3.3: Development of Quantum Theory
- Extend the concept of wave–particle duality that was observed in electromagnetic radiation to matter as well
- Understand the general idea of the quantum mechanical description of electrons in an atom, and that it uses the notion of three-dimensional wave functions, or orbitals, that define the distribution of probability to find an electron in a particular part of space
- List and describe traits of the four quantum numbers that form the basis for completely specifying the state of an electron in an atom
Bohr’s model explained the experimental data for the hydrogen atom and was widely accepted, but it also raised many questions. Why did electrons orbit at only fixed distances defined by a single quantum number n = 1, 2, 3, and so on, but never in between? Why did the model work so well describing hydrogen and one-electron ions, but could not correctly predict the emission spectrum for helium or any larger atoms? To answer these questions, scientists needed to completely revise the way they thought about matter.
Behavior in the Microscopic World
We know how matter behaves in the macroscopic world—objects that are large enough to be seen by the naked eye follow the rules of classical physics. A billiard ball moving on a table will behave like a particle: It will continue in a straight line unless it collides with another ball or the table cushion, or is acted on by some other force (such as friction). The ball has a well-defined position and velocity (or a well-defined momentum, \(p = mv\), defined by mass \(m\) and velocity \(v\)) at any given moment. In other words, the ball is moving in a classical trajectory. This is the typical behavior of a classical object.
When waves interact with each other, they show interference patterns that are not displayed by macroscopic particles such as the billiard ball. For example, interacting waves on the surface of water can produce interference patters similar to those shown on Figure \(\PageIndex{1}\). This is a case of wave behavior on the macroscopic scale, and it is clear that particles and waves are very different phenomena in the macroscopic realm.
As technological improvements allowed scientists to probe the microscopic world in greater detail, it became increasingly clear by the 1920s that very small pieces of matter follow a different set of rules from those we observe for large objects. The unquestionable separation of waves and particles was no longer the case for the microscopic world.
One of the first people to pay attention to the special behavior of the microscopic world was Louis de Broglie . He asked the question: If electromagnetic radiation can have particle-like character, can electrons and other submicroscopic particles exhibit wavelike character? In his 1925 doctoral dissertation, de Broglie extended the wave–particle duality of light that Einstein used to resolve the photoelectric-effect paradox to material particles. He predicted that a particle with mass m and velocity v (that is, with linear momentum p ) should also exhibit the behavior of a wave with a wavelength value λ , given by this expression in which h is the familiar Planck’s constant
\[\lambda=\dfrac{h}{mv}=\dfrac{h}{p} \label{6.4.1} \]
This is called the de Broglie wavelength . Unlike the other values of λ discussed in this chapter, the de Broglie wavelength is a characteristic of particles and other bodies, not electromagnetic radiation (note that this equation involves velocity [ v , m/s], not frequency [ ν , Hz]. Although these two symbols are identical, they mean very different things). Where Bohr had postulated the electron as being a particle orbiting the nucleus in quantized orbits, de Broglie argued that Bohr’s assumption of quantization can be explained if the electron is considered not as a particle, but rather as a circular standing wave such that only an integer number of wavelengths could fit exactly within the orbit (Figure \(\PageIndex{2}\)).
For a circular orbit of radius r, the circumference is 2πr, and so de Broglie’s condition is:
\[2πr=nλ \label{6.4.3} \]
with \(n=1,2,3,...\)
Since the de Broglie expression relates the wavelength to the momentum and, hence, velocity, this implies:
\[2πr=nλ=\dfrac{nh}{p}=\dfrac{nh}{mv}=\dfrac{nhr}{mvr}=\dfrac{nhr}{L} \label{6.4.3b} \]
This expression can be rearranged to give Bohr’s formula for the quantization of the angular momentum:
\[L=\dfrac{nh}{2π}=n \hbar \label{6.4.4} \]
Classical angular momentum L for a circular motion is equal to the product of the radius of the circle and the momentum of the moving particle p.
\[L=rp=rmv \;\;\; \text{(for a circular motion)} \label{6.4.5} \]
Shortly after de Broglie proposed the wave nature of matter, two scientists at Bell Laboratories, C. J. Davisson and L. H. Germer , demonstrated experimentally that electrons can exhibit wavelike behavior by showing an interference pattern for electrons travelling through a regular atomic pattern in a crystal. The regularly spaced atomic layers served as slits, as used in other interference experiments. Since the spacing between the layers serving as slits needs to be similar in size to the wavelength of the tested wave for an interference pattern to form, Davisson and Germer used a crystalline nickel target for their “slits,” since the spacing of the atoms within the lattice was approximately the same as the de Broglie wavelengths of the electrons that they used. Figure \(\PageIndex{4}\) shows an interference pattern.
The wave–particle duality of matter can be seen by observing what happens if electron collisions are recorded over a long period of time. Initially, when only a few electrons have been recorded, they show clear particle-like behavior, having arrived in small localized packets that appear to be random. As more and more electrons arrived and were recorded, a clear interference pattern that is the hallmark of wavelike behavior emerged. Thus, it appears that while electrons are small localized particles, their motion does not follow the equations of motion implied by classical mechanics, but instead it is governed by some type of a wave equation that governs a probability distribution even for a single electron’s motion. Thus the wave–particle duality first observed with photons is actually a fundamental behavior intrinsic to all quantum particles.
If an electron travels at a velocity of 1.000 × 10 7 m s –1 and has a mass of 9.109 × 10 –28 g, what is its wavelength?
Solution
We can use de Broglie’s equation to solve this problem, but we first must do a unit conversion of Planck’s constant. You learned earlier that 1 J = 1 kg m 2 /s 2 . Thus, we can write h = 6.626 × 10 –34 J s as 6.626 × 10 –34 kg m 2 /s.
\[\begin{align*} λ&=\dfrac{h}{mv} \\[4pt] &=\mathrm{\dfrac{6.626×10^{−34}\:kg\: m^2/s}{(9.109×10^{−31}\:kg)(1.000×10^7\:m/s)}}\\[4pt] &= \mathrm{7.274×10^{−11}\:m} \end{align*} \nonumber \]
This is a small value, but it is significantly larger than the size of an electron in the classical (particle) view. This size is the same order of magnitude as the size of an atom. This means that electron wavelike behavior is going to be noticeable in an atom.
Calculate the wavelength of a softball with a mass of 100 g traveling at a velocity of 35 m s –1 , assuming that it can be modeled as a single particle.
- Answer
-
1.9 × 10 –34 m.
We never think of a thrown softball having a wavelength, since this wavelength is so small it is impossible for our senses or any known instrument to detect (strictly speaking, the wavelength of a real baseball would correspond to the wavelengths of its constituent atoms and molecules, which, while much larger than this value, would still be microscopically tiny). The de Broglie wavelength is only appreciable for matter that has a very small mass and/or a very high velocity.
Werner Heisenberg considered the limits of how accurately we can measure properties of an electron or other microscopic particles. He determined that there is a fundamental limit to how accurately one can measure both a particle’s position and its momentum simultaneously. The more accurately we measure the momentum of a particle, the less accurately we can determine its position at that time, and vice versa. This is summed up in what we now call the Heisenberg uncertainty principle : It is fundamentally impossible to determine simultaneously and exactly both the momentum and the position of a particle . For a particle of mass m moving with velocity v x in the x direction (or equivalently with momentum p x ), the product of the uncertainty in the position, Δ x , and the uncertainty in the momentum, Δ p x , must be greater than or equal to \(\dfrac{ℏ}{2}\) (recall that \(ℏ=\dfrac{h}{2π}\), the value of Planck’s constant divided by 2 π ).
\[Δx×Δp_x=(Δx)(mΔv)≥\dfrac{ℏ}{2} \nonumber \]
This equation allows us to calculate the limit to how precisely we can know both the simultaneous position of an object and its momentum. For example, if we improve our measurement of an electron’s position so that the uncertainty in the position (Δ x ) has a value of, say, 1 pm (10 –12 m, about 1% of the diameter of a hydrogen atom), then our determination of its momentum must have an uncertainty with a value of at least
\[\left [Δp=mΔv=\dfrac{h}{(2Δx)} \right ]=\mathrm{\dfrac{(1.055×10^{−34}\:kg\: m^2/s)}{(2×1×10^{−12}\:m)}=5×10^{−23}\:kg\: m/s.} \nonumber \]
The value of ħ is not large, so the uncertainty in the position or momentum of a macroscopic object like a baseball is too insignificant to observe. However, the mass of a microscopic object such as an electron is small enough that the uncertainty can be large and significant.
It should be noted that Heisenberg’s uncertainty principle is not just limited to uncertainties in position and momentum, but it also links other dynamical variables. For example, when an atom absorbs a photon and makes a transition from one energy state to another, the uncertainty in the energy and the uncertainty in the time required for the transition are similarly related, as Δ E Δ t ≥ \(\dfrac{ℏ}{2}\). As will be discussed later, even the vector components of angular momentum cannot all be specified exactly simultaneously.
Heisenberg’s principle imposes ultimate limits on what is knowable in science. The uncertainty principle can be shown to be a consequence of wave–particle duality, which lies at the heart of what distinguishes modern quantum theory from classical mechanics. Recall that the equations of motion obtained from classical mechanics are trajectories where, at any given instant in time, both the position and the momentum of a particle can be determined exactly. Heisenberg’s uncertainty principle implies that such a view is untenable in the microscopic domain and that there are fundamental limitations governing the motion of quantum particles. This does not mean that microscopic particles do not move in trajectories, it is just that measurements of trajectories are limited in their precision. In the realm of quantum mechanics, measurements introduce changes into the system that is being observed.
