6.3: More Advanced Typesetting Standards
One of the strengths of the Libretexts project is that all pages share similar formatting so it is easy to mix content from different sources. This page contains general formatting rules.
- All section headings should use h1, h2, h3, etc …
- All text should be full justified.
- All number lists should use the CKE list options, this also includes bullets
- Erase all indentations
- Try to avoid underlines (use bold or italic instead)
- Full justify all text (it is automatic, but can be overwritten)
- Format figures
- Make sure all equations are in MathJax, do not import as images
Units
- Where possible, there should be universal adoption of SI units and formatting. To a large extent SI formatting has already been adopted by the contributors to LibreTexts. I note a few exceptions below.
- The energy of activation is preferably abbreviated as E a , not Ea.
- Spaces should be inserted in long numbers, for example, 34 500 J/mol is preferred to 34500 J/mol.
- Spaces should be included between numbers and units. For example, 273 K is preferred to 273K.
- Use the degree symbol when describing degrees Celsius. For example, 25 ⁰C is preferred to 25 C.
- Where ‘sec’ is used, change to ‘s’.
Figure Coding (ADVANCED)
While images can be added to pages directly as discussed previously , we have a more advanced system for tying a figure caption to the figure. For now, this is added by selecting the "AddFigureCenterCenter" template.
Once the template is embedded, there will be two spots to add the image and the caption.
This <figure> system is the HTML5 code for semantic representation of images. This allows us to tie in the figure caption to the figure so that they cannot be broken (e.g., by a page break in printing PDFs). It has several additional benefits as well. It acts as a that we can control over the entire site via CSS which allows for a standardization - this applies to the <figcaption> text that is tied to the <figure> code. It also provides a semantic background for the image so that Google and others can tie the figcaption to the figure.


27.6.0: Review Questions
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Review Questions
1 .
During an initial assessment, the patient admits to the nurse that she drinks alcohol daily and is now experiencing abdominal swelling in the RUQ. The nurse suspects enlargement of which organ?
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small intestine
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ovary
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liver
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rectum
2 .
A patient is asking why the nurse is concerned about appendicitis. The nurse explains that it is due to the pain the patient feels in which quadrant?
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LLQ
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RLQ
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RUQ
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LUQ
3 .
When examining your patient, she states she has been feeling full and has a constant acidic taste in her mouth when she lies down. What do you suspect that she might have?
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appendicitis
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anorexia
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diarrhea
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GERD
4 .
Your patient is older and has had frequent episodes of diarrhea from a possible virus. The caretaker states the patient is not taking in fluids well but is still having frequent bowel movements. What is your biggest concern?
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dehydration
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constipation
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bloating
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contractions
5 .
Your patient is complaining of hard stools when he has a bowel movement. After completing the patient interview, it is revealed he has a diet full of fast food and few whole grains, vegetables, or water. What do you suspect the constipation is due to?
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medication overuse
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lack of dietary fiber and water
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too much water and overeating of dietary fiber
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a GI virus
6 .
A nurse suspects a patient is having an acute pancreatic attack and possible hemorrhage, and needs to quickly assess the patient. What would be the best assessment technique?
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Inspect for Grey Turner sign.
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Auscultate for hyperactive bowel sounds in the LUQ.
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Palpate and percuss over the LRQ.
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Percuss in the RUQ to the midclavicular line to hear dullness.
7 .
A patient presents with a history of chronic alcohol abuse and has recently stated that he has noticed his “stomach swelling.” How would you best begin the assessment?
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Percuss for liver size in the RUQ and find the liver edge extends to the LUQ.
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Palpate for rebound tenderness of the bladder.
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Observe for borborygmi due to increased peristalsis from alcohol use.
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Observe for distended veins extending from the umbilicus.
8 .
A patient’s physical assessment reveals hypoactive bowel sounds. What does the nurse recognize that this may be a clue to?
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diarrhea
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gastroenteritis
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peritonitis
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constipation
9 .
The nurse discovers the following findings during the assessment. What should be reported to the provider?
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The patient’s abdomen appears flat and symmetric.
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There are hypoactive bowel sounds in all quadrants.
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Firmness is palpated in left lower quadrant.
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There is a scar from a previous appendectomy.
10 .
A 79-year-old female is seen in the emergency department for a “bump” she found on her lower abdomen. After various tests, the doctor tells her she has a hernia. The patient asks you what could have caused this. What answer do you provide?
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eating too much fiber in the diet
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hormone changes causing the muscles of the abdominal wall being too strong
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hormone changes causing the muscles of the abdominal wall being too weak
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undiagnosed H. pylori infection
11 .
An adult patient comes to the emergency department with abdominal distention. What finding supports the documentation of jaundice noted in the physical assessment?
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He is newly immigrated and has not had routine vaccines.
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He has yellowing of the skin and sclera.
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He is a nonsmoker.
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He is physically active and exercised regularly.
12 .
Your 65-year-old male patient comes in for a routine physical and part of the patient’s history reveals a parent who died from gastric cancer. Your patient asks what he should do help lower his risk of gastric cancer. What is your response?
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Eat more pickled food; the acid will help cut fat from the diet.
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Increase fresh fruits and vegetables in your diet.
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You are not at risk because you are past the age of diagnosis.
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You are not at risk because it only affects women.


4.1: Prelude to Capacitance
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Capacitors are important components of electrical circuits in many electronic devices, including pacemakers, cell phones, and computers. In this chapter, we study their properties, and, over the next few chapters, we examine their function in combination with other circuit elements. By themselves, capacitors are often used to store electrical energy and release it when needed; with other circuit components, capacitors often act as part of a filter that allows some electrical signals to pass while blocking others. You can see why capacitors are considered one of the fundamental components of electrical circuits.


