5.12: Explore the Material Culture Meanings of A Frame at Mt. Horeb Historical Society
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Frame
American Attributed to the Donald/Sweet Family, c.1890 Wood, walnut shells Gift of Delma Donald Woodburn Estate MHAHS 2002.001.0073 |
My object for the final project is the picture frame with photographs of Ellen Sweet and her brother George Clark Sweet, decorated with walnuts. It is currently collected at the Mount Horeb Historical Society in Dane County, Wisconsin. The date that the object was made is around 1871 to 1933. The reason why I chose this object is the materials of the decoration on frame are unconventional, which are sliced walnut or butternut shells glued to the wood frame.  These pictures were taken in 1867-1871 respectively. Madison Wisconsin Dec. 25, 1933” And there is a second note, in blue ink, reads: “Given to John S. Donald by his mother 1933.” Ellen Sweet Donald Jones gave the frame to her son John Sweet Donald (1869-1934) in 1933. The maker of the frame might be Ellen or by her daughter-in-law Vona DeCrow Donald. Another particularity of my object is that it is both the “container” and the “contained”.
In my paper, I would investigate both the “container” and the “contained”. For the “container” — photo frame, I would examine the three stages of its biography and fluid social lives. Through exploring the hand-making technique, I also found the limited role women played in the society at that time. For the “contained” — portraits of Ellen Sweet and George Clark Sweet, I would discuss how the photography tells us about the technological and intellectual development at Mount Horeb and how fashion proves that material objects are status symbols and social markers that have categorical functions and expressive functions.
The history of Mount Horeb can be traced back to late 19th century. Most of the settlers there were of English and Scottish descent. During the early days of Mount Horeb, the main industries were cheese and livestock. In 1855, the Sweet family moved from New York to live in Wisconsin on the farm. The figures in the photographs, Ellen Sweet and her brother George Clark Sweet, are two of the seven children of William Sweet and Sally/Sarah Clark Sweet. Ellen married John Strong Donald in April of 1868. Within a year John Strong Donald did from Typhoid at the age of 26. Then Ellen married John Jones, Jr. in 1882. The frame is something that Ellen gave to her son John Sweet Donald in 1933. The frame intends to display and protect the photographs in it, which also functions as a way to record the family history.
Furthermore, the picture frame indicates that women’s work was mostly restricted in domestic space during the late 19th century and early 20th century. The possible makers of the frame are Ellen or her daughter-in-law Vona DeCrow Donald, both of whom are women. The walnut shell picture frame is not a mass-produced consumer product, but a hand-made craft that most likely made at home or a small studio place. Many women in the 19th century mainly carried out home-based work and in addition, they were also responsible for the unpaid work at home like cooking, cleaning, child care and often keeping small animals and growing vegetables and fruit to help feed their families. During the second half of the 19th century and early 20th century, woman started to work in textile factories, coal and tin mines or farms. The picture frame is a historical source that records women’s engagement in a variety of social activities. In the book, Woman and the Material Culture of Needlework & Textiles, it states that “the needle has been a as a vehicle for women’s own construction of alternative discourses, discourses with eh potential to expand women’s discursive worlds and the power they wield over their own lives” Here, the frame is also a vehicle for Ellen or her daughter-in-law to show the gender’s construction by the dominant discourse. Women’s craft has been always a performance of femininity. However, although women’s work expanded to a larger range from 19th century to 20th century, the role of women was still limited to domestic work like doing hand-making craft works. Unlike today’s society, women and men’s roles were sharply defined and had clear boundaries in social activities at that time. Women were more suited to the domestic space for a conventional perspective.
Despite its maker, the frame itself also has great values to conduct the material culture studies that help us understanding the larger context and relationship between human, nature and society. By applying the biography approach to the frame, I focus on three phases during its social lives: raw material, family possession, and museum collection. Since the biography approach mainly focuses on the exceptional characteristic within its own time, thus those stages are characterized by the use of natural resources, particular human connection, symbol of local identity respectively. Recent conceptualizations of material culture studies hold the view that objects have ‘social lives’ (Appadurai, 1986) or ‘biographies’ (Kopytoff, 1986). Material objects have biographical turning points or stages in the life cycle. The turning points or stages of the photo frame evolve over time and the meanings and interpretations attached to it also flexible and fluid. Here I apply the biography approach to explore the different phases of the photo frame’s social lives. This approach traces a complete life story in an infinite time frame and highlights the exceptional and unusual features.