The Quantum–Mechanical Model of an Atom
Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, as Bohr had argued, Erwin Schrödinger extended de Broglie’s work by incorporating the de Broglie relation into a wave equation, deriving what is today known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra, and he did so without having to invoke Bohr’s assumptions of stationary states and quantized orbits, angular momenta, and energies; quantization in Schrödinger’s theory was a natural consequence of the underlying mathematics of the wave equation. Like de Broglie, Schrödinger initially viewed the electron in hydrogen as being a physical wave instead of a particle, but where de Broglie thought of the electron in terms of circular stationary waves, Schrödinger properly thought in terms of three-dimensional stationary waves, or wavefunctions , represented by the Greek letter psi, ψ . A few years later, Max Born proposed an interpretation of the wavefunction ψ that is still accepted today: Electrons are still particles, and so the waves represented by ψ are not physical waves but, instead, are complex probability amplitudes. The square of the magnitude of a wavefunction \(∣ψ∣^2\) describes the probability of the quantum particle being present near a certain location in space. This means that wavefunctions can be used to determine the distribution of the electron’s density with respect to the nucleus in an atom. In the most general form, the Schrödinger equation can be written as:
\[\hat{H}ψ=Eψ \nonumber \]
\(\hat{H}\) is the Hamiltonian operator, a set of mathematical operations representing the total energy of the quantum particle (such as an electron in an atom), ψ is the wavefunction of this particle that can be used to find the special distribution of the probability of finding the particle, and \(E\) is the actual value of the total energy of the particle.
Schrödinger’s work, as well as that of Heisenberg and many other scientists following in their footsteps, is generally referred to as quantum mechanics .
Understanding Quantum Theory of Electrons in Atoms
The goal of this section is to understand the electron orbitals (location of electrons in atoms), their different energies, and other properties. The use of quantum theory provides the best understanding to these topics. This knowledge is a precursor to chemical bonding.
As was described previously, electrons in atoms can exist only on discrete energy levels but not between them. It is said that the energy of an electron in an atom is quantized, that is, it can be equal only to certain specific values and can jump from one energy level to another but not transition smoothly or stay between these levels.
The energy levels are labeled with an n value, where n = 1, 2, 3, …. Generally speaking, the energy of an electron in an atom is greater for greater values of n . This number, n , is referred to as the principal quantum number. The principal quantum number defines the location of the energy level. It is essentially the same concept as the n in the Bohr atom description. Another name for the principal quantum number is the shell number. The shells of an atom can be thought of concentric circles radiating out from the nucleus. The electrons that belong to a specific shell are most likely to be found within the corresponding circular area. The further we proceed from the nucleus, the higher the shell number, and so the higher the energy level (Figure \(\PageIndex{5}\)). The positively charged protons in the nucleus stabilize the electronic orbitals by electrostatic attraction between the positive charges of the protons and the negative charges of the electrons. So the further away the electron is from the nucleus, the greater the energy it has.
This quantum mechanical model for where electrons reside in an atom can be used to look at electronic transitions, the events when an electron moves from one energy level to another. If the transition is to a higher energy level, energy is absorbed, and the energy change has a positive value. To obtain the amount of energy necessary for the transition to a higher energy level, a photon is absorbed by the atom. A transition to a lower energy level involves a release of energy, and the energy change is negative. This process is accompanied by emission of a photon by the atom. The following equation summarizes these relationships and is based on the hydrogen atom:
\[ \begin{align*} ΔE &=E_\ce{final}−E_\ce{initial} \\[4pt] &=−2.18×10^{−18}\left(\dfrac{1}{n^2_\ce f}−\dfrac{1}{n^2_\ce i}\right)\:\ce J \end{align*} \nonumber \]
The values n f and n i are the final and initial energy states of the electron.
The principal quantum number is one of three quantum numbers used to characterize an orbital. An atomic orbital , which is distinct from an orbit , is a general region in an atom within which an electron is most probable to reside. The quantum mechanical model specifies the probability of finding an electron in the three-dimensional space around the nucleus and is based on solutions of the Schrödinger equation. In addition, the principal quantum number defines the energy of an electron in a hydrogen or hydrogen-like atom or an ion (an atom or an ion with only one electron) and the general region in which discrete energy levels of electrons in a multi-electron atoms and ions are located.
Another quantum number is l , the angular momentum quantum number . It is an integer that defines the shape of the orbital, and takes on the values, l = 0, 1, 2, …, n – 1. This means that an orbital with n = 1 can have only one value of l , l = 0, whereas n = 2 permits l = 0 and l = 1, and so on. The principal quantum number defines the general size and energy of the orbital. The l value specifies the shape of the orbital. Orbitals with the same value of l form a subshell . In addition, the greater the angular momentum quantum number, the greater is the angular momentum of an electron at this orbital.
Orbitals with l = 0 are called s orbitals (or the s subshells). The value l = 1 corresponds to the p orbitals. For a given n , p orbitals constitute a p subshell (e.g., 3 p if n = 3). The orbitals with l = 2 are called the d orbitals , followed by the f-, g-, and h- orbitals for l = 3, 4, 5, and there are higher values we will not consider.
There are certain distances from the nucleus at which the probability density of finding an electron located at a particular orbital is zero. In other words, the value of the wavefunction ψ is zero at this distance for this orbital. Such a value of radius r is called a radial node. The number of radial nodes in an orbital is n – l – 1.
Consider the examples in Figure \(\PageIndex{7}\). The orbitals depicted are of the s type, thus l = 0 for all of them. It can be seen from the graphs of the probability densities that there are 1 – 0 – 1 = 0 places where the density is zero (nodes) for 1 s ( n = 1), 2 – 0 – 1 = 1 node for 2 s , and 3 – 0 – 1 = 2 nodes for the 3 s orbitals.
The s subshell electron density distribution is spherical and the p subshell has a dumbbell shape. The d and f orbitals are more complex. These shapes represent the three-dimensional regions within which the electron is likely to be found.
If an electron has an angular momentum ( l ≠ 0), then this vector can point in different directions. In addition, the z component of the angular momentum can have more than one value. This means that if a magnetic field is applied in the z direction, orbitals with different values of the z component of the angular momentum will have different energies resulting from interacting with the field. The magnetic quantum number , called m l, specifies the z component of the angular momentum for a particular orbital. For example, for an s orbital, l = 0, and the only value of m l is zero. For p orbitals, l = 1, and m l can be equal to –1, 0, or +1. Generally speaking, m l can be equal to – l , –( l – 1), …, –1, 0, +1, …, ( l – 1), l . The total number of possible orbitals with the same value of l (a subshell) is 2 l + 1. Thus, there is one s -orbital for ml = 0 , there are three p -orbitals for ml = 1 , five d -orbitals for ml = 2 , seven f -orbitals for ml = 3 , and so forth. The principal quantum number defines the general value of the electronic energy. The angular momentum quantum number determines the shape of the orbital. And the magnetic quantum number specifies orientation of the orbital in space, as can be seen in Figure \(\PageIndex{7}\).
Figure \(\PageIndex{8}\) illustrates the energy levels for various orbitals. The number before the orbital name (such as 2 s , 3 p , and so forth) stands for the principal quantum number, n . The letter in the orbital name defines the subshell with a specific angular momentum quantum number l = 0 for s orbitals, 1 for p orbitals, 2 for d orbitals. Finally, there are more than one possible orbitals for l ≥ 1, each corresponding to a specific value of m l . In the case of a hydrogen atom or a one-electron ion (such as He + , Li 2 + , and so on), energies of all the orbitals with the same n are the same. This is called a degeneracy, and the energy levels for the same principal quantum number, n , are called degenerate energy levels. However, in atoms with more than one electron, this degeneracy is eliminated by the electron–electron interactions, and orbitals that belong to different subshells have different energies. Orbitals within the same subshell (for example ns, np, nd, nf , such as 2 p , 3 s ) are still degenerate and have the same energy.
While the three quantum numbers discussed in the previous paragraphs work well for describing electron orbitals, some experiments showed that they were not sufficient to explain all observed results. It was demonstrated in the 1920s that when hydrogen-line spectra are examined at extremely high resolution, some lines are actually not single peaks but, rather, pairs of closely spaced lines. This is the so-called fine structure of the spectrum, and it implies that there are additional small differences in energies of electrons even when they are located in the same orbital. These observations led Samuel Goudsmit and George Uhlenbeck to propose that electrons have a fourth quantum number. They called this the spin quantum number , or m s .
The other three quantum numbers, n , l , and m l , are properties of specific atomic orbitals that also define in what part of the space an electron is most likely to be located. Orbitals are a result of solving the Schrödinger equation for electrons in atoms. The electron spin is a different kind of property. It is a completely quantum phenomenon with no analogues in the classical realm. In addition, it cannot be derived from solving the Schrödinger equation and is not related to the normal spatial coordinates (such as the Cartesian x , y , and z ). Electron spin describes an intrinsic electron “rotation” or “spinning.” Each electron acts as a tiny magnet or a tiny rotating object with an angular momentum, even though this rotation cannot be observed in terms of the spatial coordinates.
The magnitude of the overall electron spin can only have one value, and an electron can only “spin” in one of two quantized states. One is termed the α state, with the z component of the spin being in the positive direction of the z axis. This corresponds to the spin quantum number \(m_s=\dfrac{1}{2}\). The other is called the β state, with the z component of the spin being negative and \(m_s=−\dfrac{1}{2}\). Any electron, regardless of the atomic orbital it is located in, can only have one of those two values of the spin quantum number. The energies of electrons having \(m_s=−\dfrac{1}{2}\) and \(m_s=\dfrac{1}{2}\) are different if an external magnetic field is applied.
Figure \(\PageIndex{9}\) illustrates this phenomenon.  Its moment is directed up (in the positive direction of the z axis) for the \(\dfrac{1}{2}\) spin quantum number and down (in the negative z direction) for the spin quantum number of \(−\ce{1/2}\). A magnet has a lower energy if its magnetic moment is aligned with the external magnetic field (the left electron) and a higher energy for the magnetic moment being opposite to the applied field. This is why an electron with \(m_s=\dfrac{1}{2}\) has a slightly lower energy in an external field in the positive z direction, and an electron with \(m_s=−\dfrac{1}{2}\) has a slightly higher energy in the same field. This is true even for an electron occupying the same orbital in an atom. A spectral line corresponding to a transition for electrons from the same orbital but with different spin quantum numbers has two possible values of energy; thus, the line in the spectrum will show a fine structure splitting.
