12.5: Sexuality and Queer Anthropology
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- Jennifer Hasty, David G. Lewis, & Marjorie M. Snipes
- OpenStax
Learning Objectives
By the end of this section, you will be able to:
- Explain how sexuality is threaded through the life cycle and various realms of culture.
- Describe the prevalence of same-sex relationships in heteronormative societies.
- Define the concept and practices related to ritualized sexuality.
- Give two examples of transgender roles in heteronormative contexts.
Intersecting with gender, the anthropological study of sexuality explores the diversity of meanings, practices, relationships, and experiences associated with erotic interactions. Since the 1980s, the study of sexuality in anthropology has burgeoned into the dynamic subfield of queer anthropology . Anthropologists working in this subfield focus on areas of sociocultural activity distinguished from the presumed norms of heterosexuality and binary gender identities (Howe 2015).
Early Anthropological Studies of Sexuality
Cultural anthropologists have long been fascinated with sexuality.  Of central importance to marriage, kinship, and gender relations, sexuality also pervades art, religion, medicine, economics, and even politics in Trobriand culture. Malinowski charts the sexual life stages of Trobrianders, starting with sexualized games in childhood and continuing with adolescent crushes and expeditions by groups of teenage boys or girls to nearby villages in search of amorous adventures. He describes the selection of marriage partners and the frequency of extramarital sexual relations among men. Throughout his analysis, Malinowski emphasizes that all societies must regulate the primal sexual impulse. In this functional view, sexual norms and rules function to maintain order and protect the institutions of marriage and kinship.
Like Malinowski (and writing in the same time period), Margaret Mead plots the sexual life stages of women and men in Samoan culture in her most famous book, Coming of Age in Samoa (1928) .  Focusing on girls and women, Mead argues that Samoan culture had a more relaxed and open attitude toward sexuality. Throughout childhood, girls often witnessed the bodily realities of childbirth, menstruation, copulation, and death.  Free from the repression and strict sexual discipline of Euro-American culture, Samoans experienced adolescence as not a time of crisis but rather a golden era of freedom and adventure.
Shaped by the feminist movement, more contemporary approaches to gender roles and sexuality highlight structures of power in erotic relations between women and men. Over the past few decades, many Americans have become increasingly concerned about the prevalence of sexual assault on college campuses. Forms of sexual intimidation and violence can happen in many campus contexts, including offices and classrooms as well as student events and parties. An online survey conducted by researchers at the University of Oregon found that students in Greek life (fraternities and sororities) experience nonconsensual sexual contact more than three times as often as other students (Barnes et al. 2021). Anthropologist Peggy Reeves Sanday (1990) conducted ethnographic research on fraternity culture, focusing on how some young men in American fraternities engage in violent assault and criminal coercion against young women. Sanday describes how fraternity men used their privileged access to alcohol and party venues to lure insecure young women to parties where they were plied with alcohol, sometimes drugged, and then sexually assaulted by one or more fraternity members. Sanday argues that fraternity culture is often permeated with forms of verbal and physical aggression against women. Not confined to fraternities, the problem of sexual assault on campuses across the United States has prompted many universities to develop consent awareness training sessions, sexual assault response teams, and survivor support programs.
Same-Sex and Queer Studies
Though they may be provocative and enlightening, anthropological studies of heterosexuality are still focused on mainstream gender categories and norms. Even more challenging to traditional Western sensibilities are studies, first emerging in the 1970s and 1980s, that demonstrate the prevalence of same-sex erotic interactions in cultures all over the world.
A contemporary of Malinowski and Mead, renowned British anthropologist E. E. Evans-Pritchard spent his early career studying social organization and witchcraft among two different African groups, the Azande and the Nuer. Later in his career, Evans-Pritchard began thinking about the many stories he had heard in the course of his years studying African societies, particularly stories describing the prevalence of same-sex erotic practices in Zande society in precolonial times. In an article on the topic, he describes how unmarried adult warrior men, unable to marry due to the scarcity of marriageable women and forbidden to engage in adultery with other men’s wives, often took younger men as sexual partners or “wives” (1970). The warrior paid bride wealth to the parents of the younger man and performed services to the young man’s family just as he would have to the natal family of a female wife. The partners took on the roles of husband and wife, and the younger men referred to themselves as women. As the Azande did not approve of anal sex, male partners had sex “between the thighs”—that is, the older man penetrating between the thigh gap of the younger one.
Like the men, Zande women also commonly engaged in same-sex practices and relationships. In Zande culture, men were permitted to have more than one wife (a form of marriage called polygyny, as you will recall from Forming Family through Kinship ). A husband took turns sleeping with each of his wives. In a family of several wives, then, a woman would wind up sleeping alone many nights.  Zande men and women told Evans-Pritchard that lonely wives would often get together at night, cut a sweet potato or manioc root into the shape of a penis, and tie it around the waist of one of the women. With this vegetable phallus, they took turns penetrating each other. Women could also formalize a “love-friend” relationship in public, widely considered by Zande men to be a cover for same-sex relations. Unlike male-male relationships, however, women’s same-sex erotic practices were discouraged.
Sexual practices between senior and junior men have been found in many cultures, sparking controversies over questions of consent and child abuse. Studying a New Guinea group he called the “Sambia” (a pseudonym), anthropologist Gilbert Herdt (1984) described initiation rituals in which teenage boys were expected to fellate older male mentors in order to absorb the male essence that would make them into fully socialized men. Herdt termed this practice " ritualized homosexuality ," though some have argued with the application of Western categories of sexuality to describe such symbolically complex ritual practices.
While some same-sex practices are ritualized, others are more informal and less public. Some cultures construct same-sex practices as a phase associated with adolescent experimentation and tutelage. As in many parts of contemporary Africa, girls in boarding schools in Ghana are known to experiment with same-sex relationships. In Ghana, it’s called supi (possibly short for supervisor or superintendent ). In boarding high schools, a senior girl might take a junior girl as a special friend (Dankwa 2009; Gyasi-Gyamera and Søgaard 2020). Some of these bonds are fairly casual. The junior girl runs errands for the senior girl, such as fetching water or food. The senior girl provides protection and help to the junior girl (such schools could be full of difficulties, including supply shortages and bullying). Some supi relationships can become emotionally and physically intense. The two girls often exchange gifts, write each other love letters, and fondle and caress one another. They might shower together or share a bed. Supi is not limited to a special category of girls (i.e., identified lesbians) but has been widespread among schoolgirls, nearly all of whom eventually marry men and fulfill their conventional roles as wives and mothers.
In the past two decades, evangelical Christianity in Ghana has branded same-sex relationships as evils to be rooted out through ceremonies resembling exorcism. While supi is an ambiguous practice, sometimes involving sexuality and sometimes not, it has been stigmatized by evangelicals in Ghana. Christian journalists have written stories about wealthy women who snatch away young wives, referring to lesbian relationships as supi-supi . Lurid popular films such as Women in Love (1996) and Supi: The Real Woman to Woman (1996) both sensationalize and condemn women’s same-sex practices, associating them with a secret cult of mermaid worship called Mami Wata.
Many anthropological studies describe same-sex practices in societies that otherwise strongly value heterosexual marriage and fertility. In such contexts, sexuality is not so much an identity as it is a ritual, life stage, coping technique, or form of pleasure. Though sometimes shielded from public view, same-sex relations are seen as complementary to heterosexual relations in some cultural contexts, fully compatible with conventional demands for heterosexual marriage and family life. In his research on gender and sexuality in Nicaragua, for instance, Roger Lancaster (1992) found that conventionally masculine men could maintain their essentially heterosexual identities if they took the “active,” penetrative role in same-sex encounters.
With the progress of the LGBTQIA+ movement originating in the United States and western Europe, people around the world who engage in same-sex and transgender practices have formed public identities and communities, calling for the acceptance and legal recognition of their relationships.  This assertion has profound implications for notions of family and community. If heterosexual marriage and reproduction form the foundation of kinship systems based on the idea of biological descent, then same-sex relationships suggest new forms of kinship based on networks and shared values. In Families We Choose (1991), anthropologist Kath Weston explores how lesbian and gay families in the San Francisco Bay Area constructed family networks that both reflected and challenged mainstream notions of family.
Profiles in Anthropology
Esther Newton, (1940–)
Personal History: Esther Newton was born the child of an unmarried Protestant mother and an absent Jewish father. After she was born, she and her mother were ostracized from her mother’s genteel upper-class family. Her mother later remarried. Growing up in the gender-rigid, heteronormative 1940s and 1950s, Esther flouted gender norms at an early age, becoming “an anti-girl, a girl refusenik” (Newton 2018, 60). She was bullied for her unconventional dress and behavior. As a young woman, she wore men’s clothes, smoked Lucky Strike cigarettes, and dated hyperfeminine lesbian women. Thus, before she even came out as a lesbian, Newton self-consciously constructed her “butch” identity—“the first identity that had ever made sense out of my body’s situation, the first rendition of gender that ever rang true, the first look I could ever pull together” (92).
