Chapter 07. The Electronic Structure of Atoms.ipynb

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    "$$\\require{mhchem}$$\n",
    "\n",
    "### [HOME](https://wangyu16.github.io/GeneralChemistry/)\n",
    "\n",
    "# <center>General Chemistry Study Guide</center>\n",
    "\n",
    "### <center>Chapter 7. The Electronic Structure of Atoms</center>\n",
    "\n",
    "---\n",
    "\n",
    "\n",
    "##### <b>OpenStax</b> 6 Electronic Structure and Periodic Properties of Elements (6.1 to 6.4). <b>Brown</b> 6 Electronic Structure of Atoms. #####"
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    "## 1. Wave and Photon\n",
    "\n",
    "### 1.1. Wave\n",
    "\n",
    "A **wave** can be thought of as a vibrating disturbance by which energy is transmitted. \n",
    "\n",
    "The **speed** $u$ of a wave depends on the type of wave and the nature of the medium through which the wave is traveling (for example, air, water, or a vacuum). \n",
    "\n",
    "The distance between identical points on successive waves is called the **wavelength** $\\lambda$. \n",
    "\n",
    "The **frequency** $\\nu$ of the wave is the number of waves that pass through a particular point in one second. \n",
    "\n",
    "The **amplitude** is the vertical distance from the midline of a wave to the peak or trough. \n",
    "\n",
    "$$u = \\lambda\\times\\nu$$\n",
    "\n",
    "### 1.2. Electromagnetic Radiation\n",
    "\n",
    "**Electromagnetic radiation** is the emission and transmission of energy in the form of **electromagnetic waves**. An electromagnetic wave has an electric field component and a magnetic field component. \n",
    "\n",
    "**Light speed** $c$, or the speed of electromagnetic waves, in a vacuum is $3.00\\times10^8$ meters per second. The wavelength of electromagnetic waves is usually given in nanometers (nm). For light\n",
    "\n",
    "$$c = \\lambda\\times\\nu$$\n",
    "\n",
    "Types of electromagnetic radiation include:\n",
    "* $\\gamma$ ray with wavelength ~ $10^{-3}$ nm; \n",
    "* X-ray with wavelength ~ $10^{-1}$ nm;\n",
    "* Ultravialot with wavelength between X-ray and visible light;\n",
    "* Visible light with wavelength 400 ~ 700 nm;\n",
    "* Infrared with wavelength longer than visible light and shorter than microwave;\n",
    "* Microwave with wavelength ~ cm level;\n",
    "* Radio wave with wavelength ~ m level or higher. \n",
    "\n",
    "There are no clear bounderies except for visible lights. \n",
    "\n",
    "<div class=example>\n",
    "\n",
    "<b>Example:</b> (a) Calculate the frequency of the light with a wavelength 400 nm. (b) Calculate the wavelength of the light with a frequency $5.75\\times10^{14}$ Hz. <br>\n",
    "\n",
    "<b>Answer: </b><br>\n",
    "\n",
    "(a) \n",
    "$$\\lambda = \\frac{c}{\\nu} = \\frac{3.00\\times10^8\\,\\text{m/s}}{400\\times10^{-9}\\,\\text{m}}=7.50\\times10^{14}\\,\\text{Hz}$$\n",
    "\n",
    "(b)\n",
    "$$\\nu =\\frac{c}{\\lambda}=\\frac{3.00\\times10^8\\,\\text{m/s}}{5.75\\times10^{14}\\,\\text{Hz}} = 5.22\\times10^{-7}\\,\\text{m} = 522\\,\\text{nm} $$\n",
    "\n",
    "</div>\n",
    "\n",
    "### 1.3. Photon\n",
    "\n",
    "**Quantum**: the smallest quantity of energy that can be emitted (or absorbed) in the form of electromagnetic radiation which is also known as **photon**. \n",
    "\n",
    ">A photon can be considered as a particle. \n",
    "\n",
    "The energy $E$ of a single photon of energy is given by\n",
    "\n",
    "$$E = h\\nu$$\n",
    "\n",
    "where $h$ is called Planck's constant which equals to $6.63\\times10^{−34}$ J. \n",
    "\n",
    "Another form of the equation is\n",
    "\n",
    "$$E = h\\frac{c}{\\lambda}$$\n",
    "\n",
    "<div class=example>\n",
    "\n",
    "<b>Example:</b> Calculate the energy of a photon with a wavelength of 500 nm. <br>\n",
    "\n",
    "<b>Answer: </b><br>\n",
    "\\begin{align*}\n",