6.9: Steady State Temperature and the Laplacian
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Suppose we have an insulated wire, a plate, or a 3-dimensional object. We apply certain fixed temperatures on the ends of the wire, the edges of the plate, or on all sides of the 3-dimensional object. We wish to find out what is the steady state temperature distribution. That is, we wish to know what will be the temperature after long enough period of time.
We are really looking for a solution to the heat equation that is not dependent on time. Let us first solve the problem in one space variable. We are looking for a function \(u\) that satisfies
\[u_t=ku_{xx}, \nonumber \]
but such that \(u_t=0\) for all \(x\) and \(t\). Hence, we are looking for a function of \(x\) alone that satisfies \(u_{xx}=0\). It is easy to solve this equation by integration and we see that \(u=Ax+B\) for some constants \(A\) and \(B\).
Suppose we have an insulated wire, and we apply constant temperature \(T_1\) at one end (say where \(x=0\)) and \(T_2\) on the other end (at \(x=L\) where \(L\) is the length of the wire). Then our steady state solution is
\[u(x)= \dfrac{T_2-T_1}{L}x+T_1. \nonumber \]
This solution agrees with our common sense intuition with how the heat should be distributed in the wire. So in one dimension, the steady state solutions are basically just straight lines.
Things are more complicated in two or more space dimensions. Let us restrict to two space dimensions for simplicity. The heat equation in two space variables is
\[\label{eq:3} u_t=k(u_{xx}+u_{yy}), \]
or more commonly written as \( u_t=k \Delta u \) or \( u_t=k \nabla^2 u \). Here the \( \Delta \) and \( \nabla^2 \) symbols mean \( \dfrac{\partial^2}{\partial x^2}+ \dfrac{\partial^2}{\partial y^2}\). We will use \( \Delta\) from now on. The reason for using such a notation is that you can define \( \Delta\) to be the right thing for any number of space dimensions and then the heat equation is always \(u_t=k \Delta u\). The operator \( \Delta\) is called the Laplacian .
OK, now that we have notation out of the way, let us see what does an equation for the steady state solution look like. We are looking for a solution to Equation \(\eqref{eq:3}\) that does not depend on \(t\), or in other words \(u_t=0\). Hence we are looking for a function \(u(x,y)\) such that
\[ \Delta u = u_{xx}+u_{yy}=0. \nonumber \]
This equation is called the Laplace equation \(^{1}\) . Solutions to the Laplace equation are called harmonic functions and have many nice properties and applications far beyond the steady state heat problem.
Harmonic functions in two variables are no longer just linear (plane graphs). For example, you can check that the functions \(x^2-y^2\) and \(xy\) are harmonic. However, if you remember your multi-variable calculus we note that if \(u_{xx}\) is positive, \(u\) is concave up in the \(x\) direction, then \(u_{yy}\) must be negative and \(u\) must be concave down in the \(y\) direction. Therefore, a harmonic function can never have any “hilltop” or “valley” on the graph. This observation is consistent with our intuitive idea of steady state heat distribution; the hottest or coldest spot will not be inside.
Commonly the Laplace equation is part of a so-called Dirichlet problem \(^{2}\) . That is, we have a region in the \(xy\)-plane and we specify certain values along the boundaries of the region. We then try to find a solution \(u\) defined on this region such that \(u\) agrees with the values we specified on the boundary.
For simplicity, we consider a rectangular region. Also for simplicity we specify boundary values to be zero at 3 of the four edges and only specify an arbitrary function at one edge. As we still have the principle of superposition, we can use this simpler solution to derive the general solution for arbitrary boundary values by solving 4 different problems, one for each edge, and adding those solutions together. This setup is left as an exercise.
We wish to solve the following problem. Let \(h\) and \(w\) be the height and width of our rectangle, with one corner at the origin and lying in the first quadrant.
\[\begin{align} & \Delta u = 0 , & & \label{dirich:eq1} \\ & u(0,y) = 0 & & \text{for } 0 < y < h,\label{dirich:eq2} \\ & u(x,h) = 0 & & \text{for } 0 < x < w,\label{dirich:eq3} \\ & u(w,y) = 0 & & \text{for } 0 < y < h,\label{dirich:eq4} \\ & u(x,0) = f(x) & & \text{for } 0 < x < w.\label{dirich:eq5}\end{align} \]
The method we apply is separation of variables. Again, we will come up with enough building-block solutions satisfying all the homogeneous boundary conditions (all conditions except \(\eqref{dirich:eq5}\)). We notice that superposition still works for the equation and all the homogeneous conditions. Therefore, we can use the Fourier series for \(f(x)\) to solve the problem as before.
We try \(u(x,y)=X(x)Y(y)\). We plug \(u\) into the equation to get
\[X''Y+XY''=0. \nonumber \]
We put the \(X\)s on one side and the \(Y\)s on the other to get
\[ - \dfrac{X''}{X}= \dfrac{Y''}{Y}. \nonumber \]
The left hand side only depends on \(x\) and the right hand side only depends on \(y\). Therefore, there is some constant \( \lambda \) such that \( \lambda = \dfrac{-X''}{X}= \dfrac{Y''}{Y}\). And we get two equations
\[ X'' + \lambda X = 0, \\ Y'' - \lambda Y=0. \nonumber \]
Furthermore, the homogeneous boundary conditions imply that \(X(0)=X(w)=0\) and \(Y(h)=0\). Taking the equation for \(X\) we have already seen that we have a nontrivial solution if and only if \( \lambda= \lambda_n = \dfrac{n^2 \pi^2}{w^2}\) and the solution is a multiple of
\[ X_n(x) = \sin \left( \dfrac{n \pi}{w}x \right). \nonumber \]
For these given \( \lambda_n \), the general solution for \(Y\) (one for each \(n\)) is
\[\label{eq:14} Y_n(y)=A_n \cosh \left( \dfrac{n \pi}{w}y \right) + B_n \sinh \left( \dfrac{n \pi}{w}y \right). \]
We only have one condition on \(Y_n\) and hence we can pick one of \(A_n\) or \(B_n\) to be something convenient. It will be useful to have \(Y_n(0)=1\), so we let \(A_n=1\). Setting \(Y_n(h)=0\) and solving for \(B_n\) we get that
\[ B_n = \dfrac{ - \cosh \left( \dfrac{n \pi h}{w} \right)}{ \sinh \left( \dfrac{n \pi h}{w} \right)}. \nonumber \]
After we plug the \(A_n\) and \(B_n\) we into \(\eqref{eq:14}\) and simplify, we find
\[ Y_n(y) = \dfrac{ \sinh \left( \dfrac{n \pi (h-y)}{w} \right)}{ \sinh \left( \dfrac{n \pi h}{w} \right)}. \nonumber \]
We define \(u_n(x,y)=X_n(x)Y_n(y)\). And note that \(u_n\) satisfies \(\eqref{dirich:eq1}\) - \(\eqref{dirich:eq4}\).
Observe that
\[ u_n(x,0)= X_n(x)Y_n(0) = \sin \left( \dfrac{n \pi}{w}x \right). \nonumber \]
Suppose
\[ f(x)= \sum_{n=1}^{ \infty}b_n \sin \left( \dfrac{n \pi x}{w} \right). \nonumber \]
Then we get a solution of \(\eqref{dirich:eq1}\) - \(\eqref{dirich:eq5}\) of the following form.
\[ u(x,y)= \sum_{n=1}^{ \infty}b_n u_n(x,y) = \sum_{n=1}^{ \infty}b_n \sin \left( \dfrac{n \pi }{w}x \right) \left( \dfrac{ \sinh \left( \dfrac{n \pi (h-y)}{w} \right)}{ \sinh \left( \dfrac{n \pi h}{w} \right)} \right). \nonumber \]
As \(u_n\) satisfies Equation \(\eqref{dirich:eq1}\) - \(\eqref{dirich:eq4}\) and any linear combination (finite or infinite) of \(u_n\) must also satisfy \(\eqref{dirich:eq1}\) - \(\eqref{dirich:eq4}\), we see that \(u\) must satisfy Equations \(\eqref{dirich:eq1}\) - \(\eqref{dirich:eq4}\). By plugging in \(y=0\) it is easy to see that \(u\) satisfies \(\eqref{dirich:eq5}\) as well.
Example \(\PageIndex{1}\)
Suppose that we take \(w=h= \pi\) and we let \(f(x)= \pi\). We compute the sine series for the function \( \pi\)(we will get the square wave). We find that for \(0<x< \pi\) we have
\[ f(x)= \sum_ {\underset{n~ {\rm{odd}} }{n=1}}^{ \infty} \dfrac{4}{n} \sin(n x). \nonumber \]
Therefore the solution \(u(x,y)\), see Figure \(\PageIndex{2}\), to the corresponding Dirichlet problem is given as
\[ u(x,y)= \sum_ {\underset{n~ {\rm{odd}} }{n=1}}^{ \infty} \dfrac{4}{n} \sin(n x) \left( \dfrac{ \sinh(n( \pi-y)) }{ \sinh(n \pi) } \right). \nonumber \]
This scenario corresponds to the steady state temperature on a square plate of width \( \pi\) with 3 sides held at 0 degrees and one side held at \( \pi\) degrees. If we have arbitrary initial data on all sides, then we solve four problems, each using one piece of nonhomogeneous data. Then we use the principle of superposition to add up all four solutions to have a solution to the original problem.
A different way to visualize solutions of the Laplace equation is to take a wire and bend it so that it corresponds to the graph of the temperature above the boundary of your region. Cut a rubber sheet in the shape of your region—a square in our case—and stretch it fixing the edges of the sheet to the wire. The rubber sheet is a good approximation of the graph of the solution to the Laplace equation with the given boundary data.


7.5: Acids and Bases - The Lewis Definition
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Objectives
After completing this section, you should be able to
- state the Lewis definition of an acid and a base.
- identify a given compound as being a Lewis acid or Lewis base, given its Lewis structure or its Kekulé structure.
- identify an organic molecule or ion in a reaction as either an electrophile or nucleophile.
Key Terms
Make certain that you can define, and use in context, the key terms below.
- Lewis acid
- Lewis base
- Electrophile
- Nucleophile
Study Notes
The Lewis concept of acidity and basicity will be of great use to you when you study reaction mechanisms. The realization that an ion such as
is electron deficient, and is therefore a Lewis acid, should help you understand why this ion reacts with substances which are Lewis bases (e.g., H 2 O).
Introduction to Lewis Acid-Base Theory
The Brønsted acid-base theory has been used throughout the history of acid and base chemistry. However, this theory is very restrictive and focuses primarily on acids and bases acting as proton donors and acceptors. Sometimes conditions arise where the theory doesn't necessarily fit, such as in solids and gases. In 1923, G.N. Lewis from UC Berkeley proposed an alternate theory to describe acids and bases. His theory gave a generalized explanation of acids and bases based on structure and bonding. Through the use of the Lewis definition of acids and bases, chemists are now able to predict a wider variety of acid-base reactions. Lewis' theory used electrons instead of proton transfer and specifically stated that an acid is a species that accepts an electron pair while a base donates an electron pair.
