28.5: The Gothic Revival
The Gothic Revival
The Gothic Revival was an architectural movement beginning in England during the 1740s that sought to revive medieval forms.
Generalize about the Gothic Revival style architecture
Key Points
- Also termed Victorian Gothic and Neo-Gothic, the Gothic Revival sought to revive medieval forms, much like the Neoclassical style sought to revive works from classical antiquity.
- The Gothic Revival occurred as industrialization progressed, in part because there was a reaction against the use of machinery and factory production.
- While the Neoclassical style of the 18th century was associated with “radical” and liberal perspectives, the Gothic Revival was associated with “traditional” sensibilities, such as conservatism and the monarchy. The Gothic Revival style is characterized by its stone and brick structures, many of which are religious in nature, as well as having heavy decoration, pointed arches, steep gables, and large windows.
- Gothic revival cottages and smaller buildings, called “Carpenter Gothic,” also became popular.
- A. W. N. Pugin was a prominent architect, designer, artist, and critic who was involved in the Gothic Revival style.
Key Terms
- medievalism : A custom or belief from the Middle Ages.
- gable : The triangular area of external wall adjacent to two meeting sloped roofs.
- tracery : Bars or ribs, usually of stone or wood, or other material, that subdivide an opening or stand in relief against a door or wall as an ornamental feature.
The Gothic Revival was primarily an architectural movement that began in 1740s England. Also termed Victorian Gothic and Neo-Gothic, the style sought to revive medieval forms, much like the Neoclassical style sought to revive works from classical antiquity. During the 18th century, the ruins of medieval Gothic architecture began to receive newfound appreciation after having been relatively dismissed in the overall history of architecture. Some critics believe there was a kind of nostalgia for an enchanted, less rational world that was linked to the perceived superstitions of medieval Catholicism.
In England, the center of the Gothic revival, the movement was intertwined with philosophical trends associated with a reawakening of Christian traditions in response to the growth of religious nonconformism. Ultimately, the Gothic style became widespread in the third quarter of the 19th century. While the Neoclassical style of the 18th century was associated with “radical” and liberal perspectives, the Gothic Revival was associated with “traditional” sensibilities, such as conservatism and the monarchy. As industrialization progressed, there was an increasing reaction against the use of machinery and factory production. Supporters of medievalism criticized industrial society, believing the pre-industrial model to be a golden age.
Paralleling the ascendancy of Neo-Gothic styles in 19th century England, interest spread rapidly to Europe, Australia, South Africa, and the Americas. Indeed, the number of Gothic Revival and Carpenter Gothic structures built in the 19th and 20th centuries may exceed the number of authentic Gothic structures that had been built in Gothic’s original era. The Gothic Revival style is characterized by its stone and brick structures, many of which are religious in nature, as well as heavy decoration. The most fundamental element of the Gothic style of architecture is the pointed arch. Columns that support arches are smaller in Gothic buildings, and continue all the way to the roof, where they become part of the vault. In the vault, the pointed arch can be seen in three dimensions where the ribbed vaulting meets in the center of the ceiling of each bay. This ribbed vaulting is another distinguishing feature of Gothic architecture. The slender columns and lighter systems of thrust allowed for larger windows and more light in Gothic structures. The windows, tracery, carvings, and ribs make up a bewildering display of decoration where almost every surface is decorated with a profusion of shapes and patterns. Gothic revival cottages and smaller buildings also became popular and are referred to as “Carpenter Gothic.” These structures are defined by their use of Gothic elements such as pointed arches and steep gables.
A. W. N. Pugin was a prominent architect, designer, artist, and critic who was deeply involved in the Gothic Revival. The height of his work is seen in the interior design of the Palace of Westminster. Pugin designed many churches in England during his career and published a series of volumes of architectural drawings entitled Examples of Gothic Architecture and Specimens of Gothic Architecture that remained in print and were the standards for the Gothic Revival for the next century.
The Gothic Revival was supported by medievalism, which had its roots in antiquarian concerns of survivals (practices that continue from earlier times) and curiosities (unique or unusual objects). As industrialization progressed, there was a reaction against machine production and factories. Proponents of the picturesque, such as Thomas Carlyle and Augustus Pugin, took a critical view of industrial society and portrayed pre-industrial medieval society as a golden age. To Pugin, Gothic architecture was infused with the Christian values that had been supplanted by classicism and were being destroyed by industrialization.
- Curation and Revision. Provided by : Boundless.com. License : CC BY-SA: Attribution-ShareAlike
- Gothic Revival architecture. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Gothic_Revival_architecture. License : CC BY-SA: Attribution-ShareAlike
- Augustus Pugin. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Augustus_Pugin. License : CC BY-SA: Attribution-ShareAlike
- The Gothic Experience: Gothic Architecture. Provided by : Saylor. Located at : http://www.saylor.org/site/wp-content/uploads/2011/01/Gothic-Experience.pdf . License : CC BY: Attribution
- gable. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/gable. License : CC BY-SA: Attribution-ShareAlike
- tracery. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/tracery. License : CC BY-SA: Attribution-ShareAlike
- medievalism. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/medievalism. License : CC BY-SA: Attribution-ShareAlike
- Palace of Westminster Westminster Hall south. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Palace_of_Westminster_Westminster_Hall_south.jpg. License : Public Domain: No Known Copyright


1.3.6: Behavioral Approaches to Leadership
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Learning Objectives
- 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 leadersdo. 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.” 60 Jonathan Anthony calls himself an intrapreneur and corporate disorganizer, because same-old, same-old comms practices are dying in front of our eyes. 61 Apple founder 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 13.9, the Leadership Grid ® demonstrates that any combination of these two leader concerns is possible, and five styles of leadership are highlighted here.
Exhibit 13.9 Blake and Mouton’s Managerial Grid® Source:Adapted from R. McKee and B. Carlson. 1999.The Power to Change, p.16.
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?
References
59. K. Labich, 1988, 58–66.
60. C. Williams. 2017 (June 23). Leadership: Coaching has a Role to Play in Business. Central Penn Business Journal . http://www.cpbj.com/article/20170623...lay-inbusiness
61. J. Anthony. 2017. This Much We Know. (Accessed August 4, 2017). https://thismuchweknow.net/2016/09/21/ 10-ideas-and-concepts-that-describe-me-really-well/
62. E.A. Fleishman. 1953. The description of supervisory behavior. Personnel Psychology 37:1–6; E.A. Fleishman & E.F. Harris. 1962. Patterns of leadership behavior related to employee grievances and turnover. Personnel Psychology 15:43–56; A. W. Halpin & B. J. Winer. 1957. A factorial study of the leader behavior descriptions. In R.M. Stogdill & A.C. Coons (eds.) , Leader behavior: Its description and measurement . Columbus: Bureau of Business Research, Ohio State University; J.K. Hemphill & A.E. Coons. 1975. Development of the leader behavior description questionnaire. In R. M. Stogdill & A. E. Coons (eds.), Leader behavior ; S. Kerr & C. Schriesheim. 1974. Consideration, initiating structure, and organizational criteria—an update of Korman’s 1966 review. Personnel Psychology 27:555–568.
63. D. Katz & R.L. Kahn. 1952. Some recent findings in human relations research. In E. Swanson, T. Newcomb, & E. Hartley (eds.), Readings in social psychology , New York: Holt, Rinehart, & Winston; D. Katz, N. Macoby, & N. Morse. 1950. Productivity, supervision, and morale in an office situation, Ann Arbor, MI: Institute for Social Research; F.C. Mann & J. Dent. 1954. The supervisor: Member of two organizational families. Harvard Business Review 32:103–112.
64. D. G. Bowers & S. C. Seashore. 1966. Pretesting organizational effectiveness with a four-factor theory of leadership. Administrative Science Quarterly 11:238–262; Yukl, 1971; D.A. Nadler, G.D. Jenkins, Jr., C. Cammonn, and E.E. Lawler, III. 1975. The Michigan organizational assessment package progress report . Ann Arbor: Institute for Social Research, University of Michigan.
65. Bowers & Seashore, 1966.
66. R.R. Blake & J.S. Mouton. 1964. The managerial grid . Houston: Gulf; R.R. Blake & J.S. Mouton. 1981. The versatile manager: A grid profile , Homewood, IL: Dow Jones-Irwin; R.R. Blake & J.S. Mouton. 1984. The new managerial grid III . Houston: Gulf.
67. R.R. Blake & J.S. Mouton. 1981. Management by grid ® principles or situationalism: Which? Group and Organization Studies6:439–455.
68. L. L. Larson, J. G. Hunt, & R. N. Osborn. 1976. The great hi-hi leader behavior myth: A lesson from Occam’s razor. Academy of Management Journal 19:628–641.
69. D. Tjosvold. 1984. Effects of warmth and directiveness on subordinate performance on a subsequent task. Journal of Applied Psychology69:422–427; A.W. Halpin. 1957. The leader behavior and effectiveness of aircraft commanders. In R.M. Stogdill & A. E. Coons (eds.). Leader Behavior: Its description and measurement . Columbus, OH: The Ohio State University, Bureau of Business Research.; E.A. Fleishman & J. Simmons. 1970. Relationship between leadership patterns and effectiveness ratings among Israeli foremen. Personnel Psychology 23:169–172.


4.0: What Does Psychology Say?
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Sarah and Amelia love their baby Asher, but they find the transition to life with a newborn challenging. Both mothers are sleep deprived and anxious. Like most new parents, they’re also worried about making mistakes.
Asher is now eight weeks old and quite fussy. He cries until his face turns red but seems satisfied after Sarah nurses him. Amelia often places Asher in a sling and takes him for a walk. It works every time, but only briefly.
Family and friends have been offering their advice. Amelia’s father says, “You’re going to spoil that baby if you keep picking him up,” while Amelia’s mother interjects: “Turn on that cartoon with the monster trucks! He’ll like that!” Meanwhile, the mommy blogs keep saying screentime is bad for babies. Amelia looks at her son’s face as he naps in her arms. Like many caregivers, she feels overwhelmed by conflicting advice and wonders:
In this chapter, you’ll learn what current research says about questions like these.