    "E & = h\\frac{c}{\\lambda} \\\\\n",
    "& = 6.63\\times10^{-34}\\,\\text{J s}\\times\\frac{3.00\\times10^8\\,\\text{m/s}}{500\\times10^{-9}\\,\\text{m}} \\\\ \n",
    "& = 3.98\\times10^{-19}\\,\\text{J}\n",
    "\\end{align*}\n",
    "\n",
    "</div>\n",
    "\n",
    "### 1.4. The Photoelectric Effect\n",
    "\n",
    "**Photoelectric Effect**: electrons are ejected from the surface of certain metals exposed to light of at least a certain minimum frequency, called the threshold frequency. \n",
    "\n",
    "The reason: *Because electrons are bonded to nucleus with certain strength. Only photons with enough energy can \"knock out\" the electrons from a piece of metal sample.*\n",
    "\n",
    "Let $KE$ be the kinetic energy of the ejected electron and $W$ be the work function, which is a measure of how strongly the electrons are held in the metal. If the metal absorb a photon with frequancey $\\nu$, there is following correlation.\n",
    "\n",
    "$$E = h\\nu = KE + W$$ \n",
    "\n",
    "It take some energy ($W$) to knock out the electron. The rest energy will be absorbed as the kinetic energy ($KE$) of the electron. "
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    "> #### Requirements\n",
    "> 1. Understand the concepts.\n",
    "> 2. Know how to do calculations using the equations above.\n",
    "> 3. Be familiar with the types of electromagnetic waves.  \n",
    "> 4. Understand photoelectric effect and learn how to do calculations with the equation. "
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    "## 2. Bohr's Theory and Quantum Theory of Atoms\n",
    "\n",
    "### 2.1. Bohr's Theory\n",
    "\n",
    "For a hydrogen atom\n",
    "* Bohr postulated that the electron is allowed to occupy only certain orbits of specific energies.\n",
    "* An electron in any of the allowed orbits will not radiate energy.\n",
    "* Electron dropping from a higher-energy orbit to a lower one and giving up a quantum of energy (a photon) in the form of light.\n",
    "\n",
    "Energies that an electron in hydrogen atom can occupy are given by\n",
    "\n",
    "$$E_n = -R_H\\left(\\frac{1}{n^2}\\right)$$\n",
    "\n",
    "When $n = 1$, we call this the **ground state**, or the **ground level**, which refers to the lowest energy state of a system. \n",
    "\n",
    "The stability of the electron diminishes for $n = 2, 3,$ …. Each of these levels is called an **excited state**, or **excited level**, which is higher in energy than the ground state. \n",
    "\n",
    "The transition of a hydrogen atom from an excited level to another level with lower energy results in the emission of a photon of frequency $\\nu$ and energy $h\\nu$, we can write\n",
    "\n",
    "$$\\Delta E = h\\nu = R_H\\left(\\frac{1}{n_i^2}-\\frac{1}{n_f^2}\\right)$$\n",
    "\n",
    "When $n_i > n_f$, a photon is emmitted; otherwise a photon is absopted with a negtive $\\Delta E$ value. \n",
    "\n",
    "Since $n_i$ and $n_f$ can be any values, the possible emission photon frequencies could be many. The sum of all these emissions give the atomic emission spectra. \n",
    "\n",
    "**Emission spectra** is either continuous or line spectra of radiation emitted by substances.\n",
    "\n",
    "**Line spectra** are the light emission only at specific wavelengths. Atomic emission spectra are line spectra. \n",
    "\n",
    "<div class=example>\n",
    "\n",
    "<b>Example:</b> What is the wavelength of a photon (in nanometers) emitted during a transition from the ni = 5 state to the nf = 2 state in the hydrogen atom?\n",
    "<br>\n",
    "\n",
    "<b>Answer: </b><br>\n",
    "\\begin{align*}\n",
    "\\Delta E & = R_\\text{H}\\left(\\frac{1}{n_i^2} - \\frac{1}{n_f^2}\\right) \\\\\n",
    "& = 2.18\\times10^{-18}\\,\\text{J}\\times\\left(\\frac{1}{5^2}-\\frac{1}{2^2} \\right) \\\\\n",
    "& = -4.58\\times10^{-19}\\,\\text{J}\n",
    "\\end{align*}\n",