The Pauli Exclusion Principle
An electron in an atom is completely described by four quantum numbers: n , l , m l , and m s . The first three quantum numbers define the orbital and the fourth quantum number describes the intrinsic electron property called spin. An Austrian physicist Wolfgang Pauli formulated a general principle that gives the last piece of information that we need to understand the general behavior of electrons in atoms. The Pauli exclusion principle can be formulated as follows: No two electrons in the same atom can have exactly the same set of all the four quantum numbers. What this means is that electrons can share the same orbital (the same set of the quantum numbers n , l , and m l ), but only if their spin quantum numbers m s have different values. Since the spin quantum number can only have two values \(\left(±\dfrac{1}{2}\right)\), no more than two electrons can occupy the same orbital (and if two electrons are located in the same orbital, they must have opposite spins). Therefore, any atomic orbital can be populated by only zero, one, or two electrons. The properties and meaning of the quantum numbers of electrons in atoms are briefly summarized in Table \(\PageIndex{1}\).
| Name | Symbol | Allowed values | Physical meaning |
|---|---|---|---|
| principal quantum number | n | 1, 2, 3, 4, …. | shell, the general region for the value of energy for an electron on the orbital |
| angular momentum or azimuthal quantum number | l | 0 ≤ l ≤ n – 1 | subshell, the shape of the orbital |
| magnetic quantum number | m l | – l ≤ m l ≤ l | orientation of the orbital |
| spin quantum number | m s | \(\dfrac{1}{2},\:−\dfrac{1}{2}\) | direction of the intrinsic quantum “spinning” of the electron |
Indicate the number of subshells, the number of orbitals in each subshell, and the values of l and m l for the orbitals in the n = 4 shell of an atom.
Solution
For n = 4, l can have values of 0, 1, 2, and 3. Thus, s , p , d , and f subshells are found in the n = 4 shell of an atom. For l = 0 (the s subshell), m l can only be 0. Thus, there is only one 4 s orbital. For l = 1 ( p -type orbitals), m can have values of –1, 0, +1, so we find three 4 p orbitals. For l = 2 ( d -type orbitals), m l can have values of –2, –1, 0, +1, +2, so we have five 4 d orbitals. When l = 3 ( f -type orbitals), m l can have values of –3, –2, –1, 0, +1, +2, +3, and we can have seven 4 f orbitals. Thus, we find a total of 16 orbitals in the n = 4 shell of an atom.
Identify the subshell in which electrons with the following quantum numbers are found:
- n = 3, l = 1;
- n = 5, l = 3;
- n = 2, l = 0.
- Answer a
-
3 p
- Answer b
-
5 f
- Answer c
-
2s
Calculate the maximum number of electrons that can occupy a shell with (a) n = 2, (b) n = 5, and (c) n as a variable. Note you are only looking at the orbitals with the specified n value, not those at lower energies.
Solution
(a) When n = 2, there are four orbitals (a single 2 s orbital, and three orbitals labeled 2 p ). These four orbitals can contain eight electrons.
(b) When n = 5, there are five subshells of orbitals that we need to sum:
&\phantom{+}\textrm{1 orbital labeled }5s\\
&\phantom{+}\textrm{3 orbitals labeled }5p\\
&\phantom{+}\textrm{5 orbitals labeled }5d\\
&\phantom{+}\textrm{7 orbitals labeled }5f\\
&\underline{+\textrm{9 orbitals labeled }5g}\\
&\,\textrm{25 orbitals total}
\end{align*}\)
Again, each orbital holds two electrons, so 50 electrons can fit in this shell.
(c) The number of orbitals in any shell n will equal n 2 . There can be up to two electrons in each orbital, so the maximum number of electrons will be 2 × n 2
If a shell contains a maximum of 32 electrons, what is the principal quantum number, n ?
- Answer
-
n = 4
Complete the following table for atomic orbitals:
| Orbital | n | l | m l degeneracy | Radial nodes (no.) |
|---|---|---|---|---|
| 4 f | ||||
| 4 | 1 | |||
| 7 | 7 | 3 | ||
| 5 d |
Solution
The table can be completed using the following rules:
- The orbital designation is nl , where l = 0, 1, 2, 3, 4, 5, … is mapped to the letter sequence s, p , d , f , g , h , …,
- The m l degeneracy is the number of orbitals within an l subshell, and so is 2 l + 1 (there is one s orbital, three p orbitals, five d orbitals, seven f orbitals, and so forth).
- The number of radial nodes is equal to n – l – 1.
| Orbital | n | l | m l degeneracy | Radial nodes (no.) |
|---|---|---|---|---|
| 4 f | 4 | 3 | 7 | 0 |
| 4 p | 4 | 1 | 3 | 2 |
| 7 f | 7 | 3 | 7 | 3 |
| 5 d | 5 | 2 | 5 | 2 |
How many orbitals have l = 2 and n = 3?
- Answer
-
The five degenerate 3 d orbitals
Summary
Macroscopic objects act as particles. Microscopic objects (such as electrons) have properties of both a particle and a wave. Their exact trajectories cannot be determined. The quantum mechanical model of atoms describes the three-dimensional position of the electron in a probabilistic manner according to a mathematical function called a wavefunction, often denoted as ψ . Atomic wavefunctions are also called orbitals. The squared magnitude of the wavefunction describes the distribution of the probability of finding the electron in a particular region in space. Therefore, atomic orbitals describe the areas in an atom where electrons are most likely to be found.
An atomic orbital is characterized by three quantum numbers. The principal quantum number, n , can be any positive integer. The general region for value of energy of the orbital and the average distance of an electron from the nucleus are related to n . Orbitals having the same value of n are said to be in the same shell. The angular momentum quantum number, l , can have any integer value from 0 to n – 1. This quantum number describes the shape or type of the orbital. Orbitals with the same principal quantum number and the same l value belong to the same subshell. The magnetic quantum number, m l , with 2 l + 1 values ranging from – l to + l , describes the orientation of the orbital in space. In addition, each electron has a spin quantum number, m s , that can be equal to \(±\dfrac{1}{2}\). No two electrons in the same atom can have the same set of values for all the four quantum numbers.
Glossary
- angular momentum quantum number ( l )
- quantum number distinguishing the different shapes of orbitals; it is also a measure of the orbital angular momentum
- atomic orbital
- mathematical function that describes the behavior of an electron in an atom (also called the wavefunction), it can be used to find the probability of locating an electron in a specific region around the nucleus, as well as other dynamical variables
- d orbital
- region of space with high electron density that is either four lobed or contains a dumbbell and torus shape; describes orbitals with l = 2. An electron in this orbital is called a d electron
- electron density
- a measure of the probability of locating an electron in a particular region of space, it is equal to the squared absolute value of the wave function ψ
- f orbital
- multilobed region of space with high electron density, describes orbitals with l = 3. An electron in this orbital is called an f electron
- Heisenberg uncertainty principle
- rule stating that it is impossible to exactly determine both certain conjugate dynamical properties such as the momentum and the position of a particle at the same time. The uncertainty principle is a consequence of quantum particles exhibiting wave–particle duality
- magnetic quantum number ( m l )
- quantum number signifying the orientation of an atomic orbital around the nucleus; orbitals having different values of m l but the same subshell value of l have the same energy (are degenerate), but this degeneracy can be removed by application of an external magnetic field
- p orbital
- dumbbell-shaped region of space with high electron density, describes orbitals with l = 1. An electron in this orbital is called a p electron
- Pauli exclusion principle
- specifies that no two electrons in an atom can have the same value for all four quantum numbers
- principal quantum number ( n )
- quantum number specifying the shell an electron occupies in an atom
- quantum mechanics
- field of study that includes quantization of energy, wave-particle duality, and the Heisenberg uncertainty principle to describe matter
- s orbital
- spherical region of space with high electron density, describes orbitals with l = 0. An electron in this orbital is called an s electron
- shell
- set of orbitals with the same principal quantum number, n
- spin quantum number ( m s )
- number specifying the electron spin direction, either \(+\dfrac{1}{2}\) or \(−\dfrac{1}{2}\)
- subshell
- set of orbitals in an atom with the same values of n and l
- wavefunction ( ψ )
- mathematical description of an atomic orbital that describes the shape of the orbital; it can be used to calculate the probability of finding the electron at any given location in the orbital, as well as dynamical variables such as the energy and the angular momentum


3.7: Common Models of Electric Potential
By the end of this section, you will be able to:
- List and apply common models for the electric potential of continuous charge distributions.
Continuous Charge Distributions
As we saw previously in Common Models of Electric Field , there are situations where there are so many charges that the system can be modeled as a continuous distribution of charges. In other words, the superposition of electric potential as a sum of discrete contributions
\[V_p = k\sum \dfrac{q_i}{r_i}.\]
can be modeled instead as a continuous charge distribution consisting of a collection of infinitesimally separated individual points. This approach yields the integral
\[V_p = \int \dfrac{dq}{r}\]
for the potential at a point \(P\). Note that \(r\) is the distance from each individual point in the charge distribution to the point \(P\). As before, the infinitesimal charges are given by
\[\underbrace{dq = \lambda \, dl}_{\text{one dimension}}\]
\[\underbrace{dq = \sigma \, dA}_{\text{two dimensions}}\]
where \(\lambda\) is linear charge density and \(\sigma\) is the charge per unit area.
In this chapter, we will summarize several models corresponding to the most common geometric distributions of charge. In each of these models, the electric potential can be written in a closed-form expression for a specified region of space around the charge distribution. Students who are interested in the details of the calculations should refer to the section Calculating Electric Potentials of Charge Distributions in the chapter Direct Calculation of Electrical Quantities from Charge Distributions .
Electric Potentials of Common Charge Distributions
Finite Line Segment of Charge
Suppose a finite line segment of charge is located along the y-axis as illustrated in Fig. \(\PageIndex{1}\). Assume that the segment has charge density \(\lambda\) (e.g., in coulombs per meters). The electric potential along the perpendicular bisector of the segment at a distance \(x\) away from the segment is
\[ V(x) = k\lambda \ln \left[ \dfrac{L + \sqrt{L^2 + 4x^2}}{-L + \sqrt{L^2 + 4x^2}}\right] \label{Vlinesegment}\].