Example of Lewis base (oxygen atom from carbonyl) reacting with Lewis Acid (Mg 2 + ion).
The reaction of a Lewis acid and a Lewis base will produce a coordinate covalent bond. A coordinate covalent bond is just a type of covalent bond in which one reactant donates both electrons to form the bond. In this case the Lewis base donates its electrons to form a bond to the Lewis acid. The resulting product is called an addition compound, or more commonly a complex. The electron-pair flow from Lewis base to Lewis acid is shown using curved arrows much like those used for resonance structures in Section 2.5 . Curved arrows always mean that an electron pair moves from the atom at the tail of the arrow to the atom at the head of the arrow. In this case the lone pair on the Lewis base attacks the Lewis acid forming a bond. This new type of electron pair movement, along with those described in Section 2.5 will be used throughout this text to describe electron flow during reactions.
- Lewis Acid : a species that accepts an electron pair and will typically either have vacant orbitals or a polar bond involving hydrogen such that it can donate H + (which has an empty 1s orbital)
- Lewis Base : a species that donates an electron pair and will have lone-pair electrons of pi bonding electrons.
Lewis Acids
Neutral compounds of boron, aluminum, and the other Group 13 elements , (BF 3 , AlCl 3 ), which possess only six valence electrons, have a very strong tendency to gain an additional electron pair. Because these compounds are only surrounded by three electron groups, they are sp 2 hybridized, contained a vacant p orbital, and are potent Lewis acids. Trimethylamine's lone pair elections are contained in an sp 3 hybrid orbital making it a Lewis base. These two orbitals overlap, creating a covalent bond in a boron trifluoride-trimethylamine complex. The movement of electrons during this interaction is show by by an arrow.
Positive ions are often Lewis acids because they have an electrostatic attraction for electron donors. Examples include alkali and alkaline earth metals in the group IA and IIA columns. K + , Mg 2 + and Ca 2 + are sometimes seen as Lewis acidic sites in biology, for example. These ions are very stable forms of these elements because of their low electron ionization potentials. However, their positive charges do attract electron donors.The interaction between a magnesium cation (Mg +2 ) and a carbonyl oxygen is a common example of a Lewis acid-base reaction. The carbonyl oxygen (the Lewis base) donates a pair of electrons to the magnesium cation (the Lewis acid). As we will see in Chapter 19 when we begin the study of reactions involving carbonyl groups, this interaction has the very important effect of increasing the polarity of the carbon-oxygen double bond.
The eight-electron rule does not hold throughout the periodic table. In order to obtain noble gas configurations, some atoms may need eighteen electrons in their valence shell. Transition metals such as titanium, iron and nickel may have up to eighteen electrons and can frequently accept electron pairs from Lewis bases. Transition metals are often Lewis acids. For example, titanium has four valence electrons and can form four bonds in compounds such as titanium tetrakis (isopropoxide), below, or titanium tetrachloride, TiCl 4 . However, the titanium atom in that compound has only eight valence electrons, not eighteen. It can easily accept electrons from donors.
For example, when THF and TiCl 4 are combined, a Lewis acid-base complex is formed, TiCl 4 (THF) 2 . TiCl 4 (THF) 2 is a yellow solid at room temperature.
The Proton as a Common Lewis Acid
Perhaps the most common example of a Lewis acid is also the simplest. It is the hydrogen cation (H + ) or proton. It is called a proton because, in most hydrogen atoms, the only particle in the nucleus is a proton. If an electron is removed to make a cation, a proton is all that is left. A proton is a Lewis acid for a number of reasons. It has a positive charge, and so it will attract electrons, which are negative. Also, it lacks the electron configuration of its noble gas neighbor, helium. Helium has two electrons. If a Lewis base or nucleophile donates a pair of electrons to a proton, the proton will obtain a helium noble gas configuration.
There is something about hydrogen cations that is not so simple, however. They are actually not so common. Instead, protons are generally always bound to a Lewis base. Hydrogen is almost always covalently (or coordinately) bonded to another atom. Many of the other elements commonly found in compounds with hydrogen are more electronegative than hydrogen. As a result, hydrogen often has a partial positive charge making is still act as a Lewis Acid.
A acid-base reaction involving protons might better be expressed as:
The Lewis acid-base interactions we have looked at so far are slightly different here. Instead of two compounds coming together and forming a bond, we have one Lewis base replacing another at a proton. Two specific movements of electrons are shown in the reaction both of which are show by arrows. Lone pair electrons on oxygen attack the hydrogen to form an O-H bond in the product. Also, the electrons of the H-Cl bond move to become a lone pair on chlorine as the H-Cl bond breaks. These two arrows together are said to represent the mechanism of this acid-base reaction.
Lewis Bases
What makes a molecule (or an atom or ion) a Lewis base? It must have a pair of electrons available to share with another atom to form a bond. The most readily available electrons are those that are not already in bonds. Bonding electrons are low in energy. Non-bonding electrons are higher in energy and may be stabilized when they are delocalized in a new bond. Lewis bases usually have non-bonding electrons or lone pairs this makes oxygen and nitrogen compounds common Lewis bases. Lewis bases may be anionic or neutral. The basic requirement is that they have a pair of electrons to donate.
Note 1: Ammonia
Ammonia, NH 3 , has a lone pair and is a Lewis base. It can donate to compounds that will accept electrons.
Ammonia donating to an electron acceptor or Lewis acid.
Not all compounds can act as a Lewis base. For example, methane, CH 4 , has all of its valence electrons in bonding pairs. These bonding pairs are too stable to donate under normal conditions therefore methane is not a Lewis base. Neutral boron compounds also have all electrons in bonding pairs. For example, borane, BH 3 has no lone pairs; all its valence electrons are in bonds. Boron compounds are not typically Lewis bases.
Exercise \(\PageIndex{1}\)
Which of the following compounds would you expect to be Lewis bases?
a) SiH 4 b) AlH 3 c) PH 3 d) SH 2 e) - SH
- Answer
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a) No, silicon has 4 valence electrons (like carbon) and all 4 are involved in sigma bonds
b) No, aluminum has 3 valence electrons and all 3 are involved in sigma bonds
c) Yes, phosphorus has 5 valence electrons, so there is one lone pair available
d) Yes, sulfur has 6 valence electrons, so there are two lone pairs available
e) Yes, this ion has 3 lone pairs available
Lewis Acid-Base Complexes
What happens when a Lewis base donates a pair of electrons to a Lewis acid? The electron pushing formalism (arrows) we have been using to illustrate the behavior of Lewis acids and Lewis bases is meant to show the direction of electron movement from the donor to the acceptor. However, given that a bond can be thought of as a pair of electrons that are shared between two atoms (in this case, between the donor and the acceptor), these arrows also show where bonds are forming.
The electrons donated from a Lewis base to a Lewis acid form a new bond. A new, larger compound is formed from the smaller Lewis acid and Lewis base. This compound is called a Lewis acid-base complex. A simple example of Lewis acid-base complexation involves ammonia and boron trifluoride. The nitrogen atom has a lone pair and is an electron donor. The boron has no octet and is an electron acceptor. The two compounds can form a Lewis acid-base complex or a coordination complex together.
When the nitrogen donates a pair of electrons to share with the boron, the bond that forms is sometimes called a coordinate bond. A coordinate bond is any covalent bond that arose because one atom brought a pair of its electrons and donated them to another.
In organic chemistry terminology, the electron donor is called a nucleophile and the electron acceptor is called an electrophile . Ammonia is a nucleophile and boron trifluoride is an electrophile.
- Because Lewis bases are attracted to electron-deficient atoms, and because positive charge is generally associated with the nucleus of an atom, Lewis bases are sometimes refered to as "nucleophiles". Nucleophile means nucleus-loving.
- Because Lewis acids attract electron pairs, Lewis acids are sometimes called "electrophiles". Electrophile means electron-loving.
Exercise \(\PageIndex{2}\)
For the following reaction, add curved arrows (electron pushing formalism) to indicate the electron flow.
- Answer
-
Exercise \(\PageIndex{3}\)
A Lewis acid-base complex is formed between THF (tetrahyrofuran) and borane, BH 3 .
a) Which compound is the Lewis acid? Which one is the Lewis base?
b) Which atom in the Lewis acid is the acidic site? Why?
c) Which atom in the Lewis base is the basic site? Why?
d) How many donors would be needed to satisfy the acidic site?
e) Show, using arrow notation, the reaction to form a Lewis acid-base complex.
f) Borane is highly pyrophoric; it reacts violently with air, bursting into flames. Show, using arrow notation, what might be happening when borane contacts the air.
g) Borane-THF complex is much less pyrophoric than borane. Why do you suppose that is so?
Add exercises text here.
- Answer
-
a) Borane is the Lewis acid. THF has lone pair electrons so it is the Lewis base.
b) The Boron atom has an unfilled octet so it has an empty p orbital that can accept electrons.
c) The oxygen atom in THF has lone pair electrons contained in a sp 3 hybridized orbital.
d) The boron in borane has six electrons around it so it would only need one lone pair donor to reach an octet.
e) Show, using arrow notation, the reaction to form a Lewis acid-base complex.
f) The Borane initially reacts with water in air.
g) After the Borane-THF complex is formed, the boron atom has a complete octet making it less reactive.