For her undergraduate studies, Newton attended the University of Michigan, where she earned her BA with distinction in history. In Margaret Mead Made Me Gay (2000), Newton describes her reaction to reading the work of anthropologist Margaret Mead as a college student. Mead’s relativistic portrayal of the flexibility of gender categories gave Newton consolation and ignited her interest in anthropology. She went to the University of Chicago to study anthropology at the graduate level with kinship scholar David Schneider.
Area of Anthropology: For her dissertation, Newton conducted fieldwork among men who dressed as women in the American Midwest. Entitled “The ‘Drag Queens’: A Study in Urban Anthropology” (1968), this pathbreaking work described the experiences, challenges, and culture of gender-nonconforming American men in a variety of theatrical and everyday settings. Her research on this topic was later published in her book Mother Camp: Female Impersonators in America (1972), the first major anthropological study of a gay or lesbian community in the United States. In spite of its initially lukewarm reception, the book has since become a classic in LGBTQIA+ studies.
Accomplishments in the Field: Hired in 1971, Newton was a founding faculty member of the State University of New York at Purchase, also known as Purchase College. She helped establish the disciplines of anthropology, women’s studies, and gay/lesbian studies there. Newton taught at Purchase until 2006 and is now a professor emerita.
Importance of Her Work : In her memoir, My Butch Career (2018), Newton tells the story of the first half of her life, highlighting the challenges facing her generation of middle-class lesbians. She describes the difficulties of pursuing higher education and building a professional career, including the impossibility of coming out even as she studied and wrote about lesbian, gay, and gender-nonconforming communities in American society in the 1960s.
Esther Newton’s work has been translated into French, Spanish, Hebrew, Polish, and Slovak. She is the subject of the documentary film Esther Newton Made Me Gay , currently in production, which has a trailer available to view . In an interview, Newton commented, “It’s been fun being a film star” (2019).
Transgender Studies
Evans-Pritchard’s research on male-male marriage among the precolonial Azande provided an example of young men who were socially constructed as women through their wifely role in these marriages. Across the continent, in West Africa, women in precolonial Igbo society could be ritually transformed into men and then engage in female-female marriages as husbands. In Male Daughters and Female Husbands (1987), Ifi Amadiume describes how a father with no sons could make his eldest daughter into an honorary “son” who could inherit and carry on the patrilineage. This woman became a “male daughter.” If she were married, she would return to her natal compound to undergo a ceremony that transferred her into the social category of male. She would then wear men’s clothes, live in the male section of the compound, perform men’s work rather than women’s, and participate in community life as a man. She could marry women who then became her wives (thus becoming a “female husband”). Those wives would have discreet liaisons with men in the area in order to bear children, who would belong to the lineage of the female husband.
It was also possible for Igbo women who became wealthy and powerful in their communities to take a title through ritual means that allowed them to take wives of their own, just as male daughters could. Even if she were married herself, a powerful woman could have wives to do most or all of her domestic work.  Anthropologists just don’t know. Amadiume describes women joking about sex between women in such marriages, but nobody knows how common it might have been.
Building on this earlier research, a fresh area of inquiry has developed in anthropology centered on the experiences, identities, and practices of transgender and gender-nonbinary persons and communities. Transgender describes a person who transitions from a gender category ascribed at birth to a chosen gender identity. Gender nonbinary describes a person who rejects strict male and female gender categories in favor of a more flexible and contextual expression of gender. Cultural anthropologists have described a great diversity in the expression of trans identities, pointing to the prevalence of transgender practices the world over.
Taking an innovative approach, anthropologist Marcia Ochoa (2014) devised a research project on “spectacular femininity” in Venezuela by examining two communities: female beauty pageant contestants and transgender sex workers who also hold beauty pageants. Ochoa traces the emergence of the beauty pageant in Venezuela and identifies this ritual competition as a carrier of notions of modernity and nationhood.  The stylized performances of transformistas carry over into their displays on Avenida Libertador in central Caracas, the neighborhood where they conduct their trade as sex workers. In order to compete in these realms of spectacular femininity, both misses and transformistas undergo painful surgical procedures to make their bodies conform to an exaggerated ideal of Eurocentric femininity.
Ochoa’s work is pathbreaking in its ability to bring together concepts often explored separately or held in opposition: heterosexuality and non-heterosexuality, gender and sexuality, and cis and trans identities ( cisgender describes gender identity constructed on the sex assigned at birth). By juxtaposing misses and transformistas , she shows how these seemingly disparate concepts are threaded together in the complex web of Venezuelan culture.
The End of Gender?
In cultures that are strongly heteronormative with rigid two-gender systems, some people feel restricted in their gender identities and sexual practices. In many countries, efforts to create more flexibility in the expression of gender and sexuality have focused on gaining equal rights for and combating discrimination against women and LGBTQIA+ persons. In the past 50 years, this social movement has achieved great strides at national and global levels. In 2011, the United Nations Human Rights Council passed a resolution recognizing LGBTQIA+ rights. The United Nations subsequently urged all countries to pass laws to protect LGBTQIA+ persons from discrimination, hate crimes, and the criminalization of non-heterosexuality. Same-sex marriage has now been legalized in 29 countries, including the United States, Canada, Mexico, Taiwan, and most of western Europe. In many countries, however, same-sex acts and gender nonconformity are still criminalized, sometimes punishable by death.
Where progress has been made on human rights for LGBTQIA+ persons, these changes have made life much easier for many people, allowing them to feel secure in their families, their jobs, and their public lives. Some activists are concerned that such legal reforms do not go far enough, however. Gender and sexuality are not just legal issues; they are cultural issues as well. The strict heterosexual two-gender scheme common to European and American cultures is a system infused with patriarchal values, expressed in patriarchal practices and institutions. That is to say, inequality is built into the heteronormative system of gender. In order to achieve true freedom and full equality, is it necessary to get rid of categories of gender and sexuality altogether? Are gender categories inherently oppressive?
Some people think so, arguing that society should transition to more gender-blind forms of language and social relations. In the United States, a movement is underway to neutralize gender in everyday language. Whereas masculine pronouns (he/him) were previously the default way of referring to hypothetical persons or situations where gender is not specified, followed by a movement toward specifying both masculine and feminine pronouns (he or she/him or her), new conventions call for the use of third-person plural forms (they/them) as singular pronouns instead, particularly to include people who identify as neither man nor woman. For example, instead of saying, “Every person should wash his hands” or “Every person should wash his or her hands,” one might say, “Every person should wash their hands.” (Notably, this is already an accepted feature of everyday English that people commonly use without thinking about; if a housemate tells you, “Someone left a message for you,” you’re more likely to respond with “What did they want?” than with “What did he want?” or “What did he or she want?”) Moreover, a convention is evolving that allows people to specify the pronouns they would prefer, either gendered (she/her, he/him) or neutral (they/them, other).
Will changes in pronoun usage bring about greater freedom and equality in patriarchal societies? Maybe. Many languages have gender-free pronouns, such as Twi, a West African language of the Akan peoples in central Ghana. However, though matrilineal, the Akans are also patriarchal. And gender is a very fundamental aspect of identity in Akan societies, structuring norms of dress, language, behavior, and relationships throughout a person’s life. In other words, pronouns do not bear much relationship to the organization of gender in culture and social institutions. In the United States, the English language pronoun system might change to be gender neutral, but women and LGBTQIA+ people will still inhabit those cultural categories. Those categories will not just disappear.
Previous discussions of racial categories have addressed the fact that race is not a set of biological categories objectively found in nature. Rather, race, like gender, is socioculturally constructed. Even so, it is naive to pretend that race does not exist as a social reality that structures inequality in many societies. As discussed in, Social Inequalities , when people try to be “color blind,” they ignore the sociocultural reality of race and make it more difficult to recognize and remediate racial inequalities. Similarly, the fact that gender is a social construct does not mean that people can easily transition to a gender-blind society. Scholars of gender and sexuality argue that American society still grants forms of authority and privilege to heterosexual men through the cultural norms pervading public and private life. Asserting a “gender blind” perspective may obscure forms of inequality and violence that operate through gender and sexuality. Race and gender are both powerful sociocultural categories embedded in social practices and institutions. Anthropology encourages recognition of the diversity and complexity of those constructed categories alongside acknowledgment of the real histories of marginalization and struggle. Perhaps changes in pronoun use are just the beginning of more far-reaching changes to come.
Mini-Fieldwork Activity
Self-Reflection
Consider your own body. What do you do to your body on a daily or weekly basis? Why? For two nonconsecutive days, make careful note of all of the routine practices devoted to your body (including hygiene, dress, exercise, etc.). Are these practices shaped by notions of gender? Of sex or sexuality? Do these practices shape the way you think of your body as gendered? Do they influence the way you present yourself in social situations? Do you think they influence the way others interact with you? Consider how other people respond to and interact with your body (or refuse to interact with it). How are these interactions shaped by cultural notions of gender and sexuality? Are there notions of power embedded in these bodily practices? Patriarchy? Feminism? Heteronormativity?