    "The negative sign indicates that this is an emission process. (Energy is released from the process.)\n",
    "\n",
    "\\begin{align*}\n",
    "\\lambda & = \\frac{c}{\\nu} \\\\\n",
    "& = \\frac{c}{\\Delta E /h} \\\\\n",
    "& = \\frac{ch}{\\Delta E} \\\\\n",
    "& = \\frac{3.00\\times10^8\\,\\text{m/s}\\times 6.63\\times10^{-34}\\,\\text{j s}}{4.58\\times10^{-19}\\,\\text{J}} \\\\\n",
    "& = 4.34\\times10^{-7}\\,\\text{m} \\\\\n",
    "& = 434\\,\\text{nm}\n",
    "\\end{align*}\n",
    "</div>"
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    "### 2.2. The Dual Nature of the Electron\n",
    "\n",
    "> An electron can be considered as a kind of wave.\n",
    "\n",
    "The wavelength of an electron can be calculated as \n",
    "\n",
    "$$\\lambda = \\frac{h}{mu}$$\n",
    "\n",
    "<div class=example>\n",
    "\n",
    "<b>Example:</b> Calculate the wavelength of an electron moving at 68 m/s. The mass of an electron is $9.11\\times10^{-31}$ kg.\n",
    "<br>\n",
    "\n",
    "<b>Answer: </b><br>\n",
    "\n",
    "\\begin{align*}\n",
    "\\lambda & = \\frac{h}{mu} \\\\\n",
    "& = \\frac{6.63\\times10^{-34}\\,\\text{J s}}{9.11\\times10^{-31}\\,\\text{kg}\\times 68\\,\\text{m/s}} \\\\\n",
    "& = 1.1\\times10^{-5}\\,\\text{m}\n",
    "\\end{align*}\n",
    "\n",
    "</div>\n",
    "\n",
    "According to de Broglie, an electron bound to the nucleus behaves like a **standing wave**. \n",
    "\n",
    "The waves are described as standing, or stationary, because they do not travel along the string. Some points on the string, called **nodes**, do not move at all; that is, the amplitude of the wave at these points is zero. There is a node at each end, and there may be nodes between the ends. The greater the frequency of vibration, the shorter the wavelength of the standing wave and the greater the number of nodes.\n",
    "\n",
    "<img src=\"https://upload.wikimedia.org/wikipedia/commons/2/2f/Standing-waves-string.svg\" />\n",
    "\n",
    "The relation between the circumference of an allowed orbit ($2\\pi r$) and the wavelength ($\\lambda$) of the electron is given by\n",
    "\n",
    "$$2\\pi r = n\\lambda$$\n",
    "\n",
    "which means the circumference must equal to whold number of wavelengths. "
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    "### 2.3. Quantum Mechanics\n",
    "\n",
    "To fully understand the structure of atoms, **quantum mechanics** is needed. \n",
    "\n",
    "**Schrödinger equation** is the fundmental equation in quantum mechanics, by solving which one can get a wave function to describe the whole system. \n",
    "\n",
    "**Heisenberg uncertainty principle**: It is impossible to know simultaneously both the momentum (mass times velocity) and the position of a particle with certainty.\n",
    "\n",
    "Although quantum mechanics tells us that we cannot pinpoint an electron in an atom, it does define the region where the electron might be at a given time. The concept of **electron density** gives the probability that an electron will be found in a particular region of an atom. The square of the wave function, $\\psi^2$, defines the distribution of electron density in three-dimensional space around the nucleus. Regions of high electron density represent a high probability of locating the electron, whereas the opposite holds for regions of low electron density. \n",
    "\n",
    "An **atomic orbital** can be thought of as the wave function of an electron in an atom. When we say that an electron is in a certain orbital, we mean that the distribution of the electron density or the probability of locating the electron in space is described by the square of the wave function associated with that orbital. An atomic orbital, therefore, has a characteristic energy, as well as a characteristic distribution of electron density.\n",
    "\n",
    "**Pauli exclusion principle**: no two electrons in an atom can have the same four quantum numbers."