Ring of Charge
If we take a finite line segment with linear charge density \(\lambda\) and total charge \( q_0 \) and wrap it into a closed circular loop, then we get a ring of charge of radius \( R \) (Fig. \(\PageIndex{2}\)). The electric field at a distance \( z \) from the plane of the ring along its symmetry axis is
\[ V(z) = \dfrac{2\pi k \lambda R}{\sqrt{z^2 + R^2}} = k \dfrac{q_{tot}}{\sqrt{z^2 + R^2}} \label{Vring}\]
Disk of Charge
Once we know the electric field along the axis of a ring of charge, we can use this result to find the electric field along the axis of a disk of charge because a disk of charge can be modeled as a set of rings of charge of increasing radius (Fig. \(\PageIndex{3}\). The electric field for a disk of radius \(R\) and uniform surface charge density \(\sigma\) (coulombs per m\(^2\)) along the axis of the disk at a distance \(z\) from the disk is
\[ V(z) = k2\pi \sigma ( \sqrt{z^2 + R^2} - \sqrt{z^2}) \label{Vdisk} \]
Infinite Line of Charge
Consider an infinite line of charge with linear charge density \(\lambda\) aligned along the \(y\)-axis. Assume the reference potential \(V_R = 0\ \textrm{V}\) at \(x = 1\ \textrm{m}\) from the line of charge. The electric potential at a distance \(x\) in meters perpendicular to the line is
\[ V(x) = -2 k\lambda \ln x. \label{Vinfiniteline} \]
Note that unlike the result for the finite line segment Equation \ref{Vlinesegment} where the reference potential for zero voltage was taken at infinity, here the reference voltage was taken at a finite distance of 1 meter from the wire. This specification is necessary because the charge itself reaches to infinity, and therefore the potential cannot be set to zero there.
Even though no actual infinite line charge exists, the result of Equation \ref{Vinfiniteline} can still provide a simple and good approximation for the potential provided that the separation distance to the test location is small compared to the length of the line charge.


12.8: Transgenic organisms
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- Ying Liu
- City College of San Francisco
General principles of transgenesis
Transgenic organisms contain foreign DNA that has been introduced using biotechnology. Foreign DNA (the transgene ) is defined here as DNA from another species, or else recombinant DNA from the same species that has been manipulated in the laboratory then reintroduced. The terms transgenic organism and genetically modified organism ( GMO ) are generally synonymous. The process of creating transgenic organisms or cells to be come whole organisms with a permanent change to their germline has been called either transformation or transfection . (Unfortunately, both words have alternate meanings. Transformation also refers to the process of mammalian cell becoming cancerous, while transfection also refers to the process of introducing DNA into cells in culture, either bacterial or eukaryote, for a temporary use, not germ line changes.) Transgenic organisms are important research tools, and are often used when exploring a gene’s function. Transgenesis is also related to the medical practice of gene therapy, in which DNA is transferred into a patient’s cells to treat disease. Transgenic organisms are widespread in agriculture. Approximately 90% of canola, cotton, corn, soybean, and sugar beets grown in North America are transgenic. No other transgenic livestock or crops (except some squash, papaya, and alfalfa) are currently produced in North America.
To make a transgenic cell, DNA must first be transferred across the cell membrane, (and, if present, across the cell wall), without destroying the cell. In some cases, naked DNA (meaning plasmid or linear DNA that is not bound to any type of carrier ) may be transferred into the cell by adding DNA to the medium and temporarily increasing the porosity of the membrane, for example by electroporation . When working with larger cells, naked DNA can also be microinjected into a cell using a specialized needle. Other methods use vectors to transport DNA across the membrane. Note that the word “vector” as used here refers to any type of carrier, and not just plasmid vectors. Vectors for transformation/transfection include vesicles made of lipids or other polymers that surround DNA; various types of particles that carry DNA on their surface; and infectious viruses and bacteria that naturally transfer their own DNA into a host cell, but which have been engineered to transfer any DNA molecule of interest. Usually the foreign DNA is a complete expression unit that includes its own cis-regulators (e.g. promoter) as well as the gene that is to be transcribed.
When the objective of an experiment is to produce a stable (i.e. heritable) transgenic eukaryote, the foreign DNA must be incorporated into the host’s chromosomes. For this to occur, the foreign DNA must enter the host’s nucleus, and recombine with one of the host’s chromatids. In some species, the foreign DNA is inserted at a random location in a chromatid, probably wherever strand breakage and non-homologous end joining happen to occur. In other species, the foreign DNA can be targeted to a particular locus, by flanking the foreign DNA with DNA that is homologous to the host’s DNA at that locus. The foreign DNA is then incorporated into the host’s chromosomes through homologous recombination.
Furthermore, to produce multicellular organisms in which all cells are transgenic and the transgene is stably inherited, the cell that was originally transformed must be either a gamete or must develop into tissues that produce gametes. Transgenic gametes can eventually be mated to produce homozygous, transgenic offspring. The presence of the transgene in the offspring is typically confirmed using PCR or Southern blotting, and the expression of the transgene can be measured using reverse-transcription PCR (RT-PCR), RNA blotting, and Western (protein blotting).
The rate of transcription of a transgene is highly dependent on the state of the chromatin into which it is inserted (i.e. position effects ), as well as other factors. Therefore, researchers often generate several independently transformed/transfected lines with the same transgene, and then screen for the lines with the highest expression. It is also good practice to clone and sequence the transgenic locus from a newly generated transgenic organism, since errors (truncations, rearrangements, and other mutations) can be introduced during transformation/transfection.
Producing a transgenic plant
The most common method for producing transgenic plants is Agrobacterium-mediated transformation (Figure \(\PageIndex{1}\)). Agrobacterium tumifaciens is a soil bacterium that, as part of its natural pathogenesis, injects its own tumor-inducing ( T i ) plasmid into cells of a host plant. The natural T i plasmid encodes growth-promoting genes that cause a gall (i.e. tumor) to form on the plant, which also provides an environment for the pathogen to proliferate. Molecular biologists have engineered the T i plasmid by removing the tumor-inducing genes and adding restriction sites that make it convenient to insert any DNA of interest. This engineered version is called a T-DNA (transfer-DNA) plasmid; the bacterium transfers a linear fragment of this plasmid that includes the conserved “left-border (LB)”, and right-border (RB)” DNA sequences, and anything in between them (up to about 10 kb). The linear T-DNA fragment is transported into the nucleus, where it recombines with the host-DNA, probably wherever random breakages occur in the host’s chromosomes.
In Arabidopsis and a few other species, flowers can simply be dipped in a suspension of Agrobacterium, and ~1% of the resulting seeds will be transformed. In most other plant species, cells are induced by hormones to form a mass of undifferentiated tissues called a callus. The Agrobacterium is applied to a callus and a few cells are transformed, which can then be induced by other hormones to regenerate whole plants (Figure \(\PageIndex{2}\)). Some plant species are resistant (i.e. “ recalcitrant ”) to transformation by Agrobacterium. In these situations, other techniques must be used such as particle bombardment , whereby DNA is non-covalently attached to small metallic particles, which are accelerated by compressed air into callus tissue, from which complete transgenic plants can sometimes be regenerated. In all transformation methods, the presence of a selectable marker (e.g. a gene that confers antibiotic resistance or herbicide resistance) is useful for distinguishing transgenic cells from non-transgenic cells at an early stage of the transformation process.
Producing a transgenic mouse
In a commonly used method for producing a transgenic mouse, stem cells are removed from a mouse embryo, and a transgenic DNA construct is transferred into the stem cells using electroporation, and some of this transgenic DNA enters the nucleus, where it may undergo homologous recombination (Figure \(\PageIndex{3}\)). The transgenic DNA construct contains DNA homologous to either side of a locus that is to be targeted for replacement. If the objective of the experiment is simply to delete (“ knock-out ”) the targeted locus, the host’s DNA can simply be replaced by selectable marker, as shown. It is also possible to replace the host’s DNA at this locus with a different version of the same gene, or a completely different gene, depending on how the transgenic construct is made. Cells that have been transfected and express the selectable marker (i.e. resistance to the antibiotic neomycin resistance, neoR, in this example) are distinguished from unsuccessfully transfected cells by their ability to survive in the presence of the selective agent (e.g. an antibiotic). Transfected cells are then injected into early stage embryos, and then are transferred to a foster mother. The resulting pups are chimeras, meaning that only some of their cells are transgenic. Some of the chimeras will produce gametes that are transgenic, which when mated with a wild-type gamete, will produce mice that are hemizygous for the transgene. Unlike the chimeras, these hemizygotes carry the transgene in all of their cells. Through further breeding, mice that are homozygous for the transgene can be obtained.
Human gene therapy
Many different strategies for human gene therapy are under development. In theory, either the germline or somatic cells may be targeted for transfection, but most research has focused on somatic cell transfection, because of risks and ethical issues associated with germline transformation. Gene therapy approaches may be further classified as either ex vivo or in vivo , with the former meaning that cells (e.g. stem cells) are transfected in isolation before being introduced to the body, where they replace defective cells. Ex vivo gene therapies for several blood disorders (e.g. immunodeficiencies, thalassemias) are undergoing clinical trials. For in vivo therapies, the transfection occurs within the patient. The objective may be either stable integration, or non-integrative transfection. As described above, stable transfection involves integration into the host genome. In the clinical context, stable integration may not be necessary, and carries with it higher risk of inducing mutations in either the transgene or host genome). In contrast, transient transfection does not involve integration into the host genome and the transgene may therefore be delivered to the cell as either RNA or DNA. Advantages of RNA delivery include that no promoter is needed to drive expression of the transgene. Besides mRNA transgenes, which could provide a functional version of a mutant protein, there is great interest in delivery of siRNA (small-inhibitory RNAs), which can be used to silence specific genes in the host cell’s genome.
Vectors for in vivo gene therapy must be capable of delivering DNA or RNA to a large proportion of the targeted cells, without inducing a significant immune response, or having any toxic effects. Ideally, the vectors should also have high specificity for the targeted cell type. Vectors based on viruses (e.g. lentiviruses ) are being developed for in both in vivo and ex vivo gene therapies. Other, non-viral vectors (e.g. vesicles and nanoparticles) are also being developed for gene therapy as well.


3.1: Is It an Argument?
Arguments are made up of statements organized around the act of inference with the background purpose of providing an answer to an issue. This means that all arguments involve collections of statements. But not all collections of statements are arguments. Much of what is read these days, or listened to, is not an argument in the critical thinking sense. A news report on television or in a traditional newspaper is usually just that, a report or description of information that is meant to describe what has happened, where, and when. Sometimes reporters provide their own perspective on the story or an analysis of events, and these activities could be considered to be presenting an argument. The defining factor is whether reasons are being provided to answer a question, sometimes called an issue, in a particular way. In a traditional newspaper, the op-ed section is where you are most likely to find some kind of argument, although sometimes writers simply express an opinion without offering reasons or take the opportunity to rant. Beyond the news outlets, we are constantly bombarded by arguments, with advertising as one common source.
