{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Questions 87 - 96"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Q87 H$_2^+$ molecular orbitals\n",
"This question is about the energy of H$_2^+$ molecular orbitals. The definitions in 11.2 are used. If $\\varphi_1$ and $\\varphi_2$ are H atom 1S orbitals show that\n",
"\n",
"(a) $\\displaystyle \\int\\varphi_1 \\frac{q^2}{R}\\varphi_2 d\\tau=\\frac{q^2S}{R}$ where $R$ is the internuclear separation.\n",
"\n",
"(b) Calculate the resonance integral $\\displaystyle A=\\int\\varphi_1 \\frac{q^2}{r_2}\\varphi_2 d\\tau$ belonging to $H_{12}$ and thereby calculate the energy.\n",
"\n",
"(c) Use the results from the text and those just calculated and plot the energies $E_+$ and $E_-$ with internuclear separation and so reproduce figure 31. Use $\\rho=a_0R, a_0=1$ and electronic charge $q=1$.\n",
"\n",
"## Q88 Arc length\n",
"Calculate the arc length for \n",
"\n",
"(a) a circle of radius $R$, \n",
"\n",
"(b) the logarithmic or equiangular spiral $r = e^{-\\theta/a}$ from $0 \\to 2\\pi$, \n",
"\n",
"(c) the catenary $y = \\cosh(x)$ from $x = 0 \\to x_0$, and \n",
"\n",
"(d) the Archimedean spiral $r = a\\theta$ from $0 \\to 2\\pi$.\n",
"\n",
"**Strategy:** Use equation 83 or 84.\n",
"\n",
"## Q89 Area\n",
"Find the area, the $x$ and $y$ centroids and moments of inertia $I-x, I_y$, and $I_z$ of the ellipse $x^2 + y^2 = 1$.\n",
"\n",
"\n",
"## Q90 Mean value\n",
"Calculate the mean value of $r^2 = x^2 + y^2$ over the ellipse defined in the previous question.\n",
"\n",
"## Q91 Line integral\n",
"If $C$ is a line joining $(0, 0)$ to $(a, b)$ calculate $\\displaystyle \\int_C e^x\\sin(y)dx+e^x\\cos(y)dy$.\n",
"\n",
"**Strategy:** Use the two function formula and convert $dy$ into $dy/dx$ where $y$ is determined by the limits on the line, in this case a straight line from the origin to $(a, b)$.\n",
"\n",
"## Q92 Area\n",
"(a) Find the area under one arch of the cycloid that is described by the parametric equations\n",
"$x = a(t - \\sin(t)),\\, y = a(1 - cos(t))$. A description and sketch of the cycloid is given in Figure 16.\n",
"\n",
"(b) Find the length of the arch.\n",
"\n",
"\n",
"## Q93 Arc length\n",
"Calculate the arc length for curves \n",
"\n",
"(a) $r = 1$\n",
"\n",
"(b) $r = e^{-\\theta}$ from $0 \\to 2\\pi$, and **(c)** the catenary $y=\\cosh(x)$ from $x=0\\to x_0$ where $x_0 \\gt 0$.\n",
"\n",
"## Q94 Surface area\n",
"The surface area of a function $f(x)$ is given by $\\displaystyle A=2\\pi\\int_a^b f(x)\\sqrt{1+f'(x)^2}dx$.\n",
"\n",
"(a) Show that the surface area of a sphere is $4\\pi r^2$ starting with a circle of radius $r$, in which case $f(x) = r^2 - x^2$, and effectively rotating this to form the surface. The integration limits are $\\pm r$.\n",
"\n",
"(b) Work out what fraction of the earth's surface is north of the seaside town of Dunbar, Scotland that is situated at exactly lat $56.00^\\mathrm{o}$ N.\n",
"\n",
"Note: $f'(x)$ is the first derivative. Latitude is the angle from the equator to the pole.\n",
"\n",
"**Strategy:** In (a) substitute, simplify, and find a very simple integral. In (b) take the south to north axis\n",
"of the earth to be the x-axis and work out the $x$ integration limits.\n",
"\n",
"## Q95 State function\n",
"(a) In thermodynamics, what is a state variable?\n",
"\n",
"(b) The work $w$ required to expand a gas is the line integral $w = -\\int p dV$. If $T$ and $p$ are the variables to be used, this equation can be written as \n",
"\n",
"$$\\displaystyle w =\\int p\\left(\\frac{\\partial V}{\\partial T}\\right)_p dT+p\\left(\\frac{\\partial V}{\\partial p}\\right)_T dp$$\n",
"\n",
"For 1 mole of an ideal gas calculate $w$ along each of the two paths used in the example in Section 13.9 and Figure 36 and hence show that $w$ is not a state function.\n",
"\n",
"**Strategy:** Follow the example and make the integral into one in $dp$ and then $dT$ alone. Substitute for the partial derivatives and use the gas law to substitute variables to make an equation in $p$ or $T$ as necessary. Only then, work out the remaining derivative, $dp/dT$ or $dT/dp$ depending on the path taken.\n",
"\n",
"## Q96 Entropy of van der Waals gas\n",
"Calculate the entropy for an van der Walls gas whose equation of state is $(p+a/V^2)(V-b)=RT$ where $a,\\,b$ are constants. Is the entropy different to that of an ideal gas and if so why?\n",
"\n",
"**Strategy:** Calculate $(\\partial V/\\partial T)_p$ then use equation 89. "
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.9.6"
}
},
"nbformat": 4,
"nbformat_minor": 2
}