18.7: Reactions of Epoxides- Ring-opening
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Objectives
After completing this section, you should be able to
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write an equation to describe the opening of an epoxide ring under mildly acidic conditions.
- identify the product formed from the hydrolysis of an epoxide.
- write the mechanism for the opening of an epoxide ring by an aqueous acid, paying particular attention to the stereochemistry of the product.
- identify the product formed when an epoxide ring is opened by a hydrogen halide under anhydrous conditions.
- predict the major product from the acidic cleavage of a given unsymmetrical epoxide.
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write an equation to illustrate the cleavage of an epoxide ring by a base.
- identify the product formed from the reaction of a given epoxide with given base.
- explain why epoxides are susceptible to cleavage by bases, whereas other cyclic ethers are not.
Study Notes
In the discussion on base-catalyzed epoxide opening, the mechanism is essentially S N 2. While oxygen is a poor leaving group, the ring strain of the epoxide really helps to drive this reaction to completion. Indeed, larger cyclic ethers would not be susceptible to either acid-catalyzed or base-catalyzed cleavage under the same conditions because the ring strain is not as great as in the three-membered epoxide ring.
Epoxide Ring-Opening by Alcoholysis
The ring-opening reactions of epoxides provide a nice overview of many of the concepts discussed in earlier chapters of this book. Ring-opening reactions can proceed by either S N 2 or S N 1 mechanisms, depending on the nature of the epoxide and on the reaction conditions. If the epoxide is asymmetric, the structure of the product will vary according to which mechanism dominates. When an asymmetric epoxide undergoes alcoholysis in basic methanol, ring-opening occurs by an S N 2 mechanism, and the less substituted carbon is the site of nucleophilic attack, leading to what we will refer to as product B:
Conversely, when solvolysis occurs in acidic methanol, the reaction occurs by a mechanism with substantial S N 1 character, and the more substituted carbon is the site of attack. As a result, product A predominates.
These are both good examples of regioselective reactions . In a regioselective reaction, two (or more) different constitutional isomers are possible as products, but one is formed preferentially (or sometimes exclusively).
Basic Epoxide Ring-Opening by Alcoholysis
In the basic, S N 2 reaction, the leaving group is an alkoxide anion, because there is no acid available to protonate the oxygen prior to ring opening. An alkoxide is a poor leaving group ( Section 11-3 ), and thus the ring is unlikely to open without a 'push' from the nucleophile.
The nucleophile itself is potent: a deprotonated, negatively charged methoxide ion. When a nucleophilic substitution reaction involves a poor leaving group and a powerful nucleophile, it is very likely to proceed by an S N 2 mechanism.
There are two electrophilic carbons in the epoxide, but the best target for the nucleophile in an S N 2 reaction is the carbon that is least hindered . This accounts for the observed regiochemical outcome. Like in other S N 2 reactions, nucleophilic attack takes place from the backside, resulting in inversion at the electrophilic carbon.
Acid-Catalyzed Epoxide Ring-Opening by Alcoholysis
The best way to depict the acid-catalyzed epoxide ring-opening reaction is as a hybrid, or cross, between an S N 2 and S N 1 mechanism. First, the oxygen is protonated, creating a good leaving group (step 1 below). Then the carbon-oxygen bond begins to break (step 2) and positive charge begins to build up on the more substituted carbon. Recall that alkyl substituents can donate electron density through hyper conjugation and stabilize a positive charge on a carbon.
Unlike in an S N 1 reaction, the nucleophile attacks the electrophilic carbon (step 3) before a complete carbocation intermediate has a chance to form.
Attack takes place preferentially from the backside (like in an S N 2 reaction) because the carbon-oxygen bond is still to some degree in place, and the oxygen blocks attack from the front side. Notice, however, how the regiochemical outcome is different from the base-catalyzed reaction: in the acid-catalyzed process, the nucleophile attacks the more substituted carbon because it is this carbon that holds a greater degree of positive charge.
Example 18.6.1
Predict the major product(s) of the ring opening reaction that occurs when the epoxide shown below is treated with:
- ethanol and a small amount of sodium hydroxide
- ethanol and a small amount of sulfuric acid
Hint: be sure to consider both regiochemistry and stereochemistry!
- Answer
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Epoxide Ring-Opening by Hydrolysis
Epoxides may be cleaved by hydrolysis to give trans -1,2-diols (1,2 diols are also called vicinal diols or vicinal glycols) . The reaction can be preformed under acidic or basic conditions which will provide the same regioselectivity previously discussed.
Acid Catalyzed Hydrolysis
Under aqueous acidic conditions the epoxide oxygen is protonated and is subsequently attacked by a nucleophilic water. After deprotonation to reform the acid catalyst a 1,2-diol product is formed. If the epoxide is asymmetric, the incoming water nucleophile will preferably attack the more substituted epoxide carbon. The epoxide ring is opened by an S N 2 like mechanism so the two -OH groups will be trans to each other in the product.
General Reaction
Mechanism
Basic Hydrolysis
Under aqueous basic conditions the epoxide is opened by the attack of hydroxide nucleophile during an S N 2 reaction. The epoxide oxygen forms an alkoxide which is subsequently protonated by water forming the 1,2-diol product. If the epoxide is asymmetric the incoming hydroxide nucleophile will preferable attack the less substituted epoxide carbon. Because the reaction takes place by an S N 2 mechanism the two -OH groups in the product will be trans to each other.
General Reaction
Mechanism
Epoxide Ring-Opening by HX
Epoxides can also be opened by anhydrous acids (HX) to form a trans halohydrin. When both the epoxide carbons are either primary or secondary the halogen anion will attack the less substituted carbon through an S N 2 like reaction. However, if one of the epoxide carbons is tertiary, the halogen anion will primarily attack the tertiary carbon in an S N 1 like reaction.
General Reaction
Example 18.6.2
Epoxide Ring-Opening by Other Basic Nucleophiles
A wide variety of basic nucleophiles can be used for the ring opening of an epoxide including, amines, hydrides, Grignard reagents, acetylide anions, and hydride.
Reacting Grignard reagents with ethylene oxide is a particularly useful reaction because it produces a primary alcohol containing two more carbon atoms than the original Grignard reagent.
This reaction follows the same S N 2 mechanism as the opening of epoxide rings under basic conditions since Grignard reagents are both strong nucleophiles and strong bases. The first step of the mechanism of this reaction involves the S N 2 attack of the Grignard reaction to open the epoxide to form an alkoxide. The second step of the mechanism involves the protonation of the alkoxide to form an alcohol.
Additional Stereochemical Considerations of Ring-Opening
During the ring-opening of an asymmetrical epoxide, the regiochemical control of the reaction usually allows for one stereoisomer to be produced. However, if the epoxide is symmetrical, each epoxide carbon has roughly the same ability to accept the incoming nucleophile. When this occurs the product typically contains a mixture of enantiomers.
Exercises
Exercise \(\PageIndex{1}\)
Given the following, predict the product assuming only the epoxide is affected. (Remember to show stereochemistry)
- Answer
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Note that the stereochemistry has been inverted
Exercise \(\PageIndex{2}\)
Predict the product of the following, similar to above but a different nucleophile is used and not in acidic conditions. (Remember stereochemistry)
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Exercise \(\PageIndex{3}\)
Provide the structure of the product of the following reaction. Be sure to include proper stereochemistry.
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The ring side of the protonated epoxide intermediate will better stabilize a partial positive charge, so would be the more likely carbon for the chloride ion to attack.
Exercise \(\PageIndex{4}\)
Predict the product of the following reaction.
- Answer
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Exercise \(\PageIndex{5}\)
Provide the final products of the following reactions.
- Answer
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12.5: Behavioral Approaches to Leadership
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5. What are the behavioral perspectives on leadership?
The nearly four decades of research that focused on identifying the personal traits associated with the emergence of leaders and leader effectiveness resulted in two observations. First, leader traits are important—people who are endowed with the “right stuff” (drive, self-confidence, honesty, and integrity) are more likely to emerge as leaders and to be effective leaders than individuals who do not possess these characteristics. Second, traits are only a part of the story. Traits only account for part of why someone becomes a leader and why they are (or are not) effective leaders.
Still under the influence of the great man theory of leadership, researchers continued to focus on the leader in an effort to understand leadership—who emerges and what constitutes effective leadership. Researchers then began to reason that maybe the rest of the story could be understood by looking at what it is that leaders do . Thus, we now turn our attention to leader behaviors and the behavioral approaches to leadership.
It is now common to think of effective leadership in terms of what leaders do. CEOs and management consultants agree that effective leaders display trust in their employees, develop a vision, keep their cool, encourage risk, bring expertise into the work setting, invite dissent, and focus everyone’s attention on that which is important.59 William Arruda, in a Fortune article, noted that "organizations with strong coaching cultures report their revenue to be above average, compared to their peer group.” Sixty-five percent of employees “from strong coaching cultures rated themselves as highly engaged,” compared to 13 percent of employees worldwide."66 Jonathan Anthony calls himself an entrapreneur and corporate disorganizer, because same-old, same-old comms practices are dying in front of our eyes. 61 Apple found Steve Jobs believed that the best leaders are coaches and team cheerleaders. Similar views have been frequently echoed by management consultant Tom Peters.
During the late 1940s, two major research programs—The Ohio State University and the University of Michigan leadership studies—were launched to explore leadership from a behavioral perspective.
The Ohio State University Studies
A group of Ohio State University researchers, under the direction of Ralph Stogdill, began an extensive and systematic series of studies to identify leader behaviors associated with effective group performance. Their results identified two major sets of leader behaviors: consideration and initiating structure.
Consideration is the “relationship-oriented” behavior of a leader. It is instrumental in creating and maintaining good relationships (that is, addressing the group’s maintenance needs) with organizational members. Consideration behaviors include being supportive and friendly, representing people’s interests, communicating openly with group members, recognizing them, respecting their ideas, and sharing concern for their feelings.
Initiating structure involves “task-oriented” leader behaviors. It is instrumental in the efficient use of resources to attain organizational goals, thereby addressing the group’s task needs. Initiating structure behaviors include scheduling work, deciding what is to be done (and how and when to do it), providing direction to organizational members, planning, coordinating, problem-solving, maintaining standards of performance, and encouraging the use of uniform procedures.
After consideration and initiating structure behaviors were first identified, many leaders believed that they had to behave one way or the other. If they initiated structure, they could not be considerate, and vice versa. It did not take long, however, to recognize that leaders can simultaneously display any combination of both behaviors.
The Ohio State studies are important because they identified two critical categories of behavior that distinguish one leader from another. Both consideration and initiating structure behavior can significantly impact work attitudes and behaviors. Unfortunately, the effects of consideration and initiating structure are not consistent from situation to situation. 62 In some of the organizations studied, for example high levels of initiating structure increased performance. In other organizations, the amount of initiating structure seemed to make little difference. Although most organizational members reported greater satisfaction when leaders acted considerately, consideration behavior appeared to have no clear effect on performance.
