1.1: Prelude to The Nature of Light
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Our investigation of light revolves around two questions of fundamental importance:
Answers to these questions can be found in Maxwell’s equations , which predict the existence of electromagnetic waves and their behavior. Examples of light include radio and infrared waves, visible light, ultraviolet radiation, and X-rays. Interestingly, not all light phenomena can be explained by Maxwell’s theory. Experiments performed early in the twentieth century showed that light has corpuscular, or particle-like, properties. The idea that light can display both wave and particle characteristics is called wave-particle duality, which is examined in Photons and Matter Waves.
In this chapter, we study the basic properties of light. In the next few chapters, we investigate the behavior of light when it interacts with optical devices such as mirrors, lenses, and apertures.


 This area of statistics is called "Descriptive Statistics." You will learn how to calculate, and even more importantly, how to interpret these measurements and graphs.
-
- 2.0: Prelude to Graphs
- In this chapter, you will study numerical and graphical ways to describe and display your data. This area of statistics is called "Descriptive Statistics." You will learn how to calculate, and even more importantly, how to interpret these measurements and graphs. In this chapter, we will briefly look at stem-and-leaf plots, line graphs, and bar graphs, as well as frequency polygons, and time series graphs. Our emphasis will be on histograms and box plots.
-
- 2.1: Organizing Data - Frequency Distributions
- Some calculations generate numbers that are artificially precise. It is not necessary to report a value to eight decimal places when the measures that generated that value were only accurate to the nearest tenth. Round off your final answer to one more decimal place than was present in the original data. This means that if you have data measured to the nearest tenth of a unit, report the final statistic to the nearest hundredth.
Contributors and Attributions
-
Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>.


1.5: Workshop Skills Checklist
While the basic principles of editing and creating content on the LibreTexts libraries are limited, there are several skills need to master before creating and curating content. The Construction Guide and Construction Forum are great resources to consult in building these skills:
A recent Libretexts overview video is available too: youtu.be/sxLmeHMgvTg .
Activity 1
- When your instructor's account is verified go to the Conductor ( https://commons.libretexts.org/home ).
Activity 2
- Set up a Project in the Conductor
- Build a rudimentary book shell (one click, two entries)
Activity 3
- Go to the Remixer and build a few chapters/pages in your new book
- Save the book set up in the Remixer to the Library/Workbench area and go to it
Activity 4
- Rename Page (standard is "number: Title")
- Edit page (new or blank page and edit an existing page)
- Copy/past Existing Text
- Make hyperlink
- Add Box via templates
- Add equation
- Add video
- Add table
Activity 5
- Transclude (reuse) an existing page
Activity 6 (Advance)
- Add interactive component (via transclusion)
- Add Executable code (e.g. Python, R Sagemath)
Activity 7
- Add Copyright to a page
Activity 8
- Add Authorbar to a page
Activity 9
- Watch a page (little star next to social media buttons)
- Rate a Page
- Provide Feedback
Activity 10
- Export PDF


10.14: Intrinsic vs Extrinsic Rewards
- Differentiate between intrinsic and extrinsic rewards
To understand the nature of rewards, we need to distinguish between those that are intrinsic and those that are extrinsic. Intrinsic rewards are internal and reflect an individual’s interests, values and aspirations. An intrinsic reward is intangible; it might be the sense of satisfaction you get from mastering a new skill or the successful completion of a complex project or from working on a project that has personal significance or meaning.
In contrast, extrinsic reward is something that comes from an external source—for example, your instructor at school or your manager at work. Extrinsic rewards can be financial (a bonus, incentive, or commission) or non-financial (praise, a training badge, a development opportunity, or a coveted project assignment). There can also be elements of both; for example, a promotion would likely entail both a raise and a new title. Extrinsic rewards can also include intangibles such as the ability to work remotely or an invitation to participate in a mentoring program.
Dennis Coon and John Mitterer explain that “intrinsic motivation occurs when we act without any obvious external rewards. We simply enjoy an activity or see it as an opportunity to explore, learn, and actualize our potentials.” [1] Intrinsic motivation is personal; for example, you may be intrinsically motivated to learn about or create a whole new world (think: Harry Potter), to improve the world or our experience of it, or to participate in a game or other activity. When individuals are intrinsically motivated, they typically perform at relatively high levels. [2] In contrast, when a person is extrinsically motivated, his or her activity or behavior is a means to an end rather than an end in itself. [3]
The ideal is when what you find intrinsically motivating is also what you do for a living. This is the idea that Steve Jobs communicated in his legendary commencement address at Stanford: “You’ve got to find what you love. And that is as true for your work as it is for your lovers. Your work is going to fill a large part of your life, and the only way to be truly satisfied is to do what you believe is great work. And the only way to do great work is to love what you do.”
Although intrinsic motivation is personal, its characteristics are universal and include: [4]
- Autonomy: the person has control over how they activity is accomplished
- Mastery: there is an element of progress and increasing competence or growth
- Relatedness: there is a sense of community with others who are engaged in the activity
- Purpose: the activity is perceived as meaningful
A manager can’t prompt intrinsic motivation or provide an intrinsic reward, since the motivation flows from the employee’s interest in and value for a particular type of work. A manager does, however, have control of situational factors. Management consulting firm Hay Group (acquired by Korn Ferry) reports business results can vary by up to 30% based on differences in the work climate created by a manager. [5] Jobs can also be designed to better support employees, as discussed in Module 5: Workforce Planning.
- Cherry, Kendra. " Intrinsic Motivation: Why You Do Things ." VeryWell Mind. July 26, 2019. Accessed August 22, 2019. ↵
- Makki, Arooj and Momina Abid. " Influence of Intrinsic and Extrinsic Motivation on Employee's task Performance ," Studies in Asian Social Science 4, 1 (2016): 38-43. Accessed August 22, 2019. ↵
- Ibid. ↵
- W., Mike. " Intrinsic vs. Extrinsic Motivation – Clearing the Fog (not Fogg!) ." Khoros Community. February 11, 2014. Accessed August 22, 2019. ↵
- " Managing Reward: Why Line Managers are the Vital Link. " go2HR. Accessed August 22, 2019. ↵
Contributors and Attributions
- Untitled. Authored by : geralt. Provided by : Pixabay. Located at : pixabay.com/photos/finger-feedback-confirming-3530353/. License : CC0: No Rights Reserved . License Terms : Pixabay License
- Image: Intrinsic vs Extrinsic Motivation. Provided by : OpenStax CNX. Located at : https://cnx.org/contents/Sr8Ev5Og@10.24:MLADqXMi@11/10-1-Motivation . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/4abf04bf-93a...d46e68cc@10.24


26.E: Galaxies (Exercises)
For Further Exploration
Articles
Andrews, B. “What Are Galaxies Trying to Tell Us?” Astronomy (February 2011): 24. Introduction to our understanding of the shapes and evolution of different types of galaxies.
Bothun, G. “Beyond the Hubble Sequence.” Sky & Telescope (May 2000): 36. History and updating of Hubble’s classification scheme.
Christianson, G. “Mastering the Universe.” Astronomy (February 1999): 60. Brief introduction to Hubble’s life and work.
Dalcanton, J. “The Overlooked Galaxies.” Sky & Telescope (April 1998): 28. On low-brightness galaxies, which have been easy to miss.
Freedman, W. “The Expansion Rate and Size of the Universe.” Scientific American (November 1992): 76.
Hodge, P. “The Extragalactic Distance Scale: Agreement at Last?” Sky & Telescope (October 1993): 16.
Jones, B. “The Legacy of Edwin Hubble.” Astronomy (December 1989): 38.
Kaufmann, G. and van den Bosch, F. “The Life Cycle of Galaxies.” Scientific American (June 2002): 46. On galaxy evolution and how it leads to the different types of galaxies.
Martin, P. and Friedli, D. “At the Hearts of Barred Galaxies.” Sky & Telescope (March 1999): 32. On barred spirals.
Osterbrock, D. “Edwin Hubble and the Expanding Universe.” Scientific American (July 1993): 84.
Russell, D. “Island Universes from Wright to Hubble.” Sky & Telescope (January 1999) 56. A history of our discovery of galaxies.
Smith, R. “The Great Debate Revisited.” Sky & Telescope (January 1983): 28. On the Shapley-Curtis debate concerning the extent of the Milky Way and the existence of other galaxies.
Websites
ABC’s of Distance: http://www.astro.ucla.edu/~wright/distance.htm . A concise summary by astronomer Ned Wright of all the different methods we use to get distances in astronomy.
Cosmic Times 1929: http://cosmictimes.gsfc.nasa.gov/onl...mic/index.html . NASA project explaining Hubble’s work and surrounding discoveries as if you were reading newspaper articles.
Edwin Hubble: The Man Behind the Name: https://www.spacetelescope.org/about...hind_the_name/ . Concise biography from the people at the Hubble Space Telescope.
Edwin Hubble: http://apod.nasa.gov/diamond_jubilee...ge_hubble.html . An article on the life and work of Hubble by his student and successor, Allan Sandage. A bit technical in places, but giving a real picture of the man and the science.
NASA Science: Introduction to Galaxies: http://science.nasa.gov/astrophysics...-are-galaxies/ . A brief overview with links to other pages, and recent Hubble Space Telescope discoveries.