One way to determine if a passage contains an argument is to look for an issue and conclusion. Is there a question being addressed? It might be stated, or unstated, but if you can identify an issue, you have a strong clue there is an argument.
Here is a passage with the conclusion underlined, and the issue in italics:
Doomsday preppers expect the infrastructure of contemporary life to be compromised or destroyed through a catastrophe in the near future. Is it rational to be a prepper? One key issue in the ongoing debate is how likely a doomsday scenario is. It is irrational to use finite resources to prepare for an emergency that is extremely unlikely. On the other hand, it is irrational to refuse to prepare for emergencies that are very likely to happen. Given the many ways our infrastructure can crash – conventional or nuclear war, coordinated terrorist attacks, catastrophic weather events, infrastructure fragility, etc — I think an emergency is very likely to happen in the near future. That’s why I think prepping makes sense .
A second way to decide if a passage contains an argument is to look for indicator words (markers) –words or phrases which indicate that a person is using a statement as either a premise OR a conclusion.
Premise indicators are followed by sentences functioning as premises. Common premise indicators are:
- because
- since
- for
- provided that
- implies (that)
- for the reason that
- assuming that
- inasmuch as
The premise follows a premise indicator word or phrase and the conclusion often precedes the indicator. Conclusion indicators are followed by sentences functioning as the conclusion .
Common conclusion indicators are:
- so
- thus
- hence
- therefore
- it follows (that)
- consequently
- supports (that)
- suggests (that)
- we may conclude (that)
- for this reason
- implies that
- means that
The conclusion FOLLOWS the conclusion indicator word or phrase, and the premises often precede the indicator word or phrase.
The list of indicator words is not exhaustive, meaning there are more indicators than those listed here. Also, some of these words or phrases can have other uses. For this reason, they can only be treated as offering clues or hints. You can combine the hunt for indicator words with the first method – looking for an issue and conclusion. If you find what you think is a conclusion and then find or articulate the issue it relates to, you have good reason to think you have located an argument. You then need to examine if there is at least one premise that is used to provide support for the conclusion. Ask yourself, “Does the author/speaker give any reason that lends support to why they think this way?” If no easily identifiable reason is given, then it is most likely not an argument.
One key feature of fast-thinking mode is this: people EVALUATE reasoning they encounter before they ANALYZE it. If we slow ourselves down, we reverse this order. We want to make sure we understand what someone’s thinking IS before we decide if it is good or bad.
Your own reasoning as it passes through your consciousness might feel convincing. Or it might trip a feeling of doubt. Either way, if the matter is important, you might want to move into slow-thinking mode and analyze your own reasoning as a way of double-checking.
Exercise: Argument or Not?
An interactive H5P element has been excluded from this version of the text. You can view it online here:
https://minnstate.pressbooks.pub/ctar/?p=653#h5p-4


21.2A: Nose and Paranasal Sinuses
The shape of the nose is determined by the ethmoid bone and the nasal septum.
- Describe the anatomy of the nose and paranasal sinuses
Key Points
- The shape of the nose is determined by the ethmoid bone and the nasal septum, which consists mostly of cartilage and separates the nostrils.
- The nose and paranasal sinuses are part of the upper respiratory tract.
- The functions of the nose include the sense of smell and conditioning of inhaled air by warming it and making it more humid.
- Hairs inside the nose prevent large particles from entering the lungs.
- Nasal mucosa and cilia help prevent pathogens and dust from reaching the lungs. Sneezing helps remove foreign particles that irritate the nasal mucosa.
- The paranasal sinuses are air-filled spaces around the nasal cavity that have many possible functions.
- The mucosa of the upper respiratory tract contain antimicrobial proteins that are a barrier component of the innate immune system.
Key Terms
- nostril : Either of the two orifices located on the nose (or on the beak of a bird); used as a passage for air and other gases to travel the nasal passages.
- paranasal sinuses : Four air-filled spaces around the nasal cavity that perform many functions, such as draining mucus from the nose.
The nose and paranasal sinuses form much of the upper respiratory tract, along with the pharynx. The upper respiratory tract is the entrance to the respiratory system, where air first enters the body. The overall function of the upper respiratory tract is to provide a pathway for air to reach the lower respiratory tract, where gas exchange occurs.
Anatomy and Physiology of the Nose
Internal diagram of the human nose : Air flows in through the nasal passage on the right and out through the nasopharynx on the posterior side.
The external part of the human nose is the protruding part of the face that bears the nostrils. The shape of the nose is determined by the ethmoid bone and the nasal septum.
The ethmoid bone is the bone that separates the nose from the brain, and supports the shape and structure of the nasal and orbital cavities. The nasal septum is wall of cartilage that separates the right and left nostril chambers from each other. On average, the nose of a male is larger than that of a female, due to differences in facial bone structure between genders.
The interior of the nasal cavity is lined with mucous membranes, nasal hairs, and cilia (microscopic hairs), that perform many of the specialized functions of the nose. The macroscopic nasal hairs prevent large particles from reaching the lungs, while the cilia and mucus trap pathogens and dust to take them to the pharynx, where they can be destroyed by digestion.
Another function of the nose is the conditioning of inhaled air, warming it and making it more humid. Sneezing occurs from irritation of the nasal mucus, which expels foreign particles, but can also spread microbial and viral infections between humans.
Finally, the nose has an area of specialized cells that are responsible for smelling, which is considered a nervous system function rather than a respiratory system function.
Anatomy and Physiology of the Paranasal Sinuses
The paranasal sinuses are a group of four, paired, air-filled spaces, lined with respiratory epithelium (ciliated columnar epithelium). These are named according to the bones within which the sinuses lie: surrounding the nasal cavity (maxillary sinuses), above the eyes (frontal sinuses), between the eyes (ethmoid sinuses), and behind the ethmoid bone (sphenoid sinuses).
Locations of the paranasal sinuses : The paranasal sinuses are four airspaces around the nasal cavity.
The functions of the sinuses are not fully understood, but there are many possible functions. The most important function is the sinuses’ role in draining mucus from the nasal cavity to the nasopharynx, which helps regulate pressure inside the nasal cavity. This may be a component of the barrier defenses of the innate immune system because of antimicrobial proteins found in the mucosa.
Other possible sinus functions include giving resonance to the voice, supporting the structure of the skull and facial bones, heating and humidifying inhaled air, and protecting the face from injury.


7.1: Introduction
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Exhibit 7.1 (Credit: CDC/ Dawn Arlotta / US Government Works)
Learning Outcomes
After reading this chapter, you should be able to answer these questions:
- What are the traditional forms of organizational structure?
- What contemporary organizational structures are companies using?
- Why are companies using team-based organizational structures?
- What tools do companies use to establish relationships within their organizations?
- How can the degree of centralization/decentralization be altered to make an organization more successful?
- How do mechanistic and organic organizations differ?
- How does the informal organization affect the performance of the company?
- What trends are influencing the way businesses organize?
EXPLORING BUSINESS CAREERS
Elise Eberwein
EVP of People and Communications, American Airlines As executive vice president of people and communications at American Airlines, Elise Eberwein’s role within the structure of the organization might not be readily apparent. After all, you might ask, doesn’t corporate communications typically involve marketing? And what does that have to do with organizational structure? As it turns out, quite a bit at the world’s largest airline.
When American Airlines and US Airways finally got the U.S. government’s approval to merge in late 2013, it was no longer business as usual for Eberwein and her colleagues at the “new” airline. Until the merger, which basically produced the world’s largest airline with more than 6,000 daily flights and 102,900 employees, Eberwein was head of communications at US Airways—a position she held for nine years after various other jobs in the airline industry.
Exhibit 7.2 American Airlines jet. (Credit: Joao Carlos Medau/ flickr/ Attribution 2.0 Generic (CC BY 2.0))
Communications and aviation are in Eberwein’s DNA. She worked as a flight attendant at TWA before moving on to manage communications at Denver-based Frontier Airlines. Her next communications experience was at America West, which then merged with US Airways, where Eberwein served as executive vice president of people, communications, and public affairs before she took over the chief communications job at American Airlines.
Corporation communications is no longer just about marketing. The importance of an effective communications strategy cannot be understated in today’s 24/7 business environment. Corporate communication executives have taken on an expanded role in many organizations, according to a recent survey by the Korn Ferry Institute. Of the senior communications executives from Fortune 500 companies who responded to the survey, nearly 40 percent said chief communications officers report directly to the CEO. In addition, more than two-thirds of respondents believe the most important leadership characteristic for communications professionals is having a strategic mindset that goes beyond day-to-day communications activities and looks ahead to future possibilities that can be translated into achievable corporate strategies at all levels of the organization.
In a company as large as American Airlines, even after the initial two-year integration plan, there are many departments, unions, and other employees to communicate with on a daily basis, not to mention the millions of customers they serve every day. For example, American’s social media hub consists of 30 or so team members, divided into three groups: social customer service, social engagement, and social insights. The customer service group, the largest of the three, operates around the clock to address customers’ issues, including missed flight connections and lost luggage, as well as quirky questions like why American airplanes have a specific number of stripes on their tails. Reporting to Eberwein, the social media group is empowered to reach out to any company department directly to get answers for any customer.
Eberwein believes her role includes working closely with the CEO and other managers across the globe to provide consistent, detailed information to all of its stakeholders. To accomplish this feat, Eberwein and other senior managers hold a weekly Monday morning meeting to review the previous week’s operations data, revenue results, and people engagement activities. Eberwein believes establishing this regular contact with colleagues across the organization helps reinforce American’s commitment to engagement and transparent communications, which ultimately shapes the customer’s experience as well as the entire company.
Sources: “Leadership Bios: Elise Eberwein,” https://www.aa.com , accessed July 24, 2017; “By the Numbers: Snapshot of the Airline,” http://news.aa.com , accessed July 24, 2017; Richard Marshall, Beth Fowler, and Nels Olson, “The Chief Communications Officer: Survey and Findings among the Fortune 500,” https://www.kornferry.com , accessed July 24, 2017; Elise Eberwein, “Why the Chief Communications Officer Is Pivotal to the CEO, Especially a New One,” Chief Executive , http://chiefexecutive.net , September 11, 2016; Michael Slattery, “A Visit to American Airlines Social Media Hub,” Airways magazine, https://airwaysmag.com , June 10, 2016; Diana Bradley, “American Airlines CEO Discusses Comms Strategy behind US Airways Merger,” PR Week, http://www.prweek.com , May 27, 2015.