Initially, these mixed findings were disappointing to researchers and managers alike. It had been hoped that a profile of the most effective leader behaviors could be identified so that leaders could be trained in the best ways to behave. Research made clear, however, that there is no one best style of leader behavior for all situations.
The University of Michigan Studies
At about the same time that the Ohio State studies were underway, researchers at the University of Michigan also began to investigate leader behaviors. As at Ohio State, the Michigan researchers attempted to identify behavioral elements that differentiated effective from ineffective leaders. 63
The two types of leader behavior that stand out in these studies are job centered and organizational member centered. Job-centered behaviors are devoted to supervisory functions, such as planning, scheduling, coordinating work activities, and providing the resources needed for task performance. Employee-member- centered behaviors include consideration and support for organizational members. These dimensions of behavior, of course, correspond closely to the dimensions of initiating structure and consideration identified at Ohio State. The similarity of the findings from two independent groups of researchers added to their credibility. As the Ohio State researchers had done, the Michigan researchers also found that any combination of the two behaviors was possible.
The studies at Michigan are significant because they reinforce the importance of leader behavior. They also provide the basis for later theories that identify specific, effective matches of work situations and leader behaviors. Subsequent research at Michigan and elsewhere has found additional behaviors associated with effective leadership: support, work facilitation, goal emphasis, and interaction facilitation. 64
These four behaviors are important to the successful functioning of the group in that support and interaction facilitation contribute to the group’s maintenance needs, and goal emphasis and work facilitation contribute to the group’s task needs. The Michigan researchers also found that these four behaviors do not need to be brought to the group by the leader. In essence, the leader’s real job is to set the tone and create the climate that ensure these critical behaviors are present. 65
The Leadership Grid ®
Much of the credit for disseminating knowledge about important leader behaviors must go to Robert R. Blake and Jane S. Mouton, who developed a method for classifying styles of leadership compatible with many of the ideas from the Ohio State and Michigan studies. 66 In their classification scheme, concern for results (production) emphasizes output, cost effectiveness, and (in for-profit organizations) a concern for profits. Concern for people involves promoting working relationships and paying attention to issues of importance to group members. As shown in Exhibit 12.9 , the Leadership Grid® demonstrates that any combination of these two leader concerns is possible, and five styles of leadership are highlighted here.
Blake and Mouton contend that the sound (contribute and commit) leader (a high concern for results and people, or 9,9) style is universally the most effective. 67 While the Leadership Grid ® is appealing and well structured, research to date suggests that there is no universally effective style of leadership (9,9 or otherwise). 68 There are, however, well-identified situations in which a 9.9 style is unlikely to be effective. Organizational members of high-involvement organizations who have mastered their job duties require little production-oriented leader behavior. Likewise, there is little time for people-oriented behavior during an emergency. Finally, evidence suggests that the “high-high” style may be effective when the situation calls for high levels of initiating structure. Under these conditions, the initiation of structure is more acceptable, favorably affecting follower satisfaction and performance, when the leader is also experienced as warm, supportive, and considerate. 69
concept check
- What are the behavioral approaches to defining leadership?
- What roles do gender and the popular perceptions of gender roles have on views of leadership traits?


3.4: College Students and Cynicism
“It is really hard to know what is real in today’s society; there are a lot of news sources and it is difficult to trust any of them” (Anonymous student qtd. in Head et al., “How Students Engage With News,” 14).
Project Information Literacy, a nonprofit research institute that studies what it is like to be a student in the digital age, has published a number of influential reports that illustrate the rise in information cynicism among college students.
An “Increasing Avalanche” of Information
In 2018, they conducted a study of almost 6,000 students at 11 U.S. colleges (Head et al., “How Students Engage With News”), finding that:
- 82% of students said that news is necessary in a democracy
- 72% get news from social media on a daily basis
- 68% said the sheer amount of news available to them was overwhelming
- 45% found it difficult to tell real news from “fake news”
- 36% said “fake news” had made them distrust the credibility of any news
- “Deep political polarization” had made students “suspicious of biased reporting”
- Students were highly critical of an “increasing avalanche” of news that “appeals to emotions rather than conveying credible facts.”
“Super Cynical:” A Lack of Trust
Information cynicism was explored more deeply in their 2020 report based on student focus groups and faculty interviews at eight U.S. colleges (Head et al., “Information Literacy”). This study found that:
- Students were cynical almost to the point of believing their concerns and actions had little meaning, and that it was not possible to change things
- An important theme was that “no news source could be trusted at face value” and there was a pervasive belief among students that they should “rely on themselves” to decide what to believe
- Many students felt they had been taught to be critical of everything they encountered, which even extended to the authority of their teachers. One student noted: “We’re all super cynical and untrusting of information to the point that we want to find it out ourselves, so if a teacher says, ‘There’s five rows,’ then we actually look, and, yep, there’s five rows.”
Interestingly, the study found that as a whole, this lack of trust in traditional authority figures meant that trust was placed in Google as the arbiter of truth—a dangerous conclusion, given the nature of algorithms . One student who was also a parent described how he had tried to explain to his child that the bogeyman was not real, but his child had not believed him until a Google search confirmed it!
Sources
Head, Alison J., Barbara Fister, and Margy MacMillan. “ Information Literacy in the Age of Algorithms .” Project Information Literacy , 15 Jan. 2020. Licensed under CC BY-NC-SA 4.0
Head, Alison J., John Wihbey, P. Takis Metaxas, Margy MacMillan, and Dan Cohen. “ How Students Engage with the News: Five Takeaways for Educators, Journalists, and Librarians .” Project Information Literacy , 16 Oct. 2018. Licensed under CC BY-NC-SA 4.0
Image: “ Avalanche ” by Lorenzo Stella , adapted by Aloha Sargent , is licensed under CC BY 4.0
Image: “ Suspicious ” by Freepik, adapted by Aloha Sargent , from Flaticon.com


17.3: Irreducible Polynomials
A nonconstant polynomial \(f(x) \in F[x]\) is irreducible over a field \(F\) if \(f(x)\) cannot be expressed as a product of two polynomials \(g(x)\) and \(h(x)\) in \(F[x]\text{,}\) where the degrees of \(g(x)\) and \(h(x)\) are both smaller than the degree of \(f(x)\text{.}\) Irreducible polynomials function as the “prime numbers” of polynomial rings.
Example \(17.11\)
The polynomial \(x^2 - 2 \in {\mathbb Q}[x]\) is irreducible since
Solution
it cannot be factored any further over the rational numbers. Similarly, \(x^2 + 1\) is irreducible over the real numbers.
Example \(17.12\)
The polynomial \(p(x) = x^3 + x^2 + 2\) is irreducible over \({\mathbb Z}_3[x]\text{.}\) Suppose that this polynomial was reducible over \({\mathbb Z}_3[x]\text{.}\) By the division algorithm there would have to be a factor of the form \(x - a\text{,}\) where \(a\) is some element in \({\mathbb Z}_3[x]\text{.}\)
Solution
Hence, it would have to be true that \(p(a) = 0\text{.}\) However,
Therefore, \(p(x)\) has no zeros in \({\mathbb Z}_3\) and must be irreducible.
Lemma \(17.13\)
Let \(p(x) \in {\mathbb Q}[x]\text{.}\) Then
where \(r, s, a_0, \ldots, a_n\) are integers, the \(a_i\) 's are relatively prime, and \(r\) and \(s\) are relatively prime.
- Proof
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Suppose that
\[ p(x) = \frac{b_0}{c_0} + \frac{b_1}{c_1} x + \cdots + \frac{b_n}{c_n} x^n\text{,} \nonumber \]where the \(b_i\)'s and the \(c_i\)'s are integers. We can rewrite \(p(x)\) as
\[ p(x) = \frac{1}{c_0 \cdots c_n} (d_0 + d_1 x + \cdots + d_n x^n)\text{,} \nonumber \]where \(d_0, \ldots, d_n\) are integers. Let \(d\) be the greatest common divisor of \(d_0, \ldots, d_n\text{.}\) Then
\[ p(x) = \frac{d}{c_0 \cdots c_n} (a_0 + a_1 x + \cdots + a_n x^n)\text{,} \nonumber \]where \(d_i = d a_i\) and the \(a_i\)'s are relatively prime. Reducing \(d /(c_0 \cdots c_n)\) to its lowest terms, we can write
\[ p(x) = \frac{r}{s}(a_0 + a_1 x + \cdots + a_n x^n)\text{,} \nonumber \]where \(\gcd(r,s) = 1\text{.}\)
Theorem \(17.14\). Gauss's Lemma
Let \(p(x) \in {\mathbb Z}[x]\) be a monic polynomial such that \(p(x)\) factors into a product of two polynomials \(\alpha(x)\) and \(\beta(x)\) in \({\mathbb Q}[x]\text{,}\) where the degrees of both \(\alpha(x)\) and \(\beta(x)\) are less than the degree of \(p(x)\text{.}\) Then \(p(x) = a(x) b(x)\text{,}\) where \(a(x)\) and \(b(x)\) are monic polynomials in \({\mathbb Z}[x]\) with \(\deg \alpha(x) = \deg a(x)\) and \(\deg \beta(x) = \deg b(x)\text{.}\)
- Proof
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By Lemma \(17.13\), we can assume that
\begin{align*} \alpha(x) & = \frac{c_1}{d_1} (a_0 + a_1 x + \cdots + a_m x^m ) = \frac{c_1}{d_1} \alpha_1(x)\\ \beta(x) & = \frac{c_2}{d_2} (b_0 + b_1 x + \cdots + b_n x^n) = \frac{c_2}{d_2} \beta_1(x)\text{,} \end{align*}where the \(a_i\)'s are relatively prime and the \(b_i\)'s are relatively prime. Consequently,
\[ p(x) = \alpha(x) \beta(x) = \frac{c_1 c_2}{d_1 d_2} \alpha_1(x) \beta_1(x) = \frac{c}{d} \alpha_1(x) \beta_1(x)\text{,} \nonumber \]where \(c/d\) is the product of \(c_1/d_1\) and \(c_2/d_2\) expressed in lowest terms. Hence, \(d p(x) = c \alpha_1(x) \beta_1(x)\text{.}\)
If \(d = 1\text{,}\) then \(c a_m b_n = 1\) since \(p(x)\) is a monic polynomial. Hence, either \(c=1\) or \(c = -1\text{.}\) If \(c = 1\text{,}\) then either \(a_m = b_n = 1\) or \(a_m = b_n = -1\text{.}\) In the first case \(p(x) = \alpha_1(x) \beta_1(x)\text{,}\) where \(\alpha_1(x)\) and \(\beta_1(x)\) are monic polynomials with \(\deg \alpha(x) = \deg \alpha_1(x)\) and \(\deg \beta(x) = \deg \beta_1(x)\text{.}\) In the second case \(a(x) = -\alpha_1(x)\) and \(b(x) = -\beta_1(x)\) are the correct monic polynomials since \(p(x) = (-\alpha_1(x))(- \beta_1(x)) = a(x) b(x)\text{.}\) The case in which \(c = -1\) can be handled similarly.