Questions to consider:
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How does self-care benefit relationships?
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Why is community so important to healthy relationships?

Relationships are key to happy and healthy lives. According to Dr. Robert Waldinger, director of the Harvard Study of Adult Development, people with the best health outcomes were people who “leaned into relationships, with family, with friends, with community.”
Figure 13.11: Healthy relationships involve trust, respect, and support. (Credit: Garry Knight / Flickr / Attribution 2.0 Generic (CC-BY 2.0))
The quality of our relationships is important, however. What makes a relationship healthy? Relationships come in many forms: lovers, family, friends, coworkers, team members, and neighbors. Think of a relationship where you have mutual respect and trust, supporting each other in tough times, celebrating the good times, and communicating with ease and honesty. This is a healthy relationship. Do you have someone in mind? On the other hand, if communication is often tense or strained, confidences are broken, or you don’t feel listened to, appreciated, or valued, these are signs of an unhealthy relationship. Unhealthy relationships can have both immediate and longer-term health impacts. If you are unhappy in a relationship, try to improve the relationship, or end it. Do not stay in a relationship for the wrong reasons, such as fear of being alone or guilt.
If a partner tries to force you to do something sexually, harms you physically, or is verbally abusive, you are in an unhealthy relationship. Even if you believe the person loves you, it does not make up for the harm they are doing to you. End the relationship.
Take a moment to assess the health of your relationships. Who are the people who make you smile, who boost your confidence, who truly listen when you need to talk, and who want only the best for you? Investing in these relationships is likely to make you happier and healthier. Relationships are two-way streets. How committed are you to your relationships? How much effort do you put into nurturing your relationships?
Self-Care
Healthy relationships start with healthy individuals. Self-care is learning to take good care of yourself and to prioritize your own needs. Self-care involves any activity that nurtures and refuels you, such as taking a walk in the woods, going to a yoga class, attending a sporting event, reading a good book, or spending time with friends. When you are feeling calm and nourished, you are going to look forward to your day, and despite how busy it is, you will prioritize time with friends and family. If you don’t take care of and learn to love yourself, you will never be able to bring your best self to any relationship.
An important dynamic you bring to any relationship is how you feel about yourself. Self-esteem is about loving yourself and being happy for who you are. Building healthy self-esteem impacts how you see yourself, which can drastically improve your relationships. While low self-esteem won’t keep us from romantic love, it can act as a barrier to a healthy relationship. If you do not believe you are good enough, how can you expect your partner to think so?
When you feel secure in yourself, this allows you and your partner to feel more secure about the relationship. If you have insecurities, it may show in your relationship as jealousy, defensiveness, or tension that leads to unnecessary arguments. Healthy self-esteem goes hand in hand with self-confidence, and feeling confident about yourself will translate into a stronger and more satisfying relationship. If you are experiencing low self-esteem, you may give your partner too much credit or stay in a relationship that is not healthy for you. If you find yourself changing your personality for someone else, that is never a sign of a healthy relationship.
You can reverse negative self-talk and build your self-esteem. If you catch yourself thinking you are unlovable, unattractive, or not good enough, it’s important to start talking to yourself in a positive way and to celebrate all that is uniquely you.
Self-care includes self-forgiveness. We all make mistakes. A misstep isn’t the end of the world. Pick yourself up, put things in perspective, acknowledge any lessons to be learned, focus on all that makes you special, and move forward. Be kind to yourself.
The Importance of Community
The Nicoya Peninsula in Costa Rica is home to some of the highest number of centenarians (people who are 100 years old or older) in the world. Costa Ricans in general report a high level of life satisfaction. Dan Buettner, author of the Blue Zones study of the longest living populations in the world, explains that Costa Rica “is a place where religion, family, and social interaction are the main values, unlike trying to get ahead, or financial security, or status. Their cities are set up so they’re bumping into each other all day long. They walk to the markets, where they have conversations with people.”
26
In Costa Rica, multiple generations live together under the same roof or nearby where they can be involved in each other’s lives. Neighbors are like extended family, and people often stop in for a visit and go out of their way to help one another.
While this isn’t the way many of us live in the United States, the lessons from the Blue Zone study underscore the importance of community and the health benefits of connecting to and staying close to a community. What communities do you belong to? Is your dorm a community? A sports team? A club or people you volunteer with? When you start seeing the social circles you connect to as communities and prioritize your time to develop more closeness with those communities, you will experience many physical, mental, and emotional health benefits.
Figure 13.12: Joining clubs in college can be an outstanding way to join and build communities. (Credit: SupportPDX, Cerritos College / Flickr / Attribution 2.0 Generic (CC-BY 2.0))
According to an analysis of research on college students (Joe Cuseo,
The Most Potent, Research-Based Principles of College Success
), college students who have a higher sense of belonging and are more involved in their college community are more successful. Additionally, college students who are involved in extracurricular, volunteer, and part-time work experiences outside the classroom (less than 20 hours per week) earn higher grades than students who do not get involved in any out-of-class activities at all.
APPLICATION
Make a list of the communities you belong to. Your list should include formal communities—for example, sports teams, fraternities or sororities, and membership in clubs and other organizations. Your list should also include informal communities—for example, your neighbors or the people you always see at your favorite exercise class.
Next to each community, write how being a member of this community benefits you and how your involvement benefits the community. Now, make a new list of your personal interests and passions. How well do these align with the communities you already belong to? Are there new communities that would be a good fit for you?
 Find a group or club that aligns with your interests. If you can’t find one that already exists, start a new club!
Research has shown that friends provide a sense of meaning or purpose in our lives, and that having a healthy social life is important to staying physically healthy. In a meta-analysis of the research results from 148 studies of over 300,000 participants, researchers found that social relationships are important in improving our lifespan. Social support has been linked to lower blood pressure and better immune system functioning. The meta-analysis also showed that social support operates on a continuum: the greater the extent of the relationships, the lower the health risks.
27
According to a 2018 report from the American College Health Association, in a 12-month period, 63 percent of college students have felt very lonely. If you are feeling lonely or having a hard time making friends, know that the majority of people around you have also felt this way. Joining a group or a club of people who share your interests and passions is one of the best ways to make great friends and stay connected.
Sexual Health
Affection, love, and sexual intimacy all play an important role in healthy relationships, and a responsible approach to intimacy is essential for sexual health. Whether you are already sexually active or become sexually active in the future, your choices can affect your safety as well as the health and safety of your sexual partners. It’s important to understand what you can do to protect yourself from sexually transmitted infections (STIs).
Table 13.1 STI data, symptoms, diagnosis, and treatment information courtesy of Centers for Disease Control and Prevention,
CDC
[www.cdc.gov]
|
Common Sexually Transmitted Infections
|
|
Infection
|
Symptoms
|
Diagnosis and Treatment
|
|
Human papillomavirus (HPV)
|
-
HPV can be passed even when an infected person has no signs or symptoms.
-
Most people with HPV do not know they are infected.
-
Symptoms can include genital warts, abnormal Pap test results, and cancer.
|
-
There is no test for HPV.
-
There is a vaccine to prevent it.
-
There is no treatment for HPV, although there are treatments for the health problems it can cause.
-
Routine Pap tests can identify problems.
|
|
Chlamydia
|
-
Symptoms include a burning sensation when urinating and/or discharge from the penis or vagina; however, most people who have chlamydia have no symptoms.
-
In women, it can cause damage to the reproductive system.
|
-
Testing usually involves a urine sample or vaginal swab.
-
It can be cured with the right treatment.
|
|
Genital herpes
|
-
Genital herpes is caused by two types of viruses, herpes simplex type 1 and herpes simplex type 2.
-
The virus can be released through sores or through the skin even when sores are not visible.
-
Get examined by your doctor if you notice an unusual sore, a smelly discharge, or burning when urinating.
|
-
It can be diagnosed through the symptoms, testing a sample from the sore(s), or a blood test.
-
There is no cure for herpes; however, there are medicines that can prevent or shorten outbreaks.
|
|
Gonorrhea
|
-
Symptoms can include a burning sensation when urinating, abnormal discharge from the penis or vagina, and bleeding between periods.
-
Rectal infection symptoms include itching, burning, and bleeding.
|
-
Get examined by your doctor if you or your sexual partner notice any of these symptoms.
-
Testing is usually a urine sample and possibly a throat or rectum swab.
-
It can be cured with the right treatment.
-
Medication will stop the infection, but it will not undo any permanent damage caused by the disease.
|
|
Human immunodeficiency virus (HIV)
|
-
HIV damages the body's immune system cells.
-
The most advanced stage of HIV infection is commonly referred to as AIDS (acquired immunodeficiency syndrome).
-
It most often spreads through fluid exchange via unprotected sex or by sharing drug needles with an infected person.
-
Women can pass HIV to their babies during pregnancy or childbirth.
|
-
Medications such as pre-exposure prophylaxis (PrEP) can reduce risk when taken consistently and with other prevention measures.
-
People can live with the disease for many years, especially if they are diagnosed and treated early.
-
Early diagnosis is also important to reduce the risk of transmitting HIV to others.
|
How You Can Protect Yourself against STIs
The surest way to protect yourself against STIs is to practice abstinence. This means not having any vaginal, anal, or oral sex. There are many things to consider before having sex, and it’s okay to say no if you are not ready. If you do decide to have sex, you and your partner should both get tested beforehand and make sure you always use a condom. It’s not safe to stop using condoms unless you’ve both been tested, know your status, have another form of birth control, and are in a mutually monogamous relationship.
 This can help protect against STIs as long as you’ve both been tested and know you’re STI-free. Visit
this website to find a confidential STI testing location near you(opens in new window)
.
Before you have sex, talk with your partner about how you will prevent STIs and pregnancy. If you think you’re ready to have sex, you need to be ready to protect your body and your future. You should also talk to your partner ahead of time about what you will and will not do sexually. Your partner should always respect your right to say no to anything that doesn’t feel right. Sex should always be consensual and respectful.
It’s important to discuss treatment with your doctor and begin treatment as soon as possible if you find out you have an STI. If you are living with an STI, it’s important to tell your partner before you have sex. Although it may be uncomfortable to talk about your STI, open and honest conversation can help your partner make informed decisions to protect his or her health.
Footnotes
-
Blue Zones
[www.bluezones.com]
-
Holt-Lunstad,
PLoS Medicine,
PLOS One
[journals.plos.org]