National Optical Astronomy Observatories Gallery of Galaxies: www.noao.edu/image_gallery/galaxies.html. A collection of images and information about galaxies and galaxy groups of different types. Another impressive archive can be found at the European Southern Observatory site: https://www.eso.org/public/images/ar...gory/galaxies/ .
Sloan Digital Sky Survey: Introduction to Galaxies: http://skyserver.sdss.org/dr1/en/ast...s/galaxies.asp . Another brief overview.
Universe Expansion: http://hubblesite.org/newscenter/arc...leases/1999/19 . The background material here provides a nice chronology of how we discovered and measured the expansion of the universe.
Videos
Edwin Hubble (Hubblecast Episode 89): http://www.spacetelescope.org/videos/hubblecast89a/ . (5:59).
Galaxies: An Introduction: https://www.youtube.com/watch?v=HYYgangrkZg . A compilation of several short European videos that first describe galaxies in general and then focus on galaxies in Hubble telescope images (12:48).
Hubble’s Views of the Deep Universe: https://www.youtube.com/watch?v=argR2U15w-M . A 2015 public talk by Brandon Lawton of the Space Telescope Science Institute about galaxies and beyond (1:26:20).
Collaborative Group Activities
- Throughout much of the last century, the 100-inch telescope on Mt. Wilson (completed in 1917) and the 200-inch telescope on Palomar Mountain (completed in 1948) were the only ones large enough to obtain spectra of faint galaxies. Only a handful of astronomers (all male—since, until the 1960s, women were not given time on these two telescopes) were allowed to use these facilities, and in general the observers did not compete with each other but worked on different problems. Now there are many other telescopes, and several different groups do often work on the same problem. For example, two different groups have independently developed the techniques for using supernovae to determine the distances to galaxies at high redshifts. Which approach do you think is better for the field of astronomy? Which is more cost effective? Why?
- A distant relative, whom you invite to dinner so you can share all the exciting things you have learned in your astronomy class, says he does not believe that other galaxies are made up of stars. You come back to your group and ask them to help you respond. What kinds of measurements would you make to show that other galaxies are composed of stars?
- Look at the chapter thumbnail with your group. What does the difference in color between the spiral arms and the bulge of Andromeda tell you about the difference in the types of stars that populate these two regions of the galaxy? Which side of the galaxy is closer to us? Why?
- What is your reaction to reading about the discovery of the expanding universe? Discuss how the members of the group feel about a universe “in motion.” Einstein was not comfortable with the notion of a universe that had some overall movement to it, instead of being at rest. He put a kind of “fudge factor” into his equations of general relativity for the universe as a whole to keep it from moving (although later, hearing about Hubble and Humason’s work, he called it “the greatest blunder” he ever made). Do you share Einstein’s original sense that this is not the kind of universe you feel comfortable with? What do you think could have caused space to be expanding?
- In science fiction, characters sometimes talk about visiting other galaxies. Discuss with your group how realistic this idea is. Even if we had fast spaceships (traveling close to the speed of light, the speed limit of the universe) how likely are we to be able to reach another galaxy? Why?
- Despite his son’s fascination with astronomy in college, Edwin Hubble’s father did not want him to go into astronomy as a profession. He really wanted his son to be a lawyer and pushed him hard to learn the law when he won a fellowship to study abroad. Hubble eventually defied his father and went into astronomy, becoming, as you learned in this chapter, one of the most important astronomers of all time. His dad didn’t live to see his son’s remarkable achievements. Do you think he would have reconciled himself to his son’s career choice if he had? Do you or does anyone in your group or among your friends have to face a choice between the passion in your heart and what others want you to do? Discuss how people in college today are dealing with such choices.
Review Questions
- Describe the main distinguishing features of spiral, elliptical, and irregular galaxies.
- Why did it take so long for the existence of other galaxies to be established?
- Explain what the mass-to-light ratio is and why it is smaller in spiral galaxies with regions of star formation than in elliptical galaxies.
- If we now realize dwarf ellipticals are the most common type of galaxy, why did they escape our notice for so long?
- What are the two best ways to measure the distance to a nearby spiral galaxy, and how would it be measured?
- What are the two best ways to measure the distance to a distant, isolated spiral galaxy, and how would it be measured?
- Why is Hubble’s law considered one of the most important discoveries in the history of astronomy?
- What does it mean to say that the universe is expanding? What is expanding? For example, is your astronomy classroom expanding? Is the solar system? Why or why not?
- Was Hubble’s original estimate of the distance to the Andromeda galaxy correct? Explain.
- Does an elliptical galaxy rotate like a spiral galaxy? Explain.
- Why does the disk of a spiral galaxy appear dark when viewed edge on?
- What causes the largest mass-to-light ratio: gas and dust, dark matter, or stars that have burnt out?
- What is the most useful standard bulb method for determining distances to galaxies?
- When comparing two isolated spiral galaxies that have the same apparent brightness, but rotate at different rates, what can you say about their relative luminosity?
- If all distant galaxies are expanding away from us, does this mean we’re at the center of the universe?
- Is the Hubble constant actually constant?
Thought Questions
- Where might the gas and dust (if any) in an elliptical galaxy come from?
- Why can we not determine distances to galaxies by the same method used to measure the parallaxes of stars?
- Which is redder—a spiral galaxy or an elliptical galaxy?
- Suppose the stars in an elliptical galaxy all formed within a few million years shortly after the universe began. Suppose these stars have a range of masses, just as the stars in our own galaxy do. How would the color of the elliptical change over the next several billion years? How would its luminosity change? Why?
- Starting with the determination of the size of Earth, outline a sequence of steps necessary to obtain the distance to a remote cluster of galaxies. (Hint: Review the chapter on Celestial Distances.)
- Suppose the Milky Way Galaxy were truly isolated and that no other galaxies existed within 100 million light-years. Suppose that galaxies were observed in larger numbers at distances greater than 100 million light-years. Why would it be more difficult to determine accurate distances to those galaxies than if there were also galaxies relatively close by?
- Suppose you were Hubble and Humason, working on the distances and Doppler shifts of the galaxies. What sorts of things would you have to do to convince yourself (and others) that the relationship you were seeing between the two quantities was a real feature of the behavior of the universe? (For example, would data from two galaxies be enough to demonstrate Hubble’s law? Would data from just the nearest galaxies—in what astronomers call “the Local Group”—suffice?)
- What does it mean if one elliptical galaxy has broader spectrum lines than another elliptical galaxy?
- Based on your analysis of galaxies in Table \(26.3.1\) in Section 26.3, is there a correlation between the population of stars and the quantity of gas or dust? Explain why this might be.
- Can a higher mass-to-light ratio mean that there is gas and dust present in the system that is being analyzed?
Figuring for Yourself
- According to Hubble’s law, what is the recessional velocity of a galaxy that is \(10^8\) light-years away from us? (Assume a Hubble constant of 22 km/s per million light-years.)
- A cluster of galaxies is observed to have a recessional velocity of 60,000 km/s. Find the distance to the cluster. (Assume a Hubble constant of 22 km/s per million light-years.)
- Suppose we could measure the distance to a galaxy using one of the distance techniques listed in Table \(26.4.1\) in Section 26.4 and it turns out to be 200 million light-years. The galaxy’s redshift tells us its recessional velocity is 5000 km/s. What is the Hubble constant?
- Calculate the mass-to-light ratio for a globular cluster with a luminosity of 106 \(L_{\text{Sun}}\) and \(10^5\) stars. (Assume that the average mass of a star in such a cluster is 1 \(M_{\text{Sun}}\).)
- Calculate the mass-to-light ratio for a luminous star of 100 \(M_{\text{Sun}}\) having the luminosity of \(10^6\) \(L_{\text{Sun}}\).


9: Advanced Theories of Covalent Bonding
We have examined the basic ideas of bonding, showing that atoms share electrons to form molecules with stable Lewis structures and that we can predict the shapes of those molecules by valence shell electron pair repulsion (VSEPR) theory. These ideas provide an important starting point for understanding chemical bonding. But these models sometimes fall short in their abilities to predict the behavior of real substances. How can we reconcile the geometries of s, p, and d atomic orbitals with molecular shapes that show angles like 120° and 109.5°? Furthermore, we know that electrons and magnetic behavior are related through electromagnetic fields.
-
- 9.1: Valence Bond Theory and Hybrid Orbitals
- Valence bond theory describes bonding as a consequence of the overlap of two separate atomic orbitals on different atoms that creates a region with one pair of electrons shared between the two atoms. When the orbitals overlap along an axis containing the nuclei, they form a σ bond. When they overlap in a fashion that creates a node along this axis, they form a π bond.
-
- 9.2: Multiple Bonds
- Multiple bonds consist of a σ bond located along the axis between two atoms and one or two π bonds. The σ bonds are usually formed by the overlap of hybridized atomic orbitals, while the π bonds are formed by the side-by-side overlap of unhybridized orbitals. Resonance occurs when there are multiple unhybridized orbitals with the appropriate alignment to overlap, so the placement of π bonds can vary.