Now suppose that \(d \neq 1\text{.}\) Since \(\gcd(c, d) = 1\text{,}\) there exists a prime \(p\) such that \(p \mid d\) and \(p \notdivide c\text{.}\) Also, since the coefficients of \(\alpha_1(x)\) are relatively prime, there exists a coefficient \(a_i\) such that \(p \notdivide a_i\text{.}\) Similarly, there exists a coefficient \(b_j\) of \(\beta_1(x)\) such that \(p \notdivide b_j\text{.}\) Let \(\alpha_1'(x)\) and \(\beta_1'(x)\) be the polynomials in \({\mathbb Z}_p[x]\) obtained by reducing the coefficients of \(\alpha_1(x)\) and \(\beta_1(x)\) modulo \(p\text{.}\) Since \(p \mid d\text{,}\) \(\alpha_1'(x) \beta_1'(x) = 0\) in \({\mathbb Z}_p[x]\text{.}\) However, this is impossible since neither \(\alpha_1'(x)\) nor \(\beta_1'(x)\) is the zero polynomial and \({\mathbb Z}_p[x]\) is an integral domain. Therefore, \(d=1\) and the theorem is proven.
Corollary \(17.15\)
Let \(p(x) = x^n + a_{n - 1} x^{n - 1} + \cdots + a_0\) be a polynomial with coefficients in \({\mathbb Z}\) and \(a_0 \neq 0\text{.}\) If \(p(x)\) has a zero in \({\mathbb Q}\text{,}\) then \(p(x)\) also has a zero \(\alpha\) in \({\mathbb Z}\text{.}\) Furthermore, \(\alpha\) divides \(a_0\text{.}\)
- Proof
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Let \(p(x)\) have a zero \(a \in {\mathbb Q}\text{.}\) Then \(p(x)\) must have a linear factor \(x - a\text{.}\) By Gauss's Lemma, \(p(x)\) has a factorization with a linear factor in \({\mathbb Z}[x]\text{.}\) Hence, for some \(\alpha \in {\mathbb Z}\)
\[ p(x) = (x - \alpha)( x^{n - 1} + \cdots - a_0 / \alpha )\text{.} \nonumber \]Thus \(a_0 /\alpha \in {\mathbb Z}\) and so \(\alpha \mid a_0\text{.}\)
Example \(17.16\)
Let \(p(x) = x^4 - 2 x^3 + x + 1\text{.}\) We shall show that \(p(x)\) is irreducible over \({\mathbb Q}[x]\text{.}\)
Solution
Assume that \(p(x)\) is reducible. Then either \(p(x)\) has a linear factor, say \(p(x) = (x - \alpha) q(x)\text{,}\) where \(q(x)\) is a polynomial of degree three, or \(p(x)\) has two quadratic factors.
If \(p(x)\) has a linear factor in \({\mathbb Q}[x]\text{,}\) then it has a zero in \({\mathbb Z}\text{.}\) By Corollary 17.15, any zero must divide 1 and therefore must be \(\pm 1\text{;}\) however, \(p(1) = 1\) and \(p(-1)= 3\text{.}\) Consequently, we have eliminated the possibility that \(p(x)\) has any linear factors.
Therefore, if \(p(x)\) is reducible it must factor into two quadratic polynomials, say
where each factor is in \({\mathbb Z}[x]\) by Gauss's Lemma. Hence,
Since \(bd = 1\text{,}\) either \(b = d = 1\) or \(b = d = -1\text{.}\) In either case \(b = d\) and so
Since \(a + c = -2\text{,}\) we know that \(-2b = 1\text{.}\) This is impossible since \(b\) is an integer. Therefore, \(p(x)\) must be irreducible over \({\mathbb Q}\text{.}\)
Theorem \(17.17\). Eisenstein's Criterion
Let \(p\) be a prime and suppose that
If \(p \mid a_i\) for \(i = 0, 1, \ldots, n-1\text{,}\) but \(p \notdivide a_n\) and \(p^2 \notdivide a_0\text{,}\) then \(f(x)\) is irreducible over \({\mathbb Q}\text{.}\)
- Proof
-
By Gauss's Lemma, we need only show that \(f(x)\) does not factor into polynomials of lower degree in \({\mathbb Z}[x]\text{.}\) Let
\[ f(x) = (b_rx^r + \cdots + b_0)(c_s x^s + \cdots + c_0 ) \nonumber \]be a factorization in \({\mathbb Z}[x]\text{,}\) with \(b_r\) and \(c_s\) not equal to zero and \(r, s \lt n\text{.}\) Since \(p^2\) does not divide \(a_0 = b_0 c_0\text{,}\) either \(b_0\) or \(c_0\) is not divisible by \(p\text{.}\) Suppose that \(p \notdivide b_0\) and \(p \mid c_0\text{.}\) Since \(p \notdivide a_n\) and \(a_n = b_r c_s\text{,}\) neither \(b_r\) nor \(c_s\) is divisible by \(p\text{.}\) Let \(m\) be the smallest value of \(k\) such that \(p \notdivide c_k\text{.}\) Then
\[ a_m = b_0 c_m + b_1 c_{m - 1} + \cdots + b_m c_0 \nonumber \]is not divisible by \(p\text{,}\) since each term on the right-hand side of the equation is divisible by \(p\) except for \(b_0 c_m\text{.}\) Therefore, \(m = n\) since \(a_i\) is divisible by \(p\) for \(m \lt n\text{.}\) Hence, \(f(x)\) cannot be factored into polynomials of lower degree and therefore must be irreducible.
Example \(17.18\)
The polynomial
is easily seen to be irreducible over
Solution
\({\mathbb Q}\) by Eisenstein's Criterion if we let \(p = 3\text{.}\)
Eisenstein's Criterion is more useful in constructing irreducible polynomials of a certain degree over \({\mathbb Q}\) than in determining the irreducibility of an arbitrary polynomial in \({\mathbb Q}[x]\text{:}\) given an arbitrary polynomial, it is not very likely that we can apply Eisenstein's Criterion. The real value of Theorem 17.17 is that we now have an easy method of generating irreducible polynomials of any degree.
Ideals in \(F\lbrack x \rbrack\)
Let \(F\) be a field. Recall that a principal ideal in \(F[x]\) is an ideal \(\langle p(x) \rangle\) generated by some polynomial \(p(x)\text{;}\) that is,
Example \(17.19\)
The polynomial \(x^2\) in \(F[x]\) generates the ideal \(\langle x^2 \rangle\) consisting of
Solution
all polynomials with no constant term or term of degree \(1\text{.}\)
Theorem \(17.20\)
If \(F\) is a field, then every ideal in \(F[x]\) is a principal ideal.
- Proof
-
Let \(I\) be an ideal of \(F[x]\text{.}\) If \(I\) is the zero ideal, the theorem is easily true. Suppose that \(I\) is a nontrivial ideal in \(F[x]\text{,}\) and let \(p(x) \in I\) be a nonzero element of minimal degree. If \(\deg p(x)= 0\text{,}\) then \(p(x)\) is a nonzero constant and 1 must be in \(I\text{.}\) Since 1 generates all of \(F[x]\text{,}\) \(\langle 1 \rangle = I = F[x]\) and \(I\) is again a principal ideal.
Now assume that \(\deg p(x) \geq 1\) and let \(f(x)\) be any element in \(I\text{.}\) By the division algorithm there exist \(q(x)\) and \(r(x)\) in \(F[x]\) such that \(f(x) = p(x) q(x) + r(x)\) and \(\deg r(x) \lt \deg p(x)\text{.}\) Since \(f(x), p(x) \in I\) and \(I\) is an ideal, \(r(x) = f(x) - p(x) q(x)\) is also in \(I\text{.}\) However, since we chose \(p(x)\) to be of minimal degree, \(r(x)\) must be the zero polynomial. Since we can write any element \(f(x)\) in \(I\) as \(p(x) q(x)\) for some \(q(x) \in F[x]\text{,}\) it must be the case that \(I = \langle p(x) \rangle\text{.}\)
Example \(17.21\)
It is not the case that every ideal in the ring \(F[x,y]\) is a principal ideal. Consider the ideal of \(F[x, y]\) generated by the polynomials \(x\) and \(y\text{.}\)
Solution
This is the ideal of \(F[x, y]\) consisting of all polynomials with no constant term. Since both \(x\) and \(y\) are in the ideal, no single polynomial can generate the entire ideal.
Theorem \(17.22\)
Let \(F\) be a field and suppose that \(p(x) \in F[x]\text{.}\) Then the ideal generated by \(p(x)\) is maximal if and only if \(p(x)\) is irreducible.
- Proof
-
Suppose that \(p(x)\) generates a maximal ideal of \(F[x]\text{.}\) Then \(\langle p(x) \rangle\) is also a prime ideal of \(F[x]\text{.}\) Since a maximal ideal must be properly contained inside \(F[x]\text{,}\) \(p(x)\) cannot be a constant polynomial. Let us assume that \(p(x)\) factors into two polynomials of lesser degree, say \(p(x) = f(x) g(x)\text{.}\) Since \(\langle p(x) \rangle\) is a prime ideal one of these factors, say \(f(x)\text{,}\) is in \(\langle p(x) \rangle\) and therefore be a multiple of \(p(x)\text{.}\) But this would imply that \(\langle p(x) \rangle \subset \langle f(x) \rangle\text{,}\) which is impossible since \(\langle p(x) \rangle\) is maximal.
Conversely, suppose that \(p(x)\) is irreducible over \(F[x]\text{.}\) Let \(I\) be an ideal in \(F[x]\) containing \(\langle p(x) \rangle\text{.}\) By Theorem \(17.20\), \(I\) is a principal ideal; hence, \(I = \langle f(x) \rangle\) for some \(f(x) \in F[x]\text{.}\) Since \(p(x) \in I\text{,}\) it must be the case that \(p(x) = f(x) g(x)\) for some \(g(x) \in F[x]\text{.}\) However, \(p(x)\) is irreducible; hence, either \(f(x)\) or \(g(x)\) is a constant polynomial. If \(f(x)\) is constant, then \(I = F[x]\) and we are done. If \(g(x)\) is constant, then \(f(x)\) is a constant multiple of \(I\) and \(I = \langle p(x) \rangle\text{.}\) Thus, there are no proper ideals of \(F[x]\) that properly contain \(\langle p(x)\rangle\text{.}\)
Historical Note
Throughout history, the solution of polynomial equations has been a challenging problem. The Babylonians knew how to solve the equation \(ax^2 + bx + c = 0\text{.}\) Omar Khayyam (1048–1131) devised methods of solving cubic equations through the use of geometric constructions and conic sections. The algebraic solution of the general cubic equation \(ax^3 + bx^2 + cx + d = 0\) was not discovered until the sixteenth century. An Italian mathematician, Luca Pacioli (ca. 1445–1509), wrote in Summa de Arithmetica that the solution of the cubic was impossible. This was taken as a challenge by the rest of the mathematical community.