Suggested Readings
di Leonardo, Micaela, ed. 1991. Gender at the Crossroads of Knowledge: Feminist Anthropology in the Postmodern Era . Berkeley: University of California Press.
Newton, Esther. 2000. Margaret Mead Made Me Gay: Personal Essays, Public Ideas . Durham, NC: Duke University Press.
Stryker, Susan, and Stephen Whittle, ed. 2006. The Transgender Studies Reader . New York: Routledge.


1.13: Substantive Adjectives and the Article
Module 12
© 2021 Philip S. Peek, CC BY 4.0 https://doi.org/ 10.11647/OBP.0264. 13
Substantive Adjectives
A Substantive Adjective is created by using an adjective as a substitute for a noun or pronoun. Greek often uses the article and other adjectives to create substantive nouns. English uses it also but not as frequently. Some well-known examples are these:
The meek shall inherit the earth.
The Good , The Bad , The Ugly .

If feces were gold, the poor would not have any anuses.
The Article may be used in conjunction with an adverb , another noun , or a prepositional phrase to create a substantive adjective .
(1) Article with Adverb : the feminine plural article αἱ joins with the article τότε to create the noun αἱ τότε the women of the past .
(2) Article with Noun : the neuter plural article τά joins the genitive plural noun τῶν ἀνθρώ πων to create the noun τὰ τῶν ἀνθρώπων the possessions of people .
(3) Article with Prepositional Phrase : the masculine singular article ὁ joins with the prepositional phrase ἐπ ὶ τῷ ἵππῳ to create the noun ὁ ἐπὶ τῷ ἵππῳ the man on the horse .
Practice Translating Substantive Adjectives. Translate the phrases below. After translating each as best as you can, check your answers and be sure to understand why each translates as it does. Once you understand why each phrase translates as it does, read through the exercises at least two more times. Doing so helps your brain process the information as part of your working knowledge of Greek. Check your answers with those in the Answer Key .
- αἱ νῦν
- οἱ τότε
- οἱ ἐν τῇ ὁδῷ
- αἱ μάλιστα
- τὰ νῦν
- τὰ τότε
- αἱ ἐκ τ ῆς γῆς
- ὁ ἀνὰ τῷ ἵππῳ
- οἱ κάτω
- τὰ περὶ τοῦ πολέμου
- εἰς τοὺς ἐν τῇ θαλάττῃ
- παρὰ τὰ τῆς ἀρετῆς
- διὰ τῆς ἀνάγκης τῶν τοῦ πολέμου
- κατὰ τοὺς ἐν τῇ ὁδῷ
- πρὸς τὰ ἐν τῇ ἀρχῇ τοῦ χρόνου
- ὑπὸ τῷ νόμῳ τῶν ἐν ἀρχῇ
- ὁ μέν . . . ὁ δέ
- οἱ μέν . . . οἱ δέ
Vocabulary
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* ἀνάγκη , ἀνάγκης ἡ force, necessity, fate |
κάτω below |
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* ἀρετή , ἀρετῆς ἡ virtue, excellence |
* νόμος , νόμου ὁ law, custom |
|
* ἀρχή , ἀρχῆς ἡ rule, command; beginning; province |
ὁδός , ὁδοῦ ἡ road |
|
* γῆ , γῆς ἡ land, earth |
* πόλεμος , πολέμου ὁ war |
|
* θάλασσα ( θάλαττα ), θαλάσσης ἡ sea |
* χρόνος , χρόνου ὁ time |
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* ἵππος , ἵππου ὁ ἡ horse; (fem.) cavalry |
Practice Making Up Your Own Phrases. Using the vocabulary above, create your own substantives. Be creative and don’t worry about making mistakes, which are necessary for you to have success in any activity you pursue. Own the ancient Greek language and your learning of it. Make it yours.
Practice Translating. Translate the sentences below, adapted from Euripides’ Iphigeneia at Tauris ( Ἰφιγένεια ἐν Ταύροις ). Remember the meanings and functions of the cases presented in Module 7 . Nominative case endings are bolded; genitive endings are italicized; dative endings are highlighted; and accusative endings are underlined. Check your understanding with the translations in the Answer Key . Now go back and read each sentence two or three times, noticing with each rereading how much better your understanding of the sentence becomes. Make this a habit and you will improve quickly.
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Case |
Ending |
Function |
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Nominative |
- α , - ας , - η , - ηρ , - ης , - ξ , - ο , - ος , - ς , - ων |
subject of the verb |
|
Genitive |
- ας , - εως - ης , - ος , - ου , - ους , - ων |
dependence; possession; object of preposition or verb; separation |
|
Dative |
- ᾳ , - αις , - ῃ , - ι , - οις - ῳ |
indirect object of the verb; means or instrument; object of preposition |
|
Accusative |
- α , - αν , - ας , - ε , - ην , - ν , - ο , - ον , - ος , - ους |
object of the verb or preposition |
Ἰφιγένεια: ὁ Ταντάλει ος εἰς Πῖσ αν ἔρχεται καὶ θο αῖς ἵππ οις Οἰνομά ου γαμέει κόρ ην . ἡ δὲ τίκτει Ἀτρέ α . Ἀτρ έως δὲ παῖ ς Μενέλα ος Ἀγαμέμν ων τε . καὶ ἐκ αὐτ οῦ καὶ τ ῆς Τυνδαρεί ας γίγνομαι Ἰφιγένει α . ἐμ ὲ δὲ σφάζει Ἑλέν ης ἕνεχʼ ὁ πατ ὴρ Ἀρτέμιδ ι κλειν αῖς ἐν πτυχ αῖς Αὐλίδ ος . ἐνταῦθα γὰρ δὴ χιλί ων νη ῶν στόλ ον Ἑλληνικ ὸν συνάγει Ἀγαμέμν ων ἄνα ξ καὶ τ ὸν στέφαν ον Ἰλί ου ἐθέλει λαβεῖν . ὁ Ἀτρείδ ης τ οὺς γάμ ους Ἑλέν ης ἐθέλει μετελθεῖν καὶ Μενελά ῳ χάρι ν φέρειν . ὁ τ ῆς ἄνα ξ στρατηγί ας , Ἀγαμέμν ων , καίπερ ἐθέλει νῆ ας ἀφορμίζεσθαι χθον ός , οὐκ ἔχει . δειν ῆς δʼ ἀπλοί ας τυγχάνει ὥστε εἰς ἔμπυρ α ἔρχεται καὶ αὐτ ῷ λέγει ὁ Κάλχ ας ταῦτ α . Κάλχ ας λέγει ὅτι τ ῷ πατρ ὶ μὲν δεῖ ἐμ έ , κόρ ην τ ὴν Ἰφιγένει αν , σφάζειν Ἀρτέμιδ ι καὶ αὐτ ῇ δὲ λαβεῖν ἄποιν α . καὶ Κάλχ ας λέγει αὐτ ῷ ὅτι κάλλιστ ον εὔχεται παῖδ α φωσφόρ ῳ θύειν θε ᾷ . τ ὴν παῖδʼ οὖν Ἰφιγένει αν ἡ Κλυταιμήστρ α τίκτει . ὁ Κάλχ ας τ ὸ κάλλιστ ον εἰς ἐμὲ ἀναφέρει ὥστε χρὴ τ ὸν πατέρ α θύειν . ὁ δὲ ἐμὲ Ὀδυσσ έως τέχναις μητρ ὸς παραλαμβάνει ἐπὶ γάμοις Ἀχιλλ έως .