Object Biography
3 stages
raw materials, family possession, museum collection
3 features
natural resources, human connection, local identity
Before the photo frame was been made, the walnut shells and wood frame were just raw materials in nature. The uniqueness of the first stage of the frame is its origin from nature. In Marcinkus’s book Nature Fancywork, she mentions that both Aristotle and Plato considered that nature’s purpose was to serve those with an intellect. Nature and humanity were part of a whole living organism. “Organismic theory emphasized the interdependence among the parts of human boy, subordination of individual to communal purpose in family, community and state, and vital life permeating the cosmos to the lowliest stone,” argues Merchant in The Death of Nature (1980: 1). Exploring natural resources is a way to study the laws of nature, by which nature becomes something that people can conquer, discover and exploit. The use of walnut shells can be seen as something unified human and nature. The portrait and walnut shells organically integrate together, which signifies a bond between human society and wild nature.
Moving to the next stage of its biography, the frame became a memorial decorative object for the Sweet family after processed by hand-making techniques. The value of the object mostly lies in the human connection attached to the people in the photograph and the Sweet family. In the article by Sherry Turkle called The Things That Matter, she found that the photograph of her mother serves as a clue to her possible identity. The walnut photo frame also has the same function that builds a connection between the person in the photograph and the rest of the family members. The materiality and physicality of the picture frame make it a vehicle for the family members to sense the family history and conclude the possible identities of themselves. Thus, the photo frame became the thing that mattered. The photo frame carries both ideas and emotions. Thought and feeling attached to it are inseparable from our study on the material object.
Finally, now the frame is one of the museum collections at Mount Horeb Historical Society. Its value transformed from personal human connection to the symbolization of local identity that is precious for historical archaeologists to study the local and regional history. The estimate date of the frame shows that it was made 20 years or more later after the Sweet family settled in Wisconsin. In American art history, immigrants prefer to carry something precious like luxury goods or portraits to show their social status and identity. However, according to its time, the frame must not serve as something that the Sweet family eager to reveal their status and identity as new immigrants but as something functional and practical to the family. So the frame contains lots of historical evidence of the family and the region at that time. Now, as a museum, it can point toward to the artistic, cultural, scientific and historical facts of Mount Horeb, which speaks to its local identity and regional history.
According to Erving Goffman (1951), there are two principle roles for status symbols. First, status symbols have categorical functions in that they serve to ‘distinguish’ and ‘socially place’ the person who uses the symbol. So the frame and clothes serves as symbols of social and economic status. The frame, as a decorative object, is not a necessity goods, which indicates that the Sweet family was not under poor living conditions that needs to struggle with necessity goods. However, at meantime, the material of the object, walnut shell, is not considered as a luxury goods, so we can infer that the family might has a social and economic status as a upper middle class. Secondly, status symbols can serve expressive function related to a person’s style, taste or cultural values. The shape, color and materials of the frame all have expressive function that we can see how the personal style, taste, and value embody in the material object. The shape of the frame is seemingly a house. It might be the notion of a unified family or any other harmonious living environment. The style of the frame conveys a taste of simplicity and rusticity from the Sweet family. And the use of natural materials like walnut shells and wood may suggest their embracement to the nature and wild.
Material objects are also markers of aesthetics and cultural value. The clothes in the photograph also reflect the aesthetic and culture trend at that time which is “appropriate European styles in American subject matter”. During the early days of Mount Horeb, there was a rapid growing of European immigrants population due to the overcrowding in European countries. And it might be a historical explanation of the clothes that Ellen Sweet and her brother George Clark Sweet wearing in the photograph are heavily European-influenced. The tall collars that Elle Sweet was wearing are one of the features in the 1890s fashion, which is one of the fashion trends in Europe. In the 1890s, women’s fashion in America became simpler and less extravagant. American was in its “Gilded Age”. During the period, those Americans who travel to Europe tend to apply the European styles in American artistic forms like architecture, paintings, decorative arts as well as fashion. The clothes that George was wearing are the common trend shared by America and Europe, the shirts of linen or cotton with collars and neckties tied in a bow. Those elements are gentlemen at that time were often seen with.
