28.1: Introduction
Over the course of each different plant’s evolutionary history, environmental pressures can lead to modifications of plant tissues and organs in predictable ways that we can then classify. For example, plants that frequently encounter drought will experience selection for the ability to access water when there is no water in the environment. The plants that evolve water-storage will have better survival in this environment and are more likely to successfully reproduce. However, animals in this dry environment might then seek out their water-filled tissues. Plants that either hide or protect these water reserves are more likely to survive and experience less herbivory. Over time, as the environment continues to select for these defenses, they improve. And thus, over long periods of evolutionary time, one lineage of plants evolves into cacti, while another related lineage evolves in a different environment to be carnations.
There are many selective pressures on plants. limate has a strong selective effect on the anatomy and morphology of plants, particularly water availability and access to sunlight. Herbivory is another strong selective pressure, as plants cannot run away from their predators. Instead, they must evolve other deterrents while balancing the energy costs required for these adaptations with energy devoted to reproduction. A third strong selective pressure is nutrient availability, which is often determined by the soil environment. For example, serpentine soils have low levels of nitrogen and calcium. Plants in serpentine habitats often evolve the ability to trap and digest insects to absorb the missing nutrients. These “carnivorous” plants are not heterotrophic, because they do not use the trapped insects as an energy source.
Plant growth regulators (phytohormones) are small organic molecules that occur naturally in plants. Phytohormones can regulate plant physiological growth processes such as flowering, germination, stem elongation, seed dormancy, fruit ripening, and gene expression.
There are several phytohormones such as gibberellic acid that can increase stem growth parameters. Ethylene can influence fruit ripening.
The two questions you answered above are essential to understanding the evolution of secondary growth . In secondary growth, primary tissues and residual meristematic tissues produce secondary meristems, which then produce secondary tissues. Whereas primary tissues allow for vertical growth, secondary tissues allow for lateral growth: they allow stems and roots to become wider. How might this impact the ability of a plant to grow taller?
In addition to growing wider, secondary growth exchanges the living epidermis for a thick layer of dead, waterproofed cells called cork. The cork and a few other layers of tissue comprise something called the periderm, or perhaps more familiarly called bark . Consider the tradeoffs between having a living exterior with guard cells vs. a thick layer of waterproofed dead cells. How might this impact the ability of the plant to interact with the outer environment? How might this impact the ability of the outer environment to interact with the interior of the plant?
Variations on this type of growth appear in a few places, but the evolution of gymnosperms (conifers and their relatives) is when the more typical secondary growth appears in evolutionary history. As you will see in later labs, the gymnosperms evolved during a period in Earth’s history when inland seas were drying out and plants were migrating further from the water. This group of plants is specialized for growing tall (think Coast redwoods) or living in harsh, low-water environments. As a general rule, monocots do not undergo secondary growth, so this lab will only address eudicots.


26.4: Test #4
Note: This test may need to be updated and/or revised to accurately reflect current statistics, research findings, and health trends.
1. In New York State, a blood alcohol concentration (BAC) of how much constitutes Driving While Intoxicated (DWI), with the exception of commercial motor vehicle drivers?
a. .02
b. .04
c. .06
d. .08
2. Which of the following can make you sober?
a. drinking coffee
b. taking a cold shower
c. time
d. all of the above
3. Which statement is most accurate regarding how BAC is affected by certain factors?
a.  Every year, approximately how many people die in the United States as a result of using tobacco products?
a. 143,000
b. 14,000
c. 443,000
d. 44,000
5. Which of the following statements about secondhand smoke is false ?
a. It contributes to over 3,000 lung cancer deaths each year among U.S. non-smokers.
b. There are less than 25 toxic chemicals found in secondhand tobacco smoke.
c. It may increase the risk of heart disease by 25-30 percent.
d. 46,000 heart disease deaths are likely due to secondhand smoke each year in the U.S.
6. Tobacco use is responsible for about ____% of all cancer deaths annually in the U.S.
a. 15
b. 30
c. 50
d. 65
7. Place a check mark next to EACH statement that is correct regarding the use of e-cigarettes:
_____ They are FDA approved as a quit-smoking aid
_____ Extensive research shows they are safe
_____ Carcinogens may be associated with its use
_____ Nicotine addiction is still a factor
8. Which type of cancer kills more men and women than any other kind?
a. stomach
b. liver
c. lung
d. colon
9. What are three conditions or diseases that can affect a person’s liver due to excessive alcohol consumption (list them below)?
a. ____________________________________________
b. ____________________________________________
c. ____________________________________________
10. Marijuana use may increase the risk of having a heart attack.
a. True
b. False
11. Which statement below regarding prescription drugs, herbal supplements, and other available products that people consume is most accurate?
a. Herbal products are strictly regulated and approved for safety.
b. Medications are only prescribed if there is no risk for addiction.
c. Herbal and synthetic products are available that have not been regulated.
d. The FDA ensures that all products are safe before they can be sold.
12. OxyContin, a prescription drug, is most closely associated with which of the following illicit drugs?
a. cocaine
b. heroin
c. methamphetamine
d. ecstasy
13. Based on the concept of synergism, which drug combination below would be most apt to result in a severe reduction in respiration, decreased heart rate, coma, or even death if consumed together?
a. meth and cocaine
b. Ritalin and alcohol
c. Valium and alcohol
d. meth and marijuana
14. Addiction to Technology?
The popularity of the internet continues to grow because of the many ways users can interact, be entertained, informed, contacted, and occupied. Users connect through virtual worlds, blogs, games, subscription services and social networking sites. Technology gives users the ability to stay connected and contactable 24 hours a day.
However, being connected 24 hours can lead to excessive internet use, which can have a negative impact on the well-being of the user and his/her friends and family.
Do you show signs of possible excessive technology use? Circle “Yes” or “No” for each of the following questions (and be honest!). Are you…
- experiencing a decline in work/study performance as a result of time spent online? Yes No
- feeling tired as a result of late night online activity? Yes No
- missing meals or eating while online so that you can stay connected? Yes No
- lying awake at night with your mind on what’s happening online or waking up in the middle of the night to check online updates and messages (including text messages)? Yes No
- using your smart phone at inappropriate times and places such as in a movie theater or while driving? Yes No
- misleading family members about the extent of time spent on the internet? Yes No
- preferring to spend time online rather than with your family and friends? Yes No
- spending time on the internet because it’s more fun than the ‘real’ world? Yes No
- unsuccessfully trying to cut back on internet use (including social media, gaming, etc.)? Yes No
Answering yes to more than a few of these questions may mean there is a potential problem with excessive internet use. Left unchecked, this can have a negative impact on your overall well-being, relationships, and may also affect work/study performance.
It is important to strike a balance between online and offline activity!
Reflect on the previous questions and take an honest look at the amount of time you spend on electronic devices (smart phone, laptop, tablet, etc.). Discuss how you feel this affects the quality of your life (including your relationships, self-esteem, mood, time management, etc.), and whether or not this interferes with a “healthy” balance in your life. Your total response should be at least 75 words in length.
15. What is the strongest risk factor for transmitting violent behavior from one generation to the next?
a. Living with parents/caretakers who did not graduate from high school
b. Having family members who are afflicted with alcoholism and/or drug abuse
c. Witnessing violence between one’s parents/caretakers
d. Watching disproportionate amounts of violence on television
16. Approximately how many women will experience domestic violence in their lifetime?
a. 1 out of 2
b. 1 out of 3
c. 1 out of 4
d. 1 out of 10
17. Abuse that occurs in domestic violence situations tends to subside or lessen as time goes on.
a. True
b. False
18. Besides physical abuse, explain two other types of abuse that someone might use to exert power and control over his/her partner. Be sure to provide an example for each type of abuse.
19. Which statement regarding how meth use affects the brain is most accurate?
a. meth initially releases a surge of dopamine, causing a sense of euphoria
b. over time, meth destroys dopamine receptors, making it impossible to feel the same pleasure
c. both a & b
d. none of the above
20. Prescription drugs do not have the potential for dangerous outcomes (such as addiction or death) unless they are mixed with illegal drugs.
a. True
b. False
- Test #4 (except question 14). Authored by : Trina DiGregorio, M.S.. Provided by : Monroe Community College. License : CC BY: Attribution
- Question 14, Addiction to Technology?. Authored by : Commonwealth of Australia (Australian Communications and Media Authority) 2014. Located at : http://acma.gov.au/Citizen/Internet/esecurity/Staying-safe-online/excessive-internet-use . License : CC BY: Attribution


4.3: An Empire of Slavery and the Consumer Revolution
Slavery formed a cornerstone of the British Empire in the eighteenth century. Every colony had slaves, from the southern rice plantations in Charles Town, South Carolina, to the northern wharves of Boston. Slavery was more than a labor system; it also influenced every aspect of colonial thought and culture. The uneven relationship it engendered gave white colonists an exaggerated sense of their own status. English liberty gained greater meaning and coherence for whites when they contrasted their status to that of the unfree class of black slaves in British America. African slavery provided whites in the colonies with a shared racial bond and identity.
SLAVERY AND THE STONO REBELLION
The transport of slaves to the American colonies accelerated in the second half of the seventeenth century. In 1660, Charles II created the Royal African Company (Figure 4.3.1) to trade in slaves and African goods. His brother, James II, led the company before ascending the throne. Under both these kings, the Royal African Company enjoyed a monopoly to transport slaves to the English colonies. Between 1672 and 1713, the company bought 125,000 captives on the African coast, losing 20 percent of them to death on the Middle Passage, the journey from the African coast to the Americas.
The Royal African Company’s monopoly ended in 1689 as a result of the Glorious Revolution. After that date, many more English merchants engaged in the slave trade, greatly increasing the number of slaves being transported. Africans who survived the brutal Middle Passage usually arrived in the West Indies, often in Barbados. From there, they were transported to the mainland English colonies on company ships. While merchants in London, Bristol, and Liverpool lined their pockets, Africans trafficked by the company endured a nightmare of misery, privation, and dislocation.
Slaves strove to adapt to their new lives by forming new communities among themselves, often adhering to traditional African customs and healing techniques. Indeed, the development of families and communities formed the most important response to the trauma of being enslaved. Other slaves dealt with the trauma of their situation by actively resisting their condition, whether by defying their masters or running away. Runaway slaves formed what were called “maroon” communities, groups that successfully resisted recapture and formed their own autonomous groups. The most prominent of these communities lived in the interior of Jamaica, controlling the area and keeping the British away.
Slaves everywhere resisted their exploitation and attempted to gain freedom. They fully understood that rebellions would bring about massive retaliation from whites and therefore had little chance of success. Even so, rebellions occurred frequently. One notable uprising that became known as the Stono Rebellion took place in South Carolina in September 1739. A literate slave named Jemmy led a large group of slaves in an armed insurrection against white colonists, killing several before militia stopped them. The militia suppressed the rebellion after a battle in which both slaves and militiamen were killed, and the remaining slaves were executed or sold to the West Indies.