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    "> #### Requirements \n",
    "> 1. Understand Bohr's theory and atomic emission spectra. \n",
    "> 2. Remember the relation between the circumference of an allowed orbit and the wavelength of the electron. \n",
    "> 3. Be familiar with the concepts in quantum mechanics. "
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    "## 3. Electrons in atoms"
   ]
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    "### 3.1. Quantum Number\n",
    "\n",
    "#### The Principal Quantum Number ($n$)\n",
    "\n",
    "The principal quantum number ($n$) can have integral values $1, 2, 3,$ and so forth; it corresponds to the quantum number in Bohr's atomic theory. The value of $n$ determines the energy of an orbital. The principal quantum number also relates to the average distance of the electron from the nucleus in a particular orbital. The larger $n$ is, the greater the average distance of an electron in the orbital from the nucleus and therefore the larger the orbital.\n",
    "\n",
    "#### The Angular Momentum Quantum Number ($l$)\n",
    "\n",
    "The angular momentum quantum number ($l$) tells us the “shape” of the orbitals.  The values of $l$ depend on the value of the principal quantum number, $n$. For a given value of $n$, $l$ has possible integral values from $0$ to $(n − 1)$. \n",
    "\n",
    "For example, if $n=1$, the only possible value of $l$ is $0$; if $n=2$, the possible values of $l$ are $0$ and $1$. \n",
    "\n",
    "The value of $l$ is generally designated by the letters s, p, d, f, ... corresponding to the values of $0, 1, 2, 3,$ ..., respectively. \n",
    "\n",
    "#### The Magnetic Quantum Number ($m_l$)\n",
    "\n",
    "The magnetic quantum number ($m_l$) describes the orientation of the orbital in space. \n",
    "\n",
    "the value of $m_l$ depends on the value of the angular momentum quantum number, $l$. For a given value of $l$, the possible values of $m_l$ are $-l, (-l+1), ... 0, ... (+l-1), +l$. \n",
    "\n",
    "For example, if $l= 2$, the possible values of $m_l$ are $-2, -1, 0, +1, +2$. \n",
    "\n",
    "#### The Electron Spin Quantum Number ($m_s$)\n",
    "\n",
    "According to electromagnetic theory, a spinning charge generates a magnetic field, and it is this motion that causes an electron to behave like a magnet. The electron spin quantum number ($m_s$) has a value of $+1/2$ or $-1/2$. \n"
   ]
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    "> #### Requirements \n",
    "> 1. Remember everything in this part. "
   ]
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    "### 3.2. Atomic Orbitals\n",
    "\n",
    "Table 7.2 Relation Between Quantum Numbers and Atomic Orbitals\n",
    "\n",
    "|$n$ |\t$l$ |\t$m_l$ |\tNumber of Orbitals |\tAtomic Orbital Designations|\n",
    "|:---|:---|:---|:---|:---|\n",
    "|1 |\t0 |\t0 |\t1 |\t$1s$|\n",
    "|2 |\t0 |\t0 |\t1 |\t$2s$|\n",
    "|2 |\t1 |\t−1, 0, 1 |\t3 |\t$2p_x, 2p_y, 2p_z$|\n",
    "|3 |\t0 |\t0 |\t1 |\t$3s$|\n",
    "|3 |\t1 |\t−1, 0, 1 |\t3 |\t$3p_x, 3p_y, 3p_z$|\n",
    "|3 |\t2 |\t−2, −1, 0, 1, 2 |\t5 |\t$3d_{xy}, 3d_{yz}, 3d_{xz}, 3d_{x^2-y^2}, 3d_{z^2}$|\n",
    " \t \t \t \t\n",
    "\n",
    "\n",
    "All $s$ orbitals are spherical in shape but differ in size, which increases as the principal quantum number increases. \n",
    "\n",
    "<img src=\"https://cnx.org/resources/310d80319138388705df939b86ccefd800891703\" style=\"width:600px\"/>\n",
    "\n",
    "Each $p$ orbital can be thought of as two lobes on opposite sides of the nucleus. Like $s$ orbitals, $p$ orbitals increase in size from $2p$ to $3p$ to $4p$ orbital and so on.\n",
    "\n",
    "All the $3d$ orbitals in an atom are identical in energy. The $d$ orbitals for which $n$ is greater than $3 $($4d$, $5d$,…) have similar shapes.\n",
    "\n",
    "Orbitals having higher energy than $d$ orbitals are labeled $f$, $g$,… and so on. \n",
    "\n",