The photographs in the frame of Ellen Sweet and her brother George were taken in 1871 and 1867 respectively. During that time, photography was mainly documentary to capture real objects like portraits, landscape, and historical events. Photography did not step into the artistic venue until the 1900s. Thus, the function of the photograph of Ellen and George was used as a memorial portrait that can last long and be seen by generations of their family and the society. We can also infer that camera was been used by middle-class people at that time. Also as an new invent, telephone was just brought to Mount Horeb in 1895 by J. N. Dahlen, editor of Mount Horeb Times. The new technology and electronic devices like camera and telephone contribute to the popularization of communication in the community. By investigating the photograph, we can conclude that the photo frame is not only an indicator to the Sweet family history but also to the industrialization and technology development of Mount Horeb at that time. Growing interest in communication through visual imagery or sound influences the later development of Mount Horeb in a profound way.
“If you want to know your future” said Winston Churchill “look backwards first”.’ Material culture can record the history that has significant meanings to future generations. Both the “contained” and “container” of the frame carries valuable artistic, cultural and historical significance to the family, the region and the anthropological study of human society. Woodward also states that objects are material embodiment of human labor that produced them. Any object represents exploit human capacity and ultimate degradation of human creativity and identity.
To investigate how a material object fit in the large social context, we could not only acquire a clear vision of what role does the object play in the large cultural and historical context but also get a clue of our relationship with the object, the nature and the society.
In the 21st century, the advancement of manufacturing technology and transportation methods in the consumer society provides a platform for the consumer products and commodities grow rapidly. Traditional ways of making and designing gradually vanished in our horizon. The life cycle of an object has become shorter and shorter because people have easier access to get new objects. Fast-fashion and fast-food are dominant in our age. For my perspective, we still need to hold a rational and critical attitude towards our production and consumptions of material objects and treasure those things that really matter.
Bibliography:
Marcinkus, Andrea Kolasinski. Nature Fancywork : Women’s Discourse with Nature through Craft in Late Nineteenth-Century America, 2008.
Turkle, Sherry. Evocative Objects: Things We Think With. Cambridge: The MIT Press, 2007.
Woodward, Ian. Understanding Material Culture. Los Angeles ; London :Sage Publications, 2007.
Edited by Maureen Daly Goggin and Beth Fowkes Tobin. Women and the Material Culture of Needlework and Textiles, 1750-1950. Farnham, England ; Burlington, VT :Ashgate, 2009.
Edited by Fiona Candlin and Raiford Guins. The Object Reader. London ; New York :Routledge, 2009.
Sebenthal, Betty; Darrow, M.M.; Babcock, Vera; Kittleson, Ansel, Editor
The Mount Horeb centennial book : 1861-1961. Mt. Horeb Chamber of Commerce, 1961
Hays, J.E.S. Back In My Time: A Writer’s Guide to the 19th century. August 11, 2012.
backinmytime.blogspot.com/201...h-century.html
Mount Horeb — Presettlement to 1986 : a History Celebrating Mount Horeb’s Quasquicentennial. [Mount Horeb, Wis.] :Mount Horeb Area Historical Society, 1986.
Romer, Grant. Portraits in the Mid-19th Century. Pbs.org.
www.pbs.org/wgbh/amex/eastman...w/romer01.html
Woman and Work in the 19th Century. Striking Woman. Accessed April 26, 2016.
www.striking-women.org/module...y-20th-century
Men’s Portrait Gallery. Gentlemen’s Emporium. Accessed April 26, 2016.
www.gentlemansemporium.com/1...to-gallery.php


25.3: The Depths of the Great Depression
From industrial strongholds to the rural Great Plains, from factory workers to farmers, the Great Depression affected millions. In cities, as industry slowed, then sometimes stopped altogether, workers lost jobs and joined breadlines, or sought out other charitable efforts. With limited government relief efforts, private charities tried to help, but they were unable to match the pace of demand. In rural areas, farmers suffered still more. In some parts of the country, prices for crops dropped so precipitously that farmers could not earn enough to pay their mortgages, losing their farms to foreclosure. In the Great Plains, one of the worst droughts in history left the land barren and unfit for growing even minimal food to live on.