2.4: Box Plots, Quartiles, and Percentiles
- Introduce box plots
- Define quartiles
- Define percentiles
- Calculate percentiles
- Calculate values for a five-number summary
Section \(2.4\) Excel File (contains all of the data sets for this section)
Using Box Plots to Visualize Data
Frequency distributions and their graphs (bar graphs and histograms) provide insight into data by grouping observations into classes and then determining each class's frequency or relative frequency. The classes depend on the values our data takes on, and there is some freedom regarding the number of classes we might choose to separate our data into.
Another method of graphing ordinal, interval, or ratio level data, called a box plot (or a box-and-whisker plot), groups data into four classes based on order, each containing approximately \(25\%\) of the observations. Consider the following figure containing three box plots relating students' final grades in statistics across different universities.
Figure \(\PageIndex{1}\): Box plots of final statistics grades for three universities
We know that there are four classes for each box plot. Note the five vertical lines; these values correspond to the boundaries of the classes. The left-most line corresponds to the data set's smallest value, the minimum. The first class extends from the minimum to the left side of the "box" and includes \(25\%\) of the observations; we call the upper bound for this first class, the first quartile \(Q_1\) (one-quarter of the observations are less than or equal to it). The second class again needs to have \(25\%\) of the observations with a lower bound of \(Q_1\) and an upper bound \(Q_2\). We call the upper bound of the second class the second quartile \(Q_2\) , which is most commonly referred to as the median, meaning 50% of the data fall below this value. The third class again needs to have \(25\%\) of the observations with a lower bound of \(Q_2\) and an upper bound of the third quartile \(Q_3\) ; \(75\%\) of the observations are less than or equal to \(Q_3\). The final class consists of the remaining \(25\%\) of observations with lower bound \(Q_3\) and upper bound of the largest value, the maximum , of the data set. The first and fourth classes form the whiskers of the box plot, while the second and third form the box. These five numbers ( minimum , \(Q_1\), \(Q_2\), \(Q_3\), and maximum) form what we call the five-number summary of the data.
Consider Figure \(2.4.1\) above and classify each box plot as positively skewed, negatively skewed, or symmetric. Explain.
- Answer
-
We preface our answer by noting that box plots, like histograms and bar graphs from grouped frequency distributions, are formed by grouping observations. Since the graph does not provide information about each data value, we are primarily making a claim about a characteristic of the graphical representation.
Recall that a graph is positively skewed if the right tail extends further than the left tail and negatively skewed if the left tail extends further than the right tail. A graph is symmetric if we can fold it in half so that the left and right sides roughly match. The tails and the whiskers fall in similar parts of the graphs discussed. If one whisker is longer, we can say that the box plot is skewed in the direction of the longer whisker. We classify the box plot of University C as negatively skewed and the box plot of University B as positively skewed. The whiskers seem to be of equal length for University A, but that is not enough to assert symmetry. We also want the two halves of the box to be the same. This is the case for the box plot of University A. We classify the box plot of University A as symmetric.
Given a data set, we can quickly identify the minimum and maximum values. Determining the quartiles, however, presents more of a challenge. With an ordered data set, we understand where a quarter, half, and three-quarters of the data would fall. Consider the following data sets with their five-number summaries.
Figure \(\PageIndex{2}\): Three data sets with intuitive placement of the box plot boundaries
We need help determining our five-number summary for these data sets.
The first two data sets each contain \(8\) observations, while the third contains \(9\) observations. We can easily group the first two data sets in groups of \(2\) to get \(25\%\) of the observations in each class. With \(9\) observations, however, we cannot get exactly \(25\%\) in each class. However, note that there are equal numbers of observations above and below \(Q_2\) for each data set.
If we try to attach values to quartiles, we face another challenge. Let us begin with \(Q_1\). In the first data set, we see that \(Q_1\) naturally falls between \(4\) and \(6,\) but what value should we assign? Our box plots would look significantly different if we used \(6\) as opposed to \(4\). There is no easy solution to this challenge, and statisticians have developed a variety of approaches. We will provide a simple approach later in this section; please remember that it is not the only approach. However, most of the various approaches produce measures that are reasonably close to each other.
We face another challenge when we try to understand \(Q_1\) in the second data set. We would naturally assign a value of \(4\) to \(Q_1\) because the only number between \(4\) and \(4\) is \(4\). We wanted \(25\%\) of the observations to be less than or equal to \(Q_1,\) but we have \(3\) values that are less than or equal to \(4\) in our data set, that is \(37.5\%\) of our observations.
We highlight these challenges to frame our understanding appropriately. We use box plots, quartiles, and percentiles (which we will get to shortly) to get a general, intuitive feel about our data using methods that may differ from field to field, statistician to statistician, and program to program. When consuming statistics or conducting analysis, know which method is in use.
Quartiles are descriptive statistics that express at what values there will be about \(25\%,\) \(50\%,\) or \(75\%\) of the observations at or below that value. There is nothing extraordinarily unique about \(25\%,\) \(50\%,\) or \(75\%.\) We could choose \(10\%\) or \(99\%.\) When we expand our ideas to include different percentages of observations, we call them percentiles . \(Q_1\) is the \(25^{th}\) percentile. \(Q_2\) is the \(50^{th}\) percentile. \(Q_3\) is the \(75^{th}\) percentile.
Percentiles have utility beyond building summary visualizations; they help us understand how individual observations compare to the entire data set. They measure relative position within an ordered data set. For example, a test score by itself is usually difficult to interpret. For instance, if one of us had a score on a measure of shyness of \(35\) out of a possible \(50,\) we would have little idea how shy that person was compared to others. It would be helpful to know the percentage of people with equal or lower shyness scores. If \(65\%\) of the scores were at or below this person's score, then the score would be at the \(65^{th}\) percentile.
- If Helen's score was at the \(95^{th}\) percentile, what percentage of scores are at or below Helen's?
- Answer
-
The percentile means that \(95\%\) of the scores are at or below Helen's score.
- If the scores ranged from \(1\) to \(100\) on an exam and Helen earned a score of \(95,\) does this necessarily mean that her score is at the \(95^{th}\)percentile?
- Answer
-
No, the percentile gives a relative position of the scores. The number of scores at or below her score determines the percentile measure. If everyone did well and only \(70%\) of the scores fell at or below Helen's, she would be at the \(70^{th}\) percentile even though she got \(95\) out of a \(100\) points.
Calculating Percentiles
We already indicated that there are several different ways to calculate quartiles. This is because quartiles are percentiles, and there are several ways to calculate percentiles, which may lead to different values in different situations. The method that we present is one of the simplest calculations.
The \(P^{th}\) percentile is a value such that \(P\%\) of the observations fall at or below that value. We need the data to be counted and ordered from smallest to largest. Let \(n\) be the number of observations in our data set. Next, we calculate the number of observations that make up \(P\%\) of the observations. We call this number the rank \(R\) of the percentile.\[R=\frac{P}{100}\cdot n\nonumber\]Now there are two possibilities for the \(R\) value; either it will be a natural number \(\{1,\) \(2,\) \(3,\) \(4,\) \(5,\) \(...\}\) or not.
- If \(R\) is a natural number, we find the midpoint between the \(R^{th}\) and \((R+1)^{st}\) values.
- If \(R\) is not a natural number, round \(R\) up to the following natural number and take the value in that position.
The real number system has the following designations.
Natural Numbers: \(1,\) \(2,\) \(3,\) \(\dots\)
Whole Numbers: \(0,\) \(1,\) \(2,\) \(3,\) \(\dots\)
Integers: \(\dots,\) \(-3,\) \(-2,\) \(-1,\) \(0,\) \(1,\) \(2,\) \(3,\) \(\dots\)
Rational Numbers: Any number that can be written as a fraction
Irrational Numbers: Any number that cannot be written as a fraction. Examples include \(e,\) \(\pi,\) \(\sqrt{2}\)
Consider the \(20\) quiz scores shown in the table below and compute the five-number summary and \(82^{nd}\) percentile. After completing the calculation by hand, use the Section \(2.4\) Excel file to calculate each percentile using the functions \(\text{PERCENTILE.INC}\) and \(\text{PERCENTILE.EXC}.\) Compare the values.
Table \(\PageIndex{1}\): \(20\) quiz scores with corresponding rank
| Number | \(4\) | \(4\) | \(4\) | \(5\) | \(6\) | \(6\) | \(6\) | \(6\) | \(6\) | \(6\) | \(7\) | \(7\) | \(8\) | \(8\) | \(8\) | \(9\) | \(9\) | \(10\) | \(10\) | \(10\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Rank | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) | \(13\) | \(14\) | \(15\) | \(16\) | \(17\) | \(18\) | \(19\) | \(20\) |
- Answer
-
We first note that the data is already ordered from smallest to largest with \(20\) observations. A secondary row has been created to index the observations. Note that the row heading is Rank; consider how this ties back to rank \(R\) in our calculation.
Let us begin with the five-number summary.
The minimum value is \(4.\)
\(Q_1,\) the \(25^{th}\) percentile. \(R=\frac{25}{100}\cdot 20\) \(=\frac{1}{4}\cdot 20\) \(= \frac{20}{4}\) \(=5.\) Note \(5\) is a natural number. We then look at the \(5^{th}\) and \(6^{th}\) observation values, which are both \(6\) and find the midpoint between them. Thus \(Q_1\) \(=6.\) Using Excel, we get \(6;\) they are all the same.
\(Q_2,\) the \(50^{th}\) percentile. \(R=\frac{50}{100}\cdot 20\) \(=\frac{1}{2}\cdot 20\) \(=\frac{20}{2}\) \(=10.\) Note \(10\) is a natural number. We then look at the \(10^{th}\) and \(11^{th}\) observation values, which are \(6\) and \(7,\) respectively and find the midpoint between them. Thus \(Q_2\) \(=\frac{1}{2}(6+7)\) \(=\frac{13}{2}\) \(=6.5.\) Excel computes \(6.5\) and \(6.5;\) they are all the same.
\(Q_3,\) the \(75^{th}\) percentile. \(R=\frac{75}{100}\cdot 20\) \(=\frac{3}{4}\cdot 20\) \(= \frac{60}{4}\) \(=15.\) Note \(15\) is a natural number. We then look at the \(15^{th}\) and \(16^{th}\) observation values, which are \(8\) and \(9,\) respectively and find the midpoint between them. Thus \(Q_3\) \(=\frac{1}{2}(8+9)\) \(=\frac{17}{2}\) \(=8.5.\) Excel gives \(8.25\) and \(8.75;\) they are all different.
The maximum value is \(10.\)
Let us look at the \(82^{nd}\) percentile. \(R=\frac{82}{100}\cdot 20\) \(=\frac{41}{50}\cdot 20\) \(=\frac{820}{50}\) \(= \frac{82}{5}\) \(=16.4.\) Notice \(16.4\) is not a natural number. We must look at the \(17^{th}\) observation value showing the \(82^{nd}\) percentile is \(9.\) Excel calculates \(9.22\) and \(9;\) they are different and the same respectively.
We have seen that different methods of calculation can produce slightly different values. With large data sets, we generally resort to technology to produce our measures and might not have control over the precise methodology employed therein. As such, we remember that percentiles provide rough measures for the distribution of our data sets and nuance our understanding that roughly this percent of observations fall below roughly this value. When large data sets or limited time make hand computation prohibitive, we recommend using functions such as \(\text{PERCENTILE.INC}\) and \(\text{QUARTILE.INC}.\)
Consider the preceding example. If we looked at the \(83^{rd}\) percentile \((R\) \(=16.6)\) which would also be the \(17^{th}\) value, which is \(9.\) So, the \(83^{rd}\) percentile is the same as the \(82^{nd}\) percentile. This happens since there are only \(20\) data points; we cannot subdivide \(20\) indefinitely. The two percentiles would typically differ with large data sets.
Box Plots: Constructing and Interpreting
As part of the " Stroop Interference Case Study, " students in introductory statistics were presented with a page containing \(30\) colored rectangles. Their task was to name the colors as quickly as possible. Their times (in seconds) were recorded. Compare the scores for the \(15\) men and \(30\) women who participated in the experiment by making separate box plots for each gender.
Table \(\PageIndex{2}\): Women's times (left) and men's times (right)
| \(14\) | \(17\) | \(18\) | \(19\) | \(20\) | \(21\) | \(16\) | \(19\) | \(23\) | |
| \(15\) | \(17\) | \(18\) | \(19\) | \(20\) | \(22\) | \(17\) | \(20\) | \(24\) | |
| \(16\) | \(17\) | \(18\) | \(19\) | \(20\) | \(23\) | \(18\) | \(22\) | \(26\) | |
| \(16\) | \(17\) | \(18\) | \(20\) | \(20\) | \(24\) | \(19\) | \(22\) | \(26\) | |
| \(17\) | \(18\) | \(18\) | \(20\) | \(21\) | \(24\) | \(19\) | \(23\) | \(28\) |
- Answer
-
To construct box plots, we need the five-number summaries.
Table \(\PageIndex{3}\): Five number summaries for the data presented in Table \(\PageIndex{1}\)
Females Males Box Plot Component Minimum \(14\) \(16\) End of Left Whisker \(Q_1\) \(17\)
(\(R=\frac{25}{100}\cdot 30=7.5\))
\(19\)(\(R=\frac{25}{100}\cdot 15=3.75\))
Left Side of Box \(Q_2\)=median
\(18.5\)(\(R=\frac{50}{100}\cdot 30=15\))
\(22\)(\(R=\frac{50}{100}\cdot 15=7.5\))
Line in Box \(Q_3\)
\(20\)(\(R=\frac{75}{100}\cdot 30=22.5\))
\(24\)
(\(R=\frac{75}{100}\cdot 15=11.25\))
Right Side of Box Maximum \(24\) \(28\) End of Right Whisker Figure \(\PageIndex{3}\): Box plots for male and female times for naming the colors of various rectangles
The men tended to take longer than the women. About \(25\%\) of male times were longer than the maximum female time. At least \(75\%\) of the male times were longer than the median female time.
Suppose data came from a task that aims to move a computer mouse to a target on the screen as fast as possible. On \(20\) of the trials, the target was a small rectangle; on the other \(20,\) the target was a large rectangle. The time to reach the target was recorded on each trial. The box plots of the two distributions are shown below. What can we conclude by looking at the two box plots?
Figure \(\PageIndex{4}\): Box plots for the response times by small and large target
- Answer
-
We can see that although there is some overlap in times, it generally took longer to move the mouse to the small target than to the large one. The minimum time for the small target is longer than the median time of the large target. At least \(50\%\) of times for the small target are longer than all of the large target times.
Construct two data sets, treated as observations from a discrete quantitative variable, consisting of \(12\) values each so that the box plots are identical, but the bar graph of one data set is perfectly symmetric while the other is not.
- Answer
-
If the box plots are going to be identical, the five-number summaries must be the same. When arranged from least to greatest, \(Q_1\) is the average of the \(3^{rd}\) and \(4^{th}\) values, \(Q_2\) is the average of the \(6^{th}\) and \(7^{th}\) values, and \(Q_3\) is the average of the \(9^{th}\) and \(10^{th}\) values. We can construct a perfectly symmetric data set by pairing observations by proximity to the center. Since the \(6^{th}\) and \(7^{th}\) values are in the middle, they would be paired together, the \(5^{th}\) with the \(8^{th}\), and the \(4^{th}\) and \(9^{th}\), and so forth. We want the values in these positions to be equally distant from the median value.
We can start by picking the first data set that is not perfectly symmetric. We will produce such a data set if we use a pattern of one observation value followed by a different value repeated twice. We picked even numbers starting at \(0\) to ensure that our quartiles were nice values. Not all procedures would produce a data set suitable for this example because we need our second data set to be symmetric. We must check this because we want the minimum and maximum to be equally far from the median and the first and third quartiles.
\[\{0,2,2,4,6,6,8,10,10,12,14,14\}\nonumber\]
We have fixed our five number summary: \(\min=0\),\(Q_1=3\), \(Q_2=7\), \(Q_3=11\), and \(\max=14\). Our procedure produced values allowing us to produce a symmetric data set with the same five-number summary. The average of two of the same numbers is that number, so, for ease, we can start our symmetric data set with the following numbers. \[ \{0,~~,3,3,~~,7,7,~~,11,11,~~,14\}\nonumber \] All we need to do is fill in the remaining spots, ensuring symmetry and preserving order.
\[\{0,2,2,4,6,6,8,10,10,12,14,14\}\nonumber\\ \{0,1,3,3,4,7,7,10,11,11,13,14\} \]
Figure \(\PageIndex{5}\): Two identical box plots
Figure \(\PageIndex{6}\): Non-identical bar graphs of the two data sets that produced identical box plots.