2.2: The Strategy Hierarchy
Understanding the Strategy Hierarchy
In most corporations, there are several levels of management. Strategic management is the highest of these levels in the sense that it is the broadest and applies to all parts of the firm while also incorporating the longest time horizon. It gives direction to corporate values, corporate culture, corporate goals, and corporate missions. Under this broad corporate strategy there are typically business-level competitive strategies and functional unit strategies. [6]
Corporate strategy refers to the overarching strategy of the diversified firm. Such a corporate strategy answers the questions of “in which businesses should we compete?” and “how does being in these businesses create synergy and/or add to the competitive advantage of the corporation as a whole?”
Business strategy refers to the aggregated strategies of a single business firm or a strategic business unit (SBU) in a diversified corporation. According to Michael Porter, a firm must formulate a business strategy that incorporates either cost leadership, differentiation or focus in order to achieve a sustainable competitive advantage and long-term success in its chosen arenas or industries.
Functional strategies include marketing strategies, new product development strategies, human resource strategies, financial strategies, legal strategies, supply-chain strategies, and information technology management strategies. The emphasis is on short- and medium-term plans and is limited to the domain of each department’s functional responsibility. Each functional department attempts to do its part in meeting overall corporate objectives, and hence to some extent their strategies are derived from broader corporate strategies.
Many companies feel that a functional organizational structure is not an efficient way to organize activities, so they are reengineered according to processes or SBUs. A strategic business unit is a semi-autonomous unit that is usually responsible for its own budgeting, new product decisions, hiring decisions, and price setting. An SBU is treated as an internal profit centre by corporate headquarters.
An additional level of strategy called operational strategy was encouraged by Peter Drucker in his theory of Management By Objectives (MBO). It is very narrow in focus and deals with day-to-day operational activities such as scheduling criteria. Operational level strategies are informed by business level strategies which, in turn, are informed by corporate level strategies. [7]
Operations strategy categories can be broken down into many types of areas that must be addressed. The decisions made in these areas will determine whether the business strategy is executed. Below is a list of 10 critical decisions in operations management [8] :
1. Design of Goods and Services – The actual design of the product or service will have the largest impact on the cost to produce and the quality to achieve.
2. Quality – The way in which the organization will ensure that the product specifications are met. This may include the use of statistical process control, total quality management or Six Sigma.
3. Process and Capacity Design – The type of product along with its volume and variety will have the major impact on which type of process to be chosen.
4. Location – Important decisions such as how many locations and where to locate them are critical to organization success. This will be a major factor in terms of how quickly the transformation process can take place, and how quickly goods can be shipped to customers.
5. Layout Design and Strategy – Consider the placement of work centers, movement of goods, people and information How materials are delivered and used.
6. Human Resources and Job Design – Decisions regarding training for employees, how to motivate employees to achieve operational success.
7. Supply Chain Decisions – Decisions in terms of where suppliers are located and the level of supplier collaboration are major considerations that impact cost and delivery speed.
8. Inventory – How will inventories be used and controlled in the business and the supply chain
9. Scheduling – includes both how to schedule production, resources and employees in order to be effective, efficient and meet commitments to customers.
10. Maintenance – This involves maintaining equipment and machinery as well as keeping quality high and processes stable.
Source: " Operations Strategy and Competitiveness " Introduction to Operations Management Copyright © by Hamid Faramarzi and Mary Drane is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.


3.1: Bloom’s Taxonomy
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In 1956, educational psychologist Benjamin Bloom and his associates conceptualized a taxonomy to classify aspects of human learning that included three basic domains: cognitive, affective, and psychomotor. The cognitive domain of this taxonomy represents a continuum of thinking that can be useful for teachers as they develop curricula for students, but also for students as they try to determine what a professor may be asking them to do on a particular test question or writing assignment. Bloom’s Taxonomy can also help students challenge themselves as they are creating active learning study strategies and/ or developing questions for self-testing for an exam. In addition, once students have taken an exam in a particular course, conducting a post-test review of the levels of thinking required by exam questions will provide clues regarding how to effectively study for future exams.
Table 5-1: Bloom’s Taxonomy: Cognitive Domain (1956)
| Level of Cognitive Domain | Thinking Level | Generic Activity | Question Cues | Sample Questions | Example Activities of Products |
|---|---|---|---|---|---|
| Remembering | Basic thinking (lowest level) | Recalls facts, patterns, settings, etc.; remembers previously learned material | Cite, label, name, list, state, quote, reproduce, define, identify, describe(who, what, where, when) |
Who is…?
What is…? Where is…? Where was…? |
List items. Make a timeline. Recite a passage. List certain memorized facts. |
| Understanding | Basic thinking | Understands what is being communicated; grasps the meaning of material and can state in own words; can infer causes and predict consequences | Explain, restate, paraphrase, summarize, describe, illustrate, give examples, discuss, distinguish, interpret |
How would you put this into your own words?
What is an example of _____? How would you translate ____ to a visual form? What was the main idea? |
Create a flowchart to illustrate the sequence of events. Retell the story in your words. Write a summary. |
| Applying | Higher-level thinking | Uses the information in new concrete situations | Apply, classify, solve, demonstrate, calculate, illustrate(how it looks in new situation), complete, employ |
Can you apply this idea to your own life?
Can you come up with another example? What does “x” equal in this case? |
Use this method to apply to a new case study. Solve different types of math problems. |
| Analyzing | Higher-level thinking | Breaks new information into parts to understand relationships; sees patterns and organizational structure | Diagram, analyze, diagnose, conclude, outline, separate, explain(relationships), infer, find, classify, discriminate, compare, contrast, why |
Why did this happen? What were some of the motives behind _____?
What was the problem with _____? Why did ____ changes occur? Can you explain what happened when _____? What difference exists between ____ and ____? |
Construct a diagram that shows the relationships between the parts. Research the issue to find information that supports a view. Write a biography. |
| Evaluating | Higher-level thinking | Make judgement of the value of an idea, method, resource, etc.; assess the value of theories, presentations, texts; make choices based on argument; recognize subjectivity | Assess, appraise, critique, judge, weigh, recommend, convince, support, evaluate,rank, decide, select, grade, defend, justify, compare, contrast |
Is this a good or bad thing? Can you defend your position on ____?
What do you believe and why? What would you have done differently? How effective is ____? What do you think about ____? Is this a credible source? |
Debate an issue from multiple perspectives.State your opinion and evidence for your opinion. Prepare a list of criteria used to judge something and apply it. |
| Creating | Higher-level thinking | Creates something new from the elements of the old information; generalizes from given facts; relates knowledge from different areas | Create, design, compose, develop, plan, propose, integrate, invent, generalize, combine, rewrite |
What would happen if ____?
How can we improve ____? Can you design a ___ to accomplish ___? How can this idea be combined with that idea to develop a better understanding of ____? How can we solve questions? |
Invent a new machine. Write a story. Compose a new piece of music or work of art. Devise a new way to do something. |
Students who challenge themselves to engage in higher-level thinking such as Applying, Analyzing, Evaluating, and Creating are using what professors on our campus would commonly characterize as “critical thinking.”


9.3: Exercise- Evaluating the Impact of Artifact Correction
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One of our goals for artifact rejection and correction is to reduce noise that would otherwise decrease the data quality in our averaged ERPs. Another goal is to minimize the confounds that happen when the artifacts differ across groups or conditions. To evaluate these issues, we need to calculate the SME values and the averaged ERP waveforms for the corrected data, compare them with the original data, and compare them with the data after applying artifact rejection. I’ve done all this and provided the ERPsets in the Chapter_9 folder. Go ahead and load these three ERPsets ( 10_MMN_uncorrected , 10_MMN_rawblinkcorrection , and 10_MMN_blinkrejection ).
Creating the ERPsets
If you want to create these ERPsets yourself, you’ll need to add an EventList to the dataset, run BINLISTER (using BDF_MMN.txt ), and epoch the data. You’ll need to do this once for the original data (prior to correction) and once for the new dataset in which IC 1 was removed. And for the original uncorrected data, you’ll want to have two versions, one with artifact detection (step-like artifacts, threshold = 50) and one without artifact detection. This will allow you to make three ERPsets, one without correction or rejection, once with correction only, and one with rejection only. Make sure to customize the data quality parameters to add a window of 125-225 ms.
Now plot the ERP waveforms for each ERPset. To make them comparable, uncheck the auto y-scale box and put -12 12 in the Y range text box before plotting. The results for several key channels are summarized in Figure 9.2.
Large differences between standards and deviants can be seen in the Fp2 and VEOG channels when nothing was done to deal with the blinks (Figure 9.2.A). You can see a polarity reversal of the deviance effect between the Fp2 and VEOG-lower channels, consistent with blink activity. That was also true for Subject 1 in the previous chapter (see Figure 7.2.A), although the specific pattern was different. The finding of different blink-related activity for standards and deviants indicates that blinks are a confound that we need to eliminate.
The blink-related voltage deflections were eliminated by both artifact correction rejection. Note, however, that the VEOG-bipolar channel was excluded from correction, so the large deflections remain in this channel. Following both correction and rejection, the voltage is more negative for deviants than for standards in FCz in the 125-225 ms time range, which is our primary measure of the MMN. This effect was larger when no rejection or correction was performed, so part of this effect may have been coming from the blink activity that we can see during this time range in the Fp2 and VEOG electrodes. This provides even more evidence that we need to deal with the blinks to avoid confounding our measure of brain activity with blink activity (or vertical eye movements).
When artifact correction was performed, the difference between standards and deviants at Fp2 and VEOG-lower was largely eliminated. However, some difference remained in these channels when artifact rejection was performed. Which of these reflects the true pattern of brain activity? It’s difficult to be certain on the basis of these waveforms. However, given that the MMN appears to be generated primarily in the supratemporal plane (Näätänen & Kreegipuu, 2012), it should be substantially larger at FCz than at Fp2, and it should not invert in polarity between Fp2 and VEOG-lower (when referenced to P9/P10). I therefore suspect that some ocular activity escaped rejection (most likely vertical eye movements, which may fail to reach the 50 µV threshold for rejection that I used for detecting blinks). As a result, it seems likely that correction brought us “closer to the truth” than rejection in this particular case. And that’s our ultimate goal!
It's also important to assess the impact of correction and rejection on data quality. For each of the three ERPsets, you should display the data quality measures in a table. Let’s focus on the aSME values for FCz from 125-225 ms. For both the deviants (Bin 1) and the standards (Bin 2), the aSME was worst (highest) for the data without correction or rejection (Bin 1 = 0.9774, Bin 2 = 0.5688), and was improved (reduced) by rejection (0.8338, 0.5360), and was improved even more by correction (0.7802, 0.4523). This pattern makes sense because both rejection and correction minimize the uncontrolled variation produced by the blinks, but rejection reduces the number of trials whereas correction does not.
From these results, correction seems to be the better method for this particular participant in terms of both minimizing ocular confounds and maximizing data quality. That fits with my experience: When implemented correctly, ICA-based artifact correction tends to be better than rejection for dealing with blinks. And sometimes the difference is quite large, especially when a large number of trials would need to be rejected.
However, we didn’t really implement correction very well in this example. When ICA is working properly, the scalp maps of most of the ICs should look like nice gradual gradients with a unipolar pattern (a single positive or negative focus, like IC 1 in Screenshot 9.2) or a dipolar pattern (opposing positive and negative focus, like IC 7 in Screenshot 9.2).  They don’t resemble the topography we’d expect for brain activity or common artifacts, and they’re a sign that multiple sources are being mixed together rather than being separated. A few such maps are okay, especially in the latter ICs that don’t account for much variance. But you really don’t want to see a map like IC 2 in the top half of the ICs. In a later exercise, we’ll see how to improve the ICA decomposition (and make it faster as well). First, however, we need to discuss how ICA works.