The country’s most vulnerable populations, such as children, the elderly, and those subject to discrimination, like African Americans, were the hardest hit. Most white Americans felt entitled to what few jobs were available, leaving African Americans unable to find work, even in the jobs once considered their domain. In all, the economic misery was unprecedented in the country’s history.
STARVING TO DEATH
By the end of 1932, the Great Depression had affected some sixty million people, most of whom wealthier Americans perceived as the “deserving poor.” Yet, at the time, federal efforts to help those in need were extremely limited, and national charities had neither the capacity nor the will to elicit the large-scale response required to address the problem. The American Red Cross did exist, but Chairman John Barton Payne contended that unemployment was not an “Act of God” but rather an “Act of Man,” and therefore refused to get involved in widespread direct relief efforts. Clubs like the Elks tried to provide food, as did small groups of individually organized college students. Religious organizations remained on the front lines, offering food and shelter. In larger cities, breadlines and soup lines became a common sight. At one count in 1932, there were as many as eighty-two breadlines in New York City.
Despite these efforts, however, people were destitute and ultimately starving. Families would first run through any savings, if they were lucky enough to have any. Then, the few who had insurance would cash out their policies. Cash surrender payments of individual insurance policies tripled in the first three years of the Great Depression, with insurance companies issuing total payments in excess of $1.2 billion in 1932 alone. When those funds were depleted, people would borrow from family and friends, and when they could get no more, they would simply stop paying rent or mortgage payments. When evicted, they would move in with relatives, whose own situation was likely only a step or two behind. The added burden of additional people would speed along that family’s demise, and the cycle would continue. This situation spiraled downward, and did so quickly. Even as late as 1939, over 60 percent of rural households, and 82 percent of farm families, were classified as “impoverished.” In larger urban areas, unemployment levels exceeded the national average, with over half a million unemployed workers in Chicago, and nearly a million in New York City. Breadlines and soup kitchens were packed, serving as many as eighty-five thousand meals daily in New York City alone. Over fifty thousand New York citizens were homeless by the end of 1932.
Children, in particular, felt the brunt of poverty. Many in coastal cities would roam the docks in search of spoiled vegetables to bring home. Elsewhere, children begged at the doors of more well-off neighbors, hoping for stale bread, table scraps, or raw potato peelings. Said one childhood survivor of the Great Depression, “You get used to hunger. After the first few days it doesn’t even hurt; you just get weak.” In 1931 alone, there were at least twenty documented cases of starvation; in 1934, that number grew to 110. In rural areas where such documentation was lacking, the number was likely far higher. And while the middle class did not suffer from starvation, they experienced hunger as well.
By the time Hoover left office in 1933, the poor survived not on relief efforts, but because they had learned to be poor. A family with little food would stay in bed to save fuel and avoid burning calories. People began eating parts of animals that had normally been considered waste. They scavenged for scrap wood to burn in the furnace, and when electricity was turned off, it was not uncommon to try and tap into a neighbor’s wire. Family members swapped clothes; sisters might take turns going to church in the one dress they owned. As one girl in a mountain town told her teacher, who had said to go home and get food, “I can’t. It’s my sister’s turn to eat.”
For his book on the Great Depression, Hard Times , author Studs Terkel interviewed hundreds of Americans from across the country. He subsequently selected over seventy interviews to air on a radio show that was based in Chicago. Visit Studs Terkel: Conversations with America to listen to those interviews, during which participants reflect on their personal hardships as well as on national events during the Great Depression.
BLACK AND POOR: AFRICAN AMERICANS AND THE GREAT DEPRESSION
Most African Americans did not participate in the land boom and stock market speculation that preceded the crash, but that did not stop the effects of the Great Depression from hitting them particularly hard.  As the prices for cotton and other agricultural products plummeted, farm owners paid workers less or simply laid them off. Landlords evicted sharecroppers, and even those who owned their land outright had to abandon it when there was no way to earn any income.