This module focuses on the different types of organizational structure, the reasons an organization might prefer one structure over another, and how the choice of an organizational structure ultimately can impact that organization’s success.
In today’s dynamic business environment, organizational structures need to be designed so that the organization can quickly respond to new competitive threats and changing customer needs. Future success for companies will depend on their ability to be flexible and respond to the needs of customers. In this chapter, we’ll look first at how companies build organizational structures by implementing traditional, contemporary, and team-based models. Then, we’ll explore how managers establish the relationships within the structures they have designed, including determining lines of communication, authority, and power. Finally, we’ll examine what managers need to consider when designing organizational structures and the trends that are changing the choices companies make about organizational design.


6.0: Avoiding Discrimination in Recruiting
- Discuss how to avoid discrimination in the recruiting process
In order to avoid discrimination in the recruiting process, human resource personnel need to review recruiting policies and procedures to ensure compliance with federal discrimination legislation (as discussed in Module 4: Diversity in the Workplace). In particular, employers need to evaluate their advertising and use of employee recruiting, screening, testing and selection techniques to avoid not only intentional discrimination but also practices that have an “adverse impact” on protected groups.
To the advertising point, the EEOC states that “it is illegal for an employer to publish a job advertisement that shows a preference for or discourages someone from applying for a job because of his or her race, color, religion, sex (including gender identity, sexual orientation, and pregnancy), national origin, age (40 or older), disability or genetic information.” For example, a job ad that specifies “male” or “female” or language that suggests age such as “recent college graduate” or “retiree” may violate the law. [1]
With the emphasis on social and employee-supported recruiting, it’s important to note the potential danger of employee referrals. Specifically, it’s illegal for an employer to engage in recruiting practices that have a discriminatory impact. The example the EEOC cites is an employer’s reliance on word-of-mouth recruitment by its mostly Hispanic workforce, noting that if the result is that most new hires are Hispanic, that practice may violate the law. [2]
To quote from the Equal Employment Opportunity Commission’s (EEOC) Employment Tests & Selection Procedures Fact Sheet: “The use of tests and other selection procedures can be a very effective means of determining which applicants or employees are most qualified for a particular job. However, use of these tools can violate the federal anti-discrimination laws if an employer intentionally uses them to discriminate based on race, color, sex, national origin, religion, disability, or age (40 or older). Use of tests and other selection procedures can also violate the federal anti-discrimination laws if they disproportionately exclude people in a particular group by race, sex, or another covered basis, unless the employer can justify the test or procedure under the law. [3]
Title VII of the Civil Rights Act of 1964 permits the use of employment tests unless they are “designed, intended or used to discriminate because of race, color, religion, sex or national origin.” Additionally, employers are prohibited from using different standards or otherwise altering the results of employment-related tests on the basis of race, color, religion, sex, or national origin. Title VII also applies to employees and, therefore, internal recruiting and selection (for example, transfer or promotion).
Title VII prohibits both “disparate treatment” and “disparate impact” discrimination based on race, color, religion, sex, or national origin. An example of disparate treatment would be testing the reading ability of Hispanic or Latino candidates but not white candidates. To review practices for potential violation of the disparate treatment clause, ask:
- Are candidates of a different race, color, religion, sex, or national origin treated differently?
- Is there any evidence of bias, such as discriminatory statements or written comments?
- Can any difference in treatment be justified under the law?
These are also the questions that EEOC would consider in evaluating a charge of discrimination.
“Disparate impact” discrimination involves the use of tests or selection procedures that have the effect of disproportionately excluding persons based on race, color, religion, sex, or national origin and are not “job-related and consistent with business necessity.”
To review practices for potential violation of the disparate impact clause, ask:
- Do required tests disproportionately screen out members of protected groups?
- If there is a disparate impact, can that be justified as necessary for the safe and efficient performance of the job? Note that the skills tested must be specifically related to the particular job.
- Is there an alternate means of evaluating job-specific skills that is equally effective in predicting job performance that would not disproportionately exclude protected groups?
Consider the Griggs v. Duke Power Co. case decided by the Supreme Court in 1971. At issue in this case: whether Duke Power Company’s internal transfer policy, which required both a high school education and passing scores on two separate aptitude tests discriminated against African-American employees in violation of Title VII of the 1964 Civil Rights Act.
The Supreme Court found that Duke’s standardized testing requirement did constitute discrimination, finding that “Duke’s standardized testing requirement prevented a disproportionate number of African-American employees from being hired by, and advancing to higher-paying departments within, the company. Specifically, the court held that “neither the high school graduation requirement nor the two aptitude tests was directed or intended to measure an employee’s ability to learn or perform a particular job or category of jobs within the company” but rather, “the purpose of these requirements was to safeguard Duke’s long-standing policy of giving job preferences to its white employees.” This case established the legal precedent for disparate impact lawsuits based on racial discrimination. [4]
Policies and procedures should also be reviewed to ensure compliance with Title I of the Americans with Disabilities Act (ADA), which prohibits discrimination against qualified individuals with disabilities on the basis of their disabilities. [5] Review of testing procedures for compliance with the ADA is similar to the points above. Specifically, evaluate employment tests to determine whether they tend to screen out individuals with a specific type of disability or disabilities in general. Any disparate impact must be job-related and consistent with business necessity. Further, employers must ensure that tests are selected and administered in a manner that accurately reflects an applicant’s job-related aptitude or skills, not the applicant’s impairment. Finally, employers are required to make reasonable accommodations for the known physical or mental limitations of an otherwise qualified candidate unless such accommodation would impose an undue hardship.
The last piece of legislation to factor in is the ADEA, which prohibits both disparate treatment (intentional) and disparate impact discrimination based on age. [6] An example of disparate treatment would be requiring only applicants over the age of 50 to take agility or cognitive tests. Similar to Title VII and the ADA, the ADEA also prohibits employers from using tests or selection procedures that have a discriminatory impact on persons 40 or older, unless the evaluation is based on a reasonable factor other than age.
-
"
Prohibited Employment Policies/Practices
." U.S. Equal Employment Opportunity Commission. Accessed July 26, 2019. ↵
-
Ibid. ↵
-
"
Employment Tests and Selection Processes
." The U.S. Equal Employment Opportunity Commission. Accessed July 26, 2019. ↵
-
“
Griggs v. Duke Power Co.
” Encyclopædia Britannica. Encyclopædia Britannica, inc. Accessed September 17, 2019. ↵
-
Americans With Disabilities Act of 1990. Public Law 101-336. 108th Congress, 2nd session (July 26, 1990) ↵
-
Age Discrimination in Employment Act of 1967. Public Law 90-202. 90th Congress. (June 12, 1968) ↵
Contributors and Attributions
- Seal of the United States Equal Employment Opportunity Commission. Provided by : US Equal Employment Opportunity Commission. Located at : https://commons.wikimedia.org/wiki/File:Seal_of_the_United_States_Equal_Employment_Opportunity_Commission.svg . License : Public Domain: No Known Copyright


12.10: Open-Pit Mining
Open-pit mining , or open-cast mining is a surface mining technique of extracting rock or minerals from the earth by their removal from an open pit or borrow.
This form of mining differs from extractive methods that require tunneling into the earth, such as long wall mining. Open-pit mines are used when deposits of commercially useful minerals or rocks are found near the surface; that is, where the overburden (surface material covering the valuable deposit) is relatively thin or the material of interest is structurally unsuitable for tunneling (as would be the case for sand, cinder, and gravel). For minerals that occur deep below the surface—where the overburden is thick or the mineral occurs as veins in hard rock—underground mining methods extract the valued material.
Open-pit mines that produce building materials and dimension stone are commonly referred to as “quarries.”
Open-pit mines are typically enlarged until either the mineral resource is exhausted, or an increasing ratio of overburden to ore makes further mining uneconomic. When this occurs, the exhausted mines are sometimes converted to landfills for disposal of solid wastes. However, some form of water control is usually required to keep the mine pit from becoming a lake, if the mine is situated in a climate of considerableprecipitation or if any layers of the pit forming the mine border productive aquifers.
EXTRACTION
Open-cast mines are dug on benches, which describe vertical levels of the hole. These benches are usually on four to sixty meter intervals, depending on the size of the machinery that is being used. Many quarries do not use benches, as they are usually shallow.
Figure 1. Note the angled and stepped sides of the Sunrise Dam Gold Mine, Australia.
Most walls of the pit are generally dug on an angle less than vertical, to prevent and minimize damage and danger from rock falls. This depends on how weathered the rocks are, and the type of rock, and also how many structural weaknesses occur within the rocks, such as a faults, shears, joints orfoliations.
The walls are stepped. The inclined section of the wall is known as the batter, and the flat part of the step is known as the bench or berm. The steps in the walls help prevent rock falls continuing down the entire face of the wall. In some instances additional ground support is required and rock bolts, cable bolts and shotcrete are used. De-watering bores may be used to relieve water pressure by drilling horizontally into the wall, which is often enough to cause failures in the wall by itself.
A haul road is usually situated at the side of the pit, forming a ramp up which trucks can drive, carrying ore and waste rock.
Waste rock is piled up at the surface, near the edge of the open pit. This is known as the waste dump. The waste dump is also tiered and stepped, to minimize degradation.
Ore which has been processed is known as tailings, and is generally a slurry. This is pumped to a tailings dam or settling pond, where the water evaporates. Tailings dams can often be toxic due to the presence of unextracted sulfide minerals, some forms of toxic minerals in the gangue, and often cyanide which is used to treat gold ore via the cyanide leach process. This toxicity can harm the surrounding environment.
REHABILITATION
After mining finishes, the mine area must undergo rehabilitation. Waste dumps are contoured to flatten them out, to further stabilise them. If the ore contains sulfides it is usually covered with a layer of clay to prevent access of rain and oxygen from the air, which can oxidise the sulfides to produce sulfuric acid, a phenomenon known as acid mine drainage. This is then generally covered with soil, and vegetation is planted to help consolidate the material. Eventually this layer will erode, but it is generally hoped that the rate of leaching or acid will be slowed by the cover such that the environment can handle the load of acid and associated heavy metals. There are no long term studies on the success of these covers due to the relatively short time in which large scale open pit mining has existed. It may take hundreds to thousands of years for some waste dumps to become “acid neutral” and stop leaching to the environment. The dumps are usually fenced off to prevent livestock denuding them of vegetation. The open pit is then surrounded with afence, to prevent access, and it generally eventually fills up with ground water. In arid areas it may not fill due to deep groundwater levels.