30.6: Review Questions
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1.
Which transition of care intervention would the community health nurse perform when a client is discharged from the hospital to home?
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Assisting the client with activities of daily living
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Administering medications in the home setting
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Performing medication reconciliation
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Transporting client to medical appointments
2.
Which condition does the nurse recognize as increasing the risk of a poor outcome during a transition of care?
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A client being discharged from hospital to home
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A client with an identified social support system
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A client with limited health literacy
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A client with one provider
3.
The home care nurse is performing medication reconciliation with a client during the first home visit following hospital discharge. Which statement by the client requires follow-up by the nurse?
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“I understand why the doctor has stopped some of my medications.”
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“I have not seen that blue pill before.”
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“I set alarms on my phone to remind me to take my medications.”
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“I use a daily pill box to set up my medications.”
4.
The nurse, caring for a client who will be transferred from the hospital to a rehabilitation facility following a stroke, is using a care transition model to facilitate the transfer. Which explanation for the use of a care transition model will the nurse provide to the nursing student working the client?
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“Care transition models predict client outcomes during care transitions.”
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“Care transition models guide health care providers in the decision-making process.”
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“Care transition models support the coordination of care between health care settings.”
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“Care transition models focus on client education during care transitions.”
5.
Which action will the nurse take when planning discharge for a client using the IDEAL Discharge Planning guide?
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Use accurate medical terms when educating the client and family about discharge instructions
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Engage clients and their families in the discharge planning process
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Ask the client if they have any questions to evaluate understanding of the discharge teaching
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Provide the client with a standardized discharge plan of care
6.
Using the Transitional Care Model (TCM), which action would the nurse take when preparing an 81-year-old client with diabetes, hypertension, and heart failure for discharge home from the hospital following an exacerbation of heart failure?
-
Provide the client with a standardized plan of care to follow
-
Identify risk factors for readmission and develop an individualized discharge plan
-
Refer the client to a discharge coach to encourage client self-management
-
Contact the advanced practice nurse to coordinate the discharge from hospital to home
7.
The community health nurse is facilitating transitional care for an older adult client between the hospital and home. Which action is a priority responsibility of the community health nurse in this transition of care?
-
Providing direct client care in the home setting
-
Educating clients about health promotion and prevention
-
Coordinating care among health care providers
-
Administering medication in the home
8.
A nurse is concerned about the increasing rate of hospital readmissions in clients due to not filling or picking up prescriptions at the pharmacy after discharge from the hospital. Which intervention should the nurse recommend to the hospital leadership to improve adherence to the medication regimen after discharge?
-
Calling prescriptions in to the client’s pharmacy at discharge
-
Handing client prescriptions for home medications during discharge instruction
-
Providing client with home medications at discharge
-
Telling the client that medication adherence will reduce readmissions to the hospital
9.
Which intervention by the community health nurse improves care for clients during the transition from behavioral health hospitalization to outpatient care?
-
Ensuring effective communication among health care providers
-
Reducing the number of providers involved in the client’s care
-
Requiring medications to be taken under direct observation by a nurse
-
Attending follow-up appointments with the client
10.
Which health information system would the nurse utilize to promote self-management in a client with diabetes who has been having difficulty controlling blood glucose levels?
-
Decision support systems
-
Mobile health applications
-
Care coordination platforms
-
Electronic health records


12.R: Introduction to Calculus (Review)
12.1: Finding Limits - Numerical and Graphical Approaches
For the exercises 1-6, use the Figure below.
1) \(\lim \limits_{x \to -1^+}f(x)\)
- Answer
-
\(2\)
2) \(\lim \limits_{x \to -1^-}f(x)\)
3) \(\lim \limits_{x \to -1}f(x)\)
- Answer
-
does not exist
4) \(\lim \limits_{x \to 3}f(x)\)
5) At what values of \(x\) is the function discontinuous? What condition of continuity is violated?
- Answer
-
Discontinuous at \(x=-1\left (\lim \limits_{x \to a}f(x) \text{ does not exist} \right )\), \(x=3\left (\text{ jump discontinuity} \right )\), and \(x=7\left (\lim \limits_{x \to a}f(x) \text{ does not exist} \right )\).
6) Using the Table below, estimate \(\lim \limits_{x \to 0}f(x)\).
| \(x\) | \(F(x)\) |
|---|---|
| −0.1 | 2.875 |
| −0.01 | 2.92 |
| −0.001 | 2.998 |
| 0 | Undefined |
| 0.001 | 2.9987 |
| 0.01 | 2.865 |
| 0.1 | 2.78145 |
| 0.15 | 2.678 |
- Answer
-
\(3\)
For the exercises 7-9, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as \(x\) approaches \(a\). If the function has limit as \(x\) approaches \(a\), state it. If not, discuss why there is no limit.
7) \(f(x)=\begin{cases} \left | x \right |-1 & \text{ if } x\neq 1 \\ x^3 & \text{ if } x= 1 \end{cases} a=1\)
8) \(f(x)=\begin{cases} \dfrac{1}{x+1} & \text{ if } x= -2 \\ (x+1)^2 & \text{ if } x\neq -2 \end{cases} a=-2\)
- Answer
-
\(\lim \limits_{x \to -2}f(x)=1\)
9) \(f(x)=\begin{cases} \sqrt{x+3} & \text{ if } x<1 \\ -\sqrt[3]{x} & \text{ if } x>1 \end{cases} a=1\)
12.2: Finding Limits - Properties of Limits
For the exercises 1-6, find the limits if \(\lim \limits_{x \to c} f(x)=-3\) and \(\lim \limits_{x \to c} g(x)=5\).
1) \(\lim \limits_{x \to c} (f(x)+g(x))\)
- Answer
-
\(2\)
2) \(\lim \limits_{x \to c} \dfrac{f(x)}{g(x)}\)
3) \(\underset{x \to c}{\lim } (f(x)\cdot g(x))\)
- Answer
-
\(-15\)
4) \(\lim \limits_{x \to 0^+} f(x), f(x)=\begin{cases} 3x^2+2x+1 & x>0 \\ 5x+3 & x<0 \end{cases}\)
5) \(\lim \limits_{x \to 0^-} f(x), f(x)=\begin{cases} 3x^2+2x+1 & x>0 \\ 5x+3 & x<0 \end{cases}\)
- Answer
-
\(3\)
6) \(\lim \limits_{x \to 3^+} (3x-〚x〛)\)
For the exercises 7-11, evaluate the limits using algebraic techniques.
7) \(\lim \limits_{h \to 0} \left ( \dfrac{(h+6)^2-36}{h} \right )\)
- Answer
-
\(12\)
8) \(\lim \limits_{x \to 25} \left ( \dfrac{x^2-625}{\sqrt{x}-5} \right )\)
9) \(\lim \limits_{x \to 1} \left ( \dfrac{-x^2-9x}{x} \right )\)
- Answer
-
\(-10\)
10) \(\lim \limits_{x \to 4} \left ( \dfrac{7-\sqrt{12x+1}}{x-4} \right )\)
11) \(\lim \limits_{x \to 3} \left ( \dfrac{\frac{1}{3}+\frac{1}{x}}{3+x} \right )\)
- Answer
-
\(-\dfrac{1}{9}\)
12.3: Continuity
For the exercises 1-5, use numerical evidence to determine whether the limit exists at \(x=a\). If not, describe the behavior of the graph of the function at \(x=a\).
1) \(f(x)=\dfrac{-2}{x-4};\; a=4\)
2) \(f(x)=\dfrac{-2}{(x-4)^2};\; a=4\)
- Answer
-
At \(x=4\), the function has a vertical asymptote.
3) \(f(x)=\dfrac{-x}{x^2-x-6};\; a=3\)
4) \(f(x)=\dfrac{6x^2+23x+20}{4x^2-25};\; a=-\dfrac{5}{2}\)
- Answer
-
removable discontinuity at \(a=-\dfrac{5}{2}\)
5) \(f(x)=\dfrac{\sqrt{x}-3}{9-x};\; a=9\)
For the exercises 6-12, determine where the given function \(f(x)\) is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.
6) \(f(x)=x^2-2x-15\)
- Answer
-
continuous on \((-\infty, \infty)\)
7) \(f(x)=\dfrac{x^2-2x-15}{x-5}\)
8) \(f(x)=\dfrac{x^2-2x}{x^2-4x+4}\)
- Answer
-
removable discontinuity at \(x=2\). \(f(2)\) is not defined, but limits exist.
9) \(f(x)=\dfrac{x^3-125}{2x^2-12x+10}\)
10) \(f(x)=\dfrac{x^2-\frac{1}{x}}{2-x}\)
- Answer
-
discontinuity at \(x=0\) and \(x=2\). Both \(f(0)\) and \(f(2)\) are not defined.
11) \(f(x)=\dfrac{x+2}{x^2-3x-10}\)
12) \(f(x)=\dfrac{x+2}{x^3+8}\)
- Answer
-
removable discontinuity at \(x=-2\). \(f(-2)\) is not defined.
12.4: Derivatives
For the exercises 1-5, find the average rate of change \(f(x)=\dfrac{f(x+h)-f(x)}{h}\).
1) \(f(x)=3x+2\)
2) \(f(x)=5\)
- Answer
-
\(0\)
3) \(f(x)=\dfrac{1}{x+1}\)
4) \(f(x)=\ln (x)\)
- Answer
-
\(f(x)=\dfrac{\ln (x+h)-\ln (x)}{h}\)
5) \(f(x)=e^{2x}\)
For the exercises 6-7, find the derivative of the function.
6) \(f(x)=4x-6\)
- Answer
-
\(4\)
7) \(f(x)=5x^2-3x\)
8) Find the equation of the tangent line to the graph of \(f(x)\) at the indicated \(x\) value. \[f(x)=-x^3+4x;\; x=2 \nonumber \]
- Answer
-
\(y=-8x+16\)
9) For the following exercise, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable. \[f(x)=\dfrac{x}{\left | x \right |} \nonumber \]
10) Given that the volume of a right circular cone is \(V=\dfrac{1}{3}\pi r^2h\) and that a given cone has a fixed height of \(9\) cm and variable radius length, find the instantaneous rate of change of volume with respect to radius length when the radius is \(2\) cm. Give an exact answer in terms of \(π\).
- Answer
-
\(12\pi \)
Practice Test
For the exercises 1-6, use the graph of \(f\) in the Figure below.
1) \(f(1)\)
- Answer
-
\(3\)
2) \(\lim \limits_{x \to -1^+} f(x)\)
3) \(\lim \limits_{x \to -1^-} f(x)\)
- Answer
-
\(0\)
4) \(\lim \limits_{x \to -1} f(x)\)
5) \(\lim \limits_{x \to -2} f(x)\)
- Answer
-
\(-1\)
6) At what values of \(x\) is \(f\) discontinuous? What property of continuity is violated?
7) \(f(x)=\begin{cases} \dfrac{1}{3}-3 & \text{ if } x\leq 2 \\ x^3+1 & \text{ if } x>2 \end{cases} a=2\)
- Answer
-
\(\lim \limits_{x \to 2^-} f(x)=-\dfrac{5}{2}a\) and \(\lim \limits_{x \to 2^+} f(x)=9\)
Thus, the limit of the function as \(x\) approaches \(2\) does not exist.
8) \(f(x)=\begin{cases} x^3+1 & \text{ if } x<1 \\ 3x^2-1 & \text{ if } x=1\; a=1 \\ -\sqrt{x+3}+4 & \text{ if } x>1 \end{cases}\)
For the exercises 9-11, evaluate each limit using algebraic techniques.
9) \(\lim \limits_{x \to -5} \left ( \dfrac{\frac{1}{5}+\frac{1}{x}}{10+2x} \right )\)
- Answer
-
\(-\dfrac{1}{50}\)
10) \(\lim \limits_{h \to 0} \left ( \dfrac{\sqrt{h^2+25}-5}{h^2} \right )\)
11) \(\lim \limits_{h \to 0} \left ( \dfrac{1}{h}-\dfrac{1}{h^2+h} \right )\)
- Answer
-
\(1\)
For the exercises 12-13, determine whether or not the given function \(f\) is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.
12) \(f(x)=\sqrt{x^2-4}\)
13) \(f(x)=\dfrac{x^3-4x^2-9x+36}{x^3-3x^2+2x-6}\)
- Answer
-
removable discontinuity at \(x=3\)
For the exercises 14-16, use the definition of a derivative to find the derivative of the given function at \(x=a\).
14) \(f(x)=\dfrac{3}{5+2x}\)
15) \(f(x)=\dfrac{3}{\sqrt{x}}\)
- Answer
-
\(f'(x)=-\dfrac{3}{2a^{\frac{3}{2}}}\)
16) \(f(x)=2x^2+9x\)
17) For the graph in the Figure below, determine where the function is continuous/discontinuous and differentiable/not differentiable.
- Answer
-
discontinuous at \(-2,0\), not differentiable at \(-2,0, 2\).
For the exercises 18-19, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.
18) \(f(x)=\left | x-2 \right | - \left | x+2 \right |\)
19) \(f(x)=\dfrac{2}{1+e^{\frac{2}{x}}}\)
- Answer
-
not differentiable at \(x=0\) (no limit)
For the exercises 20-24, explain the notation in words when the height of a projectile in feet, \(s\), is a function of time \(t\) in seconds after launch and is given by the function \(s(t)\).
20) \(s(0)\)
21) \(s(2)\)
- Answer
-
the height of the projectile at \(t=2\) seconds
22) \(s'(2)\)
23) \(\dfrac{s(2)-s(1)}{2-1}\)
- Answer
-
the average velocity from \(t=1\) to \(t=2\)
24) \(s(t)=0\)
For the exercises 25-28, use technology to evaluate the limit.
25) \(\lim \limits_{x \to 0}\dfrac{\sin (x)}{3x}\)
- Answer
-
\(\dfrac{1}{3}\)
26) \(\lim \limits_{x \to 0}\dfrac{\tan ^2(x)}{2x}\)
27) \(\lim \limits_{x \to 0}\dfrac{\sin (x)(1-\cos (x))}{2x^2}\)
- Answer
-
\(0\)
28) Evaluate the limit by hand.
\[\lim \limits_{x \to 1}f(x), \text{ where } f(x)=\begin{cases} 4x-7 & x\neq 1 \\ x^2-4 & x= 1 \end{cases} \nonumber \]
At what value(s) of \(x\) is the function below discontinuous?
\[f(x)=\begin{cases} 4x-7 & x\neq 1 \\ x^2-4 & x= 1 \end{cases} \nonumber \]
For the exercises 29-32, consider the function whose graph appears in Figure.
29) Find the average rate of change of the function from \(x=1\) to \(x=3\).
- Answer
-
\(2\)
30) Find all values of \(x\) at which \(f'(x)=0\).
- Answer
-
\(x=1\)
31) Find all values of \(x\) at which \(f'(x)\) does not exist.
32) Find an equation of the tangent line to the graph of \(f\) the indicated point: \(f(x)=3x^2-2x-6,\; x=-2\)
- Answer
-
\(y=-14x-18\)
For the exercises 33-34, use the function \(f(x)=x(1-x)^{\frac{2}{5}}\)
33) Graph the function \(f(x)=x(1-x)^{\tfrac{2}{5}}\) by entering \(f(x)=x\left ((1-x)^2 \right )^{\tfrac{1}{5}}\) and then by entering \(f(x)=x\left ((1-x)^{\tfrac{1}{5}} \right )^2\).
34) Explore the behavior of the graph of \(f(x)\) around \(x=1\) by graphing the function on the following domains, \([0.9, 1.1], [0.99, 1.01], [0.999, 1.001]\), and \([0.9999, 1.0001]\). Use this information to determine whether the function appears to be differentiable at \(x=1\).
- Answer
-
The graph is not differentiable at \(x=1\) (cusp).
For the exercises 35-42, find the derivative of each of the functions using the definition: \(\lim \limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}\)
35) \(f(x)=2x-8\)
36) \(f(x)=4x^2-7\)
- Answer
-
\(f'(x)=8x\)
37) \(f(x)=x-\dfrac{1}{2}x^2\)
38) \(f(x)=\dfrac{1}{x+2}\)
- Answer
-
\(f'(x)=-\dfrac{1}{(2+x)^2}\)
39) \(f(x)=\dfrac{3}{x-1}\)
40) \(f(x)=-x^3+1\)
- Answer
-
\(f'(x)=-3x^2\)
41) \(f(x)=x^2+x^3\)
42) \(f(x)=\sqrt{x-1}\)
- Answer
-
\(f'(x)=-\dfrac{1}{2\sqrt{x-1}}\)
Contributors and Attributions
-
Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus .


12: Thermodynamics
Among the many capabilities of chemistry is its ability to predict if a process will occur under specified conditions. Thermodynamics, the study of relationships between the energy and work associated with chemical and physical processes, provides this predictive ability. Previous chapters in this text have described various applications of thermochemistry, an important aspect of thermodynamics concerned with the heat flow accompanying chemical reactions and phase transitions. This chapter will introduce additional thermodynamic concepts, including those that enable the prediction of any chemical or physical changes under a given set of conditions.
-
- 12.2: The Second and Third Laws of Thermodynamics
- The second law of thermodynamics states spontaneous processes increases the entropy of the universe. If a process would decrease the entropy of the universe, then the process is nonspontaneous, and if no change occurs, the system is at equilibrium. The third law of thermodynamics establishes the zero for entropy at 0 J/Kelvin for a perfect, pure crystalline solid at 0 K with only one possible microstate.
-
- 12.3: Free Energy
- Gibbs free energy (G) is a state function defined with regard to system quantities only and may be used to predict the spontaneity of a process. A negative value for ΔG indicates that the reaction will proceed in the forward direction to reach equilibrium; a positive ΔG indicates that the reaction will proceed in the reverse direction to reach equilibrium; and a ΔG of zero indicates that the system is at equilibrium. A number of approaches to the computation of free energy changes are possible.


24 .
Contrast Arnett’s and Marcia’s theories of emerging adulthood.
25 .
What are possible reasons that using electronic devices close to bedtime is associated with poorer sleep?
26 .
List and discuss some of the strategies noted in the text that will help with managing the effects of stress in one’s life.
27 .
Compare and contrast HPV, herpes, and HIV.
28 .
Why is it accurate to surmise that one’s stated sexual orientation may not fully encapsulate the range of a person’s sexual feelings?
29 .
Explain the concept of post-formal thought in your own words.
30 .
Discuss the difference in how creativity is understood between people from Eastern versus Western cultures.
31 .
Explain the difference between experiential learning and informal education and the connection between them.
32 .
Explain how low SES, being a parent, and the availability of and demand for jobs can affect a person’s job choices.