33.1: The Importance of Nurse Advocacy
By the end of this section, you should be able to:
- Examine the role of advocacy in promoting change and advancing a cause.
- Explain how nursing has historically been concerned with the social, emotional, and physical needs of people whose incomes are below the federal poverty threshold.
- Describe how the nurse advocate can advance public health and promote health equity through targeted social determinants.
Nursing is by far the largest health care profession in the United States, with over 4.6 million practicing registered nurses and 203,000 new positions created each year (American Association of Colleges of Nursing [AACN], 2023). There are many more nurses than physicians in the United States, but nurses are underrepresented in public health advocacy issues. However, given that nurses have been recognized as the most trusted profession in the United States for the past two decades (American Nurses Association [ANA], 2023b), they are well positioned to advocate for and protect clients’ civil rights, equality, health, and safety. Nurses are aware of the inequities in health care, especially among those most vulnerable. Nurses often translate health care to their clients and the public, as they possess the knowledge and skills to promote health using a compassionate approach to address health equity for all persons. Nurses educate clients on how to adapt their lifestyles to accommodate a host of health conditions. They effectively communicate the barriers to health in communities, coordinate effective health strategies, and educate persons and communities on methods to improve their health.
The Role of Nurse Advocates
Nurses can advocate for community-based changes that disrupt the barriers to equitable health care. According to the ANA (2023b), nurses advocate by promoting the rights, health, and safety of their clients. As advocates, nurses have historically championed a wide range of efforts to ensure that services, policies, and regulations meet the needs of individuals, families, and communities. Nurses promote justice, fairness, and equity in health care and address the social inequities that affect health. As discussed in Social Determinants Affecting Health Outcomes, the social determinants of health (SDOH) are the conditions in which people are born, grow, live, work, and age (U.S. Department of Health & Human Services [HHS], 2020). These determinants often guide nursing practice and health policy development, as they overwhelmingly affect population health, well-being, and quality of life. The SDOH include access to safe housing, transportation, and neighborhoods; elimination of racism , discrimination , and violence; access to education, job opportunities, and adequate income; access to nutritious foods and physical activity; access to vaccination s and clean air and water; and elimination of linguistic barriers that limit literacy . For example, a family’s access to grocery stores with quality fresh foods improves overall nutrition.
In 2020, researchers measured the confidence of 768 Midwestern nurses in discussing the SDOH with their clients. In this study, nurses expressed that they were confident asking their clients about certain determinants of health, such as access to a primary health care provider (53.7 percent) and general access to health care (46.7 percent), but were less confident discussing income (50.4 percent), civic participation (48.8 percent), crime and violence (39.1 percent), utilities (38.8 percent), and interprofessional violence (34.8 percent).
This ignited a call for enhanced skill development in SDOH interviewing to improve the working knowledge of the SDOH. This skill development improved the nurses’ ability to advocate for their clients by establishing referrals for services that promote health access and delivery for their communities (Phillips et al., 2020).
In this video, Rear Admiral and Assistant Surgeon General Jonathan Mermin discusses why a focus on the SDOH improves health outcomes and promotes health equity.
Watch the video, and then respond to the following questions.
- What are the pillars of the social determinants of health?
- What wrap-around services would the nurse explore in order to improve health equity?
- When providing care to a population, what factors outside of the health care system impact the ability to maintain health?
Nurses’ ability to advocate for populations begins with an appropriate assessment. Several tools are available to identify the social determinants of health (Tiase et al., 2022), and although using this approach has shown some success (Buitron de la Vega et al., 2019), a universal approach to assessing the SDOH remains a challenge. In some areas, nurses now screen clients in primary care clinic settings and inpatient settings to determine what social barriers prevent them from maintaining or improving their health. This has led to an increase in community-based referrals and new roles for nurses as nurse navigators (Tiase et al., 2022).
The History of Nursing Advocacy
The history of nursing advocacy involves nurses advocating for client rights, well-being, dignity, and health care improvements. Nurses have historically leveraged their unique frontline position to drive positive change. Public health nurses have identified novel ways to protect children and vulnerable populations and continue to have a strong voice for social changes that promotes health. Today’s nurses stand on the shoulders of nurses who have made history by creating positive changes in their communities. Table 33.1 highlights a few of the many nursing leaders who have advocated for significant changes in public health.
| Nursing Leaders | Changes for Which They Advocated |
|---|---|
| Florence Nightingale |
|
| Dorothea Dix |
|
| Mary Eliza Mahoney |
|
| Sojourner Truth |
|
| Clara Barton |
|
| Lavinia Dock |
|
| Lillian Wald |
|
| Mary Breckinridge |
|
| Susie Walking Bear Yellowtail |
|
| Ruth Watson Lubic |
|
Nurses must address the social, emotional, and physical needs of people whose health needs are underrepresented. For example, in the 1960s, Loretta Ford witnessed that primary care in the underserved communities surrounding Denver, Colorado, was desperately lacking. Using innovative thinking, specialized instruction, and novel curriculum design, Ford launched the first nurse practitioner program in 1965. Today, the nurse practitioner model has become so successful in meeting the needs of clients, families, and communities that several advanced nursing programs and certifications have been created to address the social determinants of health at the master’s and doctoral levels.
In the current state of health care, nurse practitioners, clinical nurse specialists, certified nurse anesthetists, and certified nurse midwives work in advanced practice clinical roles in community specialty practices such as pediatrics, maternal and family health, gerontology, mental health, and perioperative care and in a broad spectrum of institutional roles. Whether in a role as a registered nurse (RN) or an advanced practice registered nurse (APRN), every nurse has an opportunity and obligation to advocate for persons whose social, emotional, and physical needs are unmet and for persons with incomes below the federal poverty threshold in targeted social programs and strategies. However, the shortage of adequate health care professionals continues to be the most profound in rural areas in the United States as well as underserved communities and in care for older adults. The Title VIII Nursing Workforce Reauthorization Act of 2019 was developed and supported by nursing organizations to help meet the accelerating need for nurses to serve in these communities. The Title VIII act provides funding to support the education of nurses at the undergraduate and graduate levels for nurses planning to practice in these areas of greatest need. The U.S. House of Representatives passed this act in 2019, and it is currently with the Senate (Title VIII Nursing Workforce Reauthorization Act of 2019, 2019).
Nurse Advocacy: Theory and Attributes
The role of client advocate was first introduced into the code of ethics by the International Council of Nurses in the 1970s and was subsequently adopted into the code of ethics of other nursing organizations, such as the ANA (Abbasinia et al., 2020; Kalaitzidis & Jewell, 2020). The defining attributes of nurse advocacy include safeguarding , apprising , valuing mediating, and championing social justice in health care (Abbasinia et al., 2020). To advocate for persons and populations, nurses safeguard their clients by preventing medical errors and calling out incompetency or misconduct by any health care team member. Apprising may involve providing clients with education about a diagnosis, treatment, or prognosis; identifying alternatives in a treatment plan and follow-up care; and maintaining privacy. Nurses must also maintain self-control in their communications with their clients by valuing clients’ autonomy when making health care decisions and understanding their considerations related to values, culture, beliefs, and preferences. Nurses may serve as liaisons for clients and families by communicating individual preferences to other health care team members. Nurses often confront policies or rules that create inequities in the delivery of health care services or resources (Abbasinia et al., 2020). In the community setting, nurses may strengthen partnerships, develop or participate in community action efforts, and develop healthy public policies (Iriarte-Roteta et al., 2020).
As an example of the nurse’s role as an advocate, consider LGBTQIA+ youth . Transgender and nonbinary youths are two to three times more likely to experience poor mental health outcomes such as depression, anxiety, and risk for suicidal ideation, suicide attempts, and self-harm than cisgender youth (Reisner et al., 2015). A report in 2021 revealed that when nurses provided educational workshops in health care facilities and schools on LGBTQIA+ topics (such as gender-affirming health care, inclusive sex education, sexual and gender diversity education, bullying , suicide prevention , safe spaces , and support groups), health inequities and discrimination in the health care system decreased (Medina-Martinez et al., 2021). Therefore, public health nurses can advocate for the LGBTQIA community by conducting mental health and substance use screening s and promoting a gender-affirming environment. Nurses should use appropriate pronouns and gender-affirming language and demonstrate compassionate listening without making assumptions. In a recent study of 104 transgender and nonbinary youth, the receipt of gender-affirming care, including puberty blockers and gender-affirming hormones, was associated with a 60 percent reduction in moderate or severe depression as well as a 73 percent reduction in suicidality over a 12-month follow-up period (Tordoff et al., 2022).
In this video, Sarah Rosso, the executive director of the Hugh Lane Wellness Foundation and mental health advocate and ally, explains how LGBTQIA+ cultural competence supports clients’ mental health.
Watch the video, and then respond to the following questions.
- Why does Sarah advocate for LGBTQIA+ cultural competency training among the community partners and organizations that work with their organization?
- How can communities reduce the impact of suicide on LGBTQIA+ youth?
- As a new nurse, how would you approach the issue of gender dysphoria among the youth in your community?