-
- 9.3: Molecular Orbital Theory
- Molecular orbital (MO) theory describes the behavior of electrons in a molecule in terms of combinations of the atomic wavefunctions. The resulting molecular orbitals may extend over all the atoms in the molecule. Bonding molecular orbitals are formed by in-phase combinations of atomic wavefunctions, and electrons in these orbitals stabilize a molecule. Antibonding molecular orbitals result from out-of-phase combinations and electrons in these orbitals make a molecule less stable.
Contributors and Attributions
-
Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).


19.10: Summary
-
- Last updated
- Save as PDF
The first half of this chapter was focused primarily on the theoretical underpinnings of Bayesian statistics. I introduced the mathematics for how Bayesian inference works (Section 17.1), and gave a very basic overview of how Bayesian hypothesis testing is typically done (Section 17.2). Finally, I devoted some space to talking about why I think Bayesian methods are worth using (Section 17.3.
The second half of the chapter was a lot more practical, and focused on tools provided by the
BayesFactor
package. Specifically, I talked about using the
contingencyTableBF()
function to do Bayesian analogs of chi-square tests (Section 17.6, the
ttestBF()
function to do Bayesian t-tests, (Section 17.7), the
regressionBF()
function to do Bayesian regressions, and finally the
anovaBF()
function for Bayesian ANOVA.
If you’re interested in learning more about the Bayesian approach, there are many good books you could look into. John Kruschke’s book Doing Bayesian Data Analysis is a pretty good place to start (Kruschke 2011), and is a nice mix of theory and practice. His approach is a little different to the “Bayes factor” approach that I’ve discussed here, so you won’t be covering the same ground. If you’re a cognitive psychologist, you might want to check out Michael Lee and E.J. Wagenmakers’ book Bayesian Cognitive Modeling (Lee and Wagenmakers 2014). I picked these two because I think they’re especially useful for people in my discipline, but there’s a lot of good books out there, so look around!
References
Jeffreys, Harold. 1961. The Theory of Probability . 3rd ed. Oxford.
Kass, Robert E., and Adrian E. Raftery. 1995. “Bayes Factors.” Journal of the American Statistical Association 90: 773–95.
Fisher, R. 1925. Statistical Methods for Research Workers . Edinburgh, UK: Oliver; Boyd.
Johnson, Valen E. 2013. “Revised Standards for Statistical Evidence.” Proceedings of the National Academy of Sciences , no. 48: 19313–7.
Morey, Richard D., and Jeffrey N. Rouder. 2015. BayesFactor: Computation of Bayes Factors for Common Designs . http://CRAN.R-project.org/package=BayesFactor .
Gunel, Erdogan, and James Dickey. 1974. “Bayes Factors for Independence in Contingency Tables.” Biometrika , 545–57.
Kruschke, J. K. 2011. Doing Bayesian Data Analysis: A Tutorial with R and BUGS . Burlington, MA: Academic Press.
Lee, Michael D, and Eric-Jan Wagenmakers. 2014. Bayesian Cognitive Modeling: A Practical Course . Cambridge University Press.
- http://en.wikiquote.org/wiki/David_Hume
- http://en.Wikipedia.org/wiki/Climate_of_Adelaide
- It’s a leap of faith, I know, but let’s run with it okay?
- Um. I hate to bring this up, but some statisticians would object to me using the word “likelihood” here. The problem is that the word “likelihood” has a very specific meaning in frequentist statistics, and it’s not quite the same as what it means in Bayesian statistics. As far as I can tell, Bayesians didn’t originally have any agreed upon name for the likelihood, and so it became common practice for people to use the frequentist terminology. This wouldn’t have been a problem, except for the fact that the way that Bayesians use the word turns out to be quite different to the way frequentists do. This isn’t the place for yet another lengthy history lesson, but to put it crudely: when a Bayesian says “ a likelihood function” they’re usually referring one of the rows of the table. When a frequentist says the same thing, they’re referring to the same table, but to them “ a likelihood function” almost always refers to one of the columns . This distinction matters in some contexts, but it’s not important for our purposes.
- If we were being a bit more sophisticated, we could extend the example to accommodate the possibility that I’m lying about the umbrella. But let’s keep things simple, shall we?
- You might notice that this equation is actually a restatement of the same basic rule I listed at the start of the last section. If you multiply both sides of the equation by P(d), then you get P(d)P(h|d)=P(d,h), which is the rule for how joint probabilities are calculated. So I’m not actually introducing any “new” rules here, I’m just using the same rule in a different way.
- Obviously, this is a highly simplified story. All the complexity of real life Bayesian hypothesis testing comes down to how you calculate the likelihood P(d|h) when the hypothesis h is a complex and vague thing. I’m not going to talk about those complexities in this book, but I do want to highlight that although this simple story is true as far as it goes, real life is messier than I’m able to cover in an introductory stats textbook.
- http://www.imdb.com/title/tt0093779/quotes . I should note in passing that I’m not the first person to use this quote to complain about frequentist methods. Rich Morey and colleagues had the idea first. I’m shamelessly stealing it because it’s such an awesome pull quote to use in this context and I refuse to miss any opportunity to quote The Princess Bride .
- http://about.abc.net.au/reports-publications/appreciation-survey-summary-report-2013/
- http://knowyourmeme.com/memes/the-cake-is-a-lie
- In the interests of being completely honest, I should acknowledge that not all orthodox statistical tests that rely on this silly assumption. There are a number of sequential analysis tools that are sometimes used in clinical trials and the like. These methods are built on the assumption that data are analysed as they arrive, and these tests aren’t horribly broken in the way I’m complaining about here. However, sequential analysis methods are constructed in a very different fashion to the “standard” version of null hypothesis testing. They don’t make it into any introductory textbooks, and they’re not very widely used in the psychological literature. The concern I’m raising here is valid for every single orthodox test I’ve presented so far, and for almost every test I’ve seen reported in the papers I read.
- A related problem: http://xkcd.com/1478/
- Some readers might wonder why I picked 3:1 rather than 5:1, given that Johnson (2013) suggests that p=.05 lies somewhere in that range. I did so in order to be charitable to the p-value. If I’d chosen a 5:1 Bayes factor instead, the results would look even better for the Bayesian approach.
- http://www.quotationspage.com/quotes/Ambrosius_Macrobius/
- Okay, I just know that some knowledgeable frequentists will read this and start complaining about this section. Look, I’m not dumb. I absolutely know that if you adopt a sequential analysis perspective you can avoid these errors within the orthodox framework. I also know that you can explictly design studies with interim analyses in mind. So yes, in one sense I’m attacking a “straw man” version of orthodox methods. However, the straw man that I’m attacking is the one that is used by almost every single practitioner . If it ever reaches the point where sequential methods become the norm among experimental psychologists and I’m no longer forced to read 20 extremely dubious ANOVAs a day, I promise I’ll rewrite this section and dial down the vitriol. But until that day arrives, I stand by my claim that default Bayes factor methods are much more robust in the face of data analysis practices as they exist in the real world. Default orthodox methods suck, and we all know it.
-
If you’re desperate to know, you can find all the gory details in Gunel and Dickey (1974). However, that’s a pretty technical paper. The help documentation to the
contingencyTableBF()
gives this explanation: “the argumentpriorConcentration
indexes the expected deviation from the null hypothesis under the alternative, and corresponds to Gunel and Dickey’s (1974) a parameter.” As I write this I’m about halfway through the Gunel and Dickey paper, and I agree that setting a=1 is a pretty sensible default choice, since it corresponds to an assumption that you have very little a priori knowledge about the contingency table. -
In some of the later examples, you’ll see that this number is not always 0%. This is because the
BayesFactor
package often has to run some simulations to compute approximate Bayes factors. So the answers you get won’t always be identical when you run the command a second time. That’s why the output of these functions tells you what the margin for error is. - Apparently this omission is deliberate. I have this vague recollection that I spoke to Jeff Rouder about this once, and his opinion was that when homogeneity of variance is violated the results of a t-test are uninterpretable. I can see the argument for this, but I’ve never really held a strong opinion myself. (Jeff, if you never said that, I’m sorry)
- Just in case you’re interested: the “JZS” part of the output relates to how the Bayesian test expresses the prior uncertainty about the variance σ2, and it’s short for the names of three people: “Jeffreys Zellner Siow”. See Rouder et al. (2009) for details.
-
Again, in case you care … the null hypothesis here specifies an effect size of 0, since the two means are identical. The alternative hypothesis states that there
is
an effect, but it doesn’t specify exactly how big the effect will be. The r value here relates to how big the effect is expected to be according to the alternative. You can type
?ttestBF
to get more details. - Again, guys, sorry if I’ve misread you.
-
I don’t even disagree with them: it’s
not
at all obvious why a Bayesian ANOVA should reproduce (say) the same set of model comparisons that the Type II testing strategy uses. It’s precisely because of the fact that I haven’t really come to any strong conclusions that I haven’t added anything to the
lsr
package to make Bayesian Type II tests easier to produce.


1.3: Applications of Sets
A Venn diagram is a useful tool in showing the relationship between the actual elements of sets.
Consider the universal set \(U=\{1,2,3,4,5,6,8\}\).
Place the elements in sets \(A=\{3,6,8\}\) and \(B=\{1,2,6,3\}\) in the Venn diagram.
Solution
All elements in the universal set \(U\) must appear in one of the 4 regions labeled I, II, III, and IV. We must decide where each element 1, 2, 3, 4, 5, 6, and 8 go.