Scipione del Ferro (1465–1526), of the University of Bologna, solved the “depressed cubic,”
He kept his solution an absolute secret. This may seem surprising today, when mathematicians are usually very eager to publish their results, but in the days of the Italian Renaissance secrecy was customary. Academic appointments were not easy to secure and depended on the ability to prevail in public contests. Such challenges could be issued at any time. Consequently, any major new discovery was a valuable weapon in such a contest. If an opponent presented a list of problems to be solved, del Ferro could in turn present a list of depressed cubics. He kept the secret of his discovery throughout his life, passing it on only on his deathbed to his student Antonio Fior (ca. 1506–?).
Although Fior was not the equal of his teacher, he immediately issued a challenge to Niccolo Fontana (1499–1557). Fontana was known as Tartaglia (the Stammerer). As a youth he had suffered a blow from the sword of a French soldier during an attack on his village. He survived the savage wound, but his speech was permanently impaired. Tartaglia sent Fior a list of 30 various mathematical problems; Fior countered by sending Tartaglia a list of 30 depressed cubics. Tartaglia would either solve all 30 of the problems or absolutely fail. After much effort Tartaglia finally succeeded in solving the depressed cubic and defeated Fior, who faded into obscurity.
At this point another mathematician, Gerolamo Cardano (1501–1576), entered the story. Cardano wrote to Tartaglia, begging him for the solution to the depressed cubic. Tartaglia refused several of his requests, then finally revealed the solution to Cardano after the latter swore an oath not to publish the secret or to pass it on to anyone else. Using the knowledge that he had obtained from Tartaglia, Cardano eventually solved the general cubic
Cardano shared the secret with his student, Ludovico Ferrari (1522–1565), who solved the general quartic equation,
In 1543, Cardano and Ferrari examined del Ferro's papers and discovered that he had also solved the depressed cubic. Cardano felt that this relieved him of his obligation to Tartaglia, so he proceeded to publish the solutions in Ars Magna (1545), in which he gave credit to del Ferro for solving the special case of the cubic. This resulted in a bitter dispute between Cardano and Tartaglia, who published the story of the oath a year later.


30.1B: Effects of Aging on the Integumentary System
The integumentary system undergoes intrinsic and extrinsic cutaneous aging, primarily involving the dermis.
- Contrast intrinsic aging with extrinsic aging of the integumentary system
Key Points
- Intrinsic aging is caused by internal factors alone, such as diminished collagen synthesis, and is sometimes known as chronological aging.
- Extrinsic aging is caused by external factors, such as exposure to ultraviolet radiation, which can result in photodamage and may lead to skin cancer formation.
- A validated comprehensive grading scale categorizes the clinical findings of skin aging as laxity, rhytids (wrinkles), and the various facets of photoaging, including erythema/telangiectasia, dyspigmentation, solar elastosis, keratoses, and poor texture.
Key Terms
- extrinsic aging : Aging caused by external factors such as exposure to radiation or toxic substances.
- intrinsic aging : Aging caused by internal factors related to degeneration of physiologic processes.
Intrinsic and extrinsic aging are terms used to describe cutaneous aging of the skin and other parts of the integumentary system. Aging primarily involves the dermis and has limited effects on the epidermis.
Intrinsic Aging
Wrinkled Skin : Wrinkles form due to intrinsic (loss of collagen) and extrinsic (photodamage) aging.
The effects of intrinsic aging are caused solely by internal factors. Sometimes called chronological aging, intrinsic aging is an inherent degenerative process due to declining physiologic functions and capacities. This may include qualitative and quantitative changes such as diminished or defective synthesis of collagen and elastin in the dermis. As skin ages, it becomes thinner and more easily damaged. Intensifying this effect is the decreasing ability of skin to heal itself with age. Skin aging is also noted by a decrease in volume and elasticity and the increased incidence of wrinkles. Aging skin receives less blood flow and lower glandular activity. Cortisol (associated with stress) causes degradation of collagen, accelerating the aging process.
Extrinsic Aging
Skin Cancer : Image of a basal cell carcinoma caused by extrinsic aging.
Extrinsic aging of skin is caused by external factors such as ultraviolet radiation, cigarette smoking, and air pollution. Of all extrinsic causes, radiation from sunlight has the most widespread documentation of negative effects on the skin. Because of this, extrinsic aging is often referred to as photoaging, defined as skin changes caused by chronic exposure to UV light. Photodamage implies changes beyond those associated with aging alone. It is defined as cutaneous damage caused by chronic exposure to solar radiation and is associated with emergence of neoplastic (cancer) lesions. Photoaging causes two main concerns: an increased risk for skin cancer, and the appearance of damaged skin. In younger skin, sun damage will heal faster since the cells in the epidermis have a faster turnover rate, while in older adults, thinner skin and slower healing may result in damage to the dermal layer.
A validated comprehensive grading scale has categorized the clinical findings of skin aging as laxity (sagging), rhytids (wrinkles), and the various facets of photoaging, including erythema/telangiectasia (redness), dyspigmentation (brown discoloration), solar elastosis (yellowing), keratoses (abnormal growths), and poor texture.
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9.5: Assessment
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Which of the following is not a common type of incident in a SCADA/ICS?
| a. Unauthorized access to system controls | c. Vendor goes out of business and can no longer supply critical components |
| b. A worm infects a network at a nuclear power plant | d. Vendor improperly performs a security assessment, resulting in loss of system availability |
____ 2. In which phase of NIST’s SP 800-61 would organizations prioritize response to multiple threat actions?
| a. Preparation | c. Containment Eradication and Recovery |
| b. Detection and Analysis | d. Post-Incident Activity |
Matching
Match each core capability of the National Response Framework with its objective.
| A. Planning | H. Mass Care Services |
| B. Public Information and Warning | I. Mass Search and Rescue Operations |
| C. Operational Coordination | J. On-Scene Security and Protection |
| D. Critical Transportation | K. Operational Communications |
| E. Environmental Response/Health and Safety | L. Public and Private Services and Resources |
| F. Fatality Management Services | M. Public Health and Medical Services |
| G. Infrastructure Systems | N. Situational Assessment |
____ 3. Ensure the availability of guidance and resources
____ 4. Relay information on threats and hazards
____ 5. Provide life-sustaining services, including food and shelter
____ 6. Provide communications
____ 7. Establish and maintain an operational structure and process
____ 8. Provide decision-makers with information
____ 9. Deliver search and rescue operations
____ 10. Provide transportation for response
____ 11. Provide essential services
____ 12. Engage the community to develop response approaches
____ 13. Provide lifesaving medical treatment
____ 14. Stabilize infrastructure
____ 15. Provide law enforcement and security
____ 16. Body recovery and victim identification services
Match the following sections of the ICS Cyber Incident Response Plan with their contents.
| A. Overview, Goals, and Objectives | F. Response Actions |
| B. Incident Description | G. Communications |
| C. Incident Detection | H. Forensics |
| D. Incident Notification | I. Additional Sections |
| E. Incident Analysis |
____ 17. Includes media contacts
____ 18. Incident type classification
____ 19. Addresses how an incident is prioritized and escalated
____ 20. Addresses how to evaluate and analyze an incident
____ 21. Other stuff
____ 22. Discusses business objectives
____ 23. The process for collecting, examining, and analyzing incident data, with an eye to legal action
____ 24. Defines the procedures used for each type of incident
____ 25. Describes how an incident is identified and reported
Short Answer
26. Define incident containment and provide an example of how it would be applied to an incident.
27. Discuss how the response strategy for an incident that was sourced from within the organization would differ from one sourced from outside of the organization.
For the answers to these questions, email your name, the name of your college or other institution, and your position there to<EMAIL_ADDRESS>. CyberWatch West will email you a copy of the answer key.


4.7: Electrophilic addition reactions
- Draw an electrophilic addition mechanism with curly arrows showing the movement of electrons.
- Apply the electrophilic addition to reactions of alkenes and alkynes with halogen acids and water.
- Define the concepts: regioselectivity, stereoselective and stereospecific reactions, and tautomerism with examples of electrophilic addition relations of alkenes and alkynes.
Electrophilic addition mechanism
The \(\pi\)-bond of alkenes distributes electrons above and below the \(\sigma\)-bond that creates a local electron-rich (\(\delta{-}\)) region, as shown by red color region in the following electrostatic potential map of ethene.
Therefore, the alkenes are nucleophiles (\(\delta{-}\)) attracted to electrophiles (\(\delta{+}\)) and capable of donating their \(\pi\)-electrons to make a bond with them, as shown in the following reaction mechanism.
In first step of this mechanism, \(\pi\)-bond acts as an incoming nucleophile making a \(\sigma\)-bond with the electrophilic \(\delta{+}\) end of the polar reagent \(\ce{\overset{\delta{+}}{A}{-}\overset{\delta{-}}{B}}\) while the (\(\ce{\overset{\delta{-}}{B}}\)) acts as a leaving group, leaving as a nucleophile \(\ce{B^{-}}\). One of the \(\pi\)-bonded carbon that receives the electrophile \(\ce{\overset{\delta{+}}{A}}\) becomes sp 3 hybridized, and the other carbon stays sp 2 hybridized with an empty p-orbital as a carbocation. In the second step, the nucleophile (\(\ce{B^{-}}\)) attacks the electrophilic carbocation and makes a new bond. As a result, the reagent \(\ce{A-B}\) add to the \(\ce{C=C}\)-bond. Since the process begins with the addition of an electrophile \(\ce{\overset{\delta{+}}{A}}\), it is called an electrophilic addition reaction.
Examples of electrophilic addition reactions of alkenes
Halogen acids (\(\ce{\overset{\delta{+}}{H}{-}\overset{\delta{-}}{Cl}}\), \(\ce{\overset{\delta{+}}{H}{-}\overset{\delta{-}}{Br}}\), and \(\ce{\overset{\delta{+}}{H}{-}\overset{\delta{-}}{l}}\)) are among many reagents that add to alkenes by electrophilic addition mechanism, as shown in the following example.
Water (\(\ce{H2O}\)) has more polar \(\ce{\overset{\delta{+}}{H}{-}\overset{\delta{-}}{O}}\) bonds than the halogen acids, but \(\ce{-OH}\) is a bad leaving group and usually do not act as a leaving group. The bad leaving \(\ce{-OH}\) is first converted to a good leaving \(\ce{-\overset{+}{O}H2}\) by adding stronger acid like sulfuric acid (\(\ce{H2SO4}\)) in the water. Then, the hydronium ion (\(\ce{H-\overset{+}{O}H2}\)) enters the first step of the electrophilic addition reaction, as shown below.