8.3: Standard Reduction Potentials
- To use redox potentials to predict whether a reaction is spontaneous.
- To balance redox reactions using half-reactions.
In a galvanic cell, current is produced when electrons flow externally through the circuit from the anode to the cathode because of a difference in potential energy between the two electrodes in the electrochemical cell. In the Zn/Cu system, the valence electrons in zinc have a substantially higher potential energy than the valence electrons in copper because of shielding of the s electrons of zinc by the electrons in filled d orbitals. Hence electrons flow spontaneously from zinc to copper(II) ions, forming zinc(II) ions and metallic copper. Just like water flowing spontaneously downhill, which can be made to do work by forcing a waterwheel, the flow of electrons from a higher potential energy to a lower one can also be harnessed to perform work.
Because the potential energy of valence electrons differs greatly from one substance to another, the voltage of a galvanic cell depends partly on the identity of the reacting substances. If we construct a galvanic cell similar to the one in part (a) in Figure \(\PageIndex{1}\) but instead of copper use a strip of cobalt metal and 1 M Co 2 + in the cathode compartment, the measured voltage is not 1.10 V but 0.51 V. Thus we can conclude that the difference in potential energy between the valence electrons of cobalt and zinc is less than the difference between the valence electrons of copper and zinc by 0.59 V.
The measured potential of a cell also depends strongly on the concentrations of the reacting species and the temperature of the system. To develop a scale of relative potentials that will allow us to predict the direction of an electrochemical reaction and the magnitude of the driving force for the reaction, the potentials for oxidations and reductions of different substances must be measured under comparable conditions. To do this, chemists use the standard cell potential (E° cell ), defined as the potential of a cell measured under standard conditions—that is, with all species in their standard states (1 M for solutions, concentrated solutions of salts (about 1 M) generally do not exhibit ideal behavior, and the actual standard state corresponds to an activity of 1 rather than a concentration of 1 M. Corrections for non ideal behavior are important for precise quantitative work but not for the more qualitative approach that we are taking here. 1 atm for gases, pure solids or pure liquids for other substances) and at a fixed temperature, usually 25°C.
Measured redox potentials depend on the potential energy of valence electrons, the concentrations of the species in the reaction, and the temperature of the system.
Measuring Standard Electrode Potentials
It is physically impossible to measure the potential of a single electrode: only the difference between the potentials of two electrodes can be measured (this is analogous to measuring absolute enthalpies or free energies ; recall that only differences in enthalpy and free energy can be measured.) We can, however, compare the standard cell potentials for two different galvanic cells that have one kind of electrode in common. This allows us to measure the potential difference between two dissimilar electrodes. For example, the measured standard cell potential (E°) for the Zn/Cu system is 1.10 V, whereas E° for the corresponding Zn/Co system is 0.51 V. This implies that the potential difference between the Co and Cu electrodes is 1.10 V − 0.51 V = 0.59 V. In fact, that is exactly the potential measured under standard conditions if a cell is constructed with the following cell diagram:
\[Co_{(s)} ∣ Co^{2+}(aq, 1 M)∥Cu^{2+}(aq, 1 M) ∣ Cu (s)\;\;\; E°=0.59\; V \label{20.4.1} \]
This cell diagram corresponds to the oxidation of a cobalt anode and the reduction of Cu 2 + in solution at the copper cathode.
All tabulated values of standard electrode potentials by convention are listed for a reaction written as a reduction, not as an oxidation, to be able to compare standard potentials for different substances ( Table P1 ). The standard cell potential (E° cell ) is therefore the difference between the tabulated reduction potentials of the two half-reactions, not their sum:
\[E°_{cell} = E°_{cathode} − E°_{anode} \label{20.4.2} \]
In contrast, recall that half-reactions are written to show the reduction and oxidation reactions that actually occur in the cell, so the overall cell reaction is written as the sum of the two half-reactions. According to Equation \(\ref{20.4.2}\), when we know the standard potential for any single half-reaction, we can obtain the value of the standard potential of many other half-reactions by measuring the standard potential of the corresponding cell.
Although it is impossible to measure the potential of any electrode directly, we can choose a reference electrode whose potential is defined as 0 V under standard conditions. The standard hydrogen electrode (SHE) is universally used for this purpose and is assigned a standard potential of 0 V. It consists of a strip of platinum wire in contact with an aqueous solution containing 1 M H + . The [H + ] in solution is in equilibrium with H 2 gas at a pressure of 1 atm at the Pt-solution interface (Figure \(\PageIndex{2}\)). Protons are reduced or hydrogen molecules are oxidized at the Pt surface according to the following equation:
\[2H^+_{(aq)}+2e^− \rightleftharpoons H_{2(g)} \label{20.4.3} \]
One especially attractive feature of the SHE is that the Pt metal electrode is not consumed during the reaction.
Figure \(\PageIndex{3}\) shows a galvanic cell that consists of a SHE in one beaker and a Zn strip in another beaker containing a solution of Zn 2 + ions. When the circuit is closed, the voltmeter indicates a potential of 0.76 V. The zinc electrode begins to dissolve to form Zn 2 + , and H + ions are reduced to H 2 in the other compartment. Thus the hydrogen electrode is the cathode, and the zinc electrode is the anode. The diagram for this galvanic cell is as follows:
\[Zn_{(s)}∣Zn^{2+}_{(aq)}∥H^+(aq, 1 M)∣H_2(g, 1 atm)∣Pt_{(s)} \label{20.4.4} \]
The half-reactions that actually occur in the cell and their corresponding electrode potentials are as follows:
- cathode: \[2H^+_{(aq)} + 2e^− \rightarrow H_{2(g)}\;\;\; E°_{cathode}=0 V \label{20.4.5} \]
- anode: \[Zn_{(s)} \rightarrow Zn^{2+}_{(aq)}+2e^−\;\;\; E°_{anode}=−0.76\; V \label{20.4.6} \]
- overall: \[Zn_{(s)}+2H^+_{(aq)} \rightarrow Zn^{2+}_{(aq)}+H_{2(g)} \label{20.4.7} \]
We then use Equation \ref{20.4.2} to calculate the cell potential
\[\begin{align*} E°_{cell} &=E°_{cathode}−E°_{anode}\\[4pt] &=0.76\; V \end{align*} \nonumber \]
Although the reaction at the anode is an oxidation, by convention its tabulated E° value is reported as a reduction potential. The potential of a half-reaction measured against the SHE under standard conditions is called the standard electrode potential for that half-reaction. In this example, the standard reduction potential for Zn 2 + (aq) + 2e − → Zn(s) is −0.76 V, which means that the standard electrode potential for the reaction that occurs at the anode, the oxidation of Zn to Zn 2 + , often called the Zn/Zn 2 + redox couple, or the Zn/Zn 2 + couple, is −(−0.76 V) = 0.76 V. We must therefore subtract E° anode from E° cathode to obtain
\[E°_{cell}: 0 \,V − (−0.76\, V) = 0.76\, V \nonumber \]
Because electrical potential is the energy needed to move a charged particle in an electric field, standard electrode potentials for half-reactions are intensive properties and do not depend on the amount of substance involved. Consequently, E° values are independent of the stoichiometric coefficients for the half-reaction, and, most important, the coefficients used to produce a balanced overall reaction do not affect the value of the cell potential.
E° values do NOT depend on the stoichiometric coefficients for a half-reaction, because it is an intensive property.
The Standard Hydrogen Electrode (SHE): The Standard Hydrogen Electrode (SHE)(opens in new window) [youtu.be]
Standard Electrode Potentials
To measure the potential of the Cu/Cu 2 + couple, we can construct a galvanic cell analogous to the one shown in Figure \(\PageIndex{3}\) but containing a Cu/Cu 2 + couple in the sample compartment instead of Zn/Zn 2 + . When we close the circuit this time, the measured potential for the cell is negative (−0.34 V) rather than positive. The negative value of \(E°_{cell}\) indicates that the direction of spontaneous electron flow is the opposite of that for the Zn/Zn 2 + couple. Hence the reactions that occur spontaneously, indicated by a positive \(E°_{cell}\), are the reduction of Cu 2 + to Cu at the copper electrode. The copper electrode gains mass as the reaction proceeds, and H 2 is oxidized to H + at the platinum electrode. In this cell, the copper strip is the cathode, and the hydrogen electrode is the anode. The cell diagram therefore is written with the SHE on the left and the Cu 2 + /Cu couple on the right:
\[Pt_{(s)}∣H_2(g, 1 atm)∣H^+(aq, 1\; M)∥Cu^{2+}(aq, 1 M)∣Cu_{(s)} \label{20.4.8} \]
The half-cell reactions and potentials of the spontaneous reaction are as follows:
- Cathode: \[Cu^{2+}{(aq)} + 2e^− \rightarrow Cu_{(g)}\;\;\; E°_{cathode} = 0.34\; V \label{20.4.9} \]
- Anode: \[H_{2(g)} \rightarrow 2H^+_{(aq)} + 2e^−\;\;\; E°_{anode} = 0\; V \label{20.4.10} \]
- Overall: \[H_{2(g)} + Cu^{2+}_{(aq)} \rightarrow 2H^+_{(aq)} + Cu_{(s)} \label{20.4.11} \]
We then use Equation \ref{20.4.2} to calculate the cell potential
\[\begin{align*} E°_{cell} &= E°_{cathode}− E°_{anode} \\[4pt] &= 0.34\; V \end{align*} \nonumber \]
Thus the standard electrode potential for the Cu 2 + /Cu couple is 0.34 V.
Electrode Potentials and ECell: Electrode and Potentials and Ecell(opens in new window) [youtu.be]
Balancing Redox Reactions Using the Half-Reaction Method
In Section 4.4 , we described a method for balancing redox reactions using oxidation numbers. Oxidation numbers were assigned to each atom in a redox reaction to identify any changes in the oxidation states. Here we present an alternative approach to balancing redox reactions, the half-reaction method, in which the overall redox reaction is divided into an oxidation half-reaction and a reduction half-reaction, each balanced for mass and charge. This method more closely reflects the events that take place in an electrochemical cell, where the two half-reactions may be physically separated from each other.
We can illustrate how to balance a redox reaction using half-reactions with the reaction that occurs when Drano, a commercial solid drain cleaner, is poured into a clogged drain. Drano contains a mixture of sodium hydroxide and powdered aluminum, which in solution reacts to produce hydrogen gas:
\[Al_{(s)} + OH^−_{(aq)} \rightarrow Al(OH)^−_{4(aq)} + H_{2(g)} \label{20.4.12} \]
In this reaction, \(Al_{(s)}\) is oxidized to Al 3 + , and H + in water is reduced to H 2 gas, which bubbles through the solution, agitating it and breaking up the clogs.
The overall redox reaction is composed of a reduction half-reaction and an oxidation half-reaction. From the standard electrode potentials listed Table P1 , we find the corresponding half-reactions that describe the reduction of H + ions in water to H 2 and the oxidation of Al to Al 3 + in basic solution:
- reduction: \[2H_2O_{(l)} + 2e^− \rightarrow 2OH^−_{(aq)} + H_{2(g)} \label{20.4.13} \]
- oxidation: \[Al_{(s)} + 4OH^−_{(aq)} \rightarrow Al(OH)^−_{4(aq)} + 3e^− \label{20.4.14} \]
The half-reactions chosen must exactly reflect the reaction conditions, such as the basic conditions shown here. Moreover, the physical states of the reactants and the products must be identical to those given in the overall reaction, whether gaseous, liquid, solid, or in solution.
In Equation \(\ref{20.4.13}\), two H + ions gain one electron each in the reduction; in Equation \(\ref{20.4.14}\), the aluminum atom loses three electrons in the oxidation. The charges are balanced by multiplying the reduction half-reaction (Equation \(\ref{20.4.13}\)) by 3 and the oxidation half-reaction (Equation \(\ref{20.4.14}\)) by 2 to give the same number of electrons in both half-reactions:
- reduction:
\[6H_2O_{(l)} + 6e^− \rightarrow 6OH^−_{(aq)} + 3H_{2(g)} \label{20.4.15} \]
- oxidation:
\[2Al_{(s)} + 8OH^−_{(aq)} \rightarrow 2Al(OH)^−_{4(aq)} + 6e^− \label{20.4.16} \]
Adding the two half-reactions,
\[6H_2O_{(l)} + 2Al_{(s)} + 8OH^−_{(aq)} \rightarrow 2Al(OH)^−{4(aq)} + 3H_{2(g)} + 6OH^−_{(aq)} \label{20.4.17} \]
Simplifying by canceling substances that appear on both sides of the equation,
\[6H_2O_{(l)} + 2Al_{(s)} + 2OH^−_{(aq)} \rightarrow 2Al(OH)^−_{4(aq)} + 3H_{2(g)} \label{20.4.18} \]
We have a −2 charge on the left side of the equation and a −2 charge on the right side. Thus the charges are balanced, but we must also check that atoms are balanced:
\[2Al + 8O + 14H = 2Al + 8O + 14H \label{20.4.19} \]
The atoms also balance, so Equation \(\ref{20.4.18}\) is a balanced chemical equation for the redox reaction depicted in Equation \(\ref{20.4.12}\).
The half-reaction method requires that half-reactions exactly reflect reaction conditions, and the physical states of the reactants and the products must be identical to those in the overall reaction.
We can also balance a redox reaction by first balancing the atoms in each half-reaction and then balancing the charges. With this alternative method, we do not need to use the half-reactions listed in Table P1 , but instead focus on the atoms whose oxidation states change, as illustrated in the following steps:
Step 1: Write the reduction half-reaction and the oxidation half-reaction.
For the reaction shown in Equation \(\ref{20.4.12}\), hydrogen is reduced from H + in OH − to H 2 , and aluminum is oxidized from Al° to Al 3 + :
- reduction:
\[OH^−_{(aq)} \rightarrow H_{2(g)} \label{20.4.20} \]
- oxidation:
\[Al_{(s)} \rightarrow Al(OH)^−_{4(aq)} \label{20.4.21} \]
Step 2: Balance the atoms by balancing elements other than O and H. Then balance O atoms by adding H 2 O and balance H atoms by adding H + .
Elements other than O and H in the previous two equations are balanced as written, so we proceed with balancing the O atoms. We can do this by adding water to the appropriate side of each half-reaction:
- reduction:
\[OH^−_{(aq)} \rightarrow H_{2(g)} + H_2O_{(l)} \label{20.4.22} \]
- oxidation:
\[Al_{(s)} + 4H_2O_{(l)} \rightarrow Al(OH)^−_{4(aq)} \label{20.4.23} \]
Balancing H atoms by adding H + , we obtain the following:
- reduction:
\[OH^−_{(aq)} + 3H^+_{(aq)} \rightarrow H_{2(g)} + H_2O_{(l)} \label{20.4.24} \]
- oxidation:
\[Al_{(s)} + 4H_2O_{(l)} \rightarrow Al(OH)^−_{4(aq)} + 4H^+_{(aq)} \label{20.4.25} \]
We have now balanced the atoms in each half-reaction, but the charges are not balanced.
Step 3: Balance the charges in each half-reaction by adding electrons.
Two electrons are gained in the reduction of H + ions to H 2 , and three electrons are lost during the oxidation of Al° to Al 3 + :
- reduction:
\[OH^−_{(aq)} + 3H^+_{(aq)} + 2e^− \rightarrow H_{2(g)} + H_2O_{(l)} \label{20.4.26} \]
- oxidation:
\[Al_{(s)} + 4H_2O_{(l)} \rightarrow Al(OH)^−_{4(aq)} + 4H^+_{(aq)} + 3e^− \label{20.4.27} \]
Step 4: Multiply the reductive and oxidative half-reactions by appropriate integers to obtain the same number of electrons in both half-reactions.
In this case, we multiply Equation \(\ref{20.4.26}\) (the reductive half-reaction) by 3 and Equation \(\ref{20.4.27}\) (the oxidative half-reaction) by 2 to obtain the same number of electrons in both half-reactions:
- reduction:
\[3OH^−_{(aq)} + 9H^+_{(aq)} + 6e^− \rightarrow 3H_{2(g)} + 3H_2O_{(l)} \label{20.4.28} \]
- oxidation:
\[2Al_{(s)} + 8H_2O_{(l)} \rightarrow 2Al(OH)^−_{4(aq)} + 8H^+_{(aq)} + 6e^− \label{20.4.29} \]
Step 5: Add the two half-reactions and cancel substances that appear on both sides of the equation.
Adding and, in this case, canceling 8H + , 3H 2 O, and 6e − ,
\[2Al_{(s)} + 5H_2O_{(l)} + 3OH^−_{(aq)} + H^+_{(aq)} \rightarrow 2Al(OH)^−_{4(aq)} + 3H_{2(g)} \label{20.4.30} \]
We have three OH − and one H + on the left side. Neutralizing the H + gives us a total of 5H 2 O + H 2 O = 6H 2 O and leaves 2OH − on the left side:
\[2Al_{(s)} + 6H_2O_{(l)} + 2OH^−_{(aq)} \rightarrow 2Al(OH)^−_{4(aq)} + 3H_{2(g)} \label{20.4.31} \]
Step 6: Check to make sure that all atoms and charges are balanced.
Equation \(\ref{20.4.31}\) is identical to Equation \(\ref{20.4.18}\), obtained using the first method, so the charges and numbers of atoms on each side of the equation balance.
In acidic solution, the redox reaction of dichromate ion (\(\ce{Cr2O7^{2−}}\)) and iodide (\(\ce{I^{−}}\)) can be monitored visually. The yellow dichromate solution reacts with the colorless iodide solution to produce a solution that is deep amber due to the presence of a green \(\ce{Cr^{3+}(aq)}\) complex and brown \(\ce{I2(aq)}\) ions (Figure \(\PageIndex{4}\)):
\[\ce{Cr2O7^{2−}(aq) + I^{−}(aq) -> Cr^{3+}(aq) + I2(aq)} \nonumber \]
Balance this equation using half-reactions.
Given: redox reaction and Table P1
Asked for: balanced chemical equation using half-reactions