Adverbs and Verbs
|
ἀναφέρω bring up, bring back, refer |
* λαμβάνω take, receive, capture |
|
ἀφορμίζομαι loose, set loose |
* λέγω say, tell, speak |
|
γαμέω marry |
μετελθεῖν to send for, chase after, go after |
|
* γίγνομαι be, be born |
παραλαμβάνω take, take ‘x’ in acc. from ‘y’ in gen. |
|
* δεῖ it is necessary + ‘x’ in gen. or dat. or acc. + inf., δεῖ ἐλθεῖν it is necessary to come |
συνάγω gather, collect; convene |
|
* ἐθέλω ( θέλω ) wish, be willing |
σφάζω slay, slaughter |
|
ἐνταῦθα here, there, then |
τίκτω bear, give birth |
|
* ἔρχομαι come, go |
* τυγχάνω obtain, meet + gen. |
|
εὔχομαι pray, vow, promise + inf. |
* φέρω bring, bear, carry; endure bring, bear, carry; endure |
|
* ἔχω have, hold; be able + inf.; καλῶς ἔ χειν to be well |
* χρή it is necessary + inf. |
|
θύω sacrifice |
Adjectives, Nouns, Pronouns
|
Νοminative |
Genitive |
Dative |
Accusative |
English Equivalent |
|
Ἀγαμέμνων |
Ἀγαμέμνονος |
Ἀγαμέμνονι |
Ἀγαμέμνονα |
Agamemnon |
|
ἄναξ |
ἄνακτος |
ἄνακτι |
ἄνακτα |
prince, lord, king |
|
ἀπλοία |
ἀπλοίας |
ἀπλοίᾳ |
ἀπλοίαν |
failure to sail |
|
ἄποινα |
ἀποίνων |
ἀποίνοις |
ἄποινα |
ransom, payment |
|
Ἄρτεμις |
Ἀρτέμιδος |
Ἀρτέμιδι |
Ἀρτέμιδα |
Artemis |
|
Ἀτρεύς |
Ἀτρέως |
Ἀτρεῖ |
Ἀτρέα |
Atreus |
|
Ἀτρεί δης |
Ἀτρείδου |
Ἀτρείδῃ |
Ἀτρείδην |
son of Atreus |
|
Αὐλίς |
Αὐλίδος |
Αὐλίδος |
Αὐλίδα |
Aulis |
|
* αὐτή |
αὐτῆς |
αὐτῇ |
αὐτήν |
she, her, hers |
|
* αὐτός |
αὐτοῦ |
αὐτῷ |
αὐτόν |
he, him, his |
|
Ἀχιλλεύς |
Ἀχιλλέως |
Ἀχιλλεῖ |
Ἀχιλλέ α |
Akhilleus |
|
γάμοι |
γάμων |
γάμοις |
γάμους |
wedding, marriage |
|
* δεινή |
δεινῆς |
δεινῇ |
δεινήν |
awesome; terrible |
|
* ἐγώ |
ἐμοῦ |
ἐμοί |
ἐμέ |
I, me, mine |
|
Ἑλένη |
Ἑλένης |
Ἑλένῃ |
Ἑλένην |
Helen |
|
Ἑλληνικός |
Ἑλληνικοῦ |
Ἑλληνικῷ |
Ἑλληνικόν |
Greek |
|
ἔμπυρα |
ἐμπύ ρων |
ἐμπύροις |
ἔμπυρα |
burnt sacrifice |
|
* θεά |
θεᾶς |
θεᾷ |
θεάν |
goddess |
|
θοαί |
θοῶν |
θοαῖς |
θοάς |
swift |
|
Ἴλιον |
Ἰλίου |
Ἰλίῳ |
Ἴλιον |
Troy |
|
* ἵπποι |
ἵππων |
ἵπποις |
ἵππους |
horses |
|
Ἰφιγένεια |
Ἰφιγενείας |
Ἰφιγενείᾳ |
Ἰφιγένειαν |
Iphigeneia |
|
κ άλλιστος |
καλλίστου |
καλλίστῳ |
κάλλιστον |
most beautiful |
|
Κάλχας |
Κάλχαντος |
Κάλχαντι |
Κάλχαντα |
Kalkhas |
|
κλειναί |
κλεινῶν |
κλειναῖς |
κλεινάς |
famous |
|
κόρη |
κόρης |
κόρῃ |
κόρην |
maiden, girl |
|
Μενέλαος |
Μενελάου |
Μενελάῳ |
Μενέλαον |
Menelaos |
|
* μήτηρ |
μητρό ς |
μητρί |
μητέρα |
mother |
|
* νῆες |
νεῶν |
ναυσί ( ν ) |
ναῦς or νῆας |
ships |
|
Ὀδυσσεύς |
Ὀδυσσέως |
Ὀδυσσεῖ |
Ὀδυσσέα |
Odysseus |
|
Οἰνόμαος |
Οἰνομάου |
Οἰνομάῳ |
Οἰνόμαον |
Oinomaos |
|
* πα ῖς |
παιδός |
παιδί |
παῖδα |
child |
|
* πατήρ |
πατρός |
πατρί |
πατέρα |
father |
|
Πῖσα |
Πίσης |
Πίσῃ |
Πῖσαν |
Pisa |
|
πτυχαί |
πτυχῶν |
πτυχαῖς |
πτυχά ς |
folds, glens |
|
στέφανος |
στεφάνου |
στεφάνῳ |
στέ φανον |
crown, wreath, prize |
|
στόλος |
στόλου |
στόλῳ |
στόλον |
expedition |
|
στρατηγία |
στρατηγίας |
στρατηγίᾳ |
στρατηγίαν |
office; campaign |
|
Ταντάλειος |
Τανταλείου |
Τανταλείῳ |
Ταντάλειον |
Tantalean, here Pelops |
|
* ταῦτα |
τούτων |
τούτοις |
ταῦτα |
these things |
|
τέχναι |
τεχνῶν |
τέχναις |
τέχνᾱς |
skill, craft |
|
Τυνδαρεία |
Τυνδαρείας |
Τυνδαρείᾳ |
Τυνδαρείαν |
Tyndarean, here Klytaimnestra |
|
φωσφόρος |
φωσφόρου |
φωσφόρῳ |
φωσφόρον |
light-bringing |
|
χάρις |
χάριτος |
χάριτι |
χάριν |
grace, favor, thanks |
|
χθών |
χθονός |
χθονί |
χθόνα |
earth, ground |
|
χίλιαι |
χιλίων |
χιλίαις |
χιλίᾱς |
thousand |
- The asterisk indicates the top 250 most frequently occurring vocabulary, which you are to memorize.
Practice Parsing Greek Sentences. Parse each word of the sentence found below. For nouns and pronouns, give case and function. For verbs give person, number, tense, mood, and voice. For adverbs and conjunctions, identify them. For prepositional phrases, give the preposition and the preposition’s object. For adjectives, tell what noun they agree with in gender, number, and case.
δεινῆς δ ʼ ἀπλοίας τυγχάνει ὥστε εἰς ἔμπυρʼ ἔρχεται κα ὶ αὐτῷ λέγει ὁ Κάλχας ταῦτα .
Check your answers with those in the Answer Key .
Classics and Our Modern World: the Aquila Theatre and Theater of War Productions
Presenting classics from throughout time, the Aquila Theatre puts flesh and blood on the great works of the past. Past performances from the ancient world include, Herakles , the Iliad and Odyssey , and Philoktetes . Aquila seeks to reinterpret the Classics for contemporary audiences and to bring diversity to them. Aquila expands the canon and attempts to change our perceptions. It also provides extensive educational offerings and is an award-winning leader in the field of public humanities programming. For more information, follow this link:
Theater of War Productions presents dramatic readings of classical Greek tragedies and modern and contemporary works. After the performance, it holds town-hall-style discussions designed to confront social issues by discussing the themes highlighted in the plays. The open discussions invite audience members to share their perspectives and experiences and seek to break down stigmas, foster empathy, compassion, and a deeper understanding of complex issues. In an effort to reach communities directly, Theater of War Productions partners with a range of organizations and government agencies. For more information, follow this link:
Theater of War Productions . 2
Module 12 Top 250 Vocabulary to be Memorized. Like learning the alphabet and endings, memorizing vocabulary is essential to acquiring language. The better you memorize the top 250 most frequently occurring vocabulary words, the greater mastery of the language you will have.
Verbs
ἐθέλω ( θέλω ) wish, be willing
πείθω persuade + inf.; (mid. or pass.) listen to, obey + dat. or gen.
τυγχάνω happen ; meet + gen.; obtain + gen.; hit the mark, strike + gen.
φαίνω show, reveal ; (pass.) appear
χράομαι use, employ, experience + dat.


1.4.1: Effective Nuclear Charge
Z* modulates attraction
When valence electrons experience less nuclear charge than core electrons, different electrons experience different magnitudes of attraction to the nucleus. A modified form of Coulomb's Law is written below, where \(e\) is the charge of an electron, Z* is the effective nuclear charge experienced by that electron, and \(r\) is the radius (distance of the electron from the nucleus).
\[ F_{eff}=k \dfrac{Z*e^2}{r^2} \nonumber \]
This formula suggests that if we can estimate Z*, then we can predict the attractive force experienced by, and the energy of, an electron in a multi-electron atom (ex. Li).
The attraction of the nucleus to valence electrons determines the atomic or ionic size, ionization energy, electron affinity, and electronegativity. The stronger the attraction, and the stronger Z*, the closer the electrons are pulled toward the nucleus. This in turn results in a smaller size, higher ionization energy, higher electron affinity, and stronger electronegativity.
General Periodic Trends in Z*
Close inspection of Figure \(\PageIndex{3}\) and analysis of Slater's rules indicate that there are some predictable trends in Z*. The data from Figure \(\PageIndex{3}\) is plotted below in Figure \(\PageIndex{4}\) to provide a visual aid to the discussion below.
Trends in Z* for electrons in a specific shell and subshell
The Z* for electrons in a given shell and subshell generally increases as atomic number increases; this trend holds true going across the periodic table and down the periodic table. Convince yourself that this is true for any subshell by examining Figure \(\PageIndex{4}\). (CC-BY-NC-SA; Kathryn Haas)
- Do you notice any exceptions to this general trend?