    "The $f$ orbitals are important in accounting for the behavior of elements with atomic numbers greater than 57, but their shapes are difficult to represent. In general chemistry we are not concerned with orbitals having $l$ values greater than $3$ (the $g$ orbitals and beyond).\n",
    "\n",
    "The shapes of $p$, $d$, $f$ orbitals are shown below. \n",
    "\n",
    "<img src=\"https://cnx.org/resources/a90abd7d07f0304260a7d292a97f60ef7eee1b1f\" />\n",
    "\n",
    "#### The Energies of Orbitals\n",
    "\n",
    "The energy of an electron in a hydrogen atom is determined solely by its principal quantum number. Thus,\n",
    "\n",
    "$$1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f < ...$$\n",
    "\n",
    "The energy picture is more complex for many-electron atoms than for hydrogen. The energy level is shown as below. \n",
    "\n",
    "<img src=\"https://cnx.org/resources/70c2a8b2764e4a01db7425a84f714eeb4e30bc6c\" style=\"width:500px\" /> \n",
    "\n",
    "The order in which atomic subshells are filled in a many-electron atom is shown below. Start with the $1s$ orbital and move downward, following the direction of the arrows. Thus, the order goes as follows: $1s < 2s < 2p < 3s < 3p < 4s < 3d < ….$\n",
    "\n",
    "<img src=\"https://cnx.org/resources/7909e7ef9926a80d76573e73d00eec2c1be57c1e\" style=\"width:500px\" />\n",
    "\n",
    "<div class=example>\n",
    "\n",
    "<b>Example:</b> Write the four quantum numbers (all possible combinations) for an electron in a $3p$ orbital.\n",
    "<br>\n",
    "\n",
    "<b>Answer: </b><br>\n",
    "In the $3p$ orbital, the main quantum number $n=3$, the angular quantum number $l=1$, the magnetic auantum number $m_l$ could be 0 and $\\pm 1$. The electron spin quantum number $m_s$ could be $\\pm\\frac{1}{2}$. Thus, the possible combinations are: <br>\n",
    "\n",
    "(3, 1, $-1$, $+\\frac{1}{2}$) <br>\n",
    "(3, 1, $-1$, $-\\frac{1}{2}$) <br>\n",
    "(3, 1, $0$, $+\\frac{1}{2}$) <br>\n",
    "(3, 1, $0$, $-\\frac{1}{2}$) <br>\n",
    "(3, 1, $+1$, $+\\frac{1}{2}$) <br>\n",
    "(3, 1, $+1$, $-\\frac{1}{2}$)\n",
    "\n",
    "\n",
    "</div>"
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    "> #### Requirements \n",
    "> 1. Remember Table 7.2. Be familiar with $s, p, d$ orbitals. \n",
    "> 2. Remember the order in which atomic subshells are filled in a many-electron atom. "
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    "### 3.3. Electron Configuration\n",
    "\n",
    "The four quantum numbers $n$, $l$, $m_l$, and $m_s$ enable us to label completely an electron in any orbital in any atom. In a sense, we can regard the set of four quantum numbers as the “address” of an electron in an atom, somewhat in the same way that a street address, city, state, and postal ZIP code specify the address of an individual.\n",
    "\n",
    "Example. Write the four quantum numbers for an electron in a $3p$ orbital.\n",
    "\n",
    "In $3p$ orbital, the principle quantum number $n = 3$, the angular momentum Quantum number $l = 1$, the magnetic quantum number could be $m_l = -1, 0, +1$, the  electron spin quantum number could be $m_s = -1, +1$. So, all possibilities are: \n",
    "\n",
    "|$n$|$l$|$m_l$|$m_s$|\n",
    "|:---|:---|:---|:---|\n",
    "|$3$|$1$|$-1$|$+1/2$|\n",
    "|$3$|$1$|$-1$|$-1/2$|\n",
    "|$3$|$1$|$0$|$+1/2$|\n",
    "|$3$|$1$|$0$|$-1/2$|\n",
    "|$3$|$1$|$1$|$+1/2$|\n",
    "|$3$|$1$|$1$|$-1/2$|\n",
    "\n",
    "<img src=\"https://cnx.org/resources/c13c0e06b464bc81a58c4a194ad757221feb866e\" />\n",
    "\n",
    "#### H\n",
    "\n",
    "Electron configuration of hydrogen atom:\n",
    "\n",
    "<img src=\"https://cnx.org/resources/1ab48d0c386c2e3269938f6c49ab3601499f9e3b\" />\n",
    "\n",
    "The electron in a ground-state hydrogen atom must be in the $1s$ orbital. \n",
    "\n",
    "#### He\n",
    "\n",
    "There are two electrons in a helium atom. The ground-state electron configuration is $1s^2$. Both electrons are in the $1s$ orbital. However, according to Pauli exclusion principle, the four quantum numbers should not be the same. Thus, in helium, $m_s = +1/2$  for one electron; $m_s = -1/2$  for the other. Electrons that have opposite spins are said to be paired. \n",