10.15: Introduction to Situational Theories of Leadership
What you’ll learn to do: summarize the situational theories of leadership
Though much attention has been given to the traits and styles of effective leaders, another approach to the question has been developed in recent years. Rather than seeking out universal traits or behaviors that are always effective, many theorists now suggest that flexibility in leadership style is key to success. Situational theories of leadership promote the idea that the most effective style of leadership differs from situation to situation.
Contributors and Attributions
CC licensed content, Original
- Introduction to Situational Theories. Authored by : Aaron Spencer and Lumen Learning. License : CC BY: Attribution


8.3: Answer Key
Self-Test 1
- a. True
- c. All of the above
- a. True
- b. False
- c. 75%
- d. Listening
- a. True
- b. False
- d. Vocal messages
- c. The more simply something is stated, the less opportunity there is for confusion.
- c. All of the above
- b. False
- a. True
- c. Closing your eyes while an individual is speaking
- a. True
- d. Providing conditional acceptance
- a. True
- c. Employers, supervisors, and clients
- b. False
Self-Test 2
- b. False
- a. True
- d. Listening, non-verbal, and verbal communication
Self-Test 3
- c. All of the above
- c. All of the above
- c. Your employer, supervisor, co-workers, and clients
- b. To improve your performance or the quality of service
- b. False
- c. Focus on the person and not the issue.
- d. Be prepared and challenge the speaker on everything that you do not believe is correct.
Self-Test 4
- d. Assertive, non-assertive, and aggressive
- c. All of the above
- b. Failing to stand up for yourself
- d. They are easily ignored and can have their rights violated.
- c. All of the above
- b. False
- a. True
- a. True
Self-Test 5
- d. All of the above
- d. Resource allocations (e.g., differences in department budgets), perceptions, and values
- d. Remove personal opinions, be objective, look at the issue, and try and find a solution that works.
- a. True
- b. False
- a. True
- a. True
- b. False
Self-Test 6
- a. True


9.1: Kepler’s Laws
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Although humans have long been fascinated by the motion of objects in the sky, it was Johannes Kepler, in the early seventeenth century, that was the first to write down quantitative rules that described the motion of planets around the Sun. His theory was based on the extensive and detailed observations recorded by Tycho Brahe in the late sixteenth century.
Kepler proposed three laws that describe all of the data that Tycho Brahe had collected about planetary motion:
- The path of a planet around the Sun is described by an ellipse with the Sun at once of its foci.
- All planets move in such a way that the area swept by a line connecting the planet and the Sun in a given period of time is constant.
We examine these three laws in more detail in the sections that follow. It should also be noted that, even though Kepler’s Laws were derived for planets orbiting the Sun, they apply to any body that is orbiting any other body under the influence of gravity 1 .
Kepler’s First Law
Kepler noticed that the motion of all planets followed the path of an ellipse with the Sun located at one of its foci. Figure \(\PageIndex{1}\) shows a diagram of an ellipse, along with its two foci, its semi-major axis, \(s\) , its semi-minor axis, \(b\) , and its eccentricity, \(e\) . The eccentricity is a measure of how far a focus is from the center of the ellipse. A larger eccentricity thus corresponds to a “flatter” ellipse. Note that a circle is just a special case of an ellipse, with both foci located at the center of the circle.
The sun is located at one of the foci. The point of closest approach to the Sun is called the “perihelion” of the orbit (or “perigee” if the orbit is not around the Sun), and the point furthest from the Sun is called the “aphelion” of the orbit (or “apogee” if the orbit is not around the Sun), as shown in Figure \(\PageIndex{2}\) .
Exercise \(\PageIndex{1}\)
Order the ellipses from smallest eccentricity to largest eccentricity.
- Answer
-
\(A<C<B\)
Kepler’s Second Law
Kepler’s Second Law is really a statement about the speed of a planet in an elliptical orbit. It states that the area swept by a line connecting the planet and the Sun in a given period of time is fixed. This is illustrated in Figure \(\PageIndex{4}\), which shows the elliptical orbit of a planet around the Sun located at one of the foci, and the area swept out when the planet goes from \(A\) to \(B\) and from \(C\) to \(D\) .
Kepler’s Second Law states that the two areas that are shown by the grayed out sections in the figure are the same if the planet took the same amount of time to travel between points \(A\) and \(B\) as it did to travel between points \(C\) and \(D\) . Because the points \(C\) and \(D\) are further away from the Sun than points \(A\) and \(B\) , the distance between points \(C\) and \(D\) must be smaller than the distance between points \(A\) and \(B\) for the two areas to be the same. This, in turn, implies that the planet must be moving slower between \(C\) and \(D\) than between points \(A\) and \(B\) . The speed of a planet is thus greatest at the perihelion and smallest at the aphelion. As we will see in a later chapter, Kepler’s Second Law is equivalent to the statement that the angular momentum of the planet about the Sun is conserved.
Exercise \(\PageIndex{2}\)
Based on Kepler’s second law, what can you say about the speed of a planet in a circular orbit?
- Answer
-
The speed of the planet is constant.
Kepler’s Third Law
Kepler’s Third Law is quantitative and relates the orbital periods ( \(T\) ) and the semi-major axes ( \(s\) ) between any two planets in orbit around the Sun:
\[\begin{aligned} \left(\frac{T_1}{T_2}\right)^2=\left(\frac{s_1}{s_2}\right)^3\end{aligned}\]
We can re-arrange this relation so that all of the quantities related to one planet are on the same side of the equal sign:
\[\begin{aligned} \frac{T_1^2}{s_1^3}=\frac{T_2^2}{s_2^3}=\text{constant}\end{aligned}\]
In other words, the ratio between the orbital period squared and the semi-major axis cubed is a constant, independent of the particular planet. In Example 9.2.2 , we will use Newton’s Universal Theory of Gravity to evaluate the constant.
Exercise \(\PageIndex{3}\)
An object is in a circular orbit with radius \(r\) and has an orbital speed \(v\) . If you double the radius of the circular orbit, what will be the value of the orbital speed?
- \(2v\)
- \(8v\)
- \(\sqrt{8}v\)
- \(\frac{1}{\sqrt{2}}v\)
- Answer
- D.
1. In fact, they apply for any two bodies orbiting each other if the force between them is an “inverse-square” law, such as the gravitational and electric forces.


2.4: Combinatorial Proofs
Combinatorial arguments are among the most beautiful in all of mathematics. Oftentimes, statements that can be proved by other, more complicated methods (usually involving large amounts of tedious algebraic manipulations) have very short proofs once you can make a connection to counting. In this section, we introduce a new way of thinking about combinatorial problems with several examples. Our goal is to help you develop a “gut feeling” for combinatorial problems.
Let \(n\) be a positive integer. Use Figure 2.14 to explain why
\(1+2+3+\cdot \cdot \cdot + n = \frac{n(n+1)}{2}\).
- Figure 2.14: The sum of first \(n\) integers
- Solution
-
Consider an \((n+1) \times (n+1)\) array of dots as depicted in Figure 2.14. There are \((n+1)^2\) dots altogether, with exactly \(n+1\) on the main diagonal. The off-diagonal entries split naturally into two equal size parts, those above and those below the diagonal.
Furthermore, each of those two parts has \(S(n) = 1+2+3+\cdot \cdot \cdot+n\) dots. It follows that
\(S(n) = \frac{(n+1)^2-(n+1)}{2}\)
and this is obvious! Now a little algebra on the right hand side of this expression produces the formula given earlier.
Let \(n\) be a positive integer. Explain why
\(1+3+5+\cdot \cdot \cdot+2n-1 = n^2\)
- Solution
-
The left hand side is just the sum of the first \(n\) odd integers. But as suggested in Figure 2.16 , this is clearly equal to \(n^2\).
Let \(n\) be a positive integer. Explain why
\(\dbinom{n}{0}+\dbinom{n}{1}+\dbinom{n}{2}+\cdot \cdot \cdot+\dbinom{n}{n} = 2^n\).
- Solution
-
Both sides count the number of bit strings of length \(n\), with the left side first grouping them according to the number of 0's.
Let \(n\) and \(k\) be integers with \(0 \leq k \leq n\). Explain why
\(\dbinom{n}{k+1} = \dbinom{k}{k} + \dbinom{k+1}{k} + \dbinom{k+2}{k} + \cdot \cdot \cdot +\dbinom{n-1}{k}\).
- Solution
-
To prove this formula, we simply observe that both sides count the number of bit strings of length \(n\) that contain \(k+1\) 1's with the right hand side first partitioning them according to the last occurence of a 1. (For example, if the last 1 occurs in position \(k+5\), then the remaining \(k\) 1's must appear in the preceding \(k+4\) positions, giving \(C(k+4,k)\) strings of this type.) Note that when \(k=1\) (so \(k+1=2\)), we have the same formula as developed earlier for the sum of the first \(n\) positive integers.
Explain the identity
\(3^n = \dbinom{n}{0}2^0 + \dbinom{n}{1}2^1 + \dbinom{n}{2}2^2+ \cdot \cdot \cdot +\dbinom{n}{n}2^n\).
- Solution
-
Both sides count the number of \(\{0,1,2\}\)-strings of length \(n\), the right hand side first partitioning them according to positions in the string which are not 2. (For instance, if 6 of the positions are not 2, we must first choose those 6 positions in \(C(n,6)\) ways and then there are \(2^6\) ways to fill in those six positions by choosing either a 0 or a 1 for each position.)
Explain why, for each non-negative integer \(n\),
\(\dbinom{2n}{n} = \dbinom{n}{0}^2 + \dbinom{n}{1}^2 + \dbinom{n}{2}^2 + \cdot \cdot \cdot +\dbinom{n}{n}^2\).
- Solution
-
Both sides count the number of bit strings of length \(2n\) with half the bits being 0's, with the right side first partitioning them according to the number of 1's occurring in the first \(n\) positions of the string. Note that we are also using the trivial identity \(\binom{n}{k} = \binom{n}{n-k}\).


12.0: Prelude to Mendel's Experiments and Heredity
Genetics is the study of heredity. Johann Gregor Mendel set the framework for genetics long before chromosomes or genes had been identified, at a time when meiosis was not well understood. Mendel selected a simple biological system and conducted methodical, quantitative analyses using large sample sizes. Because of Mendel’s work, the fundamental principles of heredity were revealed. We now know that genes, carried on chromosomes, are the basic functional units of heredity with the capability to be replicated, expressed, or mutated. Today, the postulates put forth by Mendel form the basis of classical, or Mendelian, genetics. Not all genes are transmitted from parents to offspring according to Mendelian genetics, but Mendel’s experiments serve as an excellent starting point for thinking about inheritance.