- The element 1 appears in set \(B\) but not in set \(A\). It should be placed in III.
- The element 2 appears in set \(B\) but not in set \(A\). It should be placed in III.
- The element 3 appears in both set \(A\) and in set \(B\). It should be placed in II.
- The elements 4 and 5 appear in neither set \(A\) nor set \(B\). They should be placed in IV.
- The element 6 appears in both set \(A\) and in set \(B\). It should be placed in II.
- The element 8 appears in set \(A\) but not in set \(B\). It should be placed in I.
The completed Venn diagram is shown at the right.
Construct a Venn diagram illustrating the following sets:
\(U=\{a,b,c,d,e,f,g,h,i,j\}\) \(A=\{c,d,e,g,h,i\}\) \(B=\{a,c,d,g\}\) \(C=\{c,f,i,j\}\)
Solution
All elements in the universal set \(U\) must appear in one of 8 regions of a three-set Venn diagram.
- The element a belongs to set \(B\) but not to either of the other sets.
- The element b does not belong to any of the 3 sets but it is part of the universal set.
- The element c belongs to all three sets.
- The element d belongs to set \(A\) and set \(B\), but not to set \(C\).
- The element e belongs to set \(A\) but not to either of the other sets.
- The element f belongs to set \(C\) but not to either of the other sets.
- The element g belongs to set \(A\) and set \(B\), but not to set \(C\).
- The element h belongs to set \(A\) but not to either of the other sets.
- The element i belongs to set \(A\) and set \(C\), but not to set \(B\).
- The element j belongs to set \(C\) but not to either of the other sets.
The completed Venn diagram is shown at the right.
Sometimes we may be interested in the cardinality of the union or intersection of sets, but not know the actual elements of each set. This is common in application problems involving surveys.
A survey asks 200 people What beverage do you drink in the morning? , and offers these choices:
- Tea only
- Coffee only
- Both coffee and tea
Suppose 20 report tea only, 80 report coffee only, 40 report both. How many people drink tea in the morning? How many people drink neither tea or coffee?
Solution
This question can most easily be answered by creating a Venn diagram. We can see that we can find the number of people who drink tea by adding those who drink only tea to those who drink both: 60 people.
We can also see that those who drink neither are those not contained in the any of the three other groupings, so we can count those by subtracting from the cardinality of the universal set, 200:
So we have \(200-20-80-40=60\) people drink neither.
At a school of 500 students, there are 125 students in the drama club, 257 students who play sports, and 52 students who are in the drama club and play sports. Use a Venn diagram to illustrate this information. Then, determine how many students
- play sports but are not in the drama club
- play sports or are in the drama club or both
A survey asks Which online services have you used in the last month?
- Have used both
The results show 40% of those surveyed have used Twitter, 70% have used Facebook, and 20% have used both. How many people have used neither Twitter or Facebook?
Solution
Let \(T\) be the set of all people who have used Twitter and \(F\) be the set of all people who have used Facebook. Notice that while the cardinality of \(F\) is \(70 \%\) and the cardinality of \(T\) is \(40 \%\), the cardinality of \(F \cup T\) is not simply \(70 \%+40 \%\), since that would count those who use both services twice. To find the cardinality of \(F \cup T\), we add the cardinality of \(F\) and the cardinality of \(T\), and then subtract those in intersection that we've counted twice.
In symbols,
\(n(F \cup T)=n(F)+n(T)-n(F \cap T)\)
\(n(F \cup T)=70 \%+40 \%-20 \%\)
\(n(F \cup T)=90 \%\)
Now, to find how many people have not used either service, we're looking for the cardinality of \((F \cup T)^{\prime}\). Since the universal set contains \(100 \%\) of people and the cardinality of \(F \cup T=90 \%\), the cardinality of \((F \cup T)^{\prime}\) must be the other \(10 \%\).
The previous example illustrates two important properties.
\(n(A \cup B)=n(A)+n(B)-n(A \cap B)\)
\(n(A^{\prime})=n(U)-n(A)\)
Notice that the first property can also be written in an equivalent form by solving for the cardinality of the intersection:
\(n(A \cap B)=n(A)+n(B)-n(A \cup B)\)
In a group of 120 students, 35 were enrolled in art class and 57 were enrolled in dance class. There were 12 students who were enrolled in both classes. How many students were enrolled in neither of the classes?
- Answer
-
40
Fifty students were surveyed and asked if they were taking a social science (SS), humanities (HM) or a natural science (NS) course the next quarter.
\(\begin{array}{ll} \text{21 were taking a SS course} & \text{26 were taking a HM course} \\ \text{19 were taking a NS course} & \text{9 were taking SS and HM} \\ \text{7 were taking SS and NS} & \text{10 were taking HM and NS} \\ \text{3 were taking all three} & \text{7 were taking none} \end{array}\)
How many students are taking only a SS course?
Solution
It might help to look at a Venn diagram and complete the cardinality for each of the 8 regions of the diagram. The question here is how many elements are in region \(a\).
From the given data, we know that there are 3 students in region \(e\) and 7 students in region \(h\).
Because 7 students were taking a \(SS\) and \(NS\) course, we know that \(n(d)+n(e)=7\). Since we know there are 3 students in region \(e\), there must be \(7-3=4\) students in region \(d\).
Similarly, since there are 10 students taking \(HM\) and \(NS\), which includes regions \(e\) and \(f\), there must be \(10-3=7\) students in region \(f\).
Since 9 students were taking \(SS\) and \(HM\), there must be \(9-3=6\) students in region \(b\).
Now, we know that 21 students were taking a \(SS\) course. This includes students from regions \(a, b, d,\) and \(e .\) Since we know the number of students in all but region \(a,\) we can determine that \(21-6-4-3=8\) students are in region \(a\).
There are 8 students are taking only a \(SS\) course.
One hundred fifty people were surveyed and asked if they believed in UFOs, ghosts, and Bigfoot.
\(\begin{array}{ll} \text{43 believed in UFOs} & \text{44 believed in ghosts} \\ \text{25 believed in Bigfoot} & \text{10 believed in UFOs and ghosts} \\ \text{8 believed in ghosts and Bigfoot} & \text{5 believed in UFOs and Bigfoot} \\ \text{2 believed in all three} & \text{} \end{array}\)
How many people surveyed believed in none of these things?
- Answer
-
Starting with the intersection of all three circles, we work our way out. Since 10 people believe in UFOs and Ghosts, and 2 believe in all three, that leaves 8 that believe in only UFOs and Ghosts. Continue filling in all the regions. Once the cardinality for each region in the circles is completed, add the cardinalities to get 91 people in the union of all three sets. This leaves \(150-91=59\) people who believe in none.


3.5: Drawing Lewis Structures
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Learning Objectives
- To use Lewis dot symbols to explain the stoichiometry of a compound
We begin our discussion of the relationship between structure and bonding in covalent compounds by describing the interaction between two identical neutral atoms—for example, the H 2 molecule, which contains a purely covalent bond. Each hydrogen atom in H 2 contains one electron and one proton, with the electron attracted to the proton by electrostatic forces. As the two hydrogen atoms are brought together, additional interactions must be considered (Figure \(\PageIndex{1}\)):
- The electrons in the two atoms repel each other because they have the same charge (
- The electrons in the two atoms repel each other because they have the same charge (E > 0).
- Similarly, the protons in adjacent atoms repel each other ( E > 0).
- The electron in one atom is attracted to the oppositely charged proton in the other atom and vice versa ( E < 0). Recall that it is impossible to specify precisely the position of the electron in either hydrogen atom. Hence the quantum mechanical probability distributions must be used.
A plot of the potential energy of the system as a function of the internuclear distance (Figure \(\PageIndex{2}\)) shows that the system becomes more stable (the energy of the system decreases) as two hydrogen atoms move toward each other from r = ∞, until the energy reaches a minimum at r = r 0 (the observed internuclear distance in H 2 is 74 pm). Thus at intermediate distances, proton–electron attractive interactions dominate, but as the distance becomes very short, electron–electron and proton–proton repulsive interactions cause the energy of the system to increase rapidly. Notice the similarity between Figures \(\PageIndex{1}\) and \(\PageIndex{2}\), which described a system containing two oppositely charged ions . The shapes of the energy versus distance curves in the two figures are similar because they both result from attractive and repulsive forces between charged entities.
At long distances, both attractive and repulsive interactions are small. As the distance between the atoms decreases, the attractive electron–proton interactions dominate, and the energy of the system decreases. At the observed bond distance, the repulsive electron–electron and proton–proton interactions just balance the attractive interactions, preventing a further decrease in the internuclear distance. At very short internuclear distances, the repulsive interactions dominate, making the system less stable than the isolated atoms.
Using Lewis Dot Symbols to Describe Covalent Bonding
The valence electron configurations of the constituent atoms of a covalent compound are important factors in determining its structure, stoichiometry, and properties. For example, chlorine, with seven valence electrons, is one electron short of an octet. If two chlorine atoms share their unpaired electrons by making a covalent bond and forming Cl 2 , they can each complete their valence shell:
Each chlorine atom now has an octet. The electron pair being shared by the atoms is called a bonding pair; the other three pairs of electrons on each chlorine atom are called lone pairs. Lone pairs are not involved in covalent bonding. If both electrons in a covalent bond come from the same atom, the bond is called a coordinate covalent bond. Examples of this type of bonding are presented in Section 8.6 when we discuss atoms with less than an octet of electrons.