    "\n",
    "<img src=\"https://cnx.org/resources/089531682e67210744db450a2705221aad9db2e0\" />\n",
    "\n",
    "**Paramagnetic** substances are those that contain net unpaired spins and are attracted by a magnet. \n",
    "\n",
    "**Diamagnetic** substances do not contain net unpaired spins and are slightly repelled by a magnet. He is diamagnetic. \n",
    "\n",
    "#### Li\n",
    "\n",
    "Consider the lithium atom (Z = 3), which has three electrons. The third electron cannot go into the $1s$ orbital because it would inevitably have the same set of four quantum numbers as one of the first two electrons. Therefore, this electron “enters” the next (energetically) higher orbital, which is the $2s$ orbital. The electron configuration is $1s^22S^1$ \n",
    "\n",
    "<img src=\"https://cnx.org/resources/9961800fdc13e39e099f1b83d5665329c14d89bc\" />\n",
    "\n",
    "#### Be\n",
    "\n",
    "The ground-state electron configuration of beryllium is $1s^22s^2$. \n",
    "\n",
    "<img src=\"https://cnx.org/resources/1764b855298b21ba0c08551ca1c80c1683363b2b\" />\n",
    "\n",
    "#### B\n",
    "\n",
    "The ground-state electron configuration of boron is $1s^22s^22p^1$.\n",
    "\n",
    "<img src=\"https://cnx.org/resources/e423465bd956b3d5c839d0df74cb907bea165dda\" />\n",
    "\n",
    "#### C\n",
    "\n",
    "**Hund's rule**, which states that the most stable arrangement of electrons in subshells is the one with the greatest number of parallel spins. Thus the ground-state electron configuration of carbon is $1s^22s^22p^2$.The two $p$ orbital electrons have different $m_l$ quantum numbers.  \n",
    "\n",
    "<img src=\"https://cnx.org/resources/721b3371c8da61881a998fda6c95c6b93f168cd0\" />\n",
    "\n",
    "#### N\n",
    "\n",
    "The ground-state electron configuration of nitrogen is $1s^22s^22p^3$.\n",
    "\n",
    "\n",
    "#### O\n",
    "\n",
    "The ground-state electron configuration of oxygen is $1s^22s^22p^4$.\n",
    "\n",
    "\n",
    "#### F\n",
    "\n",
    "The ground-state electron configuration of fluorine is $1s^22s^22p^5$.\n",
    "\n",
    "#### Ne\n",
    "\n",
    "The ground-state electron configuration of neon is $1s^22s^22p^6$.\n",
    "\n",
    "<img src=\"https://cnx.org/resources/14657364ef53b5dc10ce9114912d70bd9490e7ab\" />\n",
    "\n",
    "#### General Rules\n",
    "\n",
    "1. Each shell or principal level of quantum number $n$ contains $n$ subshells. For example, if $n = 2$, then there are two subshells (two values of $l$) of angular momentum quantum numbers $0$ and $1$.\n",
    "2. Each subshell of quantum number $l$ contains ($2l + 1$) orbitals. For example, if $l = 1$, then there are three $p$ orbitals.\n",
    "3. No more than two electrons can be placed in each orbital. Therefore, the maximum number of electrons is simply twice the number of orbitals that are employed.\n",
    "4. A quick way to determine the maximum number of electrons that an atom can have in a principal level $n$ is to use the formula $2n^2$.\n",
    "\n",
    "\n",
    "<div class=example>\n",
    "\n",
    "<b>Example:</b> What is the maximum number of electrons that can be present in the principal level for which $n = 3$?\n",
    "<br>\n",
    "\n",
    "<b>Answer: </b><br>\n",
    "When $n=3$, $l$ could be 0, 1, 2, corresponding to $s$, $p$ and $d$ orbitals. <br>\n",
    "In $s$ orbital, $m_l$ must be 0. Thus, there is only one $s$ orbital.  <br>\n",
    "When $l=1$, $m_l$ has three possible values ($-1$, 0, $=1$), thus there are three $p$ orbitals. <br>\n",
    "When $l=2$, $m_l$ has five possible values ($-2$, $-1$, 0, $=1$, $+2$), thus there are five $d$ orbitals. <br>\n",
    "\n",
    "The total number of orbitals is $1+3+5=9$. Each orbital can host two electrons with $m_s = \\pm \\frac{1}{2}$, respectively. Thus, the maximum number of electrons that can be present in the principal level $n=3$ is 18. \n",
    "</div>\n",
    "\n",
    "<div class=example>\n",
    "\n",