21.9B: Internal Respiration
Cellular respiration is the metabolic process by which an organism obtains energy through the reaction of oxygen with glucose.
- Describe internal (cellular) respiration
Key Points
- Cellular respiration is the metabolic process by which an organism obtains energy by reacting oxygen with glucose to give water, carbon dioxide, and adenosine triphosphate (energy).
- The 3 steps of cellular respiration are glycolysis, the Krebs cycle, and oxidative phosphorylation.
- Carbon dioxide is a waste product of cellular respiration that comes from the carbon in glucose and the oxygen used in cellular respiration.
- Internal respiration involves gas exchange between the bloodstream and tissues, and cellular respiration.
- Gas exchange in tissues is a result of tissue surface area, gas partial pressure gradients, and the blood perfusion of those tissues.
Key Terms
- Cellular respiration : The cellular process of producing adenosine triphosphate (ATP), water, and carbon dioxide from glucose and oxygen.
- Oxidative Phosphorylation : The final step of cellular phosphorylation in which oxygen is used to a create an electron transport chain that produces a large amount of ATP from nicotinamide adenine dinucleotide (NADH).
Internal respiration refers to two distinct processes. The first is the exchange of gasses between the bloodstream and the tissues. The second is the process of cellular respiration, from which cells utilize oxygen to perform basic metabolic functions.
Gas Exchange with Tissues
Gas exchange occurs in the alveoli so that oxygen is loaded into the bloodstream and carbon dioxide is unloaded from the bloodstream. Afterwards, oxygen is brought to the left side of the heart via the pulmonary vein, which pumps it into systemic circulation.
Red blood cells carry the oxygen into the capillaries of the tissues of the body. Oxygen diffuses into the cells of the tissues, while carbon dioxide diffuses out of the cells of the tissues and into the bloodstream.
The factors that influence tissue gas exchange are similar to the factors of alveolar gas exchange, and include partial pressure gradients between the blood and the tissues, the blood perfusion of those tissues, and the surface areas of those tissues. Each of those factors generally increase gas exchange as those factors are increased (i.e., more oxygen diffusion in tissues with more blood perfusion).
Regarding the partial pressure gradients in systemic capillaries, they have a PaO 2 of 100mmHg and a PaCO 2 of 40mmHG within the capillary and a PaO 2 of 40 mmHg and PaCO 2 of 45 mmHg inside issue cells, which allows gas exchange to occur.
Cellular Respiration
Cellular respiration is the metabolic process by which an organism obtains energy through the reaction of oxygen with glucose to produce water, carbon dioxide and ATP, which is the functional source of energy for the cell. The oxygen supply for cellular respiration comes from the external respiration of the respiratory system.
Overview of cellular respiration : A diagram of cellular respiration including glycolysis, the Krebs cycle (also called the citric acid cycle), and the electron transport chain.
Cellular respiration includes three major steps, and occurs mainly in the cytoplasm of the cell and within the mitochondria of the cell. The net formula for cellular respiration is:
- Glycolysis: The breakdown of glucose into pyruvate, ATP, H 2 O, and heat.
- Krebs Cycle: Produces NADH from pyruvate.
- Oxidative Phosphorylation: Produces ATP from NADH, oxygen, and H+. The oxygen plays the role of electron receptor in an electron transport chain to produce ATP.
The net formula for cellular respiration is:
Glucose+6 Oxygen→6 Carbon Dioxide+6 Water+38 ATPGlucose+6 Oxygen→6 Carbon Dioxide+6 Water+38 ATP
The carbon dioxide waste is the result of the carbon from glucose (C 6 H 12 O 6 ) being broken down to produce the pyruvate and NADH intermediates needed to produce ATP at the end of respiration. The energy stored in ATP can then be used to drive processes that require energy, including biosynthesis, locomotion, or transportation of molecules across cell membranes.
Cellular respiration can occur anaerobically without oxygen, such as through lactic acid fermentation. Human cells may use lactic acid fermentation in muscle tissue during strenuous exercise when there isn’t enough oxygen to power the tissues. This process is very inefficient compared to aerobic respiration, as without oxidative phosphorylation, the cell cannot produce nearly as much ATP (2 ATP compared to 38 during cellular respiration).


7.3.3: Viral Structure
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- Ying Liu
- City College of San Francisco
Learning Objectives
- Describe what an animal virus consists of structurally.
-
Define the following:
- capsid
- capsomere
- nucleocapsid.
- Describe how most animal viruses obtain their envelope.
- State why some bacteriophages are more complex than typical polyhedral or helical viruses.
Since viruses are not cells, they are structurally much simpler than bacteria. An intact infectious viral particle is called a virion and consists of: a genome, a capsid, and often an envelope.
Viral Genome
The viral genome is a single or segmented, circular or linear molecule of nucleic acid functioning as the genetic material of the virus. It can be single-stranded or double-stranded DNA or RNA (but almost never both), and codes for the synthesis of viral components and viral enzymes for replication. It is also becoming recognized that viruses may play a critical role in evolution of life by serving as shuttles of genetic material between other organisms.
Viral Capsid
The capsid, or core, is a protein shell surrounding the genome and is usually composed of protein subunits called capsomeres. The capsid serves to protect and introduce the genome into host cells. Some viruses consist of no more than a genome surrounded by a capsid and are called nucleocapsid or nucleocapsid (Figure \(\PageIndex{1}\)). Attachment proteins project out from the capsid and bind the virus to susceptible host cells.
The Adenovirus and Poliomyelitis viruses are examples of naked viruses (Figure \(\PageIndex{2}\)); both exhibit polyhedral structures.
Viral Envelope
Most animal viruses also have an envelope surrounding a polyhedral or helical nucleocapsid, in which case they are called enveloped virus es (Figure \(\PageIndex{3}\)). Th e envelope may come from the host cell's nuclear membrane, vacuolar membranes (packaged by the Golgi apparatus), or outer cytoplasmic membrane.
- Transmission electron micrograph of Rubella viruses budding from an infected host cell ; courtesy of CDC.
- Transmission electron micrograph of an Influenza A virus ; courtesy of CDC.
- Transmission electron micrograph of HIV ; courtesy of CDC.
Although the envelope is usually of host cell origin, the virus does incorporate proteins of its own, often appearing as glycoprotein spikes, into the envelope. These glycoprotein spikes function in attaching the virus to receptors on susceptible host cells.
- Transmission electron micrograph showing envelope and glycoprotein spikes (gp120) of HIV ; courtesy of CDC.
- Transmission electron micrograph showing envelope and glycoprotein spikes Coronaviruses ; courtesy of CDC.
Viral Activation of Innate Immunity
To protect against infection, one of the things the body must initially do is detect the presence of microorganisms. The body does this by recognizing molecules unique to microorganisms that are not associated with human cells. These unique molecules are called pathogen-associated molecular patterns or PAMPs. (Because all microbes, not just pathogenic microbes, possess PAMPs, pathogen-associated molecular patterns are sometimes referred to as microbe-associated molecular patterns or MAMPs.)
For example, most viral genomes contain a high frequency of unmethylated cytosine-guanine dinucleotide sequences (a cytosine lacking a methyl or CH 3 group and located adjacent to a guanine). Mammalian DNA has a low frequency of cytosine-guanine dinucleotides and most are methylated. In addition, most viruses produce unique double-stranded viral RNA, and some viruses produce uracil-rich single-stranded viral RNA during portions of their life cycle. These forms of viral nucleic acids are common PAMPs associated with viruses. These PAMPs bind to pattern-recognition receptors or PRRs called toll-like receptors or TLRs found within the endosomes of phagocytic cells. Viral RNA can also bind to cytoplasmic PRRs called RIG-1 (retinoic acid-inducible gene-1)and MDR-5 (melanoma differentiation-associated gene-5).
Most of the PRRs that bind to viral components trigger the synthesis of cytokines called Type-I interferons that block viral replication within infected host cells. The TLRs for viral components are found in the membranes of the phagosomes used to degrade viruses during phagocytosis. As viruses are engulfed by phagocytes, the viral PAMPS bind to TLRs located within the phagolysosomes (endosomes ). The TLRs for viral components include:
1. TLR-3 binds double-stranded viral RNA;
2. TLR-7 binds uracil-rich single-stranded viral RNA such as in HIV;
3. TLR-8 binds single-stranded viral RNA;
4. TLR-9 binds unmethylated cytosine-guanine dinucleotide sequences (CpG DNA) found in bacterial and viral genomes.
5
.
RIG-1
(retinoic acid-inducible gene-1) and MDA-5 (melanoma differentiation-associated gene-5), are cytoplasmic sensors that both viral double-stranded and single-stranded RNA molecules produced in viral-infected cells
Bacteriophages are viruses that only infect bacteria. Some bacteriophages are structurally much more complex than typical nucleocapsid or enveloped viruses and may possess a unique tail structure composed of a base plate, tail fibers, and a contractile sheath (also see Figure \(\PageIndex{1}\)C and Figure \(\PageIndex{2}\)E ). Other bacteriophages, however, are simple icosahedrals or helical (see Figure \(\PageIndex{1}\)C) .
Electron Micrograph of a Bacteriophage with a Contractile Sheath. A = normal bacteriophage and B = bacteriophage after contraction of sheath
- Transmission electron micrograph of the bacteriophage coliphage T4 courtesy of Dennis Kunkel's Microscopy .
Exercise: Think-Pair-Share Questions
- Discuss why viruses can only replicate inside living cells.
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Most of the PRRs for viral PAMPs are found in the membranes of the phagosomes, not on the surface of the cell.
- Why do you think this is?
- Name the primary cytokines produced in response to viral PAMPs and state how they function to protect against viruses.
Summary
- Since viruses are not cells, they are structurally much simpler than bacteria.
- An intact infectious viral particle - or virion - consists of a genome, a capsid, and maybe an envelope.
- Viruses possess either DNA or RNA as their genome.
- The genome is typically surrounded by a protein shell called a capsid composed of protein subunits called capsomeres.
- Some viruses consist of no more than a genome surrounded by a capsid and are called nucleocapsid or naked viruses.
- Most animal viruses also have an envelope surrounding a polyhedral or helical nucleocapsid that is typically derived from host cell membranes by a budding process and are called enveloped viruses.
- Specific proteins or glycoproteins on the viral surface are used to attach viruses to the surface of its host cell.
- The viral nucleic acid functions as a pathogen-associated molecular pattern (PAMP).
- Binding of viral PAMPs to host cell pattern-recognition receptors (PRRs) triggers the synthesis and secretion of anti-viral cytokines called type-1 interferons that block viral replication within infected host cells.