-
Inspection of Figure \(\PageIndex{4}\) should confirm for you that the Z* increases as Z increases for electrons in any subshell (like the 1s subshell for example, which is plotted above as a red line with square points). You can see this trend as the positive slope in each series. There is one obvious exception in Period 5 in elements 39 (Y) to 41 (Nb; the Z* of 4s actually decreases across these three elements as atomic number increases. There is also an exception between Y and Zr in the 3d subshell, and between Tc and Ru in the 5s subshell.
For valence electrons:
It is useful to understand trends in valence Z* because the valence Z* determines atomic/ionic properties and chemical reactivity. The trends in the valence Z* are not simple because as atomic number increases, the valence shell and/or subshell also changes. The valence Z* is indicated in Figure \(\PageIndex{4}\) as a black line with open circles.
Down the table: As we go down a column of the periodic table, the valence Z* increases. This is a simple trend because type of subshell is consistent and there is an increase only in shell and in atomic number, Z. This trend is best illustrated by inspection of Figure \(\PageIndex{3}\).
Across the table: the trend depends on shell and subshell, but generally Z* increases across a period.
Periods 1-3 (s and p only): As we go across the table in periods 1-3, the shell stays constant as Z increases and the subshell changes from s to p. In these periods, there is a gradual increase in valence Z* as we move across any of the first three periods.
Periods 4 and 5 (s, p, and d): Now we have some more complex trends because valence subshell and shell are changing as we increase in atomic number. Notice that the valence Z* generally increases going across a period as long as the subshell isn't changing; the exception is within the 4d subshell (elements 39-44 or Y-Ru). In general, going from an \((n)s\) subshell to an \((n-1)d\) subshell, there a relatively large increase in valence Z*. And in going from an \((n-1\)d\) subshell to an \((n)p\) subshell, there is a relatively large decrease in Z*.
From one period to another: From Figure \(\PageIndex{4}\), we can see that as we increase Z by one proton, going from one period to the next, there is a relatively large decrease in Z* (from Ne to Na, for example). This is because as Z increases by a small interval, the shell number increases, and so the electrons in the valence shell are much farther from the nucleus and are more shielded by all the electrons in the lower shell numbers.
Exercises
- Compare trends in Z* and atomic size. Explain how and why atomic size depends on Z*.
- Compare trends in Z* and ionization energy. Explain how and why ionization energy depends on Z*.
- Answer
-
1. On the periodic table, atomic radius generally decreases across the periods (left to right) and increases down the groups. As atomic number increases across the periodic table, nuclear charge (Z) increases and Z* increases. In turn, the atomic radius decreases because the higher nuclear charge (and thus higher Z*) pulls electrons closer to the nucleus. Atomic radius increases down the periodic table because the shell number increases. Despite an increase in Z* going down the periodic table, larger atomic radii result from electrons occupying higher shells.
2. Ionization energies (IE) are inversely related to atomic radius; IE increases across the periods and decreases down the groups. Since the nucleus holds valence electrons more strongly (due to higher Z*) across the periods, IE increases because valence electrons are harder to remove. Down the periodic table, larger atomic radii cause electrons in valence orbitals to be shielded by core electrons. Recall that shielding reduces the nuclear charge available to electrons in higher orbital levels, resulting in a lower Z*. With more shielding and lower Z*, the valence electrons are held less tightly by the nucleus such that ionization energy decreases (i.e., valence electrons are easier to remove).
References
- Petrucci, Ralph H., William S. Harwood, F. Geoffrey Herring, and Jeffry D. Madura. General Chemistry: Principles and Modern Applications, Ninth Edition. Pearson Education Inc. Upper Saddle River, New Jersey: 2007.
- Raymond Chang. Physical Chemistry for Biological Sciences. Sausalito, California: University Science Books, 2005
- R. S. Mulliken, Electronic Structures of Molecules and Valence. II General Considerations, Physical Review, vol. 41, pp. 49-71 (1932)
- Anastopoulos, Charis (2008). Particle Or Wave: The Evolution of the Concept of Matter in Modern Physics . Princeton University Press. pp. 236–237. ISBN<PHONE_NUMBER> . http://books.google.com/?id=rDEvQZhpltEC&pg=PA236 .
Contributors and Attributions
- Sidra Ayub (UCD), Alan Chu (UCD)
-
Emily V Eames (City College of San Francisco)
-
Curated or created by Kathryn Haas


22.2: Rotation with Constant Angular Acceleration
- Derive the kinematic equations for rotational motion with constant angular acceleration
- Select from the kinematic equations for rotational motion with constant angular acceleration the appropriate equations to solve for unknowns in the analysis of systems undergoing fixed-axis rotation
- Use solutions found with the kinematic equations to verify the graphical analysis of fixed-axis rotation with constant angular acceleration
In the preceding section, we defined the rotational variables of angular displacement, angular velocity, and angular acceleration. In this section, we work with these definitions to derive relationships among these variables and use these relationships to analyze rotational motion for a rigid body about a fixed axis under a constant angular acceleration. This analysis forms the basis for rotational kinematics. If the angular acceleration is constant, the equations of rotational kinematics simplify, similar to the equations of linear kinematics discussed in Motion along a Straight Line and Motion in Two and Three Dimensions . We can then use this simplified set of equations to describe many applications in physics and engineering where the angular acceleration of the system is constant. Rotational kinematics is also a prerequisite to the discussion of rotational dynamics later in this chapter.
Kinematics of Rotational Motion
Using our intuition, we can begin to see how the rotational quantities \(\theta\), \(\omega\), \(\alpha\), and t are related to one another. For example, we saw in the preceding section that if a flywheel has an angular acceleration in the same direction as its angular velocity vector, its angular velocity increases with time and its angular displacement also increases. On the contrary, if the angular acceleration is opposite to the angular velocity vector, its angular velocity decreases with time. We can describe these physical situations and many others with a consistent set of rotational kinematic equations under a constant angular acceleration. The method to investigate rotational motion in this way is called kinematics of rotational motion .
To begin, we note that if the system is rotating under a constant acceleration, then the average angular velocity follows a simple relation because the angular velocity is increasing linearly with time. The average angular velocity is just half the sum of the initial and final values:
\[\bar{\omega} = \frac{\omega_{0} + \omega_{f}}{2} \ldotp \label{10.9}\]
From the definition of the average angular velocity, we can find an equation that relates the angular position, average angular velocity, and time:
\[\bar{\omega} = \frac{\Delta \theta}{\Delta t} \ldotp\]
Solving for \(\theta\), we have
\[\theta_{f} = \theta_{0} + \bar{\omega} t, \label{10.10}\]
where we have set t 0 = 0. This equation can be very useful if we know the average angular velocity of the system. Then we could find the angular displacement over a given time period. Next, we find an equation relating \(\omega\), \(\alpha\), and t. To determine this equation, we start with the definition of angular acceleration:
\[\alpha = \frac{d \omega}{dt} \ldotp\]
We rearrange this to get \(\alpha\)dt = d\(\omega\) and then we integrate both sides of this equation from initial values to final values, that is, from t 0 to t and \(\omega_{0}\) to \(\omega_{f}\). In uniform rotational motion, the angular acceleration is constant so it can be pulled out of the integral, yielding two definite integrals:
\[\alpha \int_{t_{0}}^{t} dt' = \int_{\omega_{0}}^{\omega_{f}} d \omega \ldotp\]
Setting t 0 = 0, we have
\[\alpha t = \omega_{f} - \omega_{0} \ldotp\]
We rearrange this to obtain
\[\omega_{f} = \omega_{0} + \alpha t, \label{10.11}\]
where \(\omega_{0}\) is the initial angular velocity. Equation \ref{10.11} is the rotational counterpart to the linear kinematics equation v f = v 0 + at. With Equation \ref{10.11}, we can find the angular velocity of an object at any specified time t given the initial angular velocity and the angular acceleration.
Let’s now do a similar treatment starting with the equation \(\omega = \frac{d \theta}{dt}\). We rearrange it to obtain \(\omega\)dt = d\(\theta\) and integrate both sides from initial to final values again, noting that the angular acceleration is constant and does not have a time dependence. However, this time, the angular velocity is not constant (in general), so we substitute in what we derived above:
\[\begin{split} \int_{t_{0}}^{t_{f}} (\omega_{0} + \alpha t') dt' & = \int_{\theta_{0}}^{\theta_{f}} d \theta; \\ \int_{t_{0}}^{t} \omega_{0} dt + \int_{t_{0}}^{t} \alpha tdt & = \int_{\theta_{0}}^{\theta_{f}} d \theta = \Bigg[ \omega_{0} t' + \alpha \left(\dfrac{(t')^{2}}{2}\right)^{2} \Bigg]_{t_{0}}^{t} = \omega_{0} t + \alpha \left(\dfrac{t^{2}}{2}\right) = \theta_{f} - \theta_{0} \ldotp \end{split}\]
where we have set t 0 = 0. Now we rearrange to obtain
\[\theta_{f} = \theta_{0} + \omega_{0} t + \frac{1}{2} \alpha t^{2} \ldotp \label{10.12}\]
Equation \ref{10.12} is the rotational counterpart to the linear kinematics equation found in Motion Along a Straight Line for position as a function of time. This equation gives us the angular position of a rotating rigid body at any time t given the initial conditions (initial angular position and initial angular velocity) and the angular acceleration.