    "<b>Example:</b> An oxygen atom has a total of eight electrons.  Write the four quantum numbers for each of the eight electrons in the ground state.\n",
    "<br>\n",
    "\n",
    "<b>Answer: </b><br>\n",
    "(1, 0, $0$, $+\\frac{1}{2}$) <br>\n",
    "(1, 0, $0$, $-\\frac{1}{2}$) <br>\n",
    "(2, 0, $0$, $+\\frac{1}{2}$) <br>\n",
    "(2, 0, $0$, $-\\frac{1}{2}$) <br>\n",
    "(2, 1, $-1$, $+\\frac{1}{2}$) <br>\n",
    "(2, 1, 0, $+\\frac{1}{2}$) <br>\n",
    "(2, 1, $+1$, $+\\frac{1}{2}$) <br>\n",
    "(2, 1, $+1$, $-\\frac{1}{2}$) <br>\n",
    "Of course, the placement of the eighth electron in the orbital labeled $m_l = +1$ is completely arbitrary.  It would be equally correct to assign it to $m_l = 0$ or $m_l = -1$.\n",
    "</div>"
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    "> #### Requirements \n",
    "> 1. Remember everything in this part. "
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    "### 3.4. The Building-Up Principle\n",
    "\n",
    "The **Aufbau principle** dictates that as protons are added one by one to the nucleus to build up the elements, electrons are similarly added to the atomic orbitals. \n",
    "\n",
    "For example, the electron configuration of argon ($Z=18$) is $1s^22s^22p^63s^23p^6$. Potassium ($Z=19$) has one more proton and one more electron than argon. The electron configuration of potassium is $1s^22s^22p^63s^23p^64s^1$. Because $3d$ orbitals have higher energy than $4s$ orbital, the outmost layer electron of potassium goes to $4s$ rather than $3d$. Since the inner layer electron configuration of potassium is the same as argon, the electron configuration of potassium can be written as $[\\ce{Ar}]4s^1$ where [Ar] denotes the “argon core.”\n",
    "\n",
    "The electron configurations of all elements except hydrogen and helium are represented by a noble gas core, which shows in brackets the noble gas element that most nearly precedes the element being considered, followed by the symbol for the highest filled subshells in the outermost shells. \n",
    "\n",
    "**Transition metals** either have incompletely filled $d$ subshells or readily give rise to cations that have incompletely filled $d$ subshells.\n",
    "\n",
    "Consider the first transition metal series, from scandium through copper. In this series additional electrons are placed in the $3d$ orbitals, according to Hund's rule. However, there are two irregularities. The electron configuration of chromium ($Z = 24$) is $[\\ce{Ar}]4s^13d^5$ and not $[\\ce{Ar}]4s^23d^4$, as we might expect. A similar break in the pattern is observed for copper, whose electron configuration is $[\\ce{Ar}]4s^13d^{10}$ rather than $[\\ce{Ar}]4s^23d^9$. The reason for these irregularities is that a slightly greater stability is associated with the half-filled ($3d^5$) and completely filled ($3d^{10}$) subshells. \n",
    "\n",
    "Following lanthanum are the 14 elements known as the **lanthanides**, or rare earth series [cerium ($Z = 58$) to lutetium ($Z = 71$)]. The rare earth metals have incompletely filled $4f$ subshells or readily give rise to cations that have incompletely filled $4f$ subshells.\n",
    "\n",
    "The last row of elements is the **actinide series**, which starts at thorium ($Z = 90$). Most of these elements are not found in nature but have been synthesized."
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    "> #### Requirements \n",
    "> 1. Understand how to represent the electron configuration using nobel gas core.\n",
    "> 2. Be familiar with the concepts and examples shown here. "
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    "<div class=practice>\n",
    "\n",
    "    \n",
    "<b>Practice:</b> <br>\n",
    "\n",
    "1. As the frequency of an electromagnetic wave increases <br>\n",
    "\n",
    "(A) its speed must increase. <br>\n",
    "(B) its wavelength must increase. <br>\n",
    "(C) its energy of one photon must increase. <br>\n",
    "(D) its amplitude must increase. <br>\n",
    "(E) its number of photons must increase. <br>\n",
    "Answer: C <br><br>\n",
    "\n",