\(\ce{H2O}\) acts as an incoming nucleophile in step#2, another water molecule removes its acidic proton in step#3. This reaction is called hydration of alkene because the result is the addition of a water molecule \(\ce{\overset{\delta{+}}{H}{-}\overset{\delta{-}}{OH}}\) to a \(\ce{C=C}\)-bond of an alkene.
Sulfuric acid is used as an acid catalyst because its conjugate base \(\ce{HSO4^{-}}\) is a poor nucleophile. \(\ce{H2O}\) is also a poor nucleophile, but being in higher concentration wins the completion in step#2. Even if \(\ce{HSO4^{-}}\) add to the carbocation in the second step, it is substituted by \(\ce{H2O}\) under the reaction conditions.
Regioselectivity of electrophilic addition reactions
In the cases of unsymmetrical alkenes, two constitutional isomers are possible, but one is produced exclusively or as a major product. The reason is that the two potential carbonation intermediates have unequal stability. The stability of carbocations increases in this order: \(\overrightarrow{\text{primary (\(\ce{R{\overset{+}{C}}H2}\)) < secondary (\(\ce{R2{\overset{+}{C}}H}\)) < tertiary (\(\ce{R3{\overset{+}{C}}}\)) }}\). The following examples show that the more stable carbocation is formed exclusively or predominantly when there is a choice.
Chemical reactions in which two constitutional isomers are possible, but one is formed exclusively or predominantly are called regioselective reactions . Electrophilic additions are regioselective reactions.
Stereoselectivity of electrophilic addition reactions
When reactants and catalysts are achiral, but the product is chiral, it is produced equally in both (R) and (S) configurations. For example, the carbocation intermediate in electrophilic addition reactions is trigonal planer. The nucleophile can attack it from either side with about equal probability resulting in a racemic mixture, as illustrated in the following example.
However, suppose one of the reactants or the catalyst is chiral and a pure enantiomer. In that case, one of the enantiomers (R or S) is usually formed exclusively or as a major product. Enzymes in biochemical systems are chiral catalysts that usually produce one enantiomer exclusively.
Chemical reactions that produce one stereoisomer exclusively or preferentially relative to the others are called stereoselective reactions .
Enzyme-catalyzed reactions are stereoselective reactions. For example, hydration fumarate, catalyzed by an enzyme fumarase, produces (S)-malate, as shown below.
The R-enantiomer is not produced in the above reaction.
Enzymes usually also react with one stereoisomer in the reactant exclusively. For example, fumarase catalyzes the hydration of fumarate but does not react with its enantiomer maleate, as illustrated below.
Chemical reactions that preferentially react with one stereoisomer among reactant and preferentially produce one stereoisomer in the products are called stereospecific reactions .
Enzyme-catalyzed reactions are often stereospecific reactions. For example, the fumarase enzyme selectively reacts with fumarate (and not with its stereoisomer maleate) and selectively produces (S)-malate (and not its stereoisomer (R)-malate).
Electrophilic addition reactions of alkynes
Alkynes have a triple \(\ce{C≡C}\)-bond, i.e., one \(\sigma\)-bond and two \(\pi\)-bonds. The electrophilic addition reactions happen on \(\pi\)-bonds of alkynes. If the reagent is in a one-to-one mole ratio, it adds to one of the \(\pi\)-bonds. If the reagent is in excess, a second addition reaction happens on the product of the first addition, as shown below.
In step#2, the nucleophile (\(\ce{Br} in this case) adds to the carbon carrying the nucleophile of step#1. This is because the carbonation intermediate of step#2 is stabilized by resonance with the lone pair of electrons on the nucleophile added in the first step.
Hydration of alkyne follows the same step#1 as for alkenes, but the intermediate enol containing \(\ce{C=C}\) and \(\ce{-OH}\) together exists in equilibrium with its structural isomer ketone, as shown in the following example.
Tautomers are constitutional isomers that are readily interconvertible. Tauomerization is the chemical reaction that interconverts the tautomers.
Enol and its isomer ketone, e.g., but-2-ene-2-ol and but-2-one in the above example, are called tautomers, and the equilibrium between the two in step#2 of the above reaction is tautomerization .


8.2: Angular Momentum
Learning Objectives
- Explain angular momentum
Nonrelativistically, the angular momentum of a particle with momentum \(p\), at a position \(r\) relative to some arbitrarily fixed point, is \(L = r×p\). When we generalize this equation to relativity, we run into a number of issues. Issues due to special relativity:
- The vector cross product only makes sense in three dimensions, so it is not well defined in special relativity (section 7.6)
- Assuming we get around issue number 1, how do we know that this quantity is conserved?
And from general relativity:
- In general relativity, only infinitesimally small spatial or spacetime displacements \(dr\) can be treated as vectors. Larger ones cannot. This is because spacetime can be curved, and vectors can’t be used to define displacements on a curved space (e.g., the surface of the earth).
- If space has a nontrivial topology, then we may not be able to define an orientation (section 7.6)
For points 3 and 4, we refer to Hawking and Ellis in section 3.5. Number 2 is addressed in section 9.3. For number 2 we will need the stress-energy tensor, which will be described in chapter 9. Lest you feel totally cheated, we will resolve issue number 1 in this section itself, but before we do that, let’s consider an interesting example that can be handled with simpler math.
The Relativistic Bohr model
If we want to see an interesting real-world example of relativistic angular momentum, we need something that rotates at relativistic velocities. At large scales we have astrophysical examples such as neutron stars and the accretion disks of black holes, but these involve gravity and would therefore require general relativity. At microscopic scales we have systems such as hadrons, nuclei, atoms, and molecules. These are quantum-mechanical, and relativistic quantum mechanics is a difficult topic that is beyond the scope of this book, but we can sidestep that issue by using the Bohr model of the atom. In the Bohr model of hydrogen, we assume that the electron has a circular orbit governed by Newton’s laws, does not radiate, and has its angular momentum quantized in units of \(\hbar \). Let’s generalize the Bohr model by applying relativity.
It will be convenient to define the constant \(\alpha = \frac{ke^2}{\hbar }\), known as the fine structure constant, where \(k\) is the Coulomb constant and \(e\) is the fundamental charge. The fine structure constant is unitless and is approximately \(1/137\). It is essentially a measure of the strength of the electromagnetic interaction, and in the Bohr model it also turns out to be the velocity of the electron (in units of \(c\)) in the ground state of hydrogen. Because this velocity is small compared to \(1\), we expect relativistic corrections in hydrogen to be small — of relative size \(α^2\). But we have an interesting opportunity to get at some additional and more exciting physics if we consider a hydrogenlike atom, i.e., an ion with \(Z\) protons in the nucleus and only one electron. Raising \(Z\) cranks up the energy scale and therefore increases the velocity as well.
Combining the Coulomb force law with the result of Example 4.5.1, for uniform circular motion, we have
\[\dfrac{kZe^2}{r} = m\gamma v^2\]
where the factor of \(γ\) is the relativistic correction. The electron’s momentum is perpendicular to the radius vector, so we assume for the moment that (as turns out to be true), the angular momentum is given by \(L = rp = mvγr\), where again a relativistic correction factor of \(γ\) appears. This is quantized, so let \(L = l\hbar\), where \(l\) is an integer. Solving these equations gives
\[v = \frac{Z\alpha }{l} \label{eq1}\]
\[r = \frac{l^2 \hbar}{mZ\alpha \gamma }\]
These differ from the nonrelativistic versions only by the factor of \(γ\) in the second equation. The electrical energy is \(U = -kZe^2/r\), and the kinetic energy \(K = m(γ - 1)\) (with \(c = 1\)). We will find it convenient to work with the (positive) binding energy in units of the mass of the electron. Call this quantity \(\varepsilon\). After some algebra, the result is
\[\varepsilon = 1 - \sqrt{1 - v^2}\]
Surprisingly, this is also the exact result given by relativistic quantum mechanics if we solve the Dirac equation for the ground state, or if we take a high-energy (nearly unbound) state with the maximum value of \(l\), as is appropriate for a semiclassical circular orbit. So we can see that even though our quantum mechanics was crude, our relativity makes some sense and gives reasonable results. For small \(Z\), a Taylor series approximation gives
\[E = v^2/2 + v^4/8 + ...\]
where the fourth-order term represents the relativistic correction.
So far so good, but now what if we crank up the value of \(Z\) to make the relativistic effects strong? A very disturbing thing happens when we make \(Z \gtrsim 137 \approx 1/\alpha\). In the ground state we get \(v > 1\) and a complex number for \(\varepsilon\). Clearly something has broken down, and our results no longer make sense. We might be inclined to dismiss this as a consequence of our crude model, but remember, our calculations happened to give the same result as the Dirac equation, which has real relativistic quantum mechanics baked in. We should take this breakdown as evidence of a real physical breakdown. The interpretation is as follows.
According to quantum mechanics, the vacuum isn’t really a vacuum. Particle-antiparticle pairs are continually popping into existence in empty space and then reannihilating one another. Their temporary creation is a violation of the conservation of mass-energy, but only a temporary violation, and this is allowed by the time-energy form of the Heisenberg uncertainty principle, \(\Delta E \Delta t \gtrsim h\), as long as \(∆t\) is short. It’s as though we steal some money, but the police don’t catch us as long as we put it back before anyone can notice. Because these particles are only temporarily in our universe, we call them virtual particles, as opposed to real particles that have a potentially permanent existence and can be detected as blips on a Geiger counter.
But when the vacuum contains an electric field that is beyond a certain critical strength, it becomes possible to create an electron-antielectron pair, let the opposite charges separate and release energy, and pay off the energy debt without having to reannihilate the particles. This is known as “sparking the vacuum.” As of this writing, only nuclei with \(Z\) up to about \(118\) have been discovered, and in any case the critical \(Z\) value of \(1/α ≈ 137\) was only a rough estimate. But by colliding heavy nuclei such as lead, one can at least temporarily form an unstable compound system with a high \(Z\), and attempts are being made to search for the predicted effect in the laboratory.
The Angular Momentum Tensor
As mentioned previously, there is no such thing as a vector cross product in four dimensions, so the nonrelativistic definition of angular momentum as \(L = r×p\) needs to be modified to be usable in relativity.
Given a position vector \(r^a\) and a momentum vector \(p^b\), we expect based both on units and the correspondence principle that a relativistic definition of angular momentum must be some kind of a product of the vectors. Based on the rules of index notation, we don’t have much leeway here. The only products we can form are \(r^a p^b\), which is a rank-\(2\) tensor, or \(r^a p_a\), a scalar. Since nonrelativistic angular momentum is a three-vector, the correspondence principle tells us that its relativistic incarnation can’t be a scalar — there simply wouldn’t be enough information in a scalar to tell us the things that the nonrelativistic angular momentum vector tells us: what axis the rotation is about, and which direction the rotation is.