Jemmy is believed to have been taken from the Kingdom of Kongo, an area where the Portuguese had introduced Catholicism. Other slaves in South Carolina may have had a similar background: Africa-born and familiar with whites. If so, this common background may have made it easier for Jemmy to communicate with the other slaves, enabling them to work together to resist their enslavement even though slaveholders labored to keep slaves from forging such communities.
In the wake of the Stono Rebellion, South Carolina passed a new slave code in 1740 called An Act for the Better Ordering and Governing of Negroes and Other Slaves in the Province, also known as the Negro Act of 1740. This law imposed new limits on slaves’ behavior, prohibiting slaves from assembling, growing their own food, learning to write, and traveling freely.
THE NEW YORK CONSPIRACY TRIALS OF 1741
Eighteenth-century New York City contained many different ethnic groups, and conflicts among them created strain. In addition, one in five New Yorkers was a slave, and tensions ran high between slaves and the free population, especially in the aftermath of the Stono Rebellion. These tensions burst forth in 1741.
That year, thirteen fires broke out in the city, one of which reduced the colony’s Fort George to ashes. Ever fearful of an uprising among enslaved New Yorkers, the city’s whites spread rumors that the fires were part of a massive slave revolt in which slaves would murder whites, burn the city, and take over the colony. The Stono Rebellion was only a few years in the past, and throughout British America, fears of similar incidents were still fresh. Searching for solutions, and convinced slaves were the principal danger, nervous British authorities interrogated almost two hundred slaves and accused them of conspiracy. Rumors that Roman Catholics had joined the suspected conspiracy and planned to murder Protestant inhabitants of the city only added to the general hysteria. Very quickly, two hundred people were arrested, including a large number of the city’s slave population.
After a quick series of trials at City Hall, known as the New York Conspiracy Trials of 1741, the government executed seventeen New Yorkers. Thirteen black men were publicly burned at the stake, while the others (including four whites) were hanged (Figure 4.3.2). Seventy slaves were sold to the West Indies. Little evidence exists to prove that an elaborate conspiracy, like the one white New Yorkers imagined, actually existed.
The events of 1741 in New York City illustrate the racial divide in British America, where panic among whites spurred great violence against and repression of the feared slave population. In the end, the Conspiracy Trials furthered white dominance and power over enslaved New Yorkers.
View the map of New York in the 1740s at the New York Public Library’s digital gallery, which allows you to zoom in and see specific events. Look closely at numbers 55 and 56 just north of the city limits to see illustrations depicting the executions.
COLONIAL GENTRY AND THE CONSUMER REVOLUTION
British Americans’ reliance on indentured servitude and slavery to meet the demand for colonial labor helped give rise to a wealthy colonial class—the gentry—in the Chesapeake tobacco colonies and elsewhere. To be “genteel,” that is, a member of the gentry, meant to be refined, free of all rudeness. The British American gentry modeled themselves on the English aristocracy, who embodied the ideal of refinement and gentility. They built elaborate mansions to advertise their status and power. William Byrd II of Westover, Virginia, exemplifies the colonial gentry; a wealthy planter and slaveholder, he is known for founding Richmond and for his diaries documenting the life of a gentleman planter (Figure 4.3.3).
The diary of William Byrd, a Virginia planter, provides a unique way to better understand colonial life on a plantation (Figure 4.3.4). What does it show about daily life for a gentleman planter? What does it show about slavery?
August 27, 1709 I rose at 5 o’clock and read two chapters in Hebrew and some Greek in Josephus. I said my prayers and ate milk for breakfast. I danced my dance. I had like to have whipped my maid Anaka for her laziness but I forgave her. I read a little geometry. I denied my man G-r-l to go to a horse race because there was nothing but swearing and drinking there. I ate roast mutton for dinner. In the afternoon I played at piquet with my own wife and made her out of humor by cheating her. I read some Greek in Homer. Then I walked about the plantation. I lent John H-ch £7 [7 English pounds] in his distress. I said my prayers and had good health, good thoughts, and good humor, thanks be to God Almighty. September 6, 1709 About one o’clock this morning my wife was happily delivered of a son, thanks be to God Almighty. I was awake in a blink and rose and my cousin Harrison met me on the stairs and told me it was a boy. We drank some French wine and went to bed again and rose at 7 o’clock. I read a chapter in Hebrew and then drank chocolate with the women for breakfast. I returned God humble thanks for so great a blessing and recommended my young son to His divine protection. . . . September 15, 1710 I rose at 5 o’clock and read two chapters in Hebrew and some Greek in Thucydides. I said my prayers and ate milk and pears for breakfast. About 7 o’clock the negro boy [ or Betty] that ran away was brought home. My wife against my will caused little Jenny to be burned with a hot iron, for which I quarreled with her. . . .
One of the ways in which the gentry set themselves apart from others was through their purchase, consumption, and display of goods. An increased supply of consumer goods from England that became available in the eighteenth century led to a phenomenon called the consumer revolution. These products linked the colonies to Great Britain in real and tangible ways. Indeed, along with the colonial gentry, ordinary settlers in the colonies also participated in the frenzy of consumer spending on goods from Great Britain. Tea, for example, came to be regarded as the drink of the Empire, with or without fashionable tea sets.
The consumer revolution also made printed materials more widely available. Before 1680, for instance, no newspapers had been printed in colonial America. In the eighteenth century, however, a flood of journals, books, pamphlets, and other publications became available to readers on both sides of the Atlantic. This shared trove of printed matter linked members of the Empire by creating a community of shared tastes and ideas.
Cato’s Letters , by Englishmen John Trenchard and Thomas Gordon, was one popular series of 144 pamphlets. These Whig circulars were published between 1720 and 1723 and emphasized the glory of England, especially its commitment to liberty. However, the pamphlets cautioned readers to be ever vigilant and on the lookout for attacks upon that liberty. Indeed, Cato’s Letters suggested that there were constant efforts to undermine and destroy it.
Another very popular publication was the English gentlemen’s magazine the Spectator , published between 1711 and 1714. In each issue, “Mr. Spectator” observed and commented on the world around him. What made the Spectator so wildly popular was its style; the essays were meant to persuade, and to cultivate among readers a refined set of behaviors, rejecting deceit and intolerance and focusing instead on the polishing of genteel taste and manners.
Novels, a new type of literature, made their first appearance in the eighteenth century and proved very popular in the British Atlantic. Daniel Defoe’s Robinson Crusoe and Samuel Richardson’s Pamela: Or, Virtue Rewarded found large and receptive audiences. Reading also allowed female readers the opportunity to interpret what they read without depending on a male authority to tell them what to think. Few women beyond the colonial gentry, however, had access to novels.
Section Summary
The seventeenth and eighteenth centuries saw the expansion of slavery in the American colonies from South Carolina to Boston. The institution of slavery created a false sense of superiority in whites, while simultaneously fueling fears of slave revolt. White response to such revolts, or even the threat of them, led to gross overreactions and further constraints on slaves’ activities. The development of the Atlantic economy also allowed colonists access to more British goods than ever before. The buying habits of both commoners and the rising colonial gentry fueled the consumer revolution, creating even stronger ties with Great Britain by means of a shared community of taste and ideas.
Review Questions
The Negro Act of 1740 was a reaction to ________.
fears of a slave conspiracy in the setting of thirteen fires in New York City
the Stono Rebellion
the Royal African Company’s monopoly
the growing power of maroon communities
B
What was the “conspiracy” of the New York Conspiracy Trials of 1741?
American patriots conspiring to overthrow the royal government
indentured servants conspiring to overthrow their masters
slaves conspiring to burn down the city and take control
Protestants conspiring to murder Catholics
C


4.3: Orbital Symmetry
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The MO picture for a molecule gets complicated when many valence AOs are involved. We can simplify the problem enormously by noting (without proof here) that orbitals of different symmetry with respect to the molecule do not interact. The symmetry operations of a molecule (which can include rotations, mirror planes, inversion centers, etc.), and the symmetry classes of bonds and orbitals in molecules, can be rigorously defined according to group theory . Here we will take a simple approach to this problem based on our intuitive understanding of the symmetry of three-dimensional objects as illustrated in Fig. \(\PageIndex{1}\).
AO’s must have the same nodal symmetry (as defined by the molecular symmetry operations), or their overlap is zero.
For example, in the HCl molecule, there is a unique symmetry axis →, which is typically defined as the Cartesian z-axis, as shown in Fig. \(\PageIndex{2}\).
We can see from this figure that the H 1s orbital is unchanged by a 180° rotation about the bond axis. However, the same rotation inverts the sign of the Cl 3p y wavefunction. Because these two orbitals have different symmetries, the Cl 3p y orbital is nonbonding and doesn’t interact with the H 1s. The same is true of the Cl 3p x orbital. The p x and p y orbitals have π symmetry (nodal plane containing the bonding axis) and are labeled π nb in the MO energy level diagram, Fig. \(\PageIndex{3}\). In contrast, the H 1s and Cl 3p z orbitals both have σ symmetry, which is also the symmetry of the clay pot shown in Fig. \(\PageIndex{1}\). Because these orbitals have the same symmetry (in the point group of the molecule), they can make the bonding and antibonding combinations shown in Fig. 2.1.1 .
The MO diagram of HCl that includes all the valence orbitals of the Cl atom is shown in Fig. \(\PageIndex{3}\). Two of the Cl valence orbitals (3p x and 3p y ) have the wrong symmetry to interact with the H 1s orbital. The Cl 3s orbital has the same (σ) symmetry as H 1s, but it is much lower in energy so there is little orbital interaction. The energy of the Cl 3s orbital is thus affected only slightly by forming the molecule. The pairs of electrons in the π nb and σ nb orbitals are therefore non-bonding .
Note that the MO result in Fig. \(\PageIndex{3}\) (1 bond and three pairs of nonbonding electrons) is the same as we would get from valence bond theory for HCl. The nonbonding orbitals are localized on the Cl atom, just as we would surmise from the valence bond picture.
In order to differentiate it from the σ bonding orbital, the σ antibonding orbital, which is empty in this case, is designated with an asterisk.


2.2: Point Forces as Vectors
A point force is any force where the point of application is considered to be a single point . In reality, most forces are technically surface forces, where the force is applied over an area, but when the area is small enough (in comparison to the bodies being analyzed) it can often be approximated as a point force. Because point forces can be represented as a single vector (rather than a field of vectors for distributed forces), they are much easier to work with in engineering analysis. For this reason, point forces are used in place of distributed forces in engineering analysis whenever possible. Below are some examples of where it is appropriate to use point forces.
In addition to the magnitude, direction, and point of application of the point force, another important term to understand is the line of action of the force. The line of action of a force is the line along which the force acts. Given the direction and point of application, one can find the line of action, but this term will be important in discussing concurrent forces and in the principle of transmissibility.
Force Vector Representation:
When vectors are drawn to form free body diagrams, the magnitude and direction are usually given in one of two formats:
- Overall magnitude and angle(s) to indicate direction (often called magnitude and direction form ).
- Magnitudes in each of the coordinate directions (often called component form ).
In either format we will need two values to fully define a force vector in a 2D system (either a magnitude and a single angle or a magnitude in each of the two coordinate axes), and three values to fully define a force vector in a 3D system (either a magnitude and two angles or a magnitude in each of the three coordinate axes). Below are some examples of force vectors in both representations.