    "2. What is the wavelength of electromagnetic radiation that has a frequency of $2.10\\times10^{14}\\,\\text{s}^{-1}$?<br>\n",
    "Answer: $1.43\\times10^{-6} m$<br><br>\n",
    "\n",
    "3. What is the energy in joules of one photon of microwave radiation with a wavelength 0.122 m?<br>\n",
    "Answer: $1.63\\times10^{-24}$ J<br>\n",
    "\n",
    "4. Calculate the frequency of the light emitted by a hydrogen atom during a transition of its electron from the $n = 6$ to the $n = 3$ principal energy level. <br>\n",
    "Answer: $2.74\\times10^{14}$ Hz<br>\n",
    "\n",
    "5. In an electron microscope, electrons are accelerated to great velocities. Calculate the wavelength of an electron traveling with a velocity of $7.0\\times10^3\\,\\text{km/s}$. The mass of an electron is $9.1\\times10^{-28}\\,\\text{g}$.<br>\n",
    "Answer: $1.04\\times10^{-10}$ m<br>\n",
    "\n",
    "6. Which one of the following sets of quantum numbers is not possible? <br>\n",
    "\n",
    "(A) $n=4, l=3, m_l=-2, m_s=+1/2$ <br>\n",
    "(B) $n=3, l=0, m_l=0, m_s=+1/2$ <br>\n",
    "(C) $n=3, l=0, m_l=1, m_s=-1/2$ <br>\n",
    "(D) $n=2, l=1, m_l=1, m_s=-1/2$ <br>\n",
    "(E) $n=2, l=0, m_l=0, m_s=+1/2$ <br>\n",
    "Answer: C <br><br>\n",
    "\n",
    "7. What is the maximum number of electrons in an atom that can have the following set of quantum numbers? $n = 4; l = 3; m_l = -2$ <br>\n",
    "Answer: 2 <br><br>\n",
    "\n",
    "8. Electrons in an orbital with $l = 3$ are in a/an <br>\n",
    "\n",
    "(A) $s$ orbital <br>\n",
    "(B) $p$ orbital <br>\n",
    "(C) $f$ orbital <br>\n",
    "(D) $d$ orbital <br>\n",
    "(E) $g$ orbital <br>\n",
    "Answer: C <br><br>\n",
    "\n",
    "9. What is the number of orbitals in a $p$ subshell?<br>\n",
    "Answer: 3 <br><br>\n",
    "\n",
    "10. The orbital diagram for a ground-state nitrogen atom is<br>\n",
    "\n",
    "(A) $1s\\ \\underline{\\uparrow\\downarrow}\\quad 2s\\ \\underline{\\uparrow\\downarrow}\\quad 2p\\ \\underline{\\uparrow\\downarrow}\\ \\underline{\\uparrow\\:\\:}\\ \\underline{\\:\\:\\:\\:}$<br>\n",
    "(B) $1s\\ \\underline{\\uparrow\\downarrow}\\quad 2s\\ \\underline{\\uparrow\\:\\:}\\quad 2p\\ \\underline{\\uparrow\\:\\:}\\ \\underline{\\uparrow\\:\\:}\\ \\underline{\\uparrow\\:\\:}$<br>\n",
    "(C) $1s\\ \\underline{\\uparrow\\downarrow}\\quad 2s\\ \\underline{\\uparrow\\downarrow}\\quad 2p\\ \\underline{\\uparrow\\:\\:}\\ \\underline{\\uparrow\\:\\:}\\ \\underline{\\uparrow\\:\\:}$<br>\n",
    "(D) $1s\\ \\underline{\\uparrow\\downarrow}\\quad 2s\\ \\underline{\\uparrow\\downarrow}\\quad 2p\\ \\underline{\\uparrow\\downarrow}\\ \\underline{\\uparrow\\:\\:}\\ \\underline{\\uparrow\\:\\:}$<br>\n",
    "(E) $1s\\ \\underline{\\uparrow\\downarrow}\\quad 2s\\ \\underline{\\uparrow\\downarrow}\\quad 2p\\ \\underline{\\uparrow\\downarrow}\\ \\underline{\\uparrow\\downarrow}\\ \\underline{\\uparrow\\:\\:}$<br>\n",
    "Answer: C <br><br>\n",
    "\n",
    "</div>"
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    "----"
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       "<p>Copyright (C) 2021 Yu Wang</p>\n",
       "\n",
       "<p><a rel=\"license\" href=\"http://creativecommons.org/licenses/by/4.0/\"><img alt=\"Creative Commons License\" style=\"border-width:0\" src=\"https://i.creativecommons.org/l/by/4.0/88x31.png\" /></a><br />This work is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by/4.0/\">Creative Commons Attribution 4.0 International License</a>.</p>\n",
       "\n",
       "<p>Contact: Dr. Yu Wang<br>\n",
       "Department of Chemistry<br>\n",
       "Institute for Materials Research and Innovation<br>\n",
       "University of Louisiana at Lafayette<br>\n",
       "Lafayette, LA 70504<br>\n",
       "Email: yuwang@louisiana.edu</p>\n"
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    "from IPython.core.display import HTML\n",
    "import requests\n",
    "copyright = requests.get(\"https://raw.githubusercontent.com/wangyu16/GeneralChemistry/master/copyright.md\")\n",
    "HTML(copyright.text)"
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