We can find an equation that is independent of time by solving for t in Equation \ref{10.11} and substituting into Equation \ref{10.12}. Equation \ref{10.12} becomes
\[\begin{split} \theta_{f} & = \theta_{0} + \omega_{0} \left(\dfrac{\omega_{f} - \omega_{0}}{\alpha}\right) + \frac{1}{2} \alpha \left(\dfrac{\omega_{f} - \omega_{0}}{\alpha}\right)^{2} \\ & = \theta_{0} + \frac{\omega_{0} \omega_{f}}{\alpha} - \frac{\omega_{0}^{2}}{\alpha} + \frac{1}{2} \frac{\omega_{f}^{2}}{\alpha} - \frac{\omega_{0} \omega_{f}}{\alpha} + \frac{1}{2} \frac{\omega_{0}^{2}}{\alpha} \\ & = \theta_{0} + \frac{1}{2} \frac{\omega_{f}^{2}}{\alpha} - \frac{1}{2} \frac{\omega_{0}^{2}}{\alpha}, \\ \theta_{f} - \theta_{0} & = \frac{\omega_{f}^{2} - \omega_{0}^{2}}{2 \alpha} \end{split}\]
or
\[\omega_{f}^{2} = \omega_{0}^{2} + 2 \alpha (\Delta \theta) \ldotp \label{10.13}\]
Equation \ref{10.10} through Equation \ref{10.13} describe fixed-axis rotation for constant acceleration and are summarized in Table 10.1.
Table 10.1 - Kinematic Equations
| Angular displacement from average angular velocity | \(\theta_{f} = \theta_{0} + \bar{\omega} t\) |
| Angular velocity from angular acceleration | \(\omega_{f} = \omega_{0} + \alpha t\) |
| Angular displacement from angular velocity and angular acceleration | \(\theta_{f} = \theta_{0} + \omega_{0} t + \frac{1}{2} \alpha t^{2}\) |
| Angular velocity from angular displacement and angular acceleration | \(\omega_{f}^{2} = \omega_{0}^{2} + 2 \alpha (\Delta \theta)\) |
Applying the Equations for Rotational Motion
Now we can apply the key kinematic relations for rotational motion to some simple examples to get a feel for how the equations can be applied to everyday situations.
A deep-sea fisherman hooks a big fish that swims away from the boat, pulling the fishing line from his fishing reel. The whole system is initially at rest, and the fishing line unwinds from the reel at a radius of 4.50 cm from its axis of rotation. The reel is given an angular acceleration of 110 rad/s 2 for 2.00 s (Figure \(\PageIndex{1}\)).
- What is the final angular velocity of the reel after 2 s?
- How many revolutions does the reel make?
Strategy
Identify the knowns and compare with the kinematic equations for constant acceleration. Look for the appropriate equation that can be solved for the unknown, using the knowns given in the problem description.
Solution
- We are given \(\alpha\) and t and want to determine \(\omega\). The most straightforward equation to use is \(\omega_{f} = \omega_{0} + \alpha t\), since all terms are known besides the unknown variable we are looking for. We are given that \(\omega_{0}\) = 0 (it starts from rest), so $\omega_{f} = 0 + (110\; rad/s^{2})(2.00\; s) = 220\; rad/s \ldotp$
- We are asked to find the number of revolutions. Because 1 rev = 2\(\pi\) rad, we can find the number of revolutions by finding θ in radians. We are given \(\alpha\) and t, and we know \(\omega_{0}\) is zero, so we can obtain \(\theta\) by using $\begin{split} \theta_{f} & = \theta_{i} + \omega_{i} t + \frac{1}{2} \alpha t^{2} \\ & = 0 + 0 + (0.500)(110\; rad/s^{2})(2.00\; s)^{2} = 220\; rad \ldotp \end{split}$Converting radians to revolutions gives $Number\; of\; rev = (220\; rad) \left(\dfrac{1\; rev}{2 \pi\; rad}\right) = 35.0\; rev \ldotp$
Significance
This example illustrates that relationships among rotational quantities are highly analogous to those among linear quantities. The answers to the questions are realistic. After unwinding for two seconds, the reel is found to spin at 220 rad/s, which is 2100 rpm. (No wonder reels sometimes make high-pitched sounds.)
In the preceding example, we considered a fishing reel with a positive angular acceleration. Now let us consider what happens with a negative angular acceleration.
Now the fisherman applies a brake to the spinning reel, achieving an angular acceleration of −300 rad/s 2 . How long does it take the reel to come to a stop?
Strategy
We are asked to find the time t for the reel to come to a stop. The initial and final conditions are different from those in the previous problem, which involved the same fishing reel. Now we see that the initial angular velocity is \(\omega_{0}\) = 220 rad/s and the final angular velocity \(\omega\) is zero. The angular acceleration is given as \(\alpha\) = −300 rad/s 2 . Examining the available equations, we see all quantities but t are known in \(\omega_{f} = \omega_{0} + \alpha t\), making it easiest to use this equation.
Solution
The equation states
\[\omega_{f} = \omega_{0} + \alpha t \ldotp\]
We solve the equation algebraically for t and then substitute the known values as usual, yielding
\[t = \frac{\omega_{f} - \omega_{0}}{\alpha} = \frac{0 - 220.0\; rad/s}{-300.0\; rad/s^{2}} = 0.733\; s \ldotp\]
Significance
Note that care must be taken with the signs that indicate the directions of various quantities. Also, note that the time to stop the reel is fairly small because the acceleration is rather large. Fishing lines sometimes snap because of the accelerations involved, and fishermen often let the fish swim for a while before applying brakes on the reel. A tired fish is slower, requiring a smaller acceleration.
A centrifuge used in DNA extraction spins at a maximum rate of 7000 rpm, producing a “g-force” on the sample that is 6000 times the force of gravity. If the centrifuge takes 10 seconds to come to rest from the maximum spin rate: (a) What is the angular acceleration of the centrifuge? (b) What is the angular displacement of the centrifuge during this time?
Figure \(\PageIndex{2}\) shows a graph of the angular velocity of a propeller on an aircraft as a function of time. Its angular velocity starts at 30 rad/s and drops linearly to 0 rad/s over the course of 5 seconds. (a) Find the angular acceleration of the object and verify the result using the kinematic equations. (b) Find the angle through which the propeller rotates during these 5 seconds and verify your result using the kinematic equations.
Strategy
- Since the angular velocity varies linearly with time, we know that the angular acceleration is constant and does not depend on the time variable. The angular acceleration is the slope of the angular velocity vs. time graph, \(\alpha = \frac{d \omega}{dt}\). To calculate the slope, we read directly from Figure \(\PageIndex{2}\), and see that \(\omega_{0}\) = 30 rad/s at t = 0 s and \(\omega_{f}\) = 0 rad/s at t = 5 s. Then, we can verify the result using \(\omega = \omega_{0} + \alpha t\).
- We use the equation \(\omega = \frac{d \theta}{dt}\); since the time derivative of the angle is the angular velocity, we can find the angular displacement by integrating the angular velocity, which from the figure means taking the area under the angular velocity graph. In other words: \(\int_{\theta_{0}}^{\theta_{f}} d \theta = \theta_{f} - \theta_{0} = \int_{t_{0}}^{t_{f}} \omega (t)dt \ldotp\) Then we use the kinematic equations for constant acceleration to verify the result.
Solution
- Calculating the slope, we get \(\alpha = \frac{\omega - \omega_{0}}{t - t_{0}} = \frac{(0 - 30.0)\; rad/s}{(5.0 - 0)\; s} = -6.0\; rad/s^{2} \ldotp\) We see that this is exactly Equation \ref{10.11} with a little rearranging of terms.
- We can find the area under the curve by calculating the area of the right triangle, as shown in Figure \(\PageIndex{3}\).
\[\Delta \theta = area(triangle) = \frac{1}{2} (30\; rad/s)(5\; s) = 75\; rad \ldotp\]
We verify the solution using Equation \ref{10.12}:
\[\theta_{f} = \theta_{0} + \omega_{0} t + \frac{1}{2} \alpha t^{2} \ldotp\]
Setting \(\theta_{0}\) = 0, we have
\[\theta_{0} = (30.0\; rad/s)(5.0\; s) + \frac{1}{2} (-6.0\; rad/s^{2})(5.0\; s)^{2} = 150.0 - 75.0 = 75.0\; rad \ldotp\]
This verifies the solution found from finding the area under the curve.
Significance
We see from part (b) that there are alternative approaches to analyzing fixed-axis rotation with constant acceleration. We started with a graphical approach and verified the solution using the rotational kinematic equations. Since \(\alpha = \frac{d \omega}{dt}\), we could do the same graphical analysis on an angular acceleration-vs.-time curve. The area under an \(\alpha\)-vs.-t curve gives us the change in angular velocity. Since the angular acceleration is constant in this section, this is a straightforward exercise.