Changing Force Vector Forms:
Because the two different forms of the vector are equivalent, we can switch between representations without changing the problem. Often in engineering problems, it will initially be easier to write the force in magnitude and angle form, but later, analysis will be easier if forces are written in component form. To switch from magnitude and direction form to component form you will use right triangles and trigonometry to determine the component of the overall magnitude in each direction. This is a simple vector decomposition, and more information on this process can be seen on the vector decomposition page. To switch back from component form into magnitude and direction form you simply use the reverse of this initial process.
Example \(\PageIndex{1}\)
The tension force on the box below is given in magnitude and direction form. Redraw the diagram with the tension force given in component form.
- Solution
Example \(\PageIndex{2}\)
The force acting on the cantilever beam shown below is given in component form. Redraw the diagram with the force given in magnitude and direction form.
- Solution
Example \(\PageIndex{3}\)
The force shown below is given in magnitude and direction form. Redraw the diagram with the force vector given in component form.
- Solution


19.1: What Makes the Climate Change?
There are two parts to climate change, the first one is known as climate forcing , which is when conditions change to give the climate a little nudge in one direction or the other. The second part of climate change, and the one that typically does most of the work, is what we call a feedback . When a climate forcing changes the climate a little, a whole series of environmental changes take place, many of which either exaggerate the initial change ( positive feedbacks ), or suppress the change ( negative feedbacks ).
An example of a climate-forcing mechanism is the increase in the amount of carbon dioxide (CO 2 ) in the atmosphere that results from our use of fossil fuels. CO 2 traps heat in the atmosphere and leads to climate warming. Warming changes vegetation patterns; contributes to the melting of snow, ice, and permafrost ; causes sea level to rise; reduces the solubility of CO 2 in sea water; and has a number of other minor effects. Most of these changes contribute to more warming. Melting of permafrost, for example, is a strong positive feedback because frozen soil contains trapped organic matter that is converted to CO 2 and methane (CH 4 ) when the soil thaws. Both these gases accumulate in the atmosphere and add to the warming effect. On the other hand, if warming causes more vegetation growth, that vegetation should absorb CO 2 , thus reducing the warming effect, which would be a negative feedback. Under our current conditions—a planet that still has lots of glacial ice and permafrost—most of the feedbacks that result from a warming climate are positive feedbacks and so the climate changes that we cause get naturally amplified by natural processes.
What is a greenhouse gas?
Throughout this chapter we’ll be talking about the role of greenhouse gases (GHGs) in controlling the climate, so it’s important to understand what greenhouse gases are and how they work. As you know, the dominant gases of the atmosphere are nitrogen (as N 2 ) and oxygen (as O 2 ). These gas molecules have only two atoms each and are not GHGs. Some of the other important gases of the atmosphere are water vapor (H 2 O), carbon dioxide (CO 2 ), and methane (CH 4 ). All of these have more than two atoms, and they are GHGs.
All molecules vibrate at various frequencies and in various ways, and some of those vibrations take place at frequencies within the range of the infrared (IR) radiation that is emitted by Earth’s surface. Gases with two atoms, such as O 2 , can only vibrate by stretching (back and forth), and those vibrations are much faster than the IR radiation (Figure \(\PageIndex{1}\)). Gases with three or more atoms (such as CO 2 ) vibrate by stretching as well, but they can also vibrate in other ways, such as by bending. Those vibrations are slower and match IR radiation frequencies.
When IR radiation interacts with CO 2 or with one of the other GHGs, the molecular vibrations are enhanced because there is a match between the wavelength of the IR light and the vibrational frequency of the molecule. This makes the molecule vibrate more vigorously, heating the surrounding air in the process. These molecules also emit IR radiation in all directions, some of which reaches Earth’s surface. The heating caused by the more vigorous vibrations of GHGs is the greenhouse effect .
Natural Climate Forcing
Natural climate forcing has been going on throughout geological time. A wide range of processes has been operating at widely different time scales, from a few years to billions of years.
The longest-term natural forcing variation is related to the evolution of the Sun. Like most other stars of a similar mass, our Sun is evolving. For the past 4.57 billion years, its rate of nuclear fusion has been increasing, and it is now emitting about 40% more energy (as light) than it did at the beginning of geological time (Figure \(\PageIndex{2}\)). A difference of 40% is big, so it’s a little surprising that the temperature on Earth has remained at a reasonable and habitable temperature for all of this time. The mechanism for that relative climate stability has been the evolution of our atmosphere from one that was dominated by CO 2 , and also had significant levels of CH 4 —both GHGs—to one with only a few hundred parts per million of CO 2 and just under 1 part per million of CH 4 . Those changes to our atmosphere have been no accident; over geological time, life and its metabolic processes have evolved and changed the atmosphere to conditions that remained cool enough to be habitable. A scientific explanation for how this could happen is known as the Gaia hypothesis .
The Gaia hypothesis
The Gaia hypothesis, developed by British scientist and environmentalist James Lovelock in the 1960s, is the theory that organisms evolve in ways that contribute to ensuring that their environment remains habitable. It does not include any sort of coordination of effort among organisms or any consciousness of a need to make changes. Gaia is not a superorganism. A way of understanding Gaia is through Lovelock’s simple Daisyworld model. A planet with a warming star is populated only by two types of daisies, white ones and black ones (Figure \(\PageIndex{3}\)). The black ones contribute to warming because they absorb solar energy, while the white ones reflect light and contribute to cooling. As the star’s luminosity gradually increases, the white daisies have better outcomes because their reflectivity cools their local environment, while the black daisies, suffering from the heat, do not reproduce as well. Over time white daisies gradually dominate the population, but eventually the star becomes so bright that even white daisies cannot compensate, and all of the daisies perish. Obviously Earth is not Daisyworld, but similar processes—such as the evolution of photosynthetic bacteria that consume CO 2 —have taken place that influence the atmosphere and moderate the climate.
Plate tectonic processes contribute to climate forcing in several different ways, and on time scales ranging from tens of millions to hundreds of millions of years. One mechanism is related to continental position. For example, we know that Gondwana (South America + Africa + Antarctica + Australia) was positioned over the South Pole between about 450 and 250 Ma, during which time there were two major glaciations (Andean-Saharan and Karoo) affecting the South polar regions (Figure \(\PageIndex{1}\)) and cooling the rest of the planet at the same time. Another mechanism is related to continental collisions. As described in Chapter 16, the collision between India and Asia, which started at around 50 Ma, resulted in massive tectonic uplift. The consequent accelerated weathering of this rugged terrain consumed CO 2 from the atmosphere and contributed to gradual cooling over the remainder of the Cenozoic. Also, as described in Chapter 16, the opening of the Drake Passage — due to plate-tectonic separation of South America from Antarctica — led to the development of the Antarctic Circumpolar Current, which isolated Antarctica from the warmer water in the rest of the ocean and thus contributed to Antarctic glaciation starting at around 35 Ma.
As we discussed in Chapter 4, volcanic eruptions don’t just involve lava flows and exploding rock fragments; various particulates and gases are also released, the important ones being sulfur dioxide and CO 2 . Sulphur dioxide is an aerosol that reflects incoming solar radiation and has a net cooling effect that is short lived (a few years in most cases, as the particulates settle out of the atmosphere within a couple of years), and doesn’t typically contribute to longer-term climate change. Volcanic CO 2 emissions can contribute to climate warming but only if a greater-than-average level of volcanism is sustained over a long time (at least tens of thousands of years). It is widely believed that the catastrophic end-Permian extinction (at 250 Ma) resulted from warming initiated by the eruption of the massive Siberian Traps over a period of at least a million years.
The large extraterrestrial impact at the end of the Cretaceous (Cretaceous-Paleogene or K-Pg boundary, a.k.a. K-T boundary) is thought to have produced a massive amount of dust, which may have remained in the atmosphere for several years, and also a great deal of CO 2 . What do you think would have been the short-term and longer-term climate-forcing implications of these two factors?
See Appendix 3 for Exercise 19.1 answers.
Earth’s orbit around the Sun is nearly circular, but like all physical systems, it has natural oscillations. First, the shape of the orbit changes on a regular time scale — close to 100,000 years — from being close to circular to being very slightly elliptical. But the circularity of the orbit is not what matters; it is the fact that as the orbit becomes more elliptical, the position of the Sun within that ellipse becomes less central or more eccentric (Figure \(\PageIndex{5}\)a). Eccentricity is important because when it is high, the Earth-Sun distance varies more from season to season than it does when eccentricity is low.
Second, Earth rotates around an axis through the North and South Poles, and that axis is at an angle to the plane of Earth’s orbit around the Sun (Figure \(\PageIndex{5}\)b). The angle of tilt (also known as obliquity ) varies on a time scale of 41,000 years. When the angle is at its maximum (24.5°), Earth’s seasonal differences are accentuated. When the angle is at its minimum (22.1°), seasonal differences are minimized. The current hypothesis is that glaciation is favoured at low seasonal differences as summers would be cooler and snow would be less likely to melt and more likely to accumulate from year to year.
Third, the direction in which Earth’s rotational axis points also varies, on a time scale of about 20,000 years (Figure \(\PageIndex{5}\)c). This variation, known as precession , means that although the North Pole is presently pointing to the star Polaris (the pole star), in 10,000 years it will point to the star Vega.
The importance of eccentricity, tilt, and precession to Earth’s climate cycles (now known as Milankovitch Cycles ) was first pointed out by Yugoslavian engineer and mathematician Milutin Milankovitch in the early 1900s. Milankovitch recognized that although the variations in the orbital cycles did not affect the total amount of insolation (light energy from the Sun) that Earth received, it did affect where on Earth that energy was strongest. Glaciations are most sensitive to the insolation received at latitudes of around 65°, and with the current configuration of continents, it would have to be 65° north (because there is almost no land at 65° south).
The most important aspects are whether the northern hemisphere is pointing toward the Sun at its closest or farthest approach, and how eccentric the Sun’s position is in Earth’s orbit. Two opposing situations are illustrated in Figure \(\PageIndex{6}\). In the upper panel, the northern hemisphere is at it farthest distance from the Sun during summer, which means cooler summers. In the lower panel, the northern hemisphere is at its closest distance to the Sun during summer, which means hotter summers. Cool summers—as opposed to cold winters—are the key factor in the accumulation of glacial ice, so the upper scenario in Figure \(\PageIndex{6}\) is the one that promotes glaciation. This factor is greatest when eccentricity is high.
Data for tilt, eccentricity, and precession over the past 400,000 years have been used to determine the insolation levels at 65° north, as shown in Figure \(\PageIndex{7}\). Also shown in Figure \(\PageIndex{7}\) are Antarctic ice-core temperatures from the same time period. The correlation between the two is clear, and it shows up in the Antarctic record because when insolation changes lead to growth of glaciers in the northern hemisphere, southern-hemisphere temperatures are also affected.