We can illustrate the formation of a water molecule from two hydrogen atoms and an oxygen atom using Lewis dot symbols:
The structure on the right is the Lewis electron structure , or Lewis structure , for H 2 O. With two bonding pairs and two lone pairs, the oxygen atom has now completed its octet. Moreover, by sharing a bonding pair with oxygen, each hydrogen atom now has a full valence shell of two electrons. Chemists usually indicate a bonding pair by a single line, as shown here for our two examples:
The following procedure can be used to construct Lewis electron structures for more complex molecules and ions:
- Arrange the atoms to show specific connections. When there is a central atom, it is usually the least electronegative element in the compound. Chemists usually list this central atom first in the chemical formula (as in CCl 4 and CO 3 2 − , which both have C as the central atom), which is another clue to the compound’s structure. Hydrogen and the halogens are almost always connected to only one other atom, so they are usually terminal rather than central.
- Determine the total number of valence electrons in the molecule or ion. Add together the valence electrons from each atom. (Recall that the number of valence electrons is indicated by the position of the element in the periodic table.) If the species is a polyatomic ion, remember to add or subtract the number of electrons necessary to give the total charge on the ion. For CO 3 2 − , for example, we add two electrons to the total because of the −2 charge.
- Place a bonding pair of electrons between each pair of adjacent atoms to give a single bond. In H 2 O, for example, there is a bonding pair of electrons between oxygen and each hydrogen.
- Beginning with the terminal atoms, add enough electrons to each atom to give each atom an octet (two for hydrogen). These electrons will usually be lone pairs.
- If any electrons are left over, place them on the central atom. We will explain later that some atoms are able to accommodate more than eight electrons.
- If the central atom has fewer electrons than an octet, use lone pairs from terminal atoms to form multiple (double or triple) bonds to the central atom to achieve an octet. This will not change the number of electrons on the terminal atoms.
Now let’s apply this procedure to some particular compounds, beginning with one we have already discussed.
The central atom is usually the least electronegative element in the molecule or ion; hydrogen and the halogens are usually terminal.
The \(H_2O\) Molecule
- Because H atoms are almost always terminal, the arrangement within the molecule must be HOH .
- Each H atom (group 1) has 1 valence electron, and the O atom (group 16) has 6 valence electrons, for a total of 8 valence electrons.
- Placing one bonding pair of electrons between the O atom and each H atom gives H:O:H, with 4 electrons left over.
- Each H atom has a full valence shell of 2 electrons.
- Adding the remaining 4 electrons to the oxygen (as two lone pairs) gives the following structure:
This is the Lewis structure we drew earlier. Because it gives oxygen an octet and each hydrogen two electrons, we do not need to use step 6.
The \(OCl^−\) Ion
- With only two atoms in the molecule, there is no central atom.
- Oxygen (group 16) has 6 valence electrons, and chlorine (group 17) has 7 valence electrons; we must add one more for the negative charge on the ion, giving a total of 14 valence electrons.
- Placing a bonding pair of electrons between O and Cl gives O:Cl, with 12 electrons left over.
- If we place six electrons (as three lone pairs) on each atom, we obtain the following structure:
Each atom now has an octet of electrons, so steps 5 and 6 are not needed. The Lewis electron structure is drawn within brackets as is customary for an ion, with the overall charge indicated outside the brackets, and the bonding pair of electrons is indicated by a solid line. OCl − is the hypochlorite ion, the active ingredient in chlorine laundry bleach and swimming pool disinfectant.
The \(CH_2O\) Molecule
1. Because carbon is less electronegative than oxygen and hydrogen is normally terminal, C must be the central atom. One possible arrangement is as follows:
2. Each hydrogen atom (group 1) has one valence electron, carbon (group 14) has 4 valence electrons, and oxygen (group 16) has 6 valence electrons, for a total of [(2)(1) + 4 + 6] = 12 valence electrons.
3. Placing a bonding pair of electrons between each pair of bonded atoms gives the following:
Six electrons are used, and 6 are left over.
4. Adding all 6 remaining electrons to oxygen (as three lone pairs) gives the following:
Although oxygen now has an octet and each hydrogen has 2 electrons, carbon has only 6 electrons.
5. There are no electrons left to place on the central atom.
6. To give carbon an octet of electrons, we use one of the lone pairs of electrons on oxygen to form a carbon–oxygen double bond:
Both the oxygen and the carbon now have an octet of electrons, so this is an acceptable Lewis electron structure. The O has two bonding pairs and two lone pairs, and C has four bonding pairs. This is the structure of formaldehyde, which is used in embalming fluid.
An alternative structure can be drawn with one H bonded to O. Formal charges , discussed later in this section, suggest that such a structure is less stable than that shown previously.
Example \(\PageIndex{1}\)
Write the Lewis electron structure for each species.
- NCl 3
- S 2 2 −
- NOCl
Given: chemical species
Asked for: Lewis electron structures
Strategy:
Use the six-step procedure to write the Lewis electron structure for each species.
Solution:
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Nitrogen is less electronegative than chlorine, and halogen atoms are usually terminal, so nitrogen is the central atom. The nitrogen atom (group 15) has 5 valence electrons and each chlorine atom (group 17) has 7 valence electrons, for a total of 26 valence electrons. Using 2 electrons for each N–Cl bond and adding three lone pairs to each Cl account for (3 × 2) + (3 × 2 × 3) = 24 electrons. Rule 5 leads us to place the remaining 2 electrons on the central N:
Nitrogen trichloride is an unstable oily liquid once used to bleach flour; this use is now prohibited in the United States.
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In a diatomic molecule or ion, we do not need to worry about a central atom. Each sulfur atom (group 16) contains 6 valence electrons, and we need to add 2 electrons for the −2 charge, giving a total of 14 valence electrons. Using 2 electrons for the S–S bond, we arrange the remaining 12 electrons as three lone pairs on each sulfur, giving each S atom an octet of electrons:
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Because nitrogen is less electronegative than oxygen or chlorine, it is the central atom. The N atom (group 15) has 5 valence electrons, the O atom (group 16) has 6 valence electrons, and the Cl atom (group 17) has 7 valence electrons, giving a total of 18 valence electrons. Placing one bonding pair of electrons between each pair of bonded atoms uses 4 electrons and gives the following:
Adding three lone pairs each to oxygen and to chlorine uses 12 more electrons, leaving 2 electrons to place as a lone pair on nitrogen:
Because this Lewis structure has only 6 electrons around the central nitrogen, a lone pair of electrons on a terminal atom must be used to form a bonding pair. We could use a lone pair on either O or Cl. Because we have seen many structures in which O forms a double bond but none with a double bond to Cl, it is reasonable to select a lone pair from O to give the following:
All atoms now have octet configurations. This is the Lewis electron structure of nitrosyl chloride, a highly corrosive, reddish-orange gas.
Exercise \(\PageIndex{1}\)
Write Lewis electron structures for CO 2 and SCl 2 , a vile-smelling, unstable red liquid that is used in the manufacture of rubber.
Answer
Using Lewis Electron Structures to Explain Stoichiometry
Lewis dot symbols provide a simple rationalization of why elements form compounds with the observed stoichiometries. In the Lewis model, the number of bonds formed by an element in a neutral compound is the same as the number of unpaired electrons it must share with other atoms to complete its octet of electrons. For the elements of Group 17 (the halogens) , this number is one; for the elements of Group 16 (the chalcogens) , it is two; for Group 15 elements , three; and for Group 14 elements four. These requirements are illustrated by the following Lewis structures for the hydrides of the lightest members of each group:
Elements may form multiple bonds to complete an octet. In ethylene, for example, each carbon contributes two electrons to the double bond, giving each carbon an octet (two electrons/bond × four bonds = eight electrons). Neutral structures with fewer or more bonds exist, but they are unusual and violate the octet rule.
Allotropes of an element can have very different physical and chemical properties because of different three-dimensional arrangements of the atoms; the number of bonds formed by the component atoms, however, is always the same. As noted at the beginning of the chapter, diamond is a hard, transparent solid; graphite is a soft, black solid; and the fullerenes have open cage structures. Despite these differences, the carbon atoms in all three allotropes form four bonds, in accordance with the octet rule.
Lewis structures explain why the elements of groups 14–17 form neutral compounds with four, three, two, and one bonded atom(s), respectively.
Elemental phosphorus also exists in three forms: white phosphorus, a toxic, waxy substance that initially glows and then spontaneously ignites on contact with air; red phosphorus, an amorphous substance that is used commercially in safety matches, fireworks, and smoke bombs; and black phosphorus, an unreactive crystalline solid with a texture similar to graphite (Figure \(\PageIndex{3}\)). Nonetheless, the phosphorus atoms in all three forms obey the octet rule and form three bonds per phosphorus atom.