Figure 2. Opencut coal mine loadout station and reclaimed land at the North Antelope Rochelle coal mine in Wyoming, United States.
Figure 3. An open-pit sulfur mine at Tarnobrzeg, Poland undergoing land rehabilitation
TYPICAL OPEN CUT GRADES
Gold is generally extracted in open-pit mines at 1 to 2 ppm (parts per million) but in certain cases, 0.75 ppm gold is economical. This was achieved by bulk heap leaching at the Peak Hill mine in western New South Wales, near Dubbo, Australia.
Nickel, generally as laterite, is extracted via open-pit down to 0.2%. Copper is extracted at grades as low as 0.15% to 0.2%, generally in massive open-pit mines in Chile, where the size of the resources and favorable metallurgy allows economies of scale.
Materials typically extracted from open-pit mines include:
- Bitumen
- Clay
- Coal
- Copper
- Coquina
- Diamonds
- Gravel and stone (stone refers to bedrock, while gravel is unconsolidated material)
- Granite
- Gritstone
- Gypsum
- Limestone
- Marble
- Metal ores, such as Copper, Iron, Gold, Silver and Molybdenum
- Uranium
- Phosphate
REFLECTION QUESTIONS
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Contributors and Attributions
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Original content from Kimberly Schulte (Columbia Basin College) and supplemented by Lumen Learning . The content on this page is copyrighted under a Creative Commons Attribution 4.0 International license.


9.4: Mk models where parameters vary across clades and/or through time
Another generalization of the Mk model we might imagine is a Mk model where rate parameters vary, either across clades or through time. There is some recent work along these lines, with two approaches that consider the possibility that rates of evolution for an Mk model vary on different branches of a phylogenetic tree (Marazzi et al. 2012; Beaulieu et al. 2013) .
We can understand how these methods work in general terms by considering a simple case where the rate of character evolution is faster in one clade than in the rest of the tree. This is the discrete-character version of the approaches for continuous characters that I discussed in Chapter 6 (O’Meara et al. 2006; Thomas et al. 2006) . The simplest way to implement a multi-rate discrete model is to directly incorporate variation across models into the pruning algorithm that is used to calculate the Mk model on a phylogenetic tree (see FitzJohn 2012 for implementation) .
One can, for example, consider a model where the overall rate of evolution varies between clades in a phylogenetic tree. To do this, we can specify the background rate of evolution using some transition matrix Q , and then assume that within our focal clade evolution can be modeled with some scalar value r , such that the new rate matrix is r Q . Given Q and r , one can calculate the likelihood for this model using the pruning algorithm, modified in such a way that the appropriate transition matrix is used along each branch in the tree; one can then maximize the likelihood of the model for all parameters (those describing Q , as well as r , which describes the relative rate of evolution in the focal clade compared to the background).
In even more general terms, we will consider the situation where we can describe the model of evolution using a set of Q matrices: Q 1 , Q 2 , …, Q n , each of which can be assigned to a particular branch in a phylogenetic tree (or be assigned to branches depending on some other character that influences the rate of the focal character; Marazzi et al. 2012) . The only limit here is that each Q matrix adds a new set of model parameters that must be estimated from the data, and it is easy to imagine this model becoming overparametized. If we imagine a model where every branch has its own Q -matrix, then we are actually describing the “no common mechanism” model (Tuffley and Steel 1997; Steel and Penny 2000) , which is statistically identical to parsimony. It should also be possible to create a method that explores all models connecting simple Mk and the no common mechanism model using the machinery of reversible-jump MCMC, although I do not think such an approach has ever been implemented (but see Huelsenbeck et al. 2004) .
One can also describe a situation where rate parameters in the Q matrix change through time. This might follow a constant pattern of increase or decrease through time, or might be related to some external driver like temperature. One can mimic models where rates change through time by changing the branch lengths of phylogenetic trees. If deep branches are lengthened relative to shallow branches, as is done by Pagel's δ , then we can fit a model where rates of evolution slow through time; conversely, lengthening shallow branches relative to deep ones creates a model where the overall rate of evolution accelerates through time (see FitzJohn 2012) .
More work could certainly be done in the area of time-varying rates of change. The most general approach is to write a set of differential equations that describe the changes in character state along single branches in the tree. Parameters in those equations can be made to vary, either through time or even in a way that is correlated with some external variable hypothesized to influence rates of change, like temperature or rainfall. Given such a model, the reverse-time approach of Maddison et al. (2007) can then be used to fit general time-varying (or even clade-varying) Mk models to data (see Uyeda et al. 2016) .


3.2.1: Trends in Size
Learning Objectives
- Describe and explain the observed trends in atomic size of the elements
The elements in groups (vertical columns) of the periodic table exhibit similar chemical behavior. This similarity occurs because the members of a group have the same number and distribution of electrons in their valence shells. However, there are also other patterns in chemical properties on the periodic table. For example, as we move down a group, the metallic character of the atoms increases. Oxygen, at the top of Group 16 (6A), is a colorless gas; in the middle of the group, selenium is a semiconducting solid; and, toward the bottom, polonium is a silver-grey solid that conducts electricity.
As we go across a period from left to right, we add a proton to the nucleus and an electron to the valence shell with each successive element. As we go down the elements in a group, the number of electrons in the valence shell remains constant, but the principal quantum number increases by one each time. An understanding of the electronic structure of the elements allows us to examine some of the properties that govern their chemical behavior. These properties vary periodically as the electronic structure of the elements changes. They are (1) size (radius) of atoms and ions, (2) ionization energies, and (3) electron affinities.
Visualizing periodic trends
Explore visualizations of the periodic trends discussed in this section (and many more trends). With just a few clicks, you can create three-dimensional versions of the periodic table showing atomic size or graphs of ionization energies from all measured elements.
Variation in Covalent Radius
The quantum mechanical picture makes it difficult to establish a definite size of an atom. However, there are several practical ways to define the radius of atoms and, thus, to determine their relative sizes that give roughly similar values. We will use the covalent radius (Figure \(\PageIndex{1}\)), which is defined as one-half the distance between the nuclei of two identical atoms when they are joined by a covalent bond (this measurement is possible because atoms within molecules still retain much of their atomic identity).
Figure \(\PageIndex{1}\) : (a) The radius of an atom is defined as one-half the distance between the nuclei in a molecule consisting of two identical atoms joined by a covalent bond. The atomic radius for the halogens increases down the group as n increases. (b) Covalent radii of the elements are shown to scale. The general trend is that radii increase down a group and decrease across a period.
We know that as we scan down a group, the principal quantum number, n , increases by one for each element. Thus, the electrons are being added to a region of space that is increasingly distant from the nucleus. Consequently, the size of the atom (and its covalent radius) must increase as we increase the distance of the outermost electrons from the nucleus. This trend is illustrated for the covalent radii of the halogens in Table \(\PageIndex{1}\) and Figure \(\PageIndex{1}\). The trends for the entire periodic table can be seen in Figure \(\PageIndex{2}\).
| Atom | Covalent radius (pm) | Nuclear charge |
|---|---|---|
| F | 64 | +9 |
| Cl | 99 | +17 |
| Br | 114 | +35 |
| I | 133 | +53 |
| At | 148 | +85 |
As shown in Figure \(\PageIndex{2}\), as we move across a period from left to right, we generally find that each element has a smaller covalent radius than the element preceding it. This might seem counterintuitive because it implies that atoms with more electrons have a smaller atomic radius. This can be explained with the concept of effective nuclear charg e, \(Z_{eff}\). This is the pull exerted on a specific electron by the nucleus, taking into account any electron–electron repulsions. For hydrogen, there is only one electron and so the nuclear charge ( Z ) and the effective nuclear charge ( Z eff ) are equal. For all other atoms, the inner electrons partially shield the outer electrons from the pull of the nucleus, and thus:
\[Z_\ce{eff}=Z−shielding\]
Shielding is determined by the probability of another electron being between the electron of interest and the nucleus, as well as by the electron–electron repulsions the electron of interest encounters. Core electrons are adept at shielding, while electrons in the same valence shell do not block the nuclear attraction experienced by each other as efficiently. Thus, each time we move from one element to the next across a period, Z increases by one, but the shielding increases only slightly. Thus, Z eff increases as we move from left to right across a period. The stronger pull (higher effective nuclear charge) experienced by electrons on the right side of the periodic table draws them closer to the nucleus, making the covalent radii smaller.
Figure \(\PageIndex{2}\) : Within each period, the trend in atomic radius decreases as Z increases; for example, from K to Kr. Within each group (e.g., the alkali metals shown in purple), the trend is that atomic radius increases as Z increases.
Video \(\PageIndex{1}\) : Learn more about shielding.
Thus, as we would expect, the outermost or valence electrons are easiest to remove because they have the highest energies, are shielded more, and are farthest from the nucleus. As a general rule, when the representative elements form cations, they do so by the loss of the ns or np electrons that were added last in the Aufbau process. The transition elements, on the other hand, lose the ns electrons before they begin to lose the ( n – 1) d electrons, even though the ns electrons are added first, according to the Aufbau principle.
Example \(\PageIndex{1}\): Sorting Atomic Radii
Predict the order of increasing covalent radius for Ge, Fl, Br, Kr.
Solution
Radius increases as we move down a group, so Ge < Fl (Note: Fl is the symbol for flerovium, element 114, NOT fluorine). Radius decreases as we move across a period, so Kr < Br < Ge. Putting the trends together, we obtain Kr < Br < Ge < Fl.
Exercise \(\PageIndex{1}\)
Give an example of an atom whose size is smaller than fluorine.
- Answer
-
Ne or He
Variation in Ionic Radii
Ionic radius is the measure used to describe the size of an ion. A cation always has fewer electrons and the same number of protons as the parent atom; it is smaller than the atom from which it is derived (Figure \(\PageIndex{3}\)). For example, the covalent radius of an aluminum atom (1 s 2 2 s 2 2 p 6 3 s 2 3 p 1 ) is 118 pm, whereas the ionic radius of an Al 3+ (1 s 2 2 s 2 2 p 6 ) is 68 pm. As electrons are removed from the outer valence shell, the remaining core electrons occupying smaller shells experience a greater effective nuclear charge Z eff (as discussed) and are drawn even closer to the nucleus.