The tensor \(r^a p^b\) also has a problem, but one that can be fixed. Suppose that in a certain frame of reference a particle of mass \(m \neq 0\) is at rest at the origin. Then its position four-vector at time \(t\) is \((t,0,0,0)\), and its energy-momentum vector is \((m,0,0,0)\). These vectors are parallel. The tensor \(r^a p^b\) is nonzero and nonconserved as time ﬂows, but clearly we want the angular momentum of an isolated particle to be conserved. Another example would be if, at a certain moment in time, we had \(r = (0,x,0,0)\) and \(p = (E,p,0,0)\), with both \(x\) and \(p\) positive. This particle’s motion is directly away from the origin, so its angular momentum should be zero by symmetry, but \(r^a p^b\) is again nonzero.
The way to fix the problem is to force the product of the position and momentum vectors to be an antisymmetric tensor:
\[L^{ab} = r^a p^b - r^b p^a\]
Antisymmetric means that \(L^{ab} = -L^{ba}\), so that elements on opposite sides of the main diagonal are the same except for opposite signs. A quick check shows that this gives the expected zero result in both of the above examples. A component such as \(L^{yz}\) measures the amount of rotation in the \(y-z\) plane. In a nonrelativistic context, we would have described this as an \(x\) component \(L_x\) of the angular momentum three-vector, because a rotation of the \(y-z\) plane about the origin is a rotation about the \(x\) axis — such a rotation keeps the \(x\) axis fixed. But in four-dimensional spacetime, a rotation in the \(y-z\) plane keeps the entire \(t-x\) plane fixed, so the notion of rotation “about an axis” breaks down. (Notice the pattern: in two dimensions we rotate about a point, in three dimensions rotation is about a line, and in four dimensions we rotate about a fixed plane.) In section 9.3, we show that \(L^{ab}\) is conserved.
If we lay the angular momentum tensor out in matrix format, it looks like this:
\[\begin{pmatrix} 0 & L^{tx} & L^{ty} & L^{tz}\\ & 0 & L^{xy} & L^{xz}\\ & & 0 & L^{yz}\\ & & & 0 \end{pmatrix}\]
The zeroes on the main diagonal are due to the antisymmetrization in the definition. I’ve left blanks below the main diagonal because although those components can be nonzero, they only contain a (negated) copy of the information given by the ones above the diagonal. We can see that there are really only \(6\) pieces of information in this \(4×4\) matrix, and we’ve already physically interpreted the triangular cluster of three space-space components on the bottom right.
Why do we have the row on the top, consisting of the time-space components, and what do they mean physically? A highbrow answer would be that this is something very deep having to do with the fact that, as described in Section 8.3, rotation and linear motion are not as cleanly separated in relativity as they are in nonrelativistic physics. A more straightforward answer is that in most situations these components are actually not very interesting. Consider a cloud of particles labeled \(i = 1\) through \(n\). Then for a representative component from the top row we have the total value
\[L^{tx} = \sum t_i p_{i}^{x} - \sum x_i E_i\]
Now suppose that we fix a certain surface of simultaneity at time \(t\). The sum becomes
\[L^{tx} = t\sum p_{i}^{x} - \sum x_i E_i\]
There is information here, but it’s not exciting information about angular momentum, it’s boring information about the position and motion of the system’s center of mass. If we fix a frame of reference in which the total momentum is zero, i.e., the center of mass frame, then we have \(\sum p_{i}^{x} = 0\). Let’s also define the position of the center of mass as the average position weighted by mass-energy, rather than the mass-weighted average, as we would do in Newtonian mechanics. Then the \(\sum x_{i}E_i\) is a constant relating to the position of the center of mass, and if we like we can make it equal zero by choosing the origin of our spatial coordinates to coincide with the center of mass.
With these choices we have a much simpler angular momentum tensor:
\[\begin{pmatrix} 0 & 0 & 0 & 0\\ & 0 & L^{xy} & L^{xz}\\ & & 0 & L^{yz}\\ & & & 0 \end{pmatrix}\]
If we wish, we can sprinkle some notational sugar on top of all of this using the Levi-Civita tensor described in optional Section 7.6. Let’s define a new tensor \(^\star L\) according to
\[^\star L_{ij} = \frac{1}{2}\epsilon _{ijkl}L^{kl}\]
Then for an observer with velocity vector \(o^µ\), the quantity \(o_{\mu }^\star L^{\mu \nu }\) has the form \((0,L^{yz},L^{zx},L^{xy}). That is, its spatial components are exactly the quantities we would have expected for the nonrelativistic angular momentum three-vector (using the correct relativistic momentum).


20.4: What Anthropologists Can Do
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- Jennifer Hasty, David G. Lewis, & Marjorie M. Snipes
- OpenStax
Learning Objectives
By the end of this section, you will be able to:
- Describe the primary areas where the anthropological approach is relevant.
- Identify the ways that anthropologists are specifically trained for today’s challenges.
- Explain how anthropological skills can help address contemporary problems.
What Anthropologists Do Today
Anthropologists are at work now to make a difference in our lives. There are various ways in which anthropologists and those utilizing an anthropological lens or framework contribute critically needed skills and resources in the 21st century.
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Research
. Sometimes referred to as
pure
or
theoretical
research, fieldwork is conducted in all kinds of settings in order to answer practical and theoretical questions that form the basis of anthropology. How do cultures change? How do artifacts and technology evolve within a culture? How do trade and exchange affect the development of cultures?
Many of the chapters in this text feature stories about anthropological research and its importance in understanding what it means to be human. Each of the subfields engages in distinct types of field research as ways to test theories and advance our knowledge of human beings. Theoretical research is the backbone of academic anthropology.
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Research and development
. Research and development are associated with practical applications, such as creating or redesigning products or services for governments or corporations. Anthropologists who work in research and development contribute what they know about human behavior and the world around us to projects that serve the interests of human organizations and the human community.
Cultural anthropologist Genevieve Bell worked for 18 years in research and development for Intel Corporation, the world’s largest semiconductor chip manufacturer. Her focus at Intel was on user experience, researching how people use technology and apply it in their lives with the goal of designing more relevant and user-friendly products. Intel valued the way Bell’s deep knowledge of human behavior and human culture helped the company better anticipate their clients’ needs. Bell’s insights helped make Intel a more competitive corporation. She has described her job as “mak[ing] sense of what makes people tick, what delights and frustrates them, and . . . us[ing] those insights to help shape next generation technology innovations. I sit happily at the intersection of cultural practices and technology adoption” ( City Eye 2017).
In a TED Salon talk titled “ 6 Big Ethical Questions about the Future of AI ,” Bell explains that the technological revolution of artificial intelligence is already in progress, affecting many aspects of our lives. She says that the challenge now is to use artificial intelligence “safely, sustainably, and responsibly.” Bell advocates for human-scale technology. Using skills and knowledge gained through her training as an anthropologist, she looks at the ways in which technology, culture, and environment interact. In her work today, she continues to use an anthropological approach: “It’s about thinking differently, asking different kinds of questions, looking holistically at the world and the systems” (Bell 2020).
Bell left Intel in 2017 to serve as a distinguished professor at the Australian National University College of Engineering and Computer Science, where she serves as the director of the School of Cybernetics and continues to research the interface between culture and technology.
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Public policy
. Anthropologists are involved in public policy making all over the world. Anthropological skills and outlooks are increasingly valuable to the development of principles and regulatory measures that increase public safety and resolve real-world problems. Applying a holistic approach to these issues allows government and nongovernment organizations to avoid some problems and better anticipate future challenges.
The American Anthropological Association (AAA) has identified five public policy areas that would greatly benefit from an anthropological approach. In each of these areas, the AAA hopes to involve more anthropologists in public policy in the 21st century and to work collectively to message international, national, and local agencies about the importance of anthropological knowledge and involvement:
- Social and cultural aspects of health : identifying ways in which categories of race, ethnicity, gender, socioeconomic status, and age hinder medical delivery.
- Culture and diversity in education : understanding the diversities that affect educational delivery and the gaps that exist in current educational policies due to such things as changing demographics and new information technologies.
- An interdisciplinary approach to the environment : focusing on the ways in which anthropological knowledge contributes to understanding the human dimensions of the environment and interfacing with federal agencies actively seeking to support this type of environmental research.
- Economic, social, and cultural aspects of the information revolution : examining the human dimensions of the information revolution and the impact that it is having on our work and personal lives.
- Globalization and its impact on public policy : specifically, focusing on issues of conflict and war and the effects of globalization on transnational communities.
One of the challenges that anthropologists face is better educating governments and corporations about the skills they can bring to understanding and addressing contemporary problems. Working collaboratively within and beyond the discipline is important for advancing an awareness of the possibilities that anthropologists offer as public policy advocates.
Profiles in Anthropology
Gillian Tett (1967-)
Personal History: Gillian Tett is a British author and journalist who trained in anthropology. She studied at Clare College, Cambridge University, where she earned her PhD in social anthropology after conducting doctoral research in Tajikistan, in what was then the Soviet Union. Tett intentionally chose to turn her anthropological gaze outside of the university setting, where she believed her training would have greater impact.
Area of Anthropology and Importance of Her Work: Though is trained as a social anthropologist, Gillian Tett works for the Financial Times , a global daily newspaper, as chair of the editorial board and editor at large in addition to her role as a journalist. Her articles on finance, business, and political economy appear in the Financial Times and in various leading newspapers and media outlets. She forecast early warnings about the 2008 economic downturn, applying her anthropological knowledge and skills to understand emerging global economic patterns, and she participates frequently in conferences on finance and global economics. Tett also contributes to new directions in anthropology; at the joint 2019 American Anthropological Association and Canadian Anthropology Society/La société canadienne d’anthropologie Annual Meeting, she served as a discussant in a presidential session on the topic of breaking down silos in anthropology,
Accomplishments in the Field: Tett has earned various commendations and awards in and outside of the field of anthropology, including the British Press Award for Business and Finance Journalist of the Year in both 2008 and 2009. She was awarded the President’s Medal of the British Academy in 2011, given in recognition of “academic-related service activity” beyond the academy. Her book Fool’s Gold: How Unrestrained Greed Corrupted a Dream, Shattered Global Markets, and Unleashed a Catastrophe (2009), which takes a cultural anthropological approach to analyzing the global economy and financial system, was a New York Times best seller and was chosen as the 2009 Financial Book of the Year by Spear’s magazine. In 2014, Tett received the Royal Anthropological Institute’s Marsh Award for Anthropology in the World, which “recognises an outstanding individual based outside academia, one who has shown how to apply anthropology or anthropological ideas to the better understanding of the world’s problems” (Royal Anthropological Institute 2021). Her latest book is Anthro-Vision: A New Way to See in Business and Life (2021), published by Simon & Schuster.