Ocean currents are important to climate, and currents also have a tendency to oscillate. Glacial ice cores show clear evidence of changes in the Gulf Stream (and other parts of the thermohaline circulation system) that affected global climate on a time scale of about 1,500 years during the last glaciation. The east-west changes in sea-surface temperature and surface pressure in the equatorial Pacific Ocean—known as the El Niño Southern Oscillation or ENSO—varies on a much shorter time scale of between two and seven years. These variations tend to garner the attention of the public because they have significant climate implications in many parts of the world. The past 65 years of ENSO index values are shown in Figure \(\PageIndex{8}\). The strongest El Niños in recent decades were in 1983 and 1998, and those were both very warm years from a global perspective. During a strong El Niño, the equatorial Pacific sea-surface temperatures are warmer than normal and heat the atmosphere above the ocean, which leads to warmer-than-average global temperatures.
Climate Feedbacks
As already stated, climate feedbacks are critically important in amplifying weak climate forcings into full-blown climate changes. When Milankovitch published his theory in 1924, it was widely ignored, partly because it was evident to climate scientists that the forcing produced by the orbital variations was not strong enough to drive the significant climate changes of the glacial cycles. Those scientists did not recognize the power of positive feedbacks. It wasn’t until 1973, 15 years after Milankovitch’s death, that sufficiently high-resolution data were available to show that the Pleistocene glaciations were indeed driven by the orbital cycles, and it became evident that the orbital cycles were just the forcing that initiated a range of feedback mechanisms that made the climate change.
Since Earth still has a very large volume of ice — mostly in the continental ice sheets of Antarctica and Greenland, but also in alpine glaciers and permafrost — melting is one of the key feedback mechanisms. Melting of ice and snow leads to several different types of feedbacks, an important one being a change in albedo . Albedo is a measure of the reflectivity of a surface. Earth’s various surfaces have widely differing albedos, expressed as the percentage of light that reflects off a given material. This is important because most solar energy that hits a very reflective surface is not absorbed and therefore does little to warm Earth. Water in the oceans or on a lake is one of the darkest surfaces, reflecting less than 10% of the incident light, while clouds and snow or ice are among the brightest surfaces, reflecting 70% to 90% of the incident light (Figure \(\PageIndex{9}\)).
When a forest is harvested, there are significant changes to the rate of biological uptake of CO 2 , but there is also a change in albedo. Using Figure \(\PageIndex{9}\), and assuming that a clear-cut has an albedo similar to sand, what is the albedo-only implication of clear-cutting for climate change? Be aware that this implication applies only to the albedo change caused by clear-cutting. Clear-cutting reduces the capacity of the area to absorb CO 2 , and also has numerous negative ecological and geological implications.
See Appendix 3 for Exercise 19.2 answers.
When sea ice melts, as it has done in the Arctic Ocean at a disturbing rate over the past decade, the albedo of the area affected changes dramatically, from around 80% down to less than 10%. This is a positive feedback because much more solar energy is absorbed by the water than by the pre-existing ice, and the temperature increase is amplified. The same applies to ice and snow on land, but the difference in albedo is not as great.
When ice and snow on land melt, sea level rises. (Sea level is also rising because the oceans are warming and that increases their volume.) A higher sea level means a larger proportion of the planet is covered with water, and since water has a lower albedo than land, more heat is absorbed and the temperature goes up a little more. Since the last glaciation, sea-level rise has been about 125 m; a huge area that used to be land is now flooded by heat-absorbent seawater. During the current period of anthropogenic climate change, sea level has risen only about 20 cm, and although that doesn’t make a big change to albedo, sea-level rise is accelerating.
Most of northern Canada has a layer of permafrost that ranges from a few centimeters to hundreds of meters in thickness; the same applies in Alaska, Russia, and Scandinavia. Permafrost is a mixture of soil and ice (Figure \(\PageIndex{11}\)), and it also contains a significant amount of trapped organic carbon that is released as CO 2 and CH 4 when the permafrost breaks down. Because the amount of carbon stored in permafrost is in the same order of magnitude as the amount released by burning fossil fuels, this is a feedback mechanism that has the potential to equal or surpass the forcing that has unleashed it.
In some polar regions, including northern Canada, permafrost includes methane hydrate (see section 18.3), a highly concentrated form of CH 4 trapped in solid form. Breakdown of permafrost releases this CH 4 . Even larger reserves of methane hydrate exist on the sea floor, and while it would take significant warming of ocean water down to a depth of hundreds of meters, this too is likely to happen in the future if we don’t limit our impact on the climate. There is strong isotopic evidence that the Paleocene-Eocene thermal maximum (see Figure \(\PageIndex{1}\)) was caused, at least in part, by a massive release of sea-floor methane hydrate.
There is about 45 times as much carbon in the ocean (as dissolved bicarbonate ions, HCO 3 − ) as there is in the atmosphere (as CO 2 ), and there is a steady exchange of carbon between the two reservoirs. But the solubility of CO 2 in water decreases as the temperature goes up. In other words, the warmer it gets, the more of that oceanic bicarbonate gets transferred to the atmosphere as CO 2 . That makes CO 2 solubility another positive feedback mechanism.
Vegetation growth responds positively to both increased temperatures and elevated CO 2 levels, and so in general, it represents a negative feedback to climate change because the more the vegetation grows, the more CO 2 is taken from the atmosphere. But it’s not quite that simple because when trees grow bigger and more vigorously, forests become darker (they have lower albedo) so they absorb more heat. Furthermore, climate warming isn’t necessarily good for vegetation growth; some areas have become too hot, too dry, or even too wet to support the plant community that was growing there, and it might take centuries for something to replace it successfully.
All of these positive (and negative) feedbacks work both ways. For example, during climate cooling, growth of glaciers leads to higher albedos, and formation of permafrost results in storage of carbon that would otherwise have returned quickly to the atmosphere.
Image Descriptions
Figure \(\PageIndex{2}\) image description: Our sun is roughly 4.5 billion years old. Over the next 5 billion years, it will continue getting warmer until it becomes a red giant. Six billion years from now, it will become a planetary nebula and then a white dwarf. [Return to Figure \(\PageIndex{2}\)]
Figure \(\PageIndex{5}\) image description:
- The eccentricity of the Sun within the Earth’s orbit goes through a 100,000 year cycle. The Earth’s orbit around the sun can be very circular, but when it becomes more oval-shaped, the Sun is offset from the centre of the ellipse. This means that sometimes the Earth will be farther from the sun than normal and sometimes it will be closer to the sun than normal.
- The angle of tilt (or obliquity) or the Earth’s axis goes through a 41,000 year cycle. When comparing the North Pole to a line perpendicular to the plane of the earth’s orbit, the obliquity can range from 22.1 ° to 24.5 °.
- The Earth’s axial precession operates on a 20,000 year cycle.
[Return to Figure \(\PageIndex{5}\)]
Figure \(\PageIndex{9}\) image description: The albedo of Earth’s surfaces: Water=around 5%; Forest=10% to 15%; Sand=30%; Old snow and ice=70%; Cloud=75% to 85%. Fresh snow=90%. Earth’s average albedo is 30%. [Return to Figure \(\PageIndex{9}\)]
Media Attributions
- Figure \(\PageIndex{1}\): © Steven Earle. CC BY.
- Figure \(\PageIndex{2}\): “Solar Life Cycle” by Oliverbeatson. Public domain.
- Figure \(\PageIndex{3}\): © Steven Earle. CC BY.
- Figure \(\PageIndex{4}\): “Coast Impact” by Don Davis, NASA. Public domain.
- Figure \(\PageIndex{5}\)A: “Eccentricity zero” by NASA. Adapted by Steven Earle. Public domain.
- Figure \(\PageIndex{5}\)B: “ Earth obliquity range ” by NASA. Public domain.
- Figure \(\PageIndex{5}\)C: “Earth Precession” by NASA. Public domain.
- Figure \(\PageIndex{6}\): © Steven Earle. CC BY.
- Figure \(\PageIndex{7}\): © Steven Earle. CC BY. Based on data from Valerie Masson-Delmotte, “EPICA Dome C Ice Core 800KYr Deuterium Data and Temperature Estimates,” WDCA Contribution Series Number : 2007 -091, NOAA/NCDC Paleoclimatology Program, Boulder CO, USA. Retrieved from: ftp://ftp.ncdc.noaa.gov/pub/data/pal...uttemp2007.txt and from Berger, A. and Loutre, M.F. (1991). Insolation values for the climate of the last 10 million years. Quaternary Science Reviews, 10, 297-317.
- Figure \(\PageIndex{8}\): “ Multivariate ENSO Index (MEI) ” by NOAA. Adapted by Steven Earle. Public domain.
- Figure \(\PageIndex{9}\): © Steven Earle. CC BY.
- Figure \(\PageIndex{10}\): “Clearcutting-Oregon” © Calibas. CC BY-SA .
- Figure \(\PageIndex{11}\): “ Collapsed permafrost block of coastal tundra on Alaska’s Arctic Coast ” by the USGS . Public domain.


4.2: Simple Decision Making and Forecasting from Condition Assessment
In many cases, asset management decisions are made without complicated deterioration models at all. The simplest approaches simply use the existing component condition, a linear projection of the component condition over time, or a projection based upon the past history of similar components (using a graph such as those illustrated in Figure 4.1.3). These approaches are discussed in this section.
Using the existing condition can be augmented with simple decision rules. For example, ‘if the condition is x or lower, then rehabilitation is desirable.’ Components with only two condition states defined are particularly amenable to this approach. For example, an incandescent light either works or is burnt out. The maintenance rule might be to only replace lights when they burn out.
A subset of this simplified method includes a “run to failure” approach. Unfortunately, this “fix it when it breaks approach” is a widely applied and expensive approach to infrastructure management. There are scenarios, like window air conditioners, when it simply doesn’t make sense to replace before failure. As we will explore, it is nearly always more cost-effective to replace before failure when considering major infrastructure systems.
Using existing conditions has the advantage of eliminating any costs associated with deterioration modelling. However, the amount of effort may fluctuate considerably as many components cross over the trigger condition for action and this may not be compatible with budget constraints. Also, if deterioration has significant costs, waiting until deterioration occurs may not be the best approach.
Simple linear extrapolation is another approach that is inexpensive to employ for deterioration models. In this approach, c where ct is the condition at time t, and Δt is some desired time period in the future.
\[c_{t+Δt} = c_t + (c_t - c_{t-1})\]
As an example, suppose the component condition is now 3, last year the condition was 4, then the forecast for next year is \(3 + (3-4)*1 = 3 – 1 = 2\) and the forecast for the following year (two years from now) would be \(3 + (3-4)*2 = 1\). More complicated forms of extrapolation could also be used, but linear extrapolation is the most common for infrastructure deterioration.
A single year maybe two short of a period to effectively capture deterioration, so a moving average of multiple years might be used instead. In this case, \(c_t\) would be the average condition for the current period of years (which might be the past three years). This approach would be useful for very slowly deteriorating infrastructure components. Moving averages of this type are common for smoothing fluctuating time series histories such as stock prices.
Finally, forecasts of component deterioration might be based upon simple historical records. For example, Figure 4.1.3 shows the average deterioration trajectory of different bridge decks under specific conditions. An infrastructure manager might assume that a particular bridge deck with a particular condition would simply follow this trajectory in the future. Even without a formal database, infrastructure managers might have their mental model of expected deterioration and make subjective forecasts based upon their experience.