Strategy:
Follow the steps to balance the redox reaction using the half-reaction method.
Solution
From the standard electrode potentials listed in Table P1 , we find the half-reactions corresponding to the overall reaction:
- reduction: \[\ce{Cr2O7^{2−}(aq) + 14H^{+}(aq) + 6e^{−} -> 2Cr^{3+}(aq) + 7H2O(l)} \nonumber \]
- oxidation: \[\ce{2I^{−}(aq) -> I2(aq) + 2e^{−}} \nonumber \]
Balancing the number of electrons by multiplying the oxidation reaction by 3,
- oxidation: \[\ce{6I^{−}(aq) -> 3I2(aq) + 6e^{−}} \nonumber \]
Adding the two half-reactions and canceling electrons,
\[\ce{Cr2O^{2−}7(aq) + 14H^{+}(aq) + 6I^{−}(aq) -> 2Cr^{3+}(aq) + 7H2O(l) + 3I2(aq)} \nonumber \]
We must now check to make sure the charges and atoms on each side of the equation balance:
\[\begin{align*} (−2) + 14 + (−6) &= +6 \\[4pt] +6 &\overset{\checkmark}{=} +6 \end{align*} \nonumber \]
and atoms
\[\ce{2Cr + 7O + 14H + 6I} \overset{\checkmark}{=} \ce{2Cr + 7O + 14H + 6I} \nonumber \]
Both the charges and atoms balance, so our equation is balanced.
We can also use the alternative procedure, which does not require the half-reactions listed in Table P1 .
Step 1: Chromium is reduced from \(\ce{Cr^{6+}}\) in \(\ce{Cr2O7^{2−}}\) to \(\ce{Cr^{3+}}\), and \(\ce{I^{−}}\) ions are oxidized to \(\ce{I2}\). Dividing the reaction into two half-reactions,
- reduction: \[Cr_2O^{2−}_{7(aq)} \rightarrow Cr^{3+}_{(aq)} \nonumber \]
- oxidation: \[I^−_{(aq)} \rightarrow I_{2(aq)} \nonumber \]
Step 2: Balancing the atoms other than oxygen and hydrogen,
- reduction: \[Cr_2O^{2−}_{7(aq)} \rightarrow 2Cr^{3+}_{(aq)} \nonumber \]
- oxidation: \[2I^−_{(aq)} \rightarrow I_{2(aq)} \nonumber \]
We now balance the O atoms by adding H 2 O—in this case, to the right side of the reduction half-reaction. Because the oxidation half-reaction does not contain oxygen, it can be ignored in this step.
- reduction: \[Cr_2O^{2−}_{7(aq)} \rightarrow 2Cr^{3+}_{(aq)} + 7H_2O_{(l)} \nonumber \]
Next we balance the H atoms by adding H + to the left side of the reduction half-reaction. Again, we can ignore the oxidation half-reaction.
- reduction: \[Cr_2O^{2−}_{7(aq)} + 14H^+_{(aq)} \rightarrow 2Cr^{3+}_{(aq)} + 7H_2O_{(l)} \nonumber \]
Step 3: We must now add electrons to balance the charges. The reduction half-reaction (2Cr +6 to 2Cr +3 ) has a +12 charge on the left and a +6 charge on the right, so six electrons are needed to balance the charge. The oxidation half-reaction (2I − to I 2 ) has a −2 charge on the left side and a 0 charge on the right, so it needs two electrons to balance the charge:
- reduction: Cr 2 O 7 2 − (aq) + 14H + (aq) + 6e − → 2Cr 3 + (aq) + 7H 2 O(l)
- oxidation: 2I − (aq) → I 2 (aq) + 2e −
Step 4: To have the same number of electrons in both half-reactions, we must multiply the oxidation half-reaction by 3:
- oxidation: 6I − (aq) → 3I 2 (s) + 6e −
Step 5: Adding the two half-reactions and canceling substances that appear in both reactions,
\[\ce{Cr2O7^{2−}(aq) + 14H^{+}(aq) + 6I^{−}(aq) → 2Cr^{3+}(aq) + 7H2O(l) + 3I2(aq)} \nonumber \]
Step 6: This is the same equation we obtained using the first method. Thus the charges and atoms on each side of the equation balance.
Copper is found as the mineral covellite (\(\ce{CuS}\)). The first step in extracting the copper is to dissolve the mineral in nitric acid (\(\ce{HNO3}\)), which oxidizes sulfide to sulfate and reduces nitric acid to \(\ce{NO}\):
\[\ce{CuS(s) + HNO3(aq) \rightarrow NO(g) + CuSO4(aq)} \nonumber \]
Balance this equation using the half-reaction method.
- Answer
-
\[\ce{3CuS(s) + 8HNO3(aq) -> 8NO(g) + 3CuSO4(aq) + 4H2O(l)} \nonumber \]
Calculating Standard Cell Potentials
The standard cell potential for a redox reaction (E° cell ) is a measure of the tendency of reactants in their standard states to form products in their standard states; consequently, it is a measure of the driving force for the reaction, which earlier we called voltage. We can use the two standard electrode potentials we found earlier to calculate the standard potential for the Zn/Cu cell represented by the following cell diagram:
\[ Zn{(s)}∣Zn^{2+}(aq, 1 M)∥Cu^{2+}(aq, 1 M)∣Cu_{(s)} \label{20.4.32} \]
We know the values of E° anode for the reduction of Zn 2 + and E° cathode for the reduction of Cu 2 + , so we can calculate \(E°_{cell}\):
- cathode: \[Cu^{2+}_{(aq)} + 2e^− \rightarrow Cu_{(s)} \;\;\; E°_{cathode} = 0.34\; V \label{20.4.33} \]
- anode: \[Zn_{(s)} \rightarrow Zn^{2+}(aq, 1 M) + 2e^−\;\;\; E°_{anode} = −0.76\; V \label{20.4.34} \]
- overall: \[Zn_{(s)} + Cu^{2+}_{(aq)} \rightarrow Zn^{2+}_{(aq)} + Cu_{(s)} \label{20.4.35} \]
\[E°_{cell} = E°_{cathode} − E°_{anode} = 1.10\; V \nonumber \]
This is the same value that is observed experimentally. If the value of \(E°_{cell}\) is positive, the reaction will occur spontaneously as written. If the value of \(E°_{cell}\) is negative, then the reaction is not spontaneous, and it will not occur as written under standard conditions; it will, however, proceed spontaneously in the opposite direction. As we shall see in Section 20.9 , this does not mean that the reaction cannot be made to occur at all under standard conditions. With a sufficient input of electrical energy, virtually any reaction can be forced to occur. Example \(\PageIndex{2}\) and its corresponding exercise illustrate how we can use measured cell potentials to calculate standard potentials for redox couples.
A positive \(E°_{cell}\) means that the reaction will occur spontaneously as written. A negative \(E°_{cell}\) means that the reaction will proceed spontaneously in the opposite direction.
A galvanic cell with a measured standard cell potential of 0.27 V is constructed using two beakers connected by a salt bridge. One beaker contains a strip of gallium metal immersed in a 1 M solution of GaCl 3 , and the other contains a piece of nickel immersed in a 1 M solution of NiCl 2 . The half-reactions that occur when the compartments are connected are as follows:
cathode: Ni 2 + (aq) + 2e − → Ni(s)
anode: Ga(s) → Ga 3 + (aq) + 3e −
If the potential for the oxidation of Ga to Ga 3 + is 0.55 V under standard conditions, what is the potential for the oxidation of Ni to Ni 2 + ?
Given: galvanic cell, half-reactions, standard cell potential, and potential for the oxidation half-reaction under standard conditions
Asked for: standard electrode potential of reaction occurring at the cathode
Strategy:
- Write the equation for the half-reaction that occurs at the anode along with the value of the standard electrode potential for the half-reaction.
- Use Equation \(\ref{20.4.2}\) to calculate the standard electrode potential for the half-reaction that occurs at the cathode. Then reverse the sign to obtain the potential for the corresponding oxidation half-reaction under standard conditions.
Solution
A We have been given the potential for the oxidation of Ga to Ga 3 + under standard conditions, but to report the standard electrode potential, we must reverse the sign. For the reduction reaction Ga 3 + (aq) + 3e − → Ga(s), E° anode = −0.55 V.
B Using the value given for \(E°_{cell}\) and the calculated value of E° anode , we can calculate the standard potential for the reduction of Ni 2 + to Ni from Equation \(\ref{20.4.2}\):
\[\begin{align*} E°_{cell} &= E°_{cathode} − E°_{anode} \\[4pt] 0.27\, V &= E^o°_{cathhode} − (−0.55\, V) \\[4pt] E^°_{cathode} &= −0.28 \,V \end{align*} \nonumber \]
This is the standard electrode potential for the reaction Ni 2 + (aq) + 2e − → Ni(s). Because we are asked for the potential for the oxidation of Ni to Ni 2 + under standard conditions, we must reverse the sign of E° cathode . Thus E° = −(−0.28 V) = 0.28 V for the oxidation. With three electrons consumed in the reduction and two produced in the oxidation, the overall reaction is not balanced. Recall, however, that standard potentials are independent of stoichiometry.
A galvanic cell is constructed with one compartment that contains a mercury electrode immersed in a 1 M aqueous solution of mercuric acetate \(\ce{Hg(CH_3CO_2)_2}\) and one compartment that contains a strip of magnesium immersed in a 1 M aqueous solution of \(\ce{MgCl2}\). When the compartments are connected, a potential of 3.22 V is measured and the following half-reactions occur:
- cathode: \(\ce{Hg^{2+} (aq) + 2e^{−} → Hg(l)}\)
- anode: \(\ce{Mg(s) → Mg^{2+}(aq) + 2e^{−}}\)
If the potential for the oxidation of \(\ce{Mg}\) to \(\ce{Mg^{2+}}\) is 2.37 V under standard conditions, what is the standard electrode potential for the reaction that occurs at the cathode?
- Answer
-
0.85 V
Reference Electrodes and Measuring Concentrations
When using a galvanic cell to measure the concentration of a substance, we are generally interested in the potential of only one of the electrodes of the cell, the so-called indicator electrode , whose potential is related to the concentration of the substance being measured. To ensure that any change in the measured potential of the cell is due to only the substance being analyzed, the potential of the other electrode, the reference electrode , must be constant. You are already familiar with one example of a reference electrode: the SHE. The potential of a reference electrode must be unaffected by the properties of the solution, and if possible, it should be physically isolated from the solution of interest. To measure the potential of a solution, we select a reference electrode and an appropriate indicator electrode. Whether reduction or oxidation of the substance being analyzed occurs depends on the potential of the half-reaction for the substance of interest (the sample) and the potential of the reference electrode.
The potential of any reference electrode should not be affected by the properties of the solution to be analyzed, and it should also be physically isolated.
There are many possible choices of reference electrode other than the SHE. The SHE requires a constant flow of highly flammable hydrogen gas, which makes it inconvenient to use. Consequently, two other electrodes are commonly chosen as reference electrodes. One is the silver–silver chloride electrode , which consists of a silver wire coated with a very thin layer of AgCl that is dipped into a chloride ion solution with a fixed concentration. The cell diagram and reduction half-reaction are as follows:
\[Cl^−_{(aq)}∣AgCl_{(s)}∣Ag_{(s)} \label{20.4.36} \]
\[AgCl_{(s)}+e^− \rightarrow Ag_{(s)} + Cl^−_{(aq)} \nonumber \]
If a saturated solution of KCl is used as the chloride solution, the potential of the silver–silver chloride electrode is 0.197 V versus the SHE. That is, 0.197 V must be subtracted from the measured value to obtain the standard electrode potential measured against the SHE.
A second common reference electrode is the saturated calomel electrode (SCE) , which has the same general form as the silver–silver chloride electrode. The SCE consists of a platinum wire inserted into a moist paste of liquid mercury (Hg 2 Cl 2 ; called calomel in the old chemical literature) and KCl. This interior cell is surrounded by an aqueous KCl solution, which acts as a salt bridge between the interior cell and the exterior solution (part (a) in Figure \(\PageIndex{5}\). Although it sounds and looks complex, this cell is actually easy to prepare and maintain, and its potential is highly reproducible. The SCE cell diagram and corresponding half-reaction are as follows:
\[Pt_{(s)} ∣ Hg_2Cl_{2(s)}∣KCl_{(aq, sat)} \label{20.4.37} \]
\[Hg_2Cl_{2(s)} + 2e^− \rightarrow 2Hg_{(l)} + 2Cl^−{(aq)} \label{20.4.38} \]
At 25°C, the potential of the SCE is 0.2415 V versus the SHE, which means that 0.2415 V must be subtracted from the potential versus an SCE to obtain the standard electrode potential.
One of the most common uses of electrochemistry is to measure the H + ion concentration of a solution. A glass electrode is generally used for this purpose, in which an internal Ag/AgCl electrode is immersed in a 0.10 M HCl solution that is separated from the solution by a very thin glass membrane (part (b) in Figure \(\PageIndex{5}\). The glass membrane absorbs protons, which affects the measured potential. The extent of the adsorption on the inner side is fixed because [H + ] is fixed inside the electrode, but the adsorption of protons on the outer surface depends on the pH of the solution. The potential of the glass electrode depends on [H + ] as follows (recall that pH = −log[H + ]):
\[E_{glass} = E′ + (0.0591\; V \times \log[H^+]) = E′ − 0.0591\; V \times pH \label{20.4.39} \]
The voltage E′ is a constant that depends on the exact construction of the electrode. Although it can be measured, in practice, a glass electrode is calibrated; that is, it is inserted into a solution of known pH, and the display on the pH meter is adjusted to the known value. Once the electrode is properly calibrated, it can be placed in a solution and used to determine an unknown pH.
Ion-selective electrodes are used to measure the concentration of a particular species in solution; they are designed so that their potential depends on only the concentration of the desired species (part (c) in Figure \(\PageIndex{5}\)). These electrodes usually contain an internal reference electrode that is connected by a solution of an electrolyte to a crystalline inorganic material or a membrane, which acts as the sensor. For example, one type of ion-selective electrode uses a single crystal of Eu-doped \(LaF_3\) as the inorganic material. When fluoride ions in solution diffuse to the surface of the solid, the potential of the electrode changes, resulting in a so-called fluoride electrode. Similar electrodes are used to measure the concentrations of other species in solution. Some of the species whose concentrations can be determined in aqueous solution using ion-selective electrodes and similar devices are listed in Table \(\PageIndex{1}\).
| Species | Type of Sample |
|---|---|
| H + | laboratory samples, blood, soil, and ground and surface water |
| NH 3 /NH 4 + | wastewater and runoff water |
| K + | blood, wine, and soil |
| CO 2 /HCO 3 − | blood and groundwater |
| F − | groundwater, drinking water, and soil |
| Br − | grains and plant extracts |
| I − | milk and pharmaceuticals |
| NO 3 − | groundwater, drinking water, soil, and fertilizer |
Summary
Redox reactions can be balanced using the half-reaction method. The standard cell potential is a measure of the driving force for the reaction. \(E°_{cell} = E°_{cathode} − E°_{anode} \nonumber \] The flow of electrons in an electrochemical cell depends on the identity of the reacting substances, the difference in the potential energy of their valence electrons, and their concentrations. The potential of the cell under standard conditions (1 M for solutions, 1 atm for gases, pure solids or liquids for other substances) and at a fixed temperature (25°C) is called the standard cell potential (E° cell ). Only the difference between the potentials of two electrodes can be measured. By convention, all tabulated values of standard electrode potentials are listed as standard reduction potentials. The overall cell potential is the reduction potential of the reductive half-reaction minus the reduction potential of the oxidative half-reaction (E° cell = E° cathode − E° anode ). The potential of the standard hydrogen electrode (SHE) is defined as 0 V under standard conditions. The potential of a half-reaction measured against the SHE under standard conditions is called its standard electrode potential. The standard cell potential is a measure of the driving force for a given redox reaction. All E° values are independent of the stoichiometric coefficients for the half-reaction. Redox reactions can be balanced using the half-reaction method, in which the overall redox reaction is divided into an oxidation half-reaction and a reduction half-reaction, each balanced for mass and charge. The half-reactions selected from tabulated lists must exactly reflect reaction conditions. In an alternative method, the atoms in each half-reaction are balanced, and then the charges are balanced. Whenever a half-reaction is reversed, the sign of E° corresponding to that reaction must also be reversed. If \(E°_{cell}\) is positive, the reaction will occur spontaneously under standard conditions. If \(E°_{cell}\) is negative, then the reaction is not spontaneous under standard conditions, although it will proceed spontaneously in the opposite direction. The potential of an indicator electrode is related to the concentration of the substance being measured, whereas the potential of the reference electrode is held constant. Whether reduction or oxidation occurs depends on the potential of the sample versus the potential of the reference electrode. In addition to the SHE, other reference electrodes are the silver–silver chloride electrode; the saturated calomel electrode (SCE); the glass electrode, which is commonly used to measure pH; and ion-selective electrodes, which depend on the concentration of a single ionic species in solution. Differences in potential between the SHE and other reference electrodes must be included when calculating values for E°.