Formal Charges
It is sometimes possible to write more than one Lewis structure for a substance that does not violate the octet rule, as we saw for CH 2 O, but not every Lewis structure may be equally reasonable. In these situations, we can choose the most stable Lewis structure by considering the formal charge on the atoms, which is the difference between the number of valence electrons in the free atom and the number assigned to it in the Lewis electron structure. The formal charge is a way of computing the charge distribution within a Lewis structure; the sum of the formal charges on the atoms within a molecule or an ion must equal the overall charge on the molecule or ion. A formal charge does not represent a true charge on an atom in a covalent bond but is simply used to predict the most likely structure when a compound has more than one valid Lewis structure.
To calculate formal charges, we assign electrons in the molecule to individual atoms according to these rules:
- Nonbonding electrons are assigned to the atom on which they are located.
- Bonding electrons are divided equally between the bonded atoms.
For each atom, we then compute a formal charge:
\( \begin{matrix}
formal\; charge= & valence\; e^{-}- & \left ( non-bonding\; e^{-}+\frac{bonding\;e^{-}}{2} \right )\\
& ^{\left ( free\; atom \right )} & ^{\left ( atom\; in\; Lewis\; structure \right )}
\end{matrix} \label{8.5.1} \) (atom in Lewis structure)
To illustrate this method, let’s calculate the formal charge on the atoms in ammonia (NH 3 ) whose Lewis electron structure is as follows:
A neutral nitrogen atom has five valence electrons (it is in group 15). From its Lewis electron structure, the nitrogen atom in ammonia has one lone pair and shares three bonding pairs with hydrogen atoms, so nitrogen itself is assigned a total of five electrons [2 nonbonding e − + (6 bonding e − /2)]. Substituting into Equation \(\ref{8.5.2}\), we obtain
\[ formal\; charge\left ( N \right )=5\; valence\; e^{-}-\left ( 2\; non-bonding\; e^{-} +\dfrac{6\; bonding\; e^{-}}{2} \right )=0 \label{8.5.2} \]
A neutral hydrogen atom has one valence electron. Each hydrogen atom in the molecule shares one pair of bonding electrons and is therefore assigned one electron [0 nonbonding e − + (2 bonding e − /2)]. Using Equation \(\ref{8.5.2}\) to calculate the formal charge on hydrogen, we obtain
\[ formal\; charge\left ( H \right )=1\; valence\; e^{-}-\left ( 0\; non-bonding\; e^{-} +\dfrac{2\; bonding\; e^{-}}{2} \right )=0 \label{8.5.3} \]
The hydrogen atoms in ammonia have the same number of electrons as neutral hydrogen atoms, and so their formal charge is also zero. Adding together the formal charges should give us the overall charge on the molecule or ion. In this example, the nitrogen and each hydrogen has a formal charge of zero. When summed the overall charge is zero, which is consistent with the overall charge on the NH 3 molecule.
An atom, molecule, or ion has a formal charge of zero if it has the number of bonds that is typical for that species.
Typically, the structure with the most charges on the atoms closest to zero is the more stable Lewis structure. In cases where there are positive or negative formal charges on various atoms, stable structures generally have negative formal charges on the more electronegative atoms and positive formal charges on the less electronegative atoms. The next example further demonstrates how to calculate formal charges.
Example \(\PageIndex{2}\): The Ammonium Ion
Calculate the formal charges on each atom in the NH 4 + ion.
Given: chemical species
Asked for: formal charges
Strategy:
Identify the number of valence electrons in each atom in the NH 4 + ion. Use the Lewis electron structure of NH 4 + to identify the number of bonding and nonbonding electrons associated with each atom and then use Equation \(\ref{8.5.2}\) to calculate the formal charge on each atom.
Solution:
The Lewis electron structure for the NH 4 + ion is as follows:
The nitrogen atom shares four bonding pairs of electrons, and a neutral nitrogen atom has five valence electrons. Using Equation \(\ref{8.5.1}\), the formal charge on the nitrogen atom is therefore
\[ formal\; charge\left ( N \right )=5-\left ( 0+\dfrac{8}{2} \right )=0 \nonumber \]
Each hydrogen atom in has one bonding pair. The formal charge on each hydrogen atom is therefore
\[ formal\; charge\left ( H \right )=1-\left ( 0+\dfrac{2}{2} \right )=0 \nonumber \]
The formal charges on the atoms in the NH 4 + ion are thus
Adding together the formal charges on the atoms should give us the total charge on the molecule or ion. In this case, the sum of the formal charges is 0 + 1 + 0 + 0 + 0 = +1.
Exercise \(\PageIndex{2}\)
Write the formal charges on all atoms in BH 4 − .
Answer
If an atom in a molecule or ion has the number of bonds that is typical for that atom (e.g., four bonds for carbon), its formal charge is zero.


18.1: On the Equality of the Sexes
TO THE
EDITORS
OF THE
MASSACHUSETTS MAGAZINE,
GENTLEMEN,
The following E SSAY is yielded to the patronage of Candour.—If it hath been anticipated, the testimony of many respectable persons, who saw it in manuscripts as early as the year 1779, can obviate the imputation of plagiarism.
|
THAT minds are not alike, full well I know,
This truth each day’s experience will show; To heights surprising some great spirits soar, With inborn strength mysterious depths explore; Their eager gaze surveys the path of light, Confest it stood to Newton’s piercing sight. Deep science, like a bashful maid retires, And but the ardent breast her worth inspires; By perseverance the coy fair is won. And Genius, led by Study, wears the crown. But some there are who wish not to improve Who never can the path of knowledge love, Whose souls almost with the dull body one, With anxious care each mental pleasure shun; Weak is the level’d, enervated mind, And but while here to vegetate design’d. The torpid spirit mingling with its clod, Can scarcely boast its origin from God; Stupidly dull—they move progressing on— They eat, and drink, and all their work is done. While others, emulous of sweet applause, Industrious seek for each event a cause, Tracing the hidden springs whence knowledge flows, Which nature all in beauteous order shows. Yet cannot I their sentiments imbibe, Who this distinction to the sex ascribe, As if a woman’s form must needs enrol, A weak, a servile, an inferiour soul; And that the guise of man must still proclaim, Greatness of mind, and him, to be the same: Yet as the hours revolve fair proofs arise, Which the bright wreath of growing fame supplies; And in past times some men have sunk so low , That female records nothing less can show. But imbecility is still confin’d, And by the lordly sex to us consign’d; They rob us of the power t’improve, And then declare we only trifles love; Yet haste the era, when the world shall know, That such distinctions only dwell below; The soul unfetter’d, to no sex confin’d, Was for the abodes of cloudless day design’d. Mean time we emulate their manly fires, Though erudition all their thoughts inspires, Yet nature with equality imparts And noble passions, swell e’en female hearts . |
Is it upon mature consideration we adopt the idea, that nature is thus partial in her distributions? Is it indeed a fact, that she hath yielded to one half of the human species so unquestionable a mental superiority? I know that to both sexes elevated understandings, and the reverse, are common. But, suffer me to ask, in what the minds of females are so notoriously deficient, or unequal. May not the intellectual powers be ranged under these four heads – imagination, reason, memory and judgment. The province of imagination hath long since been surrendered to us, and we have been crowned and undoubted sovereigns of the regions of fancy. Invention is perhaps the most arduous effort of the mind; this branch of imagination hath been particularly ceded to us, and we have been time out of mind invested with that creative faculty. Observe the variety of fashions (here I bar the contemptuous smile) which distinguish and adorn the female world: how continually are they changing, insomuch that they almost render the wise man’s assertion problematical, and we are ready to say, there is something new under the sun. Now what a playfulness, what an exuberance of fancy, what strength of inventine imagination, doth this continual variation discover? Again, it hath been observed, that if the turpitude of the conduct of our sex, hath been ever so enormous, so extremely ready are we, that the very first thought presents us with an apology, so plausible, as to produce our actions even in an amiable light. Another instance of our creative powers, is our talent for slander; how ingenious are we at inventive scandal? what a formidable story can we in a moment fabricate merely from the force of a prolifick imagination? how many reputations, in the fertile brain of a female, have been utterly despoiled? how industrious are we at improving a hint? suspicion how easily do we convert into conviction, and conviction, embellished by the power of eloquence, stalks abroad to the surprise and confusion of unsuspecting innocence. Perhaps it will be asked if I furnish these facts as instances of excellency in our sex. Certainly not; but as proofs of a creative faculty, of a lively imagination. Assuredly great activity of mind is thereby discovered, and was this activity properly directed, what beneficial effects would follow. Is the needle and kitchen sufficient to employ the operations of a soul thus organized? I should conceive not, Nay, it is a truth that those very departments leave the intelligent principle vacant, and at liberty for speculation. Are we deficient in reason? we can only reason from what we know, and if an opportunity of acquiring knowledge hath been denied us, the inferiority of our sex cannot fairly be deduced from thence. Memory, I believe, will be allowed us in common, since everyone’s experience must testify, that a loquacious old woman is as frequently met with, as a communicative man; their subjects are alike drawn from the fund of other times, and the transactions of their youth, or of maturer life, entertain, or perhaps fatigue you, in the evening of their lives.