32.6: Fission
Learning Objectives
By the end of this section, you will be able to:
- Define nuclear fission.
- Discuss how fission fuel reacts and describe what it produces.
- Describe controlled and uncontrolled chain reactions.
Nuclear fission is a reaction in which a nucleus is split (or fissured ). Controlled fission is a reality, whereas controlled fusion is a hope for the future. Hundreds of nuclear fission power plants around the world attest to the fact that controlled fission is practical and, at least in the short term, economical, as seen in Figure \(\PageIndex{1}\). Whereas nuclear power was of little interest for decades following TMI and Chernobyl (and now Fukushima Daiichi), growing concerns over global warming has brought nuclear power back on the table as a viable energy alternative. By the end of 2009, there were 442 reactors operating in 30 countries, providing 15% of the world’s electricity. France provides over 75% of its electricity with nuclear power, while the US has 104 operating reactors providing 20% of its electricity. Australia and New Zealand have none. China is building nuclear power plants at the rate of one start every month.
Fission is the opposite of fusion and releases energy only when heavy nuclei are split. As noted previously, energy is released if the products of a nuclear reaction have a greater binding energy per nucleon (\(BE/A\)) than the parent nuclei. Figure \(\PageIndex{2}\) shows that \(BE/A\) is greater for medium-mass nuclei than heavy nuclei, implying that when a heavy nucleus is split, the products have less mass per nucleon, so that mass is destroyed and energy is released in the reaction. The amount of energy per fission reaction can be large, even by nuclear standards. The graph in Figure \(\PageIndex{2}\) shows \(BE/A\) to be about 7.6 MeV/nucleon for the heaviest nuclei (\(A\) about 240), while \(BE/A\) is about 8.6 MeV/nucleon for nuclei having \(A\) about 120. Thus, if a heavy nucleus splits in half, then about 1 MeV per nucleon, or approximately 240 MeV per fission, is released. This is about 10 times the energy per fusion reaction, and about 100 times the energy of the average \(\alpha\), \(\beta\), or \(\gamma\) decay.
Example \(\PageIndex{1}\): Calculating Energy Released by Fission
Calculate the energy released in the following spontaneous fission reaction:
\[\ce{^{238}U} \rightarrow \ce{^{95}Sr} + \ce{^{140}Xe} + 3n \nonumber\]
given the atomic masses to be \(m(^{238}U) = 238.050784 \, u, \, m(^{95}Sr) = 94.919388 \, u\), \(m(^{140}Xe) = 139.921610 \, u\), and \(m(n) = 1.008665 \, u\).
Strategy
As always, the energy released is equal to the mass destroyed times \(c^2\), so we must find the difference in mass between the parent \(^{238}U\) and the fission products.
Solution
The products have a total mass of
\[\begin{align} m_{products} &= 94.919388 \, u + 139.921610 \, u + 3(1.008665 \, u) \nonumber\\[5pt] &= 237.866993 \, u. \nonumber\end{align} \nonumber\]
The mass lost is the mass of \(^{238}U\) minus \(m_{products}\), or
\[\begin{align*} \Delta m &= 238.050784 \, u -<PHONE_NUMBER> \, u \nonumber\\[5pt] &= 0.183791 \, u\nonumber\end{align*}\]
so the energy released is
\[\begin{align*} E &= (\Delta m)c^2 \nonumber \\[5pt] &= (0.183791 \, u)\dfrac{931.5 \, MeV/c^2}{u}c^2 \nonumber \\[5pt] &= 171.2 \, MeV. \nonumber\end{align*}\]
Discussion
A number of important things arise in this example. The 171-MeV energy released is large, but a little less than the earlier estimated 240 MeV. This is because this fission reaction produces neutrons and does not split the nucleus into two equal parts. Fission of a given nuclide, such as \(^{238}U\), does not always produce the same products. Fission is a statistical process in which an entire range of products are produced with various probabilities. Most fission produces neutrons, although the number varies with each fission. This is an extremely important aspect of fission, because neutrons can induce more fission , enabling self-sustaining chain reactions.
Spontaneous fission can occur, but this is usually not the most common decay mode for a given nuclide. For example, \(^{238}U\) can spontaneously fission, but it decays mostly by \(\alpha\) emission. Neutron-induced fission is crucial as seen in Figure \(\PageIndex{2}\). Being chargeless, even low-energy neutrons can strike a nucleus and be absorbed once they feel the attractive nuclear force. Large nuclei are described by a liquid drop model with surface tension and oscillation modes, because the large number of nucleons act like atoms in a drop. The neutron is attracted and thus, deposits energy, causing the nucleus to deform as a liquid drop. If stretched enough, the nucleus narrows in the middle. The number of nucleons in contact and the strength of the nuclear force binding the nucleus together are reduced. Coulomb repulsion between the two ends then succeeds in fissioning the nucleus, which pops like a water drop into two large pieces and a few neutrons. Neutron-induced fission can be written as
\[n + ^AX \rightarrow FF_1 + FF_2 + xn,\]
where \(FF_1\) and \(FF_2\) are the two daughter nuclei, called fission fragments , and \(x\) is the number of neutrons produced. Most often, the masses of the fission fragments are not the same. Most of the released energy goes into the kinetic energy of the fission fragments, with the remainder going into the neutrons and excited states of the fragments. Since neutrons can induce fission, a self-sustaining chain reaction is possible, provided more than one neutron is produced on average — that is, if \(x > 1\) in \(n + ^AX \rightarrow FF_1 + FF_2 + xn\). This can also be seen in Figure \(\PageIndex{3}\).
An example of a typical neutron-induced fission reaction is
\[\ce{n} + \ce{_{92}^{235}U} \rightarrow \ce{_{56}^{142}Ba} + \ce{_{36}^{142}Kr} + \ce{3n}.\]
Note that in this equation, the total charge remains the same (is conserved): \(92 + 0 = 56 + 36\). Also, as far as whole numbers are concerned, the mass is constant: \(1 + 235 = 142 + 91 + 3\). This is not true when we consider the masses out to 6 or 7 significant places, as in the previous example.
Not every neutron produced by fission induces fission. Some neutrons escape the fissionable material, while others interact with a nucleus without making it fission. We can enhance the number of fissions produced by neutrons by having a large amount of fissionable material. The minimum amount necessary for self-sustained fission of a given nuclide is called its critical mass . Some nuclides, such as \(^{239}Pu\), produce more neutrons per fission than others, such as \(^{235}U\). Additionally, some nuclides are easier to make fission than others. In particular, \(^{235}U\) and \(^{239}Pu\) are easier to fission than the much more abundant \(^{238}U\). Both factors affect critical mass, which is smallest for \(^{239}Pu\).
The reason \(^{235}U\) and \(^{239}Pu\) are easier to fission than \(^{238}U\) is that the nuclear force is more attractive for an even number of neutrons in a nucleus than for an odd number. Consider that \(_{92}^{235}U_{143}\) has 143 neutrons, and \(_{94}^{239}P_{145}\) has 145 neutrons, whereas \(_{92}^{238}U_{146}\) has 146. When a neutron encounters a nucleus with an odd number of neutrons, the nuclear force is more attractive, because the additional neutron will make the number even. About 2-MeV more energy is deposited in the resulting nucleus than would be the case if the number of neutrons was already even. This extra energy produces greater deformation, making fission more likely. Thus, \(^{235}U\) and \(^{239}Pu\) are superior fission fuels. The isotope \(^{235}U\) is only 0.72 % of natural uranium, while \(^{238}U\) is 99.27%, and \(^{239}Pu\) does not exist in nature. Australia has the largest deposits of uranium in the world, standing at 28% of the total. This is followed by Kazakhstan and Canada. The US has only 3% of global reserves.
Most fission reactors utilize \(^{235}U\), which is separated from \(^{238}U\) at some expense. This is called enrichment. The most common separation method is gaseous diffusion of uranium hexafluoride \((UF_6)\) through membranes. Since \(^{235}U\) has less mass than \(^{238}U\), its \(UF_6\) molecules have higher average velocity at the same temperature and diffuse faster. Another interesting characteristic of \(^{235}U\) is that it preferentially absorbs very slow moving neutrons (with energies a fraction of an eV), whereas fission reactions produce fast neutrons with energies in the order of an MeV. To make a self-sustained fission reactor with \(^{235}U\), it is thus necessary to slow down (“thermalize”) the neutrons. Water is very effective, since neutrons collide with protons in water molecules and lose energy. Figure \(\PageIndex{4}\) shows a schematic of a reactor design, called the pressurized water reactor.
Control rods containing nuclides that very strongly absorb neutrons are used to adjust neutron flux. To produce large power, reactors contain hundreds to thousands of critical masses, and the chain reaction easily becomes self-sustaining, a condition called criticality . Neutron flux should be carefully regulated to avoid an exponential increase in fissions, a condition called supercriticality . Control rods help prevent overheating, perhaps even a meltdown or explosive disassembly. The water that is used to thermalize neutrons, necessary to get them to induce fission in \(^{235}U\), and achieve criticality, provides a negative feedback for temperature increases. In case the reactor overheats and boils the water to steam or is breached, the absence of water kills the chain reaction. Considerable heat, however, can still be generated by the reactor’s radioactive fission products. Other safety features, thus, need to be incorporated in the event of a loss of coolant accident, including auxiliary cooling water and pumps.
Example \(\PageIndex{3}\): Calculating Energy from a Kilogram of Fissionable Fuel
Calculate the amount of energy produced by the fission of 1.00 kg of \(\ce{^{235}U}\), given the average fission reaction of \(\ce{^{235}U}\) produces 200 MeV.
Strategy
The total energy produced is the number of \(^{235}U\) atoms times the given energy per \(^{235}U\) fission. We should therefore find the number of \(^{235}U\) atoms in 1.00 kg.
Solution
The number of \(^{235}U\) atoms in 1.00 kg is Avogadro’s number times the number of moles. One mole of \(^{235}U\) has a mass of 235.04 g; thus, there are \((1000 \, g)/(235.04 \, g/mol) = 4.25 \, mol.\) The number of \(^{235}U\) atoms is therefore,
\[(4.25 \, mol)(6.02 \times 10^{23} \, ^{235}U/mol) = 2.56 \times 10^{24} \, ^{235}U. \nonumber\]
So the total energy released is
\[\begin{align} E &= (2.56 \times 10^{24} \, ^{235}U) \left(\dfrac{200 \, MeV}{^{235}U} \right) \left(\dfrac{1.60 \times 10^{-13} \, J}{MeV}\right) \nonumber \\[5pt] &= 8.21 \times 10^{13} \, J. \nonumber \end{align}\nonumber\]
Discussion
This is another impressively large amount of energy, equivalent to about 14,000 barrels of crude oil or 600,000 gallons of gasoline. But, it is only one-fourth the energy produced by the fusion of a kilogram mixture of deuterium and tritium. Even though each fission reaction yields about ten times the energy of a fusion reaction, the energy per kilogram of fission fuel is less, because there are far fewer moles per kilogram of the heavy nuclides. Fission fuel is also much more scarce than fusion fuel, and less than 1% of uranium (the \(^{235}U)\) is readily usable.