In cities, African Americans fared no better. Unemployment was rampant, and many whites felt that any available jobs belonged to whites first. In some Northern cities, whites would conspire to have African American workers fired to allow white workers access to their jobs. Even jobs traditionally held by black workers, such as household servants or janitors, were now going to whites. By 1932, approximately one-half of all black Americans were unemployed. Racial violence also began to rise. In the South, lynching became more common again, with twenty-eight documented lynchings in 1933, compared to eight in 1932. Since communities were preoccupied with their own hardships, and organizing civil rights efforts was a long, difficult process, many resigned themselves to, or even ignored, this culture of racism and violence. Occasionally, however, an incident was notorious enough to gain national attention.
One such incident was the case of the Scottsboro Boys (Figure 25.3.1). In 1931, nine black boys, who had been riding the rails, were arrested for vagrancy and disorderly conduct after an altercation with some white travelers on the train.  The case, which was tried in Scottsboro, Alabama, reignited decades of racial hatred and illustrated the injustice of the court system. Despite significant evidence that the women had not been raped at all, along with one of the women subsequently recanting her testimony, the all-white jury quickly convicted the boys and sentenced all but one of them to death. The verdict broke through the veil of indifference toward the plight of African Americans, and protests erupted among newspaper editors, academics, and social reformers in the North. The Communist Party of the United States offered to handle the case and sought retrial; the NAACP later joined in this effort. In all, the case was tried three separate times. The series of trials and retrials, appeals, and overturned convictions shone a spotlight on a system that provided poor legal counsel and relied on all-white juries. In October 1932, the U.S. Supreme Court agreed with the Communist Party’s defense attorneys that the defendants had been denied adequate legal representation at the original trial, and that due process as provided by the Fourteenth Amendment had been denied as a result of the exclusion of any potential black jurors. Eventually, most of the accused received lengthy prison terms and subsequent parole, but avoided the death penalty. The Scottsboro case ultimately laid some of the early groundwork for the modern American civil rights movement. Alabama granted posthumous pardons to all defendants in 2013.
Read Voices from Scottsboro for the perspectives of both participants and spectators in the Scottsboro case, from the initial trial to the moment, in 1976, when one of the women sued for slander.
ENVIRONMENTAL CATASTROPHE MEETS ECONOMIC HARDSHIP: THE DUST BOWL
Despite the widely held belief that rural Americans suffered less in the Great Depression due to their ability to at least grow their own food, this was not the case. Farmers, ranchers, and their families suffered more than any group other than African Americans during the Depression.
From the turn of the century through much of World War I, farmers in the Great Plains experienced prosperity due to unusually good growing conditions, high commodity prices, and generous government farming policies that led to a rush for land. As the federal government continued to purchase all excess produce for the war effort, farmers and ranchers fell into several bad practices, including mortgaging their farms and borrowing money against future production in order to expand. However, after the war, prosperity rapidly dwindled, particularly during the recession of 1921. Seeking to recoup their losses through economies of scale in which they would expand their production even further to take full advantage of their available land and machinery, farmers plowed under native grasses to plant acre after acre of wheat, with little regard for the long-term repercussions to the soil. Regardless of these misguided efforts, commodity prices continued to drop, finally plummeting in 1929, when the price of wheat dropped from two dollars to forty cents per bushel.
Exacerbating the problem was a massive drought that began in 1931 and lasted for eight terrible years. Dust storms roiled through the Great Plains, creating huge, choking clouds that piled up in doorways and filtered into homes through closed windows. Even more quickly than it had boomed, the land of agricultural opportunity went bust, due to widespread overproduction and overuse of the land, as well as to the harsh weather conditions that followed, resulting in the creation of the Dust Bowl(Figure 25.3.2).
Livestock died, or had to be sold, as there was no money for feed. Crops intended to feed the family withered and died in the drought. Terrifying dust storms became more and more frequent, as “black blizzards” of dirt blew across the landscape and created a new illness known as “dust pneumonia.” In 1935 alone, over 850 million tons of topsoil blew away. To put this number in perspective, geologists estimate that it takes the earth five hundred years to naturally regenerate one inch of topsoil; yet, just one significant dust storm could destroy a similar amount. In their desperation to get more from the land, farmers had stripped it of the delicate balance that kept it healthy. Unaware of the consequences, they had moved away from such traditional practices as crop rotation and allowing land to regain its strength by permitting it to lie fallow between plantings, working the land to death.