16.7: Heart Dissection
The heart dissection is probably one of the most difficult dissections you will do. Part of the reason it is so difficult to learn is that the heart is not perfectly symmetrical, but it is so close that it becomes difficult to discern which side you are looking at (dorsel, ventral, left or right). Finding the vessels is directly related to being able to orient the heart correctly and figuring out which side you are looking at.
The heart is also difficult because the fatty tissue that surrounds the heart can obscure the openings to the vessels. This means that you really must experience the heart with your hands and feel your way to find the openings. Many people will be squeamish about this, and because the heart is slippery, it is easy to drop. Don't be shy with the heart, use your fingers to feel your way through the dissection.
-
Step One: Orientation
When you first remove your heart from the bag, you will see a lot of fatty tissue surrounding it. It is usually a waste of time to try to remove this tissue. Grab some colored pencils to help you identify and mark the vessels you find.
There are a few clues to help you figure out the left and the right side, but often the packaging and preserving process can cause the heart to be misshapen. If you are lucky, the heart will be nicely preserved and you will see that the front (ventral) side of the heart has a couple of key features: 1) a large pulmonary trunk that extends off the top of it 2) the flaps of the auricles covering the top of the atria. 3) the curve of the entire front side, whereas the backside is much flatter.
The first image shows the front side of the heart, often identified by the anterior interventricular artery that runs cross it at an angle (yellow).
The auricle is the flap that covers the atrium, it looks like an ear. The pulmonary trunk is the located at the front of the heart and enters at an angle.
Step 2: Locate the Aorta
Use your fingers to probe around the top of the heart. Four major vessels can be found entering the heart: the pulmonary trunk, aorta, superior vena cava, and the pulmonary vein. Remember that if you are looking at the back of the heart, then the right and left sides are the same as your right and left hand. This picture was on the board the day of the dissection so that you could glance up and recall which vessel entered which part of the heart.
If you find the pulmonary vein, the aorta should be situated a little bit behind it. It may be covered by fat, so use your fingers to poke around until you find the opening. Push your finger all the way in and you will feel inside of the left ventricle. The left ventricle has a very thick wall, unlike the right ventricle. Insert your finger through the pulmonary vessel to feel the left ventricle and you will notice and feel that it is much thinner than the left side of the heart.
With your fingers or probes in the aorta and the pulmonary trunk you should notice that they criss-cross each other, with the pulmonary trunk in the front. At this point, you may want to use your colored pencils to mark these vessels so that you don't get them confused when you are searching for the other two openings that top of the heart.
Step 3: Locate the Veins
The two major veins that enter the heart can be found on the backside, as both enter the atria. On the left side, you should be able to find the opening of the pulmonary vein as it enters the left atrium. The superior vena cava enters the right atrium. In many preserved hearts, the heart was cut at these points, so you won't see the vessels themselves, you will just find the openings. Again, use your fingers to feel around the heart to find the openings. If you've marked the aorta and pulmonary then you won't mistake them for the veins you are looking for. This picture shows all of the vessels labeled.
Sometimes, the aorta still has its branches attached to it. There are three vessels that branch from the aorta: the brachiocephalic, left common carotid and the left subclavian. The majority of the time, these vessels are not visible because the aorta was cut too close to the main part of the heart when the heart was removed from the animal. Occassionally, you can find the brachiocephalic artery attached, as it is in this photo.
Step 4: Make the Incisions
Now that you have all of the vessels located and marked, you can now open the heart to view the inner chambers. Use the superior vena cava and pulmonary vein as guides for where to cut. You are basically going to be cutting each side of the heart so that you can look inside. (Some dissections will ask you to make a coronal cut where a single cut opens the entire back side of the heart). The heart below is marked to show you where the two incisions should be made.
Cut the heart in half to expose the chambers. My students affectionally call these two variations the "hot dog cut" as pictured above because it looks like a hot dog bun, or the "hamburger cut, where the heart is cut into the front and the back half, as shown below.Figure \(\PageIndex{2}\): Copy and Paste Caption here. (Copyright; author via source)
Step 5: Viewing the Chambers
At this point it is helpful to have two hands, one to hold the heart apart so you can take a peak inside of it and another to use a probe to locate the specific parts. Your colored pencils you used to mark the heart in step 2 can also now be used to see where those vessels connect within the heart. For instance, the aorta pencil can now be seen ending in the left ventricle.
You can also now see how much thicker the walls of the left ventricle are compared to the right ventricle.
Figure \(\PageIndex{10}\): The opened heart
The other obvious structures seen within the heart are the chordae tendinae which are attached to papillary muscles. These tendons hold the heart valves in place, sometimes they are called the "heartstrings". The valves were probably cut when the heart was opened, but if you follow the "cords" they should lead you to a thin flap that is the atrioventricular (bicuspid) valve. You can find a similar valve on the right side of the heart (tricuspid).
Image shows the left atrioventricular valve (bicuspid) and the chordae tendinae.
Figure \(\PageIndex{11}\): Displaying the chordae tendinae


1.4: Atomic Structure - Electron Configurations
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Objective
After completing this section, you should be able to write the ground-state electron configuration for each of the elements up to and including atomic number 36.
Key Terms
Make certain that you can define, and use in context, the key terms below.
- ground-state electronic configuration
- Hund's rule
- Pauli exclusion principle
- aufbau principle
The electron configuration of an atom is the representation of the arrangement of electrons distributed among the orbital shells and subshells. Commonly, the electron configuration is used to describe the orbitals of an atom in its ground state, but it can also be used to represent an atom that has ionized into a cation or anion by compensating with the loss of or gain of electrons in their subsequent orbitals. Many of the physical and chemical properties of elements can be correlated to their unique electron configurations. The valence electrons, electrons in the outermost shell, are the determining factor for the unique chemistry of the element.
Electron Configurations
The electron configuration of an atom is the representation of the arrangement of electrons distributed among the orbital shells and subshells. Commonly, the electron configuration is used to describe the orbitals of an atom in its ground state, but it can also be used to represent an atom that has ionized into a cation or anion by compensating with the loss of or gain of electrons in their subsequent orbitals. Many of the physical and chemical properties of elements can be correlated to their unique electron configurations. The valence electrons, electrons in the outermost shell, are the determining factor for the unique chemistry of the element.
Before assigning the electrons of an atom into orbitals, one must become familiar with the basic concepts of electron configurations. Every element on the periodic table consists of atoms, which are composed of protons, neutrons, and electrons. Electrons exhibit a negative charge and are found around the nucleus of the atom in electron orbitals, defined as the volume of space in which the electron can be found within 95% probability. The four different types of orbitals (s,p,d, and f) have different shapes, and one orbital can hold a maximum of two electrons. The p, d, and f orbitals have different sublevels, thus can hold more electrons.
As stated, the electron configuration of each element is unique to its position on the periodic table. The energy level is determined by the period and the number of electrons is given by the atomic number of the element. Orbitals on different energy levels are similar to each other, but they occupy different areas in space. The 1s orbital and 2s orbital both have the characteristics of an s orbital (radial nodes, spherical volume probabilities, can only hold two electrons, etc.) but, as they are found in different energy levels, they occupy different spaces around the nucleus. Each orbital can be represented by specific blocks on the periodic table. The s-block is the region of the alkali metals including helium (Groups 1 & 2), the d-block are the transition metals (Groups 3 to 12), the p-block are the main group elements from Groups 13 to 18, and the f-block are the lanthanides and actinides series.
Using the periodic table to determine the electron configurations of atoms is key, but also keep in mind that there are certain rules to follow when assigning electrons to different orbitals. The periodic table is an incredibly helpful tool in writing electron configurations. For more information on how electron configurations and the periodic table are linked, visit the Connecting Electrons to the Periodic Table module.
Rules for Assigning Electron Orbitals
Pauli Exclusion Principle
The Pauli exclusion principle states that no two electrons can have the same four quantum numbers. The first three (n, l, and m l ) may be the same, but the fourth quantum number must be different. A single orbital can hold a maximum of two electrons, which must have opposing spins; otherwise they would have the same four quantum numbers, which is forbidden. One electron is spin up (m s = +1/2) and the other would spin down (m s = -1/2). This tells us that each subshell has double the electrons per orbital. The s subshell has 1 orbital that can hold up to 2 electrons, the p subshell has 3 orbitals that can hold up to 6 electrons, the d subshell has 5 orbitals that hold up to 10 electrons, and the f subshell has 7 orbitals with 14 electrons.
Example 1: Hydrogen and Helium
The first three quantum numbers of an electron are n=1, l=0, m l =0. Only two electrons can correspond to these, which would be either m s = -1/2 or m s = +1/2. As we already know from our studies of quantum numbers and electron orbitals, we can conclude that these four quantum numbers refer to the 1s subshell. If only one of the m s values are given then we would have 1s 1 (denoting hydrogen) if both are given we would have 1s 2 (denoting helium). Visually, this is be represented as:
As shown, the 1s subshell can hold only two electrons and, when filled, the electrons have opposite spins.