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Applied or practicing anthropology
. Anthropologists are engaged in wide-ranging work on the ground in real-life situations, helping address numerous current and emerging needs in communities around the world. Many work within nongovernmental agencies. Some anthropologists are already engaged in efforts pertaining to the COVID-19 pandemic, gathering preliminary data and working to streamline access to treatment and preventative measures.
In 2014, the WHO reached out to sociocultural anthropologists to help address an outbreak of the Ebola virus in Mali. They sought the help of these anthropologists as liaisons to connect with the local people and lessen their anxieties about the disease, help those recovering cope with the stigma of having had Ebola, and build a bridge between the community and the health system. They also sought anthropological direction on how best to interact with local people while respecting their culture and traditions. The WHO described some of the roles of the anthropologists who aided in this project:
The social anthropologists have also helped train teams searching for Ebola patients and monitoring Ebola contacts, teaching them to make allowances for local culture and the rules of hospitality and politeness when visiting families. These factors are key to getting the message across and being heard by members of the community. (World Health Organization 2015)
The global emergency of COVID-19 mobilized a number of anthropologists, especially those in the applied field of medical anthropology. Medical anthropologist Mark Nichter (2020), who has studied emerging diseases and global health for much of his career, was returning from fieldwork in India and Indonesia when COVID-19 cases started being diagnosed in the United States. He traveled from Asian countries, where people were wearing masks and showing a high level of concern for the disease, into Europe and then the United States, where there seemed to be little concern. These different attitudes prompted him to think about other pandemics he had experienced as a medical anthropologist and about how complex these global events can be. Deeply aware of issues of social inequality, he worried about the poor infrastructure conditions in so many countries and the dense populations in refugee camps. What would happen in water-insecure areas where accessing any kind of water, especially clean water for handwashing, was difficult? He wondered just how bad this was going to be as a global event.
During lockdown in the United States, Nichter used his training as a medical anthropologist to create positive change within his community. He first developed a COVID-19 primer, explaining health concepts about COVID-19 and methods of slowing and preventing transmission in everyday terms to help professors and teachers educate themselves and their students. The primer quickly began circulating on campuses in the United States and around the world. Nichter also worked with fellow anthropologists in a special working group supported by the American Anthropological Association to identify research areas of critical need. Many of these research areas concerned structural threats and areas where mortality data were revealing disparities, indicating that certain populations were more vulnerable than others. Third, Nichter began advocating and working for COVID-19 testing resources, the development of contact tracing, and symptom monitoring to better contain outbreaks within communities. Lastly, he helped develop a health care worker support network with both online and grassroots resources, knowing that frontline workers would be those most taxed by the pandemic. Nichter advocates for what he calls anticipatory anthropology . In the context of medical anthropology, anticipatory anthropology acts to shore up the fault lines that have emerged in the global health system, working toward creating stronger resistance to the next health care emergency. “COVID-19 provides an opportunity to build alliances and momentum for significant health care reform” (Nichter 2020).
Anthropological skills are increasingly vital to developing and communicating culturally relevant messages. While global health initiatives are very prominent within the field of applied and practicing anthropology, the range of interventions is wide. Applied anthropology projects might involve improved farming techniques and heirloom seed banks, better educational services, and even work on the front lines with persons displaced by war, migration, or climate emergencies.
Anthropological Skills and Resources
Anthropologists are trained to look at the larger context and understand how smaller, local environments fit into overarching forces. They aim to hold a multicultural perspective that represents various constituencies and to interact with people around them with the goal of better understanding where they are coming from and what things mean to them. Anthropologists gather and analyze data that reflects real life on the ground and in the streets . The central anthropological specialty is an unfettered interest in human beings.
In 2020, career research and employment website Zippia interviewed a group of teaching and practicing anthropologists about the anthropological skills they believe are most valuable in today’s job market. The two quotes below illustrate the breadth of career preparation that anthropology provides:
Organizations are looking for people who can articulate the value of their experiences. Anthropology provides a broad array of skills. Some [are] more general, such as critical thinking and written and oral communication and teamwork. Some skills are more specific, such as survey and excavation for archaeology positions, research design, data analysis skills (qualitative and quantitative), and familiarity with research ethics. — John Ziker
Young graduates need to think quickly and with skepticism, read situations from multiple angles, and have openness to variable solutions. This means that they need skills in understanding pluralistic vantage points, judging where information comes from and who it benefits and who it hurts, and being gifted at recognizing and acknowledging their own biases. Anthropology teaches these skills as it prepares graduates for work in a wide array of fields. — Suzanne Morrissey (Stark et al. 2020)
Anthropologists and anthropology students, undergraduate and graduate, fit into a wide array of careers and contribute valuable skills and resources to their communities everywhere. As people specialists, anthropologists understand how to approach diverse peoples, elicit information about and from them, and work with that information to understand broader situations. Some of the broadly applicable skills that different anthropologists have include interviewing; excavating; mapping; analyzing data using various types of methodologies, including mixed methods (combining qualitative and quantitative methods); applying ethics in difficult, emerging situations; and engaging with new technologies in the sciences. All of these are 21st-century skills and resources. However, the most advantageous of an anthropologist’s skills is an attitude of respect and dignity toward diverse peoples everywhere. In our global world, this may be the most important asset of all. As anthropologist Tim Ingold says, anthropologists “study ... with people” and “learn from them, not just about them” (2018, 32).
How Anthropology Can Lead in the Future
Career and employment trends today align with what anthropologists do, whether or not one is a full-time practicing anthropologist. Students heading into any fields that address the human condition, past or present, will benefit from studies in anthropology. Within colleges and universities across the world, there is a reemergence of transdisciplinary approaches that utilize methods and perspectives from multiple disciplines to study and propose solutions to complex problems. This educational model, sometimes called the matrix model (National Academy of Sciences, National Academy of Engineers, and Institute of Medicine 2005), has resulted in the development of new interdisciplinary degree programs such as the biomedical informatics program at Stanford University; the Indigenous food, energy, and water systems program at the University of Arizona; and the science, medicine, and technology in culture program at Union College. Training in anthropological holism is the ideal foundation for working in teams with multiple interests and a shared focus on the larger context. Specifically, the four-field approach in anthropology prepares researchers to apply a keen perception of the ways in which biology and culture interact and influence each other.
With the increasing prominence of social media and grassroots communication across cultures, it is important that emerging leaders have the ability to interview people, elicit relevant information from them, and analyze what they think, do, and desire. Anthropologists are trained to interact with others, seek connections and patterns in what they observe, and analyze the symbolic significance of what they find.
Anthropologists are also trained to work in the field, wherever and whatever the field may be , taking their offices and research labs into the communities in which they work and live. Accustomed to being flexible and adaptable to the needs of the situation and letting the field dictate how best to accomplish their work, anthropologists have the skills, technology, and experience to work well in a global community.
In the 20th century, academia sought to become ever more specialized, constructing departments, specialties, and subspecialties to home in on very particular subjects such as a disease, a genre of literature, or a type of religion. This approach was an advance over the more generalist approach that was common in the 19th century, in which academics were trained in very broad fields such as medicine, ancient history, or culture. Now, in the 21st century, the shift is toward a more complex and multifaceted understanding of how we live and the challenges we face. Many anthropology programs today provide vocational skills and workplace training. There is a growing awareness that we need to develop the ability to think both generally and systematically (such as in an ecosystemic approach) while also seeking to understand the particularities of specific challenges. Anthropology, with its holistic approach, mixed methodology analyses, and deep, abiding appreciation of diversity and the dignity of all people, is situated at the crossroads of what comes next. This is how anthropology can guide us as we move into the future.
As Geertz said, “We have turned out to be rather good at waddling in” (1985, 624). Anthropological skills are based on flexibility and adaptation to a changing world, open-mindedness and openness to new ideas, and a willingness to engage with complex issues in order to find solutions to problems facing our world today. The anthropological skillset is critical in the 21st century.
You can read more about the important work of anthropologists today in the Profile features in each chapter. Through research and work such as the examples featured there, anthropologists are changing the world.
Mini-Fieldwork Activity
Global Challenges
Choose three global challenges, and research more about them. Consider how these three global challenges are linked to one another and to long-standing historical inequities. Collect information on the current state of each problem in the United States and worldwide, what measures are being taken to mitigate the problem, and whether there are any local initiatives in your own community. Consider both campus and community organizations. Using what you have learned about anthropology, propose three anthropological skills that you could employ to help address each of these challenges.


5.18: Run-on Sentences
- Recognize run-on sentences
- Revise for run-on sentences
Run-on sentences occur when two or more independent clauses are improperly joined. Often, in a run-on sentence, the independent clauses are simply joined without any punctuation or a coordinating conjunction. Another type of run-on that you’ve probably heard of is the comma splice , in which two independent clauses are joined by a comma without a coordinating conjunction ( and , or , but , etc.).
Let’s look at a few examples of run-on sentences:
- Often, choosing a topic for a paper is the hardest part it’s a lot easier after that.
- Sometimes, books do not have the most complete information, it is a good idea then to look for articles in specialized periodicals.
- She loves skiing but he doesn’t.
All three of these have two independent clauses. Each clause should be separated from another with a period, a semicolon, or a comma and a coordinating conjunction (remember, FANBOYS):
- Often, choosing a topic for a paper is the hardest part. It’s a lot easier after that.
- Sometimes, books do not have the most complete information; it is a good idea then to look for articles in specialized periodicals.
- She loves skiing, but he doesn’t.
A run-on sentence should not just be defined as a sentence that goes on and on. Not every long sentence is a run-on sentence. For example, look at this quote from The Great Gatsby :
If you look at the punctuation, you’ll see that this quote is a single sentence. F. Scott Fitzgerald used commas and semicolons in such a way that, despite its great length, it’s grammatically sound, as well. Length is no guarantee of a run-on sentence.
Can you spot a run-on sentence? Click through this interactive to test yourself.
https://lumenlearning.h5p.com/content/1290910801743442618/embed
Common Causes of Run-on Sentences
We often write run-on sentences because we sense that the sentences involved are closely related, and dividing them with a period just doesn’t seem right. We may also write them because the parts seem too short to need any division, as in “She loves skiing but he doesn’t.” However, “She loves skiing” and “he doesn’t” are both independent clauses, so they need to be divided by a comma and a coordinating conjunction—not just a coordinating conjunction by itself.
Another common cause of run-on sentences is mistaking adverbial conjunctions for coordinating conjunctions. For example, if we were to write, “She loved skiing, however he didn’t,” we would have produced a comma splice. The correct sentence would be “She loved skiing; however, he didn’t.”
Fixing Run-on Sentences
Before you can fix a run-on sentence, you’ll need to identify the problem. When you write and revise, carefully look at each part of every sentence. Are the parts independent clauses, or are they dependent clauses or phrases? Remember, only independent clauses can stand on their own. This also means they have to stand on their own; they can’t run together without correct punctuation.