For farmers, the results were catastrophic. Unlike most factory workers in the cities, in most cases, farmers lost their homes when they lost their livelihood. Most farms and ranches were originally mortgaged to small country banks that understood the dynamics of farming, but as these banks failed, they often sold rural mortgages to larger eastern banks that were less concerned with the specifics of farm life. With the effects of the drought and low commodity prices, farmers could not pay their local banks, which in turn lacked funds to pay the large urban banks. Ultimately, the large banks foreclosed on the farms, often swallowing up the small country banks in the process. It is worth noting that of the five thousand banks that closed between 1930 and 1932, over 75 percent were country banks in locations with populations under 2,500. Given this dynamic, it is easy to see why farmers in the Great Plains remained wary of big city bankers.
For farmers who survived the initial crash, the situation worsened, particularly in the Great Plains where years of overproduction and rapidly declining commodity prices took their toll. Prices continued to decline, and as farmers tried to stay afloat, they produced still more crops, which drove prices even lower. Farms failed at an astounding rate, and farmers sold out at rock-bottom prices. One farm in Shelby, Nebraska was mortgaged at $4,100 and sold for $49.50. One-fourth of the entire state of Mississippi was auctioned off in a single day at a foreclosure auction in April 1932.
Not all farmers tried to keep their land. Many, especially those who had arrived only recently, in an attempt to capitalize on the earlier prosperity, simply walked away (Figure 25.3.3). In hard-hit Oklahoma, thousands of farmers packed up what they could and walked or drove away from the land they thought would be their future. They, along with other displaced farmers from throughout the Great Plains, became known asOkies. Okies were an emblem of the failure of the American breadbasket to deliver on its promise, and their story was made famous in John Steinbeck’s novel, The Grapes of Wrath .
Experience the Interactive Dust Bowl to see how decisions compounded to create peoples’ destiny. Click through to see what choices you would make and where that would take you.
Now we are facing a fourth year of failure. There can be no wheat for us in 1935 in spite of all our careful and expensive work in preparing ground, sowing and re-sowing our allocated acreage. Native grass pastures are permanently damaged, in many cases hopelessly ruined, smothered under by drifted sand. Fences are buried under banks of thistles and hard packed earth or undermined by the eroding action of the wind and lying flat on the ground. Less traveled roads are impassable, covered deep under by sand or the finer silt-like loam.  Impossible it seems not to grieve that the work of hands should prove so perishable. —Caroline Henderson, Shelton, Oklahoma, 1935
Much like other farm families whose livelihoods were destroyed by the Dust Bowl, Caroline Henderson describes a level of hardship that many Americans living in Depression-ravaged cities could never understand. Despite their hard work, millions of Americans were losing both their produce and their homes, sometimes in as little as forty-eight hours, to environmental catastrophes. Lacking any other explanation, many began to question what they had done to incur God’s wrath. Note in particular Henderson’s references to “dead,” “dying,” and “perishable,” and contrast those terms with her depiction of the “careful and expensive work” undertaken by their own hands. Many simply could not understand how such a catastrophe could have occurred.
CHANGING VALUES, CHANGING CULTURE
In the decades before the Great Depression, and particularly in the 1920s, American culture largely reflected the values of individualism, self-reliance, and material success through competition. Novels like F. Scott Fitzgerald’s The Great Gatsby and Sinclair Lewis’s Babbit portrayed wealth and the self-made man in America, albeit in a critical fashion. In film, many silent movies, such as Charlie Chaplin’s The Gold Rush , depicted the rags-to-riches fable that Americans so loved. With the shift in U.S. fortunes, however, came a shift in values, and with it, a new cultural reflection. The arts revealed a new emphasis on the welfare of the whole and the importance of community in preserving family life. While box office sales briefly declined at the beginning of the Depression, they quickly rebounded. Movies offered a way for Americans to think of better times, and people were willing to pay twenty-five cents for a chance to escape, at least for a few hours.