“But our judgment is not so strong—we do not distinguish so well.”—Yet it may be questioned, from what doth this superiority, in this determining faculty of the soul, proceed. May we not trace its source in the difference of education, and continued advantages? Will it be said that the judgment of a male of two years old, is more sage than that of a female’s of the same age? I believe the reverse is generally observed to be true. But from that period what partiality! how is the one exalted, and the other depressed, by the contrary modes of education which are adopted! the one is taught to aspire, and the other is early confined and limitted. As their years increase, the sister must be wholly domesticated, while the brother is led by the hand through all the flowery paths of science. Grant that their minds are by nature equal, yet who shall wonder at the apparent superiority, if indeed custom becomes second nature ; nay if it taketh place of nature, and that it doth the experience of each day will evince. At length arrived at womanhood, the uncultivated fair one feels a void, which the employments allotted her are by no means capable of filling. What can she do? to books she may not apply; or if she doth, to those only of the novel kind, lest she merit the appellation of a learned lady; and what ideas have been affixed to this term, the observation of many can testify. Fashion, scandal, and sometimes what is still more reprehensible, are then called in to her relief; and who can say to what lengths the liberties she takes may proceed. Meantimes she herself is most unhappy; she feels the want of a cultivated mind. Is she single, she in vain seeks to fill up time from sexual employments or amusements. Is she united to a person whose soul nature made equal to her own, education hath set him so far above her, that in those entertainments which are productive of such rational felicity, she is not qualified to accompany him. She experiences a mortifying consciousness of inferiority, which embitters every enjoyment. Doth the person to whom her adverse fate hath consigned her, possess a mind incapable of improvement, she is equally wretched, in being so closely connected with an individual whom she cannot but despise. Now, was she permitted the same instructors as her brother, (with an eye however to their particular departments) for the employment of a rational mind an ample field would be opened. In astronomy she might catch a glimpse of the immensity of the Deity, and thence she would form amazing conceptions of the august and supreme Intelligence. In geography she would admire Jehovah in the midst of his benevolence; thus adapting this globe to the various wants and amusements of its inhabitants. In natural philosophy she would adore the infinite majesty of heaven, clothed in condescension; and as she traversed the reptile world, she would hail the goodness of a creating God.  Fashions, in their variety, would then give place to conjectures, which might perhaps conduce to the improvements of the literary world; and there would be no leisure for slander or detraction. Reputation would not then be blasted, but serious speculations would occupy the lively imaginations of the sex. Unnecessary visits would only be indulged by way of relaxation, or to answer the demands of consanguinity and friendship. Females would become discreet, their judgments would be invigorated, and their partners for life being circumspectly chosen, an unhappy Hymen would then be as rare, as is now the reverse.
Will it be urged that those acquirements would supersede our domestick duties. I answer that every requisite in female economy is easily attained; and, with truth I can add, that when once attained, they require no further mental attention . Nay, while we are pursuing the needle, or the superintendency of the family, I repeat, that our minds are at full liberty for reflection; that imagination may exert itself in full vigor; and that if a just foundation is early laid, our ideas will then be worthy of rational beings. If we were industrious we might easily find time to arrange them upon paper, or should avocations press too hard for such an indulgence, the hours allotted for conversation would at least become more refined and rational. Should it still be vociferated, “Your domestick employments are sufficient” – I would calmly ask, is it reasonable, that a candidate for immortality, for the joys of heaven, an intelligent being, who is to spend an eternity in contemplating the works of the Deity, should at present be so degraded, as to be allowed no other ideas, than those which are suggested by the mechanism of a pudding, or the sewing the seams of a garment? Pity that all such censurers of female improvement do not go one step further, and deny their future existence; to be consistent they surely ought.
Yes, ye lordly, ye haughty sex, our souls are by nature equal to yours; the same breath of God animates, enlivens, and invigorates us; and that we are not fallen lower than yourselves, let those witness who have greatly towered above the various discouragements by which they have been so heavily oppressed; and though I am unacquainted with the list of celebrated characters on either side, yet from the observations I have made in the contracted circle in which I have moved, I dare confidently believe, that from the commencement of time to the present day, there hath been as many females, as males, who, by the mere force of natural powers, have merited the crown of applause; who, thus unassisted , have seized the wreath of fame. I know there are who assert, that as the animal power of the one sex are superiour, of course their mental faculties also must be stronger; thus attributing strength of mind to the transient organization of this earth born tenement. But if this reasoning is just, man must be content to yield the palm to many of the brute creation, since by not a few of his brethren of the field, he is far surpassed in bodily strength. Moreover, was this argument admitted, it would prove too much, for occular demonstration evinceth, that there are many robust masculine ladies, and effeminate gentlemen. Yet I fancy that Mr. Pope, though clogged with an enervated body, and distinguished by a diminutive stature, could nevertheless lay claim to greatness of soul; and perhaps there are many other instances which might be adduced to combat so unphilosophical an opinion. Do we not often see, that when the clay built tabernacle is well nigh dissolved, when it is just ready to mingle with the parent soil, the immortal inhabitant aspires to, and even attaineth heights the most sublime, and which were before wholly unexplored. Besides, were we to grant that animal strength proved any thing, taking into consideration the accustomed impartiality of nature, we should be induced to imagine, that she had invested the female mind with superiour strength as an equivalent for the bodily powers of man. But waving this however palpable advantage, for equality only, we wish to contend.
I AM aware that there are many passages in the sacred oracles which seem to give the advantage to the other sex; but I consider all these as wholly metaphorical. Thus David was a man after God’s own heart, yet see him enervated by his licentious passions! behold him following Uriah to the death, and shew me wherein could consist the immaculate Being’s complacency. Listen to the curses which Job bestoweth upon the day of his nativity, and tell me where is his perfection, where his patience– literally it existed not. David and Job were types of him who was to come; and the superiority of man, as exhibited in scripture, being also emblematical, all arguments deduced from thence, of course fall to the ground. The exquisite delicacy of the female mind proclaimeth the exactness of its texture, while its nice sense of honour announceth its innate, its native grandeur. And indeed, in one respect, the preeminence seems to be tacitly allowed us; for after an education which limits and confines, and employments and recreations which naturally tend to enervate the body, and debilitate the mind; after we have from early youth been adorned with ribbons, and other gewgaws, dressed out like the ancient victims previous to a sacrifice, being taught by the care of our parents in collecting the most showy materials that the ornamenting our exteriour ought to be the principal object of our attention; after, I say, fifteen years thus spent, we are introduced into the world, amid the united adulation of every beholder. Praise is sweet to the soul; we are immediately intoxicated by large draughts of flattery, which being plentifully administered, is to the pride of our hearts, the most acceptable incense. It is expected that with the other sex we should commence immediate war, and that we should triumph over the machinations of the most artful. We must be constantly upon our guard; prudence and discretion must be our characteristiks; and we must rise superiour to, and obtain a complete victory over those who have been long adding to the native strength of their minds, by an unremitted study of men and books, and who have, moreover, conceived from the loose characters which they have seen portrayed in the extensive variety of their reading, a most contemptible opinion of the sex. Thus unequal, we are, notwithstanding, forced to the combat, and the infamy which is consequent upon the smallest deviation in our conduct, proclaims the high idea which was formed of our native strength; and thus, indirectly at least, is the preference acknowledged to be our due. And if we are allowed an equality of acquirements, let serious studies equally employ our minds, and we will bid our souls arise to equal strengths. We will meet upon even ground, the despot man; we will rush with alacrity to the combat, and, crowned by success, we shall then answer the exalted expectations, which are formed. Though sensibility, soft compassion, and gentle commiseration, are inmates in the female bosom, yet against every deep laid art, altogether fearless of the event, we will set them in array; for assuredly the wreath of victory will encircle the spotless brow. If we meet an equal, a sensible friend, we will reward him with the hand of amity, and through life we will be assiduous to promote his happiness; but from every deep laid scheme, for our ruin, retiring into ourselves, amid the flowery paths of science, we will indulge in all the refined and sentimental pleasures of contemplation: And should it still be urged, that the studies thus insisted upon would interfere with our more peculiar department, I must further reply, that early hours, and close application, will do wonders; and to her who is from the first dawn of reason taught to fill up time rationally, both the requisites will be easy. I grant that niggard fortune is too generally unfriendly to the mind; and that much of that valuable treasure, time, is necessarily expended upon the wants of the body; but it should be remembered; that in embarrassed circumstances our companions have as little leisure for literary improvements, as is afforded to us; for most certainly their provident care is at least as requisite as our exertions. Nay, we have even more leisure for sedentary pleasures, as our avocations are more retired, much less laborious, and, as hath been observed, by no means require that avidity of attention which is proper to the employments of the other sex. In high life, or, in other words, where the parties are in possession of affluence, the objection respecting time is wholly obviated, and of course falls to the ground; and it may also be repeated, that many of those hours which are at present swallowed up in fashion and scandal, might be redeemed, were we habituated to useful reflections. But in one respect, O ye arbiters of our fate! we confess that the superiority is indubitably yours; you are by nature formed for our protectors; we pretend not to vie with you in bodily strength; upon this point we will never contend for victory. Shield us then, we beseech you, from external evils, and in return we will transact your domestick affairs. Yes, your, for are you not equally interested in those matters with ourselves? Is not the elegancy of neatness as agreeable to your sight as to ours; is not the well favoured viand equally delightful to your taste; and doth not your sense of hearing suffer as much, from the discordant sounds prevalent in an ill regulated family, produced by the voices of children and many et ceteras ?