There are five ways you can fix a run-on sentence or comma splice:
- Change the comma in the comma splice to a semicolon (or add a semicolon between the independent clauses).
- Change the comma in the comma splice to a colon, but only if the two clauses are very closely related and the second one seems to be explaining or clarifying the first.
- Add a coordinating conjunction (FANBOYS) after the comma in the comma splice.
- Change the comma to a period and separate the two independent clauses into two separate sentences.
- Rewrite the sentence entirely so the run-on sentence is no longer a problem.
Here is an example of a sentence with a comma splice:
- Jill has been waiting for years for this doctor’s appointment, it’s very frustrating to have it postponed.
Now let’s look at how we can revise the comma splice using the techniques learned above:
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Changing to a semicolon
- Jill has been waiting for years for this doctor’s appointment; it’s very frustrating to have it postponed.
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Changing to a colon
- Jill has been waiting for years for this doctor’s appointment: it’s very frustrating to have it postponed.
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Adding a coordinating conjunction
- Jill has been waiting for years for this doctor’s appointment, so it’s very frustrating to have it postponed.
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Changing to a period and separating into two sentences
- Jill has been waiting for years for this doctor’s appointment. It’s very frustrating to have it postponed.
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Rewriting the sentence
- Jill has been waiting for years for this doctor’s appointment, making it very frustrating to have it postponed.
Let’s take a look at a few other run-on sentences and their revisions:
- Most of the hours I’ve earned toward my associate’s degree do not transfer, however, I do have at least some hours the university will accept.
This is a comma splice. The adverbial conjunction “however” is being treated like a coordinating conjunction (remember, FANBOYS). There are two easy fixes to this problem. The first is to turn the comma before “however” into a period:
- Most of the hours I’ve earned toward my associate’s degree do not transfer. However, I do have at least some hours the university will accept.
If this feels like too hard of a stop between ideas, you can change the comma into a semicolon instead:
- Most of the hours I’ve earned toward my associate’s degree do not transfer; however, I do have at least some hours the university will accept.
Let’s take a look at another sentence.
- The opposite is true of stronger types of stainless steel they tend to be more susceptible to rust.
This second sentence is a run-on as well. “The opposite is true of stronger types of stainless steel” and “they tend to be more susceptible to rust” are both independent clauses. Since the two clauses are very closely related, and the second clarifies the information provided in the first, the best solution is to insert a colon between the two clauses (although a semicolon is grammatically correct as well):
The opposite is true of stronger types of stainless steel: they tend to be more susceptible to rust.
Let’s try another one.
- Some people were highly educated professionals, others were from small villages in underdeveloped countries.
In this last example, once again, we have two independent clauses. The two clauses provide contrasting information. Adding a conjunction could help the reader move from one kind of information to another. However, you may want that sharp contrast. Here are two revision options:
- Some people were highly educated professionals, while others were from small villages in underdeveloped countries.
- Some people were highly educated professionals. Others were from small villages in underdeveloped countries.
Watch this video for additional tips on how to identify and fix run-on sentences and comma splices.
You can view the transcript for “Run-ons and comma splices” here (opens in new window) .
https://assessments.lumenlearning.co...essments/20249
Review tips on how to spot (and avoid!) run-on sentences in the following interactive.
https://lumenlearning.h5p.com/content/1290907184330046558/embed
Contributors and Attributions
- Revision, Adaptation, and Original Content. Provided by : Lumen Learning. License : CC BY: Attribution
- Punctuation: Commas. Authored by : David McMurrey. Located at : https://www.prismnet.com/~hcexres/textbook/gram1.html . License : CC BY: Attribution
- Run-On Sentences Activity. Authored by : Excelsior OWL. Located at : https://owl.excelsior.edu/writing-refresher/grammar-refresher/run-on-sentences/ . License : CC BY: Attribution
- Run-on Sentences Interactive. Authored by : Excelsior Online Writing Lab. Located at : https://owl.excelsior.edu/esl-wow/editing-polishing/check-punctuation/esl-run-on-sentences/ . License : CC BY: Attribution
- Run-ons and comma splices. Authored by : Khan Academy. Provided by : Khan Academy. Located at : https://www.youtube.com/watch?v=Fh45mhVsZrU&feature=youtu.be . License : CC BY: Attribution . License Terms : Standard YouTube license


5.3: Procedure, flowchart, and datasheets for separation and confirmation of group III cations
|
Chemical |
Hazard |
|---|---|
|
0.1M ammonium chloride (\(\ce{NH4Cl}\)) |
Toxic and irritant |
|
0.1M Chromium(III) chloride |
Toxic and irritant |
|
0.1M Iron(III) chloride |
Toxic and corrosive |
|
0.1M Nickel(II) chloride |
Toxic, irritant, and suspected carcinogen |
- *Hazards of 6M ammonia, 6M hydrochloric acid, 6M nitric acid, 3% hydrogen peroxide, and 1M thioacetamide are listed in chapter 2 in the commonly used reagent section. Caution! Used heavy metal ion solutions are disposed of in a labeled metal waste disposal container, do not drain these solutions down the drain.
- Used heavy metal ion solutions or precipitates are disposed of in a labeled metal waste disposal container, do not drain these solutions down the drain or in the regular trash.
- Take 15 drops of the fresh test solution if group I and group II cations are not present in the sample or take the supernatant of step 1 of group II cations analysis . Add 2 drops of 6M \(\ce{HCl}\). Then add 6M \(\ce{NH3}\) drop by drop while stirring till the solution turns basic. Use red litmus paper to test -when it turns blue the solution is basic. Add 5 more drops of 6M \(\ce{NH3}\) after the solution becomes alkaline to make \(\ce{NH3}\)/\(\ce{NH4^{+}}\) buffer. Record the observations in the datasheet.
- Add 10 drops of 1M thioacetamide to the solution from step 1, stir, and heat in a boiling water bath for 10 min. Then centrifuge for 2 min and decant. Keep the supernatant for group IV cations analysis and keep the precipitate for group III cations. Record the observations in the datasheet.
- Wash the group III precipitates from step 2 by re-suspending in 15 drops of 0.1M \(\ce{NH4Cl}\). Then centrifuge, decant, and discard the supernatant that is just the washing liquid. Re-suspend the precipitates in 10 drops of 6M \(\ce{HCl}\), heat for 2 min in a boiling water bath, centrifuge for two minutes, and keep the supernatant that may contain \(\ce{Fe^{2+}}\) and/or \(\ce{Cr^{3+}}\) and keep the precipitate , if there is any. Record the observations in the datasheet.
- Wash the precipitate of step 3 by re-suspending it in 15 drops of distilled water. Then centrifuge for 1 min, decant and discard the supernatant that is just the washing liquid. Re-suspend the precipitate after adding 4 drops of 6M \(\ce{HCl}\) and 6 drops of 6M \(\ce{HNO3}\) (i.e., aqua regia). Heat the suspension in a boiling water bath for 2 min, then centrifuge for 2 min, decant, and discard the precipitate which is solid sulfur that contains no ions in it, but keep the supernatant for nickel analysis.
- Use the cotton-plug technique to aspirate clear supernatant if it is not already a clear solution. Add 6M \(\ce{NH3}\) to the clear supernatant drop by drop till the solution turns alkaline. Use red litmus paper to test -when it turns blue the solution is alkaline. If the solution turns turbid at this stage, centrifuge for 1 min, decant and discard the precipitate, but keep the supernatant .
- Add 5 drops of dimethylglyoxime to the clear solution of step 5, stir, and leave for a minute. If a bright red precipitate is formed at this stage, it confirms \(\ce{Ni^{2+}}\) is present in the test sample. Discard the mixture in the metal waste container and record the observations in the datasheet.
- Inspect the supernatant of step 3, if it is not clear make it clear using the cotton plug technique. Add 6M \(\ce{NH3}\) drop-by-drop to the clear supernatant while stirring till the mixture until it turns alkaline and has pH in the range of 9 to 10. Use a pH paper (not a litmus paper) to determine the pH. Add 5 drops of 3% \(\ce{H2O2}\) to the alkaline solution, stir, and leave for half-min. Heat the mixture in a boiling water bath to destroy excess \(\ce{H2O2}\) till the oxygen gas bubbles stop evolving from the solution. It may take about 3 min or more. Centrifuge for 1 min and test again for pH with a pH paper -if pH is less than 9, repeat this step 7 from the beginning, otherwise decant and keep the supernatant for analysis of chromium ions and keep the precipitate , there is any, for analysis of iron ions. Record the observations in the datasheet.
- Dissolve the precipitate from step 7 in 5 drops of 6M \(\ce{HCl}\) under stirring. Add 5 drops of distilled water to the solution followed by 5 drops of 0.1M potassium thiocyanate (\(\ce{KSCN}\)) and stir to mix. If the solution color changes to deep red, it confirms iron ions are present in the test sample. Discard the mixture in a metal waste container, and record the observations in the datasheet.
- To the supernatant of step 7 add 6M \(\ce{HNO3}\) drop by drop till the solution becomes acidic with pH ~3. Use pH paper(not litmus paper) to determine the pH. Then add 1 drop of 3% \(\ce{H2O2}\), mix and leave for half-min. Then heat in boiling water bath to destroy excess \(\ce{H2O2}\) till oxygen bubbles stop forming in the mixture. It may take about 3 min or more. Cool the mixture by placing it in a room temperature water bath.
- Add 6M \(\ce{NaOH}\) drop by drop to the solution of step 9 at room temperature till the solution is basic, i.e., turn red litmus paper to blue. The formation of gray-green precipitate at this stage confirms \(\ce{Cr^{3+}}\) is present in the test sample. Caution: \(\ce{H2O2}\) decomposes slower in the acidic medium than in the basic medium. If the solution turns dark blue or yellow, it indicates chromium is present as \(\ce{CrO5}\) or \(\ce{CrO4^{2-}}\) and \(\ce{H2O2}\) was not destroyed completely. In this case, add 6M \(\ce{HNO3}\) drop by drop with stirring till the color fades away. Then repeat the addition of 6M NaOH till the solution turns basic and observe. The formation of gray-green precipitate at this stage confirms \(\ce{Cr^{3+}}\) is present in the test sample. Discard the mixture in a metal waster container and record the observations in the datasheet.
- Step number refers to the corresponding step number in the procedure sub-section.
- In “the expected chemical reaction and expected observations column”, write an overall net ionic equation of the reaction that will happen if the ion being processed in the step was present, write the expected color change of the solution, the expected precipitate formed and its expected color, etc.
- In the “the actual observations and conclusion” column write the color change, the precipitate formed and its color, etc. that is actually observed as evidence, and state the specific ion as present or absent.
- In “the overall conclusion” row write one by one symbol of the ions being tested with a statement “present” or “absent” followed by evidence/s to support your conclusion.