Even more than escapism, other films at the close of the decade reflected on the sense of community and family values that Americans struggled to maintain throughout the entire Depression. John Ford’s screen version of Steinbeck’s The Grapes of Wrath came out in 1940, portraying the haunting story of the Joad family’s exodus from their Oklahoma farm to California in search of a better life. Their journey leads them to realize that they need to join a larger social movement—communism—dedicated to bettering the lives of all people.  The greater lesson learned was one of the strength of community in the face of individual adversity.
Another trope was that of the hard-working everyman against greedy banks and corporations. This was perhaps best portrayed in the movies of Frank Capra, whose Mr . Smith Goes to Washington was emblematic of his work. In this 1939 film, Jimmy Stewart plays a legislator sent to Washington to finish out the term of a deceased senator. While there, he fights corruption to ensure the construction of a boy’s camp in his hometown rather than a dam project that would only serve to line the pockets of a few. He ultimately engages in a two-day filibuster, standing up to the power players to do what’s right. The Depression era was a favorite of Capra’s to depict in his films, including It’s a Wonderful Life , released in 1946. In this film, Jimmy Stewart runs a family-owned savings and loan, which at one point faces a bank run similar to those seen in 1929–1930. In the end, community support helps Stewart retain his business and home against the unscrupulous actions of a wealthy banker who sought to bring ruin to his family.
They used to tell me I was building a dream, and so I followed the mob
When there was earth to plow or guns to bear, I was always there, right on the job
They used to tell me I was building a dream, with peace and glory ahead
Why should I be standing in line, just waiting for bread?
Once I built a railroad, I made it run, made it race against time
Once I built a railroad, now it’s done, Brother, can you spare a dime?
Once I built a tower up to the sun, brick and rivet and lime
Once I built a tower, now it’s done, Brother, can you spare a dime?
—Jay Gorney and “Yip” Harburg
“Brother, Can You Spare a Dime?” first appeared in 1932, written for the Broadway musical New Americana by Jay Gorney, a composer who based the song’s music on a Russian lullaby, and Edgar Yipsel “Yip” Harburg, a lyricist who would go on to win an Academy Award for the song “Over the Rainbow” from The Wizard of Oz (1939).
With its lyrics speaking to the plight of the common man during the Great Depression and the refrain appealing to the same sense of community later found in the films of Frank Capra, “Brother, Can You Spare a Dime?” quickly became the de facto anthem of the Great Depression. Recordings by Bing Crosby, Al Jolson, and Rudy Vallee all enjoyed tremendous popularity in the 1930s.
For more on “Brother Can You Spare a Dime?” and the Great Depression, visit ArtsEdge to explore the Kennedy Center’s digital resources and learn the “Story Behind the Song.”
Finally, there was a great deal of pure escapism in the popular culture of the Depression. Even the songs found in films reminded many viewers of the bygone days of prosperity and happiness, from Al Dubin and Henry Warren’s hit “We’re in the Money” to the popular “Happy Days are Here Again.” The latter eventually became the theme song of Franklin Roosevelt’s 1932 presidential campaign. People wanted to forget their worries and enjoy the madcap antics of the Marx Brothers, the youthful charm of Shirley Temple, the dazzling dances of Fred Astaire and Ginger Rogers (Figure 25.3.4), or the comforting morals of the Andy Hardy series. The Hardy series—nine films in all, produced by MGM from 1936 to 1940—starred Judy Garland and Mickey Rooney, and all followed the adventures of a small-town judge and his son. No matter what the challenge, it was never so big that it could not be solved with a musical production put on by the neighborhood kids, bringing together friends and family members in a warm display of community values.
All of these movies reinforced traditional American values, which suffered during these hard times, in part due to declining marriage and birth rates, and increased domestic violence.  While the birth rate was dropping, surveys in Fortune magazine in 1936–1937 found that two-thirds of college students favored birth control, and that 50 percent of men and 25 percent of women admitted to premarital sex, continuing a trend among younger Americans that had begun to emerge in the 1920s. Contraceptive sales soared during the decade, and again, culture reflected this shift. Blonde bombshell Mae West was famous for her sexual innuendoes, and her flirtatious persona was hugely popular, although it got her banned on radio broadcasts throughout the Midwest. Whether West or Garland, Chaplin or Stewart, American film continued to be a barometer of American values, and their challenges, through the decade.