One nuclide already mentioned is \(^{239}Pu\) which has a 24,120-y half-life and does not exist in nature. Plutonium-239 is manufactured from \(^{238}U\) in reactors, and it provides an opportunity to utilize the other 99% of natural uranium as an energy source. The following reaction sequence, called breeding , produces \(^{239}Pu\). Breeding begins with neutron capture by \(^{238}U\).
\[\ce{^{238}U + n \rightarrow ^{239}U + \gamma}.\]
Uranium-239 then \(\beta^-\) decays:
\[\ce{^{239}U \rightarrow ^{239}Np + \beta^- + \nu_e } \,(t_{1/2} = 23 \, min).\]
Neptunium-239 also \(\beta^-\) decays:
\[\ce{^{239}Np \rightarrow ^{239}Pu + \beta^- + \nu_e } \,(t_{1/2} = 2.4 \, d).\]
Plutonium-239 builds up in reactor fuel at a rate that depends on the probability of neutron capture by \(^{238}U\) (all reactor fuel contains more \(^{238}U\) than \(^{235}U\)). Reactors designed specifically to make plutonium are called breeder reactors . They seem to be inherently more hazardous than conventional reactors, but it remains unknown whether their hazards can be made economically acceptable. The four reactors at Chernobyl, including the one that was destroyed, were built to breed plutonium and produce electricity. These reactors had a design that was significantly different from the pressurized water reactor illustrated above.
Plutonium-239 has advantages over \(^{235}U\) as a reactor fuel — it produces more neutrons per fission on average, and it is easier for a thermal neutron to cause it to fission. It is also chemically different from uranium, so it is inherently easier to separate from uranium ore. This means \(^{239}Pu\) has a particularly small critical mass, an advantage for nuclear weapons.
PHET EXPLORATIONS: NUCLEAR FISSION
Start a chain reaction , or introduce non-radioactive isotopes to prevent one. Control energy production in a nuclear reactor!
Summary
- Nuclear fission is a reaction in which a nucleus is split.
- Fission releases energy when heavy nuclei are split into medium-mass nuclei.
- Self-sustained fission is possible, because neutron-induced fission also produces neutrons that can induce other fissions, \(n + ^AX \rightarrow FF_1 + FF_2 + xn\), where \(FF_1\) and \(FF_2\) are the two daughter nuclei, or fission fragments, and x is the number of neutrons produced.
- A minimum mass, called the critical mass, should be present to achieve criticality.
- More than a critical mass can produce supercriticality.
- The production of new or different isotopes (especially \(^{239}Pu\)) by nuclear transformation is called breeding, and reactors designed for this purpose are called breeder reactors.
Glossary
- breeder reactors
- reactors that are designed specifically to make plutonium
- breeding
- reaction process that produces \(^{239}Pu\)
- criticality
- condition in which a chain reaction easily becomes self-sustaining
- critical mass
- minimum amount necessary for self-sustained fission of a given nuclide
- fission fragments
- a daughter nuclei
- liquid drop model
- a model of nucleus (only to understand some of its features) in which nucleons in a nucleus act like atoms in a drop
- nuclear fission
- reaction in which a nucleus splits
- neutron-induced fission
- fission that is initiated after the absorption of neutron
- supercriticality
- an exponential increase in fissions


7.1: Introduction to Demonstrate Reporting and Documentation of Client Data
Learning Objectives
- Measure temperature, pulse, and respiration
- Measure weight and height
- Recognize normal and abnormal blood pressure readings
- Identify normal and abnormal vital signs
- Measure intake and output
- Document client data
- Prioritize the reporting of data
- Recognize signs of client discomfort
Nursing assistants collect and document client data such as vital signs, height, weight, intake, and output. While performing cares and collecting data, nursing assistants spend a lot of time with residents and may observe subtle changes in behavior, mood, mobility, and cognition, as well as signs of discomfort. They report these observations promptly to nurses to ensure safe, quality, holistic care. This chapter will discuss collection of this data, as well as prioritization of data that should be immediately reported to the nurse.


5.4: Captions and Titles
Learning Outcome
- Describe the impact of descriptive captions and titles when incorporating graphics into a message
Most of the time, putting the data into graph form isn’t quite enough. In fact, it’s just the start.
You’ll find most communicators title their graphs according to the content they display. It might say “Year-over-year performance” or “Weekly Average Ticket Sales.” But if your message is about how weekly average ticket sales are down 10 percent compared to last year, you might consider calling your graph “Average Ticket Sales are Decreasing.”
The video below is a demonstration about how to make a graph tell your story by making it easier to read and making the title active:
Captions usually indicate the source of information. If your sources and communications are not produced by people within your company, then this is an important step. Captioning the source gives your information credibility and strengthens your story.
You can also use captions to convey other pertinent information. You may want to include the sample size of a survey the graph is illustrating or additional background information about the data (as shown in Figure 1). Using captions in this manner helps the reader draw the right conclusion.
A caption on a graph can go badly when there is more information captioned than is of interest to your audience, or if information included in the caption would be better displayed elsewhere.
Let’s take a look at this line graph displaying the percent of world GDP (Figure 2):
Imagine if the graph above, which shows the percent of the world’s gross domestic product from 1700–2000, looked more like Figure 3:
Now the title is included in the caption, as well as some information about British colonization. How could we make it a little easier for the audience to absorb that information?
Figure 4 is even better. Now the graph is titled, and those instances of colonization are marked on the timeline. However, let’s imagine the author is focusing on India in their report or presentation. In that case, the note about the colonization of the United States isn’t quite relevant and should be trimmed out (Figure 5).
Much better! Now the graph shows only the information relevant to the point the creator is trying to make.
Visual media should always make a point clearer, so make sure your graph’s format, titles and captions are working for you rather than against you.
Contributors and Attributions
- Captions and Titles. Authored by : Freedom Learning Group. Provided by : Lumen Learning. License : CC BY: Attribution
- Unemployment graph from Patterns of Unemployment. Provided by : OpenStax College. Located at : https://cnx.org/contents/vEmOH-_p@4.44:82qMsJce/Patterns-of-Unemployment . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/donate/download/4061c...cb31@10.49/pdf


2.4: Intonation- Statements and Questions
Intonation: Statements and Questions
Rules to Remember
- In statements, the voice usually falls at the end of the sentence.
- In wh- questions that ask for new information, the voice usually falls at the end of the question.
- In yes/no question, the voice usually rises at the end of the question.
Watch the Video
Rachel's English - Questions and Intonation: Up or Down?
Answer the Questions
Listening Practice: End of Sentence Intonation
Listen to each word. Does the speaker's intonation rise or fall? Write "rise" or "fall" next to each word. End of Sentence intonation.m4a
| Number | Word | Does the intonation rise or fall? |
|---|---|---|
| 1 Example | car? | rise |
| 2 | yes. | |
| 3 | pets? | |
| 4 | no. | |
| 5 | job? | |
| 6 | farmer. | |
| 7 | free time? | |
| 8 | cooking. |
Listening Practice: Intonation and Questions
Listen to each sentence. Does the speaker's intonation rise or fall at the end of each sentence? Write "rise" or "fall" next to each sentence. Listening Practice - Intonation and Quesitons.m4a
| Number | Sentence | Does the intonation rise or fall? |
|---|---|---|
| 1 Example | What's your name? | fall |
| 2 | Where did you grow up? | |
| 3 | Where are you living now? | |
| 4 | What's your job? | |
| 5 | What do you do each day? | |
| 6 | Why do you like your job? | |
| 7 | What do you not like about your job? | |
| 8 | Why is your job important to you? | |
| 9 | What do you like to do in your free time? | |
| 10 | Do you have any pets? | |
| 11 | Do you have a car? | |
| 12 | Are you a morning person? | |
| 13 | Do you eat meat? | |
| 14 | Are you a vegan? | |
| 15 | What does that mean? | |
| 16 | Why are you vegan? |
Listening Practice: Yes/No Questions
Listen to each sentence. Does the speaker's intonation rise or fall at the end of the sentence? Write "rise" or "fall" next to each sentence. Intonation in Yes No Questions.m4a
| Number | Sentence | Does the intonation rise or fall? |
|---|---|---|
| 1 Example | Do you have any pets? | rise |
| 2 | Do you have any brothers or sisters? | |
| 3 | Do you have a car? | |
| 4 | Are you a morning person? | |
| 5 | Do you eat meat? | |
| 6 | Are you a vegan? | |
| 7 | Do you have children? |
Pair Work
Work with a partner. Take turns asking and answering the questions above.
Write
Write 3 new yes/no questions to ask your teacher. Do not copy any of the questions you've already seen in this unit.
Listening Practice: Wh- Questions
Listen to each sentence. Does the speaker's intonation rise or fall at the end of the sentence? Write "rise" or "fall" next to each sentence. Intonation in WH questions.m4a
| Number | Sentence | Does the intonation rise or fall? |
|---|---|---|
| 1 Example | What's your name? | fall |
| 2 | Where did you grow up? | |
| 3 | Where are you living now? | |
| 4 | What do you usually eat for dinner? | |
| 5 | What's your job? | |
| 6 | Why do you like your job? | |
| 7 | What do you like to do in your free time? |
Pair Work
Work with a partner. Take turns asking and answering the questions above.
Write
Write 3 new Why- questions to ask your teacher. Do not copy any of the questions you've already seen in this unit.
Recording: Yes/No and Wh- Questions
Make a recording of yourself asking each question. Make sure that you use the correct intonation for yes/no and wh- questions. You will receive one point for each question.
____ / 15 points
- What's your name?
- Where did you grow up?
- Where are you living now?
- What's your job?
- What do you do each day?
- Why do you like your job?
- What do you not like about your job?
- Why is your job important to you?
- What do you like to do in your free time?
- Do you have a car?
- Are you a morning person?

- Are you a vegan?
- What does that mean?
- Why are you vegan?


14.4: Vectorized Arithmetic Operators
As with NumPy ndarrays , you can apply arithmetic operators like + and * to entire Serieses at a time, which is not only easy code to write but also runs blazing fast. But the Pandas Series is even smarter than that.
Figure \(\PageIndex{1}\): Two Series es in memory
Consider the memory picture in Figure 12.1. Here we have two Series es, one pointed to by a salaries variable and the other by raises , which are of different sizes and which have overlapping, but not identical, sets of keys. What do you suppose Pandas would do if we executed this code?
Code \(\PageIndex{1}\) (Python):
new_salaries = salaries + raises
The answer, happily, is the smartest possible thing it could do. Pandas gets neither confused nor stifled by the fact that the keys are in different orders in the two Series es, and instead it does what you surely want: add corresponding elements, with matching keys, and produce a new Series with all of those sums.