9.6: Assignment- Practice Quoting
Responsible academic writing involves a good deal of direct quotation from sources. Let’s practice that now, to make sure that we’ve got the finer points figured out by the time the essay is due.
Find a quote from the original article that you think will serve you well in your Source Evaluation Essay.
Assignment Requirements
- Include an introductory “signal” phrase that cues us in on the context for the quote, and then the quote itself. Follow this quote with a phrase or sentence in your own words that summarizes, interprets, or explains the quote in your own words.
This 3-step process is sometimes referred to as a “quote sandwich” and is useful for every time you’d like to incorporate a quote into an academic essay you’re writing.
If you have any questions at all about using quotations in your writing, please include them in your submission.
This work is just a draft–you may choose to use it or not in the final version of your essay at your discretion.
Contributors and Attributions
- Composition II. Authored by : Alexis McMillan-Clifton. Provided by : Tacoma Community College. Located at : http://www.tacomacc.edu . Project : Kaleidoscope Open Course Initiative. License : CC BY: Attribution
- Image of sandwich. Authored by : James Yu. Located at : https://flic.kr/p/ki52 . License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike


8.1.5: Nutrients Work Together
It is important to have a well-balanced diet. While each of the individual nutrients discussed above are important, it is important that a person take in a combination of all of them to make a well-balanced diet. Together they work to keep the body working at its optimum (best) level.
Select the nutrient category to which the food in the list belongs.
A. Protein B. Carbohydrate C. Fat
1. Butter
2. Tuna fish
3. Steak
4. Noodles
5. Margarine
6. Canola oil
7. Peanut butter
8. Tortillas
9. Eggs
10. Olive oil
11. Beef
12. Bread
13. Crackers
14. Hamburger
15. Shrimp
16. Mashed potatoes
17. Oatmeal
18. Rice
19. Vegetable oil
20. Chicken
21. Salad dressing
22. Hot dogs
23. Popcorn
- Answer
-
1. C
2. A
3. A
4. B
5. C
6. C
7. A
8. B
9. A
10. C
11. A
12. B
13. B
14. A
15. A
16. B
17. B
18. B
19. C
20. A
21. C
22. A
23. B
FEEDBACK :
Protein sources include meats, poultry, fish, peanut butter, and eggs. Carbohydrate sources include pasta, rice, potatoes, popcorn, and grains. Fats include oils, butter, margarine, and salad dressings.