Section Summary
The Great Depression affected huge segments of the American population—sixty million people by one estimate. But certain groups were hit harder than the rest. African Americans faced discrimination in finding employment, as white workers sought even low-wage jobs like housecleaning. Southern blacks moved away from their farms as crop prices failed, migrating en masse to Northern cities, which had little to offer them. Rural Americans were also badly hit. The eight-year drought that began shortly after the stock market crash exacerbated farmers’ and ranchers’ problems. The cultivation of greater amounts of acreage in the preceding decades meant that land was badly overworked, and the drought led to massive and terrible dust storms, creating the region’s nickname, the Dust Bowl. Some farmers tried to remain and buy up more land as neighbors went broke; others simply fled their failed farms and moved away, often to the large-scale migrant farms found in California, to search for a better life that few ever found. Maltreated by Californians who wished to avoid the unwanted competition for jobs that these “Okies” represented, many of the Dust Bowl farmers were left wandering as a result.
There was very little in the way of public assistance to help the poor. While private charities did what they could, the scale of the problem was too large for them to have any lasting effects. People learned to survive as best they could by sending their children out to beg, sharing clothing, and scrounging wood to feed the furnace. Those who could afford it turned to motion pictures for escape. Movies and books during the Great Depression reflected the shift in American cultural norms, away from rugged individualism toward a more community-based lifestyle.
Review Questions
Which of the following hardships did African Americans not typically face during the Great Depression?
lower farm wages in the South
the belief that white workers needed jobs more than their black counterparts
white workers taking historically “black” jobs, such as maids and janitors
widespread race riots in large urban centers
D
Which of the following was not a key factor in the conditions that led to the Dust Bowl?
previous overcultivation of farmland
decreasing American demand for farm produce
unfavorable weather conditions
poor farming techniques regarding proper irrigation and acreage rotation
B
What did the popular movies of the Depression reveal about American values at that time? How did these values contrast with the values Americans held before the Depression?
American films in the 1930s served to both assuage the fears and frustrations of many Americans suffering through the Depression and reinforce the idea that communal efforts—town and friends working together—would help to address the hardships. Previous emphasis upon competition and individualism slowly gave way to notions of “neighbor helping neighbor” and seeking group solutions to common problems. The Andy Hardy series, in particular, combined entertainment with the concept of family coming together to solve shared problems. The themes of greed, competition, and capitalist-driven market decisions no longer commanded a large audience among American moviegoers.
Glossary
- Dust Bowl
- the area in the middle of the country that had been badly overfarmed in the 1920s and suffered from a terrible drought that coincided with the Great Depression; the name came from the “black blizzard” of topsoil and dust that blew through the area
- Scottsboro Boys
- a reference to the infamous trial in Scottsboro, Alabama in 1931, where nine African American boys were falsely accused of raping two white women and sentenced to death; the extreme injustice of the trial, particularly given the age of the boys and the inadequacy of the testimony against them, garnered national and international attention


17: Evolution and the Origin of Species
The theory of evolution is the unifying theory of biology, meaning it is the framework within which biologists ask questions about the living world. Its power is that it provides direction for predictions about living things that are borne out in experiment after experiment. The Ukrainian-born American geneticist Theodosius Dobzhansky famously wrote that “nothing makes sense in biology except in the light of evolution. " He meant that the tenet that all life has evolved and diversified from a common ancestor is the foundation from which we approach all questions in biology.
-
- 17.2: Understanding Evolution
- Evolution by natural selection describes a mechanism for how species change over time. That species change had been suggested and debated well before Darwin began to explore this idea. The view that species were static and unchanging was grounded in the writings of Plato, yet there were also ancient Greeks who expressed evolutionary ideas.
Contributors and Attributions
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Connie Rye (East Mississippi Community College), Robert Wise (University of Wisconsin, Oshkosh), Vladimir Jurukovski (Suffolk County Community College), Jean DeSaix (University of North Carolina at Chapel Hill), Jung Choi (Georgia Institute of Technology), Yael Avissar (Rhode Island College) among other contributing authors. Original content by OpenStax (CC BY 4.0; Download for free at<EMAIL_ADDRESS>).


8: Gender and Education
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At the end of this chapter, you will be able to do the following:
Decades of sociological research has been dedicated to educational attainment and differences in educational