Hund's Rule
When assigning electrons in orbitals, each electron will first fill all the orbitals with similar energy (also referred to as degenerate) before pairing with another electron in a half-filled orbital. Atoms at ground states tend to have as many unpaired electrons as possible. When visualizing this processes, think about how electrons are exhibiting the same behavior as the same poles on a magnet would if they came into contact; as the negatively charged electrons fill orbitals they first try to get as far as possible from each other before having to pair up.
Example 2: Oxygen and Nitrogen
If we look at the correct electron configuration of the Nitrogen (Z = 7) atom, a very important element in the biology of plants: 1s 2 2s 2 2p 3
We can clearly see that p orbitals are half-filled as there are three electrons and three p orbitals. This is because Hund's Rule states that the three electrons in the 2p subshell will fill all the empty orbitals first before filling orbitals with electrons in them. If we look at the element after nitrogen in the same period, oxygen (Z = 8) its electron configuration is: 1s 2 2s 2 2p 4 (for an atom).
Oxygen has one more electron than nitrogen and as the orbitals are all half filled the electron must pair up.
Occupation of Orbitals
Electrons fill orbitals in a way to minimize the energy of the atom. Therefore, the electrons in an atom fill the principal energy levels in order of increasing energy (the electrons are getting farther from the nucleus). The relative energy of the orbitals is shown in Figure \(\PageIndex{2}\)
The order of levels filled is then:
1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, and 7p
The general order in which orbitals are filled is depicted in Figure \(\PageIndex{3}\). Subshells corresponding to each value of n are written from left to right on successive horizontal lines, where each row represents a row in the periodic table. The order in which the orbitals are filled is indicated by the diagonal lines running from the upper right to the lower left. Accordingly, the 4 s orbital is filled prior to the 3 d orbital because of shielding and penetration effects. Consequently, the electron configuration of potassium, which begins the fourth period, is [Ar]4 s 1 , and the configuration of calcium is [Ar]4 s 2 . Five 3 d orbitals are filled by the next 10 elements, the transition metals, followed by three 4 p orbitals. Notice that the last member of this row is the noble gas krypton ( Z = 36), [Ar]4 s 2 3 d 10 4 p 6 = [Kr], which has filled 4 s , 3 d , and 4 p orbitals. The fifth row of the periodic table is essentially the same as the fourth, except that the 5 s , 4 d , and 5 p orbitals are filled sequentially.
The Aufbau Process
Aufbau comes from the German word "aufbauen" meaning "to build." When writing electron configurations, orbitals are built up from atom to atom. When writing the electron configuration for an atom, orbitals are filled in order of increasing atomic number. However, there are some exceptions to this rule.
Example 3: 3 rd row elements
Following the pattern across a period from B (Z=5) to Ne (Z=10), the number of electrons increases and the subshells are filled. This example focuses on the p subshell, which fills from boron to neon.
- B (Z=5) configuration: 1s 2 2s 2 2p 1
- C (Z=6) configuration:1s 2 2s 2 2p 2
- N (Z=7) configuration:1s 2 2s 2 2p 3
- O (Z=8) configuration:1s 2 2s 2 2p 4
- F (Z=9) configuration:1s 2 2s 2 2p 5
- Ne (Z=10) configuration:1s 2 2s 2 2p 6
The Number of Valence Electrons
The number of valence electrons of an element can be determined by the periodic table group (vertical column) in which the element is categorized. With the exception of groups 3–12 (the transition metals ), the units digit of the group number identifies how many valence electrons are associated with a neutral atom of an element listed under that particular column. For example in group 16, the units digit is 6 and elements in this group have 6 valence electrons.
| Periodic table group | Valence electrons |
|---|---|
| Group 1: alkali metals | 1 |
| Group 2: alkaline earth metals | 2 |
| Groups 3-12: transition metals | 2* (The 4s shell is complete and cannot hold any more electrons) |
| Group 13: boron group | 3 |
| Group 14: carbon group | 4 |
| Group 15: pnictogens | 5 |
| Group 16: chalcogens | 6 |
| Group 17: halogens | 7 |
| Group 18: noble gases | 8** |
* The general method for counting valence electrons is generally not useful for transition metals. Instead the modified d electron count method is used.
** Except for helium , which has only two valence electrons.
The electron configuration of an element is the arrangement of its electrons in its atomic orbitals. By knowing the electron configuration of an element, we can predict and explain a great deal of its chemistry.
Example 1.3.1
Draw an orbital diagram and use it to derive the electron configuration of phosphorus, Z = 15. What is its valence electron configuration?
Given: atomic number
Asked for: orbital diagram and valence electron configuration for phosphorus
Strategy:
- Locate the nearest noble gas preceding phosphorus in the periodic table. Then subtract its number of electrons from those in phosphorus to obtain the number of valence electrons in phosphorus.
- Referring to Figure 1.3.1, draw an orbital diagram to represent those valence orbitals. Following Hund’s rule, place the valence electrons in the available orbitals, beginning with the orbital that is lowest in energy. Write the electron configuration from your orbital diagram.
- Ignore the inner orbitals (those that correspond to the electron configuration of the nearest noble gas) and write the valence electron configuration for phosphorus.
Solution:
A Because phosphorus is in the third row of the periodic table, we know that it has a [Ne] closed shell with 10 electrons. We begin by subtracting 10 electrons from the 15 in phosphorus.
B The additional five electrons are placed in the next available orbitals, which Figure 1.2.5 tells us are the 3 s and 3 p orbitals:
Because the 3 s orbital is lower in energy than the 3 p orbitals, we fill it first:
Hund’s rule tells us that the remaining three electrons will occupy the degenerate 3 p orbitals separately but with their spins aligned:
The electron configuration is [Ne]3 s 2 3 p 3 .
C We obtain the valence electron configuration by ignoring the inner orbitals, which for phosphorus means that we ignore the [Ne] closed shell. This gives a valence-electron configuration of 3 s 2 3 p 3 .
Exercise 1.3.1
Draw an orbital diagram and use it to derive the electron configuration of chlorine, Z = 17. What is its valence electron configuration?
- Answer
- [Ne]3 s 2 3 p 5 ; 3 s 2 3 p 5
The sixth row of the periodic table will be different from the preceding two because the 4 f orbitals, which can hold 14 electrons, are filled between the 6 s and the 5 d orbitals. The elements that contain 4 f orbitals in their valence shell are the lanthanides. When the 6 p orbitals are finally filled, we have reached the next (and last known) noble gas, radon ( Z = 86), [Xe]6 s 2 4 f 14 5 d 10 6 p 6 = [Rn]. In the last row, the 5 f orbitals are filled between the 7 s and the 6 d orbitals, which gives the 14 actinide elements. Because the large number of protons makes their nuclei unstable, all the actinides are radioactive.
Example 1.3.2
Write the electron configuration of mercury ( Z = 80), showing all the inner orbitals.
Given: atomic number
Asked for: complete electron configuration
Strategy:
Using the orbital diagram in Figure \(\PageIndex{3}\) and the periodic table as a guide, fill the orbitals until all 80 electrons have been placed.
Solution:
By placing the electrons in orbitals following the order shown in Figure \(\PageIndex{3}\) and using the periodic table as a guide, we obtain
| 1 s 2 | row 1 | 2 electrons |
| 2 s 2 2 p 6 | row 2 | 8 electrons |
| 3 s 2 3 p 6 | row 3 | 8 electrons |
| 4 s 2 3 d 10 4 p 6 | row 4 | 18 electrons |
| 5 s 2 4 d 10 5 p 6 | row 5 | 18 electrons |
| row 1–5 | 54 electrons |
After filling the first five rows, we still have 80 − 54 = 26 more electrons to accommodate. According to Figure 1.3.1, we need to fill the 6 s (2 electrons), 4 f (14 electrons), and 5 d (10 electrons) orbitals. The result is mercury’s electron configuration:
1 s 2 2 s 2 2 p 6 3 s 2 3 p 6 4 s 2 3 d 10 4 p 6 5 s 2 4 d 10 5 p 6 6 s 2 4 f 14 5 d 10 = Hg = [Xe]6 s 2 4 f 14 5 d 10
with a filled 5 d subshell, a 6 s 2 4 f 14 5 d 10 valence shell configuration, and a total of 80 electrons. (You should always check to be sure that the total number of electrons equals the atomic number.)
Summary
Based on the Pauli principle and a knowledge of orbital energies obtained using hydrogen-like orbitals, it is possible to construct the periodic table by filling up the available orbitals beginning with the lowest-energy orbitals (the aufbau principle ), which gives rise to a particular arrangement of electrons for each element (its electron configuration ). Hund’s rule says that the lowest-energy arrangement of electrons is the one that places them in degenerate orbitals with their spins parallel. For chemical purposes, the most important electrons are those in the outermost principal shell, the valence electrons .