CONSTANTIA.
By way of supplement to the foregoing pages, I subjoin the following extract from a letter, wrote to a friend in the December of 1780
AND now assist me, O thou genius of my sex, while I undertake the arduous task of endeavouring to combat that vulgar, that almost universal errour, which hath, it seems, enlisted even Mr. P— under its banners. The superiority of your sex hath, I grant, been time out of mind esteemed a truth incontrovertible; in consequence of which persuasion, every plan of education hath been calculated to establish this favourite tenet. Not long since, weak and presuming as I was, I amused myself with selecting some arguments from nature, reason, and experience; against this so generally received idea, I confess that to sacred testimonies I had not recourse. I held them to be merely metaphorical, and thus regarding them, I could not persuade myself that there was any propriety in bringing them to decide in this very important debate . However, as you, sir, confine yourself entirely to the sacred oracles, I mean to bend the whole of my artillery against those supposed proofs, which you have from thence provided, and from which you have formed an intrenchment apparently so invulnerable. And first, to begin with our great progenitors; but here, suffer me to premise, that it is for mental strength I mean to contend, for with respect to animal powers, I yield them undisputed to that sex, which enjoys them in common with the lion, the tyger, and many other beasts of prey; therefore your observations respecting the rib under the arm, at a distance from the head, &c.&c. in no sort militate against my view. Well, but the woman was first in the transgression. Strange how blind self love renders you men; were you not wholly absorbed in a partial admiration of your own abilities, you would long since have acknowledged the force of what I am now going to urge. It is true some ignoramuses have absurdly enough informed us, that the beauteous fair of paradise, was seduced from her obedience, by a malignant demon, in the guise of a baleful serpent ; but we, who are better informed, know that the fallen spirit presented himself to her view, a shining angel still; for thus, saith the criticks in the Hebrew tongue, ought the word to be rendered. Let us examine her motive—Hark! the seraph declares that she shall attain a perfection of knowledge; for is there aught which is not comprehended under one or other of the terms good and evil . It doth not appear that she was governed by any one sensual appetite; but merely by a desire of adorning her mind; a laudable ambition fired her soul, and a thirst for knowledge impelled the predilection so fatal in its consequences. Adam could not plead the same deception; assuredly he was not deceived; nor ought we to admire his superiour strength, or wonder at his sagacity, when we so often confess that example is much more influential than precept. His gentle partner stood before him, a melancholy instance of the direful effects of disobedience; he saw her not possessed of that wisdom which she had fondly hoped to obtain, but he beheld the once blooming female, disrobed of that innocence, which had heretofore rendered her so lovely. To him then deception became impossible, as he had proof positive of the fallacy of the argument, which the deceiver had suggested. What then could be his inducement to burst the barriers, and to fly directly in the face of that command, which immediately from the mouth of deity he had received, since, I say, he could not plead that fascinating stimulous, the accumulation of knowledge, as indisputable conviction was so visibly portrayed before him. What mighty cause impelled him to sacrifice myriads of beings yet unborn, and by one impious act, which he saw would be productive of such fatal effects, entail undistinguished ruin upon a race of beings, which he was yet to produce. Blush, ye vaunters of fortitude; ye boasters of resolution; ye haughty lords of the creation; blush when ye remember, that he was influenced by no other motive than a bare pusilianimous attachment to a woman! by sentiments so exquisitely soft, that all his sons have, from that period, when they have designed to degrade them, described as highly feminine. Thus it should seem, that all the arts of the grand deceiver (since means adequate to the purpose are, I conceive, invariably pursued) were requisite to mislead our general mother, while the father of mankind forfeited his own, and relinquished the happiness of posterity, merely in compliance with the blandishments of a female. The subsequent subjection the apostle Paul explains as a figure; after enlarging upon the subject, he adds, “This is a great mystery; but I speak concerning Christ and the church.” Now we know with what consummate wisdom the unerring father of eternity hath formed his plans; all the types which he hath displayed, he hath permitted materially to fail, in the very virtue for which they were famed. The reason for this is obvious, we might otherwise mistake his economy, and render that honour to the creature, which is due only to the creator. I know that Adam was a figure of him who was to come. The grace contained in this figure, is the reason of my rejoicing, and while I am very far from prostrating before the shadow, I yield joyfully in all things the preeminence to the second federal head. Confiding faith is prefigured by Abraham, yet he exhibits a contrast to affiance, when he says of his fair companion, she is my sister. Gentleness was the characteristick of Moses, yet he hesitated not to reply to Jehovah himself, with unsaintlike tongue he murmured at the waters of strife, and with rash hands he break the tables, which were inscribed by the finger of divinity. David, dignified with the title of the man after God’s own heart, and yet how stained was his life. Solomon was celebrated for wisdom, but folly is write in legible characters upon his almost every action. Lastly, let us turn our eyes to man in the aggregate. He is manifested as the figure of strength, but that we may not regard him as any thing more than a figure, his soul is formed in no sort superiour, but every way equal to the mind of her who is the emblem of weakness and whom he hails the gentle companion of his better days.


25.4: Configurations of Aldoses
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- Steven Farmer, Dietmar Kennepohl, & William Reusch
- University of Illinois Springfield
Objectives
After completing this section, you should be able to
- draw the structures of all possible aldotetroses, aldopentoses, and aldohexoses, without necessarily being able to assign names to the individual compounds.
- draw the Fischer projection of D-glyceraldehyde, D-ribose and D-glucose from memory.
The four chiral centers in glucose indicate there may be as many as sixteen (2 4 ) stereoisomers having this constitution. These would exist as eight diastereomeric pairs of enantiomers, and the initial challenge was to determine which of the eight corresponded to glucose. This challenge was accepted and met in 1891 by the German chemist Emil Fischer. His successful negotiation of the stereochemical maze presented by the aldohexoses was a logical tour de force, and it is fitting that he received the 1902 Nobel Prize for chemistry for this accomplishment. At the time Fischer undertook the glucose project it was not possible to establish the absolute configuration of an enantiomer. Consequently, Fischer made an arbitrary choice for (D)-glucose and established a network of related aldose configurations that he called the D -family. The mirror images of these configurations were then designated the L -family of aldoses. To illustrate using present day knowledge, Fischer projection formulas and names for the D-aldose family (three to six-carbon atoms) are shown below, with the asymmetric carbon atoms (chiral centers) colored red.
Aldotrioses : 3 carbon sugars with one chiral center. Aldotrioses have two (2 1 ) possible stereoisomers. A pair of enantiomers called D-gylceraldehyde and L-glyceraldehyde.
Aldotetroses : 4 carbon sugars with two chiral centers. Aldotetroses have four (2 2 ) possible stereoisomers. Two pairs of D/L enantiomers called erythrose and threose.
Aldopentoses : 5 carbon sugars with three chiral centers. Aldopentoses have eight (2 3 ) possible stereoisomers. Four pairs of D/L enantiomers called ribose, arabinose, xylose, and lyxose.
Aldohexoses: 6 carbon sugars with four chiral centers. Aldohexoses have sixteen (2 4 ) possible stereoisomers. five pairs of D/L enantiomers called allose, altrose, glucose, mannose, gulose, idose, galactose, and talose.
Below are the Fischer projects 3-6 carbon aldoses. Starting with the three carbon aldose D-glyceraldehyde, each additional carbon adds a new chiral center and doubles the number of possible stereoisomers of the D-aldoses. Remember that only D-aldoses are represented below. Each D-aldose has an L-aldose enantiomer which is not shown. The L-aldose versions can be draw by inverting all of the chiral centers in the D-aldose's Fischer projection as discussed in the previous section.
Worked Example \(\PageIndex{1}\)
Draw the Fisher projection of L-erythrose and L-Glucose
Solution
Use the Fischer projection provided above and reverse all of the chiral centers to provide the L-sugar. Note that in both cases the D sugars have the OH going to the right on the chiral center furthest away from the carbonyl. The L-sugars have the OH going to the left.
Worked Example \(\PageIndex{2}\)
Please draw the Fischer projection fo the following aldopentose and determine if the sugars is D or L.
Solution
First, rotate the model so that the carbonyl is at the top. This is requirement of a Fischer projection. Next rotate the model so that the H and OH of the chiral carbon just below the carbonyl are facing towards you. In this orientation, a dash/wedge model will have every other set of bonds going into the plane of the page. This is not the correct orientation of a Fischer project so they must be modified. The H and OH bonds need to be coming out of the plane of the page in a Fisher projects. When converting bonds from going into the page to going out of the page the orientation of the H and OH are reversed. Remember that the last -CH 2 OH of a sugar is achiral so the orientation does not need to be shown. Once the bonds are oriented correctly the wedge bonds can be converted to those of a Fischer projection.
Exercise \(\PageIndex{1}\)
For the following model of a sugar, please draw its Fischer projection and name it.
- Answer
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D-Mannose
Exercise \(\PageIndex{2}\)
How many heptose stereoisomers would there expected to be? How many would be D-Sugars?
- Answer
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There would be 2 5 = 32 heptose stereoisomers. Half of these would be D-sugars or 16.
Exercise \(\PageIndex{3}\)
Draw the Fischer projection of the following sugars.
- L-Ribose
- L-Galactose
- L-Talose
- Answer
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a)
b)
c)