The actual result in this case is in Figure 12.2.2, and the output is here:
Code \(\PageIndex{2}\) (Python):
new_salaries = salaries + raises
print(new_salaries)
Figure \(\PageIndex{2}\): The result of + ’ing two Series es that don’t have all the same keys.
| Dwight 71.5
| Jim NaN
| Michael 129.5
| Pam 69.0
| Robert NaN
| Ryan NaN
| dtype: float64
Convince yourself that Dwight ’s $68,000 salary got added to his $3,500 raise, that Michael ’s $113,000 salary was added to his $16,500 raise, etc.
Don’t get freaked out by those NaN entries just yet. The special value “ NaN ” stands for “ not a number ,” and basically means that Pandas has to throw up its hands in that case. And with good cause. Jim has a current salary of $100,200 in the first Series, but has no value at all in the second one (no raise for Jim this year? Haven’t decided what his raise will be yet? Something else?) So Pandas does the safe thing, shrugs, and says “dunno.” We say that the Jim entry in the new_salaries Series is a missing value . The same is true for Robert and Ryan , each of whom was present in only one of the two operands.
Now I know what you’re thinking: “can’t Pandas just assume the salary and/or raise is 0 if there’s a missing one?” The answer is that yes it can, but it won’t do so unless you give the go-ahead. Pandas is being cautious here, and doesn’t want to introduce errors into your data stream by false assumptions. (Maybe in your company, for instance, there’s a default entry-level salary that every employee receives who’s unspecified in the salary Series . Or maybe the yearly raise is always assumed to be a flat 2.5% cost-of-living raise unless explicitly specified.)
If we do want Pandas to assume a certain default value, we have to change tactics a bit and go with the add() function (or sub() , mul() , or div() ):
Code \(\PageIndex{3}\) (Python):
new_salaries = pd.Series.add(salaries, raises, fill_value=0)
print(new_salaries)
| Dwight 71.5
| Jim 100.2
| Michael 129.5
| Pam 69.0
| Robert 100.0
| Ryan 68.0
| dtype: float64
The fill_value argument is the important one here: it specifies what default value to use if one of the addends is missing a key from the other. Now the result is as in Figure 12.2.3. You can, of course, choose a fill_value other than zero, if you wish.
As with NumPy arrays, we can add (or subtract, or multiply, ...) a single atomic value to a series as well:
Code \(\PageIndex{4}\) (Python):
cost_of_living_increase = salaries * .025
print(cost_of_living_increase)
Figure \(\PageIndex{3}\): Using add() instead, and passing a fill_value .
| Micheal 2.825
| Dwight 1.700
| Pam 1.675
| Jim 2.505
| Ryan 1.700
| dtype: float64
Code \(\PageIndex{5}\) (Python):
salaries = salaries + cost_of_living_increase
print(salaries)
| Micheal 115.825
| Dwight 69.700
| Pam 68.675
| Jim 102.705
| Ryan 69.700
| dtype: float64
It can sometimes be useful to do string concatenation as well, for instance if we had employee first names and last names in two Series es with their employee ID as the index:
Code \(\PageIndex{6}\) (Python):
firsts = pd.Series(['Hannibal', 'Clarice', 'Multiple', 'Buffalo'], index=[666, 1993, 47, 988])
lasts = pd.Series(['Starling', 'Crawford', 'Lecter', 'Bill', 'Miggs'], index=[1993, 1650, 666, 988, 47])
print(firsts + " " + lasts)
|
47 Multiple Miggs
|
666 Hannibal Lecter
|
988 Buffalo Bill
|
1650 NaN
|
1993 Clarice Starling
|
dtype: object


3.1: Iterated Integrals and Area
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In the previous chapter we found that we could differentiate functions of several variables with respect to one variable, while treating all the other variables as constants or coefficients. We can integrate functions of several variables in a similar way. For instance, if we are told that \(f_x(x,y) = 2xy\), we can treat \(y\) as staying constant and integrate to obtain \(f(x,y)\):
\[\begin{align*}
f(x,y) &= \int f_x(x,y) \,dx\\
&= \int 2xy \,dx \\
&= x^2y + C.
\end{align*}\]
Make a careful note about the constant of integration, \(C\). This "constant'' is something with a derivative of \(0\) with respect to \(x\), so it could be any expression that contains only constants and functions of \(y\). For instance, if \(f(x,y) = x^2y+ \sin y + y^3 + 17\), then \(f_x(x,y) = 2xy\). To signify that \(C\) is actually a function of \(y\), we write:
\[f(x,y) = \int f_x(x,y) \,dx = x^2y+C(y).\]
Using this process we can even evaluate definite integrals.
Example \(\PageIndex{1}\): Integrating functions of more than one variable
Evaluate the integral \(\displaystyle \int_1^{2y} 2xy \,dx.\)
Solution
We find the indefinite integral as before, then apply the Fundamental Theorem of Calculus to evaluate the definite integral:
\[\begin{align*}
\int_1^{2y} 2xy \,dx &= x^2y\Big|_1^{2y}\\
&= (2y)^2y - (1)^2y \\
&= 4y^3-y.
\end{align*}\]
We can also integrate with respect to \(y\). In general,
\[\int_{h_1(y)}^{h_2(y)} f_x(x,y) \,dx = f(x,y)\Big|_{h_1(y)}^{h_2(y)} = f\big(h_2(y),y\big)-f\big(h_1(y),y\big),\]
and
\[\int_{g_1(x)}^{g_2(x)} f_y(x,y) \,dy = f(x,y)\Big|_{g_1(x)}^{g_2(x)} = f\big(x,g_2(x)\big)-f\big(x,g_1(x)\big).\]
Note that when integrating with respect to \(x\), the bounds are functions of \(y\) (of the form \(x=h_1(y)\) and \(x=h_2(y)\)) and the final result is also a function of \(y\). When integrating with respect to \(y\), the bounds are functions of \(x\) (of the form \(y=g_1(x)\) and \(y=g_2(x)\)) and the final result is a function of \(x\). Another example will help us understand this.
Example \(\PageIndex{2}\): Integrating functions of more than one variable
Evaluate \(\displaystyle \int_1^x\big(5x^3y^{-3}+6y^2\big) \,dy\).
Solution
We consider \(x\) as staying constant and integrate with respect to \(y\):
\[\begin{align*}
\int_1^x\big(5x^3y^{-3}+6y^2\big) \,dy & = \left(\frac{5x^3y^{-2}}{-2}+\frac{6y^3}{3}\right)\Bigg|_1^x \\
&= \left(-\frac52x^3x^{-2}+2x^3\right) - \left(-\frac52x^3+2\right) \\
&= \frac92x^3-\frac52x-2.
\end{align*}\]
Note how the bounds of the integral are from \(y=1\) to \(y=x\) and that the final answer is a function of \(x\).
In the previous example, we integrated a function with respect to \(y\) and ended up with a function of \(x\). We can integrate this as well. This process is known as iterated integration , or multiple integration .
Example \(\PageIndex{3}\): Integrating an integral
Evaluate \(\displaystyle \int_1^2\left(\int_1^x\big(5x^3y^{-3}+6y^2\big) \,dy\right) \,dx.\)
Solution
We follow a standard "order of operations'' and perform the operations inside parentheses first (which is the integral evaluated in Example \(\PageIndex{2}\).)
\[\begin{align*}
\int_1^2\left(\int_1^x\big(5x^3y^{-3}+6y^2\big) \,dy\right) \,dx &= \int_1^2 \left(\left[\frac{5x^3y^{-2}}{-2}+\frac{6y^3}{3}\right]\Bigg|_1^x\right) \,dx \\
&= \int_1^2 \left(\frac92x^3-\frac52x-2\right) \,dx \\
&= \left(\frac98x^4-\frac54x^2-2x\right)\Bigg|_1^2\\
&= \frac{89}8.
\end{align*}\]
Note how the bounds of \(x\) were \(x=1\) to \(x=2\) and the final result was a number.
The previous example showed how we could perform something called an iterated integral; we do not yet know why we would be interested in doing so nor what the result, such as the number \(89/8\), means . Before we investigate these questions, we offer some definitions.
Definition: Iterated Integration
Iterated integration is the process of repeatedly integrating the results of previous integrations. Integrating one integral is denoted as follows.
Let \(a\), \(b\), \(c\) and \(d\) be numbers and let \(g_1(x)\), \(g_2(x)\), \(h_1(y)\) and \(h_2(y)\) be functions of \(x\) and \(y\), respectively. Then:
- \(\displaystyle \int_c^d\int_{h_1(y)}^{h_2(y)} f(x,y) \,dx \,dy = \int_c^d\left(\int_{h_1(y)}^{h_2(y)} f(x,y) \,dx\right) \,dy.\)
- \(\displaystyle \int_a^b\int_{g_1(x)}^{g_2(x)} f(x,y) \,dy \,dx = \int_a^b\left(\int_{g_1(x)}^{g_2(x)} f(x,y) \,dy\right) \,dx.\)
Again make note of the bounds of these iterated integrals.
With \(\displaystyle \int_c^d\int_{h_1(y)}^{h_2(y)} f(x,y) \,dx \,dy\), \(x\) varies from \(h_1(y)\) to \(h_2(y)\), whereas \(y\) varies from \(c\) to \(d\). That is, the bounds of \(x\) are curves , the curves \(x=h_1(y)\) and \(x=h_2(y)\), whereas the bounds of \(y\) are constants , \(y=c\) and \(y=d\). It is useful to remember that when setting up and evaluating such iterated integrals, we integrate "from curve to curve, then from point to point.''
We now begin to investigate why we are interested in iterated integrals and what they mean.
Area of a plane region
Consider the plane region \(R\) bounded by \(a\leq x\leq b\) and \(g_1(x)\leq y\leq g_2(x)\), shown in Figure \(\PageIndex{1}\). We learned in Section 7.1 (in Calculus I) that the area of \(R\) is given by
\[\int_a^b \big(g_2(x)-g_1(x)\big) \,dx.\]
We can view the expression \(\big(g_2(x)-g_1(x)\big)\) as
\[\big(g_2(x)-g_1(x)\big) = \int_{g_1(x)}^{g_2(x)} 1 \,dy =\int_{g_1(x)}^{g_2(x)} \,dy,\nonumber\]
meaning we can express the area of \(R\) as an iterated integral:
\[\text{area of }R = \int_a^b \big(g_2(x)-g_1(x)\big) \,dx = \int_a^b\left(\int_{g_1(x)}^{g_2(x)} \,dy\right) \,dx =\int_a^b\int_{g_1(x)}^{g_2(x)} \,dy \,dx.\]
In short: a certain iterated integral can be viewed as giving the area of a plane region.
A region \(R\) could also be defined by \(c\leq y\leq d\) and \(h_1(y)\leq x\leq h_2(y)\), as shown in Figure \(\PageIndex{2}\). Using a process similar to that above, we have
\[\text{the area of }R = \int_c^d\int_{h_1(y)}^{h_2(y)} \,dx \,dy.\]
We state this formally in a theorem.