Figure \(\PageIndex{3}\) : The radius for a cation is smaller than the parent atom (Al), due to the lost electrons; the radius for an anion is larger than the parent (S), due to the gained electrons.
Cations with larger charges are smaller than cations with smaller charges (e.g., V 2+ has an ionic radius of 79 pm, while that of V 3+ is 64 pm). Proceeding down the groups of the periodic table, we find that cations of successive elements with the same charge generally have larger radii, corresponding to an increase in the principal quantum number, n .
An anion (negative ion) is formed by the addition of one or more electrons to the valence shell of an atom. This results in a greater repulsion among the electrons and a decrease in \(Z_{eff}\) per electron. Both effects (the increased number of electrons and the decreased Z eff ) cause the radius of an anion to be larger than that of the parent atom ( Figure \(\PageIndex{3}\)). For example, a sulfur atom ([Ne]3 s 2 3 p 4 ) has a covalent radius of 104 pm, whereas the ionic radius of the sulfide anion ([Ne]3 s 2 3 p 6 ) is 170 pm. For consecutive elements proceeding down any group, anions have larger principal quantum numbers and, thus, larger radii.
Atoms and ions that have the same electron configuration are said to be isoelectronic . Examples of isoelectronic species are N 3– , O 2– , F – , Ne, Na + , Mg 2+ , and Al 3+ (1 s 2 2 s 2 2 p 6 ). Another isoelectronic series is P 3– , S 2– , Cl – , Ar, K + , Ca 2+ , and Sc 3+ ([Ne]3 s 2 3 p 6 ). For atoms or ions that are isoelectronic, the number of protons determines the size. The greater the nuclear charge, the smaller the radius in a series of isoelectronic ions and atoms.
Summary
Video \(\PageIndex{2}\) : A summary on atomic radius.
Electron configurations allow us to understand many periodic trends. Covalent radius increases as we move down a group because the n level (orbital size) increases. Covalent radius mostly decreases as we move left to right across a period because the effective nuclear charge experienced by the electrons increases, and the electrons are pulled in tighter to the nucleus. Anionic radii are larger than the parent atom, while cationic radii are smaller, because the number of valence electrons has changed while the nuclear charge has remained constant.
Glossary
- covalent radius
- one-half the distance between the nuclei of two identical atoms when they are joined by a covalent bond
- effective nuclear charge
- charge that leads to the Coulomb force exerted by the nucleus on an electron, calculated as the nuclear charge minus shielding
- isoelectronic
- group of ions or atoms that have identical electron configurations
Contributors and Attributions
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Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).
- Adelaide Clark, Oregon Institute of Technology
- Fuse School, Open Educational Resource free of charge, under a Creative Commons License: Attribution-NonCommercial CC BY-NC (View License Deed: https://creativecommons.org/licenses/by-nc/4.0/ )
- Guillotined Chemistry is a production of Mark Anticole and available for free on Youtube.


6.3: Weighted Averages
Weighted, tapered, or windowed averages are straightforward generalizations of simple averages. They take the form
\[x=\sum_{n=1}^{N} w_{n} u_{n} \nonumber \]
with the constraint that the “weights in the window,” \(w_n\), sum to 1:
\[\sum_{n=1}^{N} w_{n}=1 \nonumber \]
When \(w_{n}=\frac{1}{N}\) then \(x\) is the simple average studied in the section on "Simple Averages" .
There are many windows that are commonly used in engineering practice. For \(N\) odd, the standard triangular window is
\[w_{n}=\frac{2}{N+1}(1-\frac{2}{N+1}\Bigg |\frac{N+1}{2}-n|) \nonumber \]
This window, illustrated in Figure 1, weights the input \(u_{(N+1) / 2}\) by \(\dfrac{2}{N+1}\) and the inputs \(u_1\) and \(u_N\) by \(\left(\dfrac{2}{N+1}\right)^{2}\). The most general triangular window takes the form
\[w_{n}=\alpha(1-\beta|\frac{N+1}{2}-n|) ; \alpha, \beta>0, \quad N \text { odd. } \nonumber \]
Determine the constraints on \(\alpha\) and \(\beta\) to make the general triangular window a valid window (i.e., \(\sum_{n=1}^{N} w_{n}=1\)). Show that \(\alpha=\frac{2}{N+1}=\beta\) is a valid solution. Propose another solution that you like.
You are taking three 3-credit courses, one 5-credit course, and one 2-credit course. Write down the weighted average for computing your GPA in a system that awards 4.0 points for an A, 3.0 points for a B,..., and (horrors!) 0 points for an F.


3.1: Gender Socialization
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In Chapter 1, we learned that gender is achieved, rather than ascribed. We discussed gender is something we do rather than something we are . So, if gender is something we learn, how do we learn it? This can be achieved through gender socialization , or the shaping of individual behavior and perceptions in such a way that the individual conforms to socially prescribed expectations for males and females . 23 In other words, our gender roles are socially proscribed expectations and attitudes assigned to and associated with one's biological sex. A gender role is a set of societal norms dictating the types of behaviors that are generally considered acceptable, appropriate, or desirable for people based on their actual or perceived sex.
Gender is so taken for granted, that we don’t often recognize our gendered behaviors. Since we do gender every day, it’s not something we challenge or question very often. Until, that is, someone or something challenges our assumptions and taken for granted positions on the topic. In fact, it’s so taken for granted, most people believe (and reinforce) the idea that gender is something we are born with, rather than something we create and recreate. Therefore, gender is a human production that exists only when people do it. 24
Think about this for a moment: How many people do you know who want to know the sex of a fetus? Why? Does the parents knowing their baby’s sex affect the health of the baby? No. Does knowing the baby’s sex have any correlation with the happiness of the baby? No. Does knowing the baby’s sex before birth help improve its development? No. Then why? So we know what color to paint the nursery, of course! I mean, how could we possibly put a female baby in a blue nursery? That would be insanity!
According to a study conducted in 2001 by a team of doctors at Harvard Medical School in Boston with over 1,300 participants, about 58% of parents-to-be wanted to know the sex of the fetus before the birth. 25 So sex is a pretty important variable for most parents-to-be. But why? So they can start planning, of course! Boy or girl, pink or blue? How should they decorate the nursery? What toys should they play with? What books should they read? Historically, and even today, a lot of the answers to these questions will be mostly shaped by the sex of the baby. Does the baby know its sex? Or that it’s in pink or blue? Of course not! So who cares? Many parents form gendered expectations for their child before it is even born, after determining the child's sex. The child thus arrives to gender-specific clothes, games, and even ambitions. And, enter gender socialization: Our primary caregivers will be the most influential in our gender socialization in our primary years (we’ll discuss other influences later on). While various socializing agents —parents, teachers, peers, movies, television, music, books, and religion—teach and reinforce gender roles throughout the lifespan, parents probably exert the greatest influence, especially on their very young offspring.
Figure \(3.1.1\). Bar Graph of “Parents Who Want to Know the Sex of Their Baby”
Adults perceive and treat female and male infants differently. Parents and guardians probably do this in response to their having been recipients of gender expectations as young children. Sociologists have found that in the U.S., traditionally, fathers teach boys how to fix and build things; mothers teach girls how to cook, sew, and keep house. 26 Sound old-fashioned and out of date? I hope so! And certainly sociologists accept and acknowledge exceptions, but the truth is, today children still receive parental approval more often for conforming to gender expectations and adopting culturally accepted and conventional roles. Despite social revelations that girls are not too fragile to play sports and boys can benefit from learning to manage household responsibilities, we still find ourselves surrounded by limited gender expectations and persistent gender inequalities. Additional socializing agents, such as media, peers, siblings, etc, reinforce all of this. In other words, learning gender roles occurs within a social context, the values of the parents and society being passed along to the children. This results in children adopting a gender identity early in life, resulting in them also developing gender-role preferences . 27 Gender identity is one’s concept of self as female, male, or neither . Gender-role preference is one’s preference for the culturally prescribed roles associated with gender identity .
Gender roles adopted during childhood normally continue into adulthood. People have certain presumptions about decision‐making, child‐rearing practices, financial responsibilities, and so forth. At work, people also have presumptions about power, the division of labor, and organizational structures. None of this is meant to imply that gender roles are good or bad; rather, this is an acknowledgement that they exist and shape our perceptions of reality. Gender roles are realities in almost everyone's life, but since they are not biologically determined, our “realities” surrounding gender can differ from generation to generation, from group to group, even from individual to individual.
Gendered social arrangements also dictate or create external means of control of how females and males should act, and they are often justified by religion and cultural morés. In Western culture, alternatives to our constructed gendered norms has largely been virtually unthinkable. 28 While there is no “essential” gender for human beings, society and culture holds the individual responsible for reproducing the expected gendered norms assigned to them. The individual is expected to recreate the already prescribed gendered behaviors laid out for them, and, in turn, they themselves become the re- creators of what it means to be a women or a man in their society. “If we fail to do gender appropriately, we as individuals may be called to account (for our character, motives, and predispositions).” 29
23
Hammond, Ron, Cheney, Paul.
Introduction to Sociology
.
24
Doing Gender
. Candace
West
; Don H.
Zimmerman
. Gender and Society, Vol. 1, No. 2. (Jun., 1987), pp. 125-151.
25
Shipp, T. D., Shipp, D. Z., Bromley, B., Sheahan, R., Cohen, A., Lieberman, E. and Benacerraf, B. (2004), What Factors Are Associated with Parents’ Desire To Know the Sex of Their Unborn Child?. Birth, 31: 272–279. doi:10.1111/j.0730- 7659.2004.00319.x
26
The Social Construction of Gender, Margaret L. Andersen and Dana Hysock,
Thinking about Women
, Allyn & Bacon, 2009
27
Ibid
28
Foucault, Michael. 1972.
The Archeology of Knowledge and the Discourse on Language
. NY, New York. Pantheon.
29
Doing Gender
. Candace
West
; Don H.
Zimmerman
. Gender and Society, Vol. 1, No. 2. (Jun., 1987), pp. 146.