titles": ["1. 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