9.3: Exercises
1. Let \(F\) be the group of all functions from \([0,1]\) to \(\mathbb{R}\text{,}\) under pointwise addition. Let
\begin{equation*} N=\{f\in F: f(1/4)=0\}. \end{equation*}
Prove that \(F/N\) is a group that's isomorphic to \(\mathbb{R}\text{.}\)
2. Let \(N=\{1,-1\}\subseteq \mathbb{R}^*\text{.}\) Prove that \(\mathbb{R}^*/N\) is a group that's isomorphic to \(\mathbb{R}^+\text{.}\)
3. Let \(n\in \mathbb{Z}^+\) and let \(H=\{A\in GL(n,\mathbb{R})\,:\, \det A =\pm 1\}\text{.}\) Identify a group familiar to us that is isomorphic to \(GL(n,\mathbb{R})/H\text{.}\)
4. Let \(G\) and \(G'\) be groups with respective normal subgroups \(N\) and \(N'\text{.}\) Prove or disprove: If \(G/N\simeq G'/N'\) then \(G\simeq G'\text{.}\)


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active transport
-
involves moving solutes and ions across a cell membrane from an area of lower concentration to an area of higher concentration
-
anasarca
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severe and generalized
edema
throughout the body
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BUN to creatinine ratio
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a comparison of BUN to creatinine to determine if alterations in BUN and creatinine are related to changes in fluid volume or kidney filtration
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Catheter-associated urinary tract infection (CAUTI)
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a urinary tract infection that develops directly related to the presence of an IUC
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dialysis
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procedure to remove waste products and excess fluids from the blood when the kidneys cannot
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diffusion
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the movement of molecules from an area of higher concentration to an area of lower concentration to equalize the concentration of solutes throughout an area
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edema
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swelling caused by excessive buildup of fluid in tissue spaces or a body cavity
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enophthalmos
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sunken eyes
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extracellular fluid (ECF)
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fluid located outside of cells
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filtration
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occurs when
hydrostatic pressure
pushes fluids and solutes through a permeable membrane so they can be excreted
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fluid balance
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the difference between intake and output
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fluid volume deficit (FVD)
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results from a loss of body fluids or fluid that becomes unavailable in the body for use
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fluid volume excess (FVE)
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increased fluid retained in the intravascular compartment
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functional urinary incontinence
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the inability to access a toilet due to physical or cognitive challenges
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homeostasis
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the maintenance of equilibrium of two or more interdependent elements
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hypercalcemia
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serum calcium level >10.5 mg/dL
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hyperchloremia
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serum chloride level >105 mEq/L
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hyperkalemia
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serum potassium level > 5.1 mEq/L
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hypermagnesemia
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magnesium level >2.2 mg/dL
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hyperphosphatemia
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phosphate level >4.8 mg/dL
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hypertonic solution
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has a higher concentration of dissolved particles than does blood
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hypervolemia
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excessive fluid volume
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hypocalcemia
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total serum calcium level <8.5 mg/dL or an ionized calcium level <4.6 mg/dL
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hypochloremia
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serum chloride level <97 mEq/L
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hypokalemia
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serum potassium level <3.7 mEq/L
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hypomagnesemia
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serum magnesium level <1.7 mg/dL
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hyponatremia
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serum sodium level of <136 mEq/L
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hypophosphatemia
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serum phosphate level <2.5 mg/dL.
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hypotonic hyponatremia
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serum sodium level is decreased due to dilution
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hypotonic solution
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has a lower concentration of dissolved solutes than does blood
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hypovolemia
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deficient fluid volume
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indwelling urinary catheter (IUC)
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a urinary catheter that stays in the bladder for a time
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intermittent catheterization
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used for the temporary relief of urinary retention
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interstitial fluid
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fluid outside of blood vessels and between the cells
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intracellular fluid (ICF)
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found inside cells; comprise protein, water, electrolytes, and solutes
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intravascular fluid
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fluid in the vascular system
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isotonic solution
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has a similar concentration of dissolved particles as blood
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mixed urinary incontinence
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two or more types of urinary incontinence that occur in the same individual
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oncotic pressure
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pressure inside the vascular compartment created by protein content of the blood (in the form of albumin) that holds water inside the blood vessels
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orthostatic vital signs (also known as postural vital signs)
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assessing a patient’s blood pressure when the patient is lying supine and comparing it to the patient’s blood pressure when either sitting upright or standing upright
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osmolality
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a measure of the concentration of dissolved solutes in the blood
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osmosis
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water movement through a semipermeable membrane, from an area of lesser
solute
concentration to an area of greater solute concentration, in an attempt to equalize the solute concentrations on either side of the membrane
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overflow incontinence
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when bladder contraction fails to fully empty the bladder and urine leaks due to the high pressure from urine retention
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peritoneal dialysis
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uses the capillaries in the peritoneal cavity to exchange waste products
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post-void residual (PVD)
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determined by assessing the amount of urine that remains in the bladder immediately after the patient attempts to empty their bladder
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skin turgor
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the elasticity of the skin assessed by pinching the skin into a tent shape and assessing how quickly the skin returns to its normal position
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solute
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soluble particle
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stress incontinence
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urine leakage during times of increased intrabdominal pressure
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third spacing
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type of hypovolemia in which total body fluid may be adequate or even excessive, but fluid has moved out of the intravascular space and into the interstitial space
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tonicity
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the osmotic gradient between two solutions
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transcellular fluid
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fluid in cerebrospinal, synovial, and intrapleural areas, for example, and the gastrointestinal system
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urge incontinence
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an overactive bladder that contracts and causes urine leakage when the bladder is not full
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urinary incontinence (UI)
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unintentional and uncontrollable urine leakage
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urine specific gravity
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a urine test that commonly measures hydration status by measuring the concentration of particles in urine


5.6: Nomenclature- Naming Compounds
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Nomenclature is the process of naming chemical compounds so that they can be easily identified as separate chemicals. The primary function of chemical nomenclature is to ensure that a spoken or written chemical name leaves no ambiguity concerning which chemical compound the name refers to—each chemical name should refer to a single substance. A less important aim is to ensure that each substance has a single name, although a limited number of alternative names is acceptable in some cases. Preferably, the name also conveys some information about the structure or chemistry of a compound. A common name will often suffice to identify a chemical compound in a particular set of circumstances. To be more generally applicable, the name should indicate at least the chemical formula. To be more specific still, the three-dimensional arrangement of the atoms may need to be specified.
Wikipedia (CC-BY-SA-3.0)


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9.8.3: Course Objectives for Part I
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Suggested Course Objectives for Part I—Dominant and Minority Groups
Upon completion of this unit, students will be able to:
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Read about, find, identify, interpret, understand, explain, critique, define, present factually and persuasively, and write about and/or discuss dominant and minority groups and differentiate between them.
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Read about, find, identify, interpret, understand, explain, critique, define, present factually and persuasively, and write about and/or discuss prejudice and discrimination and the various forms they take.
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Read about, find, identify, interpret, understand, explain, critique, define, present factually and persuasively, and write about and/or discuss immigration, immigration theories, and the ongoing national debate concerning immigration.
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Find, interpret, understand, analyze, critique, and write about and/or discuss statistical and historical information concerning race relations in the past and present in the US.
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Understand, analyze, critique, and write about and/or discuss popular media’s interpretation of stratification/inequality by race and ethnicity.
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Find, read, interpret, understand, analyze, critique, and write about and/or discuss immigration law and immigration studies found in peer reviewed scientific journal articles.
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Find and interpret demographic data about immigration from the US Census and other valid and reliable sources.
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Understand, analyze, critique, and write about and/or discuss popular media’s interpretation of immigration and immigration policy.
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Find, interpret, understand, analyze, critique, and write about and/or discuss Congressional policies concerning immigration and immigrants in the US.