THEOREM \(\PageIndex{1}\): Area of a plane region
- Let \(R\) be a plane region bounded by \(a\leq x\leq b\) and \(g_1(x)\leq y\leq g_2(x)\), where \(g_1\) and \(g_2\) are continuous functions on \([a,b]\). The area \(A\) of \(R\) is $$A = \int_a^b\int_{g_1(x)}^{g_2(x)} \,dy \,dx.$$
- Let \(R\) be a plane region bounded by \(c\leq y\leq d\) and \(h_1(y)\leq x\leq h_2(y)\), where \(h_1\) and \(h_2\) are continuous functions on \([c,d]\). The area \(A\) of \(R\) is $$A = \int_c^d\int_{h_1(y)}^{h_2(y)} \,dx \,dy.$$
The following examples should help us understand this theorem.
Example \(\PageIndex{4}\): Area of a rectangle
Find the area \(A\) of the rectangle with corners \((-1,1)\) and \((3,3)\), as shown in Figure \(\PageIndex{3}\).
Solution
Multiple integration is obviously overkill in this situation, but we proceed to establish its use.
The region \(R\) is bounded by \(x=-1\), \(x=3\), \(y=1\) and \(y=3\). Choosing to integrate with respect to \(y\) first, we have
\[A = \int_{-1}^3\int_1^3 1 \,dy \,dx = \int_{-1}^3 \left(y\ \Big|_1^3\right) \,dx = \int_{-1}^3 2 \,dx = 2x\Big|_{-1}^3=8.\nonumber\]
We could also integrate with respect to \(x\) first, giving:
\[A = \int_1^3\int_{-1}^3 1 \,dx \,dy =\int_1^3 \left(x\ \Big|_{-1}^3\right) \,dy = \int_1^3 4 \,dy = 4y\Big|_1^3 = 8.\nonumber\]
Clearly there are simpler ways to find this area, but it is interesting to note that this method works.
Example \(\PageIndex{5}\): Area of a triangle
Find the area \(A\) of the triangle with vertices at \((1,1)\), \((3,1)\) and \((5,5)\), as shown in Figure \(\PageIndex{4}\).
Solution
The triangle is bounded by the lines as shown in the figure. Choosing to integrate with respect to \(x\) first gives that \(x\) is bounded by \(x=y\) to \(x = \frac{y+5}2\), while \(y\) is bounded by \(y=1\) to \(y=5\). (Recall that since \(x\)-values increase from left to right, the leftmost curve, \(x=y\), is the lower bound and the rightmost curve, \(x=(y+5)/2\), is the upper bound.) The area is
\[\begin{align*}
A &= \int_1^5\int_{y}^{\frac{y+5}2} \,dx \,dy \\
&= \int_1^5\left(x\ \Big|_y^{\frac{y+5}2}\right) \,dy \\
&= \int_1^5 \left(-\frac12y+\frac52\right) \,dy \\
&= \left(-\frac14y^2+\frac52y\right)\Big|_1^5\\
&=4.
\end{align*}\]
We can also find the area by integrating with respect to \(y\) first. In this situation, though, we have two functions that act as the lower bound for the region \(R\), \(y=1\) and \(y=2x-5\). This requires us to use two iterated integrals. Note how the \(x\)-bounds are different for each integral:
\[\begin{align*}
A &= \int_1^3\int_1^x 1 \,dy \,dx &+& & &\int_3^5\int_{2x-5}^x1 \,dy \,dx\\
&= \int_1^3\big(y\big)\Big|_1^x \,dx & + & & & \int_3^5\big(y\big)\Big|_{2x-5}^x \,dx\\
&= \int_1^3\big(x-1\big) \,dx & + & & & \int_3^5\big(-x+5\big) \,dx \\
&= 2 & + & & & 2 \\
&=4.
\end{align*}\]
As expected, we get the same answer both ways.
Example \(\PageIndex{6}\): Area of a plane region
Find the area of the region enclosed by \(y=2x\) and \(y=x^2\), as shown in Figure \(\PageIndex{5}\).
Solution
Once again we'll find the area of the region using both orders of integration.
Using \(\,dy \,dx\):
\[\int_0^2\int_{x^2}^{2x}1 \,dy \,dx = \int_0^2(2x-x^2) \,dx = \big(x^2-\frac13x^3\big)\Big|_0^2 = \frac43.\]
Using \(\,dx \,dy\):
\[\int_0^4\int_{y/2}^{\sqrt{y}} 1 \,dx \,dy = \int_0^4 (\sqrt{y}-y/2) \,dy = \left(\frac23y^{3/2} - \frac14y^2\right)\Big|_0^4 = \frac43.\]
Changing Order of Integration
In each of the previous examples, we have been given a region \(R\) and found the bounds needed to find the area of \(R\) using both orders of integration. We integrated using both orders of integration to demonstrate their equality.
We now approach the skill of describing a region using both orders of integration from a different perspective. Instead of starting with a region and creating iterated integrals, we will start with an iterated integral and rewrite it in the other integration order. To do so, we'll need to understand the region over which we are integrating.
The simplest of all cases is when both integrals are bound by constants. The region described by these bounds is a rectangle (see Example \(\PageIndex{4}\)), and so:
\[\int_a^b\int_c^d 1 \,dy \,dx = \int_c^d\int_a^b1 \,dx \,dy.\]
When the inner integral's bounds are not constants, it is generally very useful to sketch the bounds to determine what the region we are integrating over looks like. From the sketch we can then rewrite the integral with the other order of integration.
Examples will help us develop this skill.
Example \(\PageIndex{7}\): Changing the order of integration
Rewrite the iterated integral \(\displaystyle \int_0^6\int_0^{x/3} 1 \,dy \,dx\) with the order of integration \(\,dx \,dy\).
Solution
We need to use the bounds of integration to determine the region we are integrating over.
The bounds tell us that \(y\) is bounded by \(0\) and \(x/3\); \(x\) is bounded by 0 and 6. We plot these four curves: \(y=0\), \(y=x/3\), \(x=0\) and \(x=6\) to find the region described by the bounds. Figure \(\PageIndex{6}\) shows these curves, indicating that \(R\) is a triangle.
To change the order of integration, we need to consider the curves that bound the \(x\)-values. We see that the lower bound is \(x=3y\) and the upper bound is \(x=6\). The bounds on \(y\) are \(0\) to \(2\). Thus we can rewrite the integral as \(\displaystyle \int_0^2\int_{3y}^6 1 \,dx \,dy.\)
Example \(\PageIndex{8}\): Changing the order of integration
Change the order of integration of \(\displaystyle \int_0^4\int_{y^2/4}^{(y+4)/2}1 \,dx \,dy\).
Solution
We sketch the region described by the bounds to help us change the integration order. \(x\) is bounded below and above (i.e., to the left and right) by \(x=y^2/4\) and \(x=(y+4)/2\) respectively, and \(y\) is bounded between 0 and 4. Graphing the previous curves, we find the region \(R\) to be that shown in Figure \(\PageIndex{7}\).
To change the order of integration, we need to establish curves that bound \(y\). The figure makes it clear that there are two lower bounds for \(y\): \(y=0\) on \(0\leq x\leq 2\), and \(y=2x-4\) on \(2\leq x\leq 4\). Thus we need two double integrals. The upper bound for each is \(y=2\sqrt{x}\). Thus we have
\[\int_0^4\int_{y^2/4}^{(y+4)/2}1 \,dx \,dy = \int_0^2\int_0^{2\sqrt{x}} 1 \,dy \,dx + \int_2^4\int_{2x-4}^{2\sqrt{x}}1 \,dy \,dx.\nonumber\]
This section has introduced a new concept, the iterated integral. We developed one application for iterated integration: area between curves. However, this is not new, for we already know how to find areas bounded by curves.
In the next section we apply iterated integration to solve problems we currently do not know how to handle. The "real" goal of this section was not to learn a new way of computing area. Rather, our goal was to learn how to define a region in the plane using the bounds of an iterated integral. That skill is very important in the following sections.


6.4: How the Brain Processes Information to Make Decisions - Reflective and Reactive Systems
- Understand the two systems of decision-making in the brain.
The human brain processes information for decision-making using one of two routes: a reflective system and a reactive (or reflexive) system (Facione & Facione, 2007; Lieberman, 2003). The reflective system is logical, analytical, deliberate, and methodical, while the reactive system is quick, impulsive, and intuitive, relying on emotions or habits to provide cues for what to do next. Research in neuropsychology suggests that the brain can only use one system at a time for processing information and that the two systems are directed by different parts of the brain. The prefrontal cortex is more involved in the reflective system, and the basal ganglia and amygdala (more primitive parts of the brain, from an evolutionary perspective) are more involved in the reactive system (Darlow & Sloman, 2010).
Reactive Decision-Making
We tend to assume that the logical, analytical route leads to superior decisions, but whether this is accurate depends on the situation. The quick, intuitive route can be lifesaving; when we suddenly feel intense fear, a fight-or-flight response kicks in that leads to immediate action without methodically weighing all possible options and their consequences. Additionally, experienced managers can often make decisions very quickly because experience or expertise has taught them what to do in a given situation. These managers might not be able to explain the logic behind their decision and will instead say they just went with their “gut” or did what “felt” right. Because the manager has faced a similar situation in the past and has figured out how to deal with it, the brain shifts immediately to the quick, intuitive decision-making system (Gladwell, 2005).
Reflective Decision-Making
The quick route is not always the best decision-making path to take, however. When faced with novel and complex situations, it is better to process available information logically, analytically, and methodically. As a manager, you need to think about whether a situation requires not a fast, “gut” reaction but some serious thought prior to making a decision. It is especially important to pay attention to your emotions because strong emotions can make it difficult to process information rationally. Successful managers recognize the effects of emotions and know to wait and address a volatile situation after their emotions have calmed down. Intense emotions—whether positive or negative—tend to pull us toward the quick, reactive route of decision-making.  This speaks to the power our emotions exert on our decision-making. Big decisions should generally not be made impulsively but reflectively.
The Role of Emotions
Being aware of the role emotions play in decision-making does not mean that we should ignore them. Emotions can serve as powerful signals about what we should do, especially in situations with ethical implications. You can read more about this particular type of decision-making in the Ethics in Practice box later in this chapter. Thinking through how we feel about the possible options and why we feel that way can greatly enhance our decision-making (George, 2000). Effective decision-making, then, relies on both logic and emotions. For this reason, the concept of emotional intelligence has become popular as a characteristic of effective managers. Emotional intelligence is the ability to recognize, understand, pay attention to, and manage one’s own emotions and the emotions of others. It involves self-awareness and self-regulation—essentially, this is a toggling back and forth between emotions and logic so that we analyze and understand our own emotions and then exert the necessary control to manage them as appropriate for the situation. Emotional intelligence also involves empathy—the ability to understand other peoples’ emotions (and an interest in doing so). Finally, emotional intelligence involves social skills to manage the emotional aspects of relationships with others. Managers who are aware of their own emotions can think through what their emotions mean in a given situation and use that information to guide their decision-making. Managers who are aware of the emotions of others can also utilize that information to help groups function more effectively and engage in better group decision-making. While emotional intelligence seems to come easily to some people, it is something that we can develop and improve on with practice. A model of emotional intelligence is presented in Exhibit 6.2.
- Explain the two systems used by the brain in decision-making.
- What is emotional intelligence, and why is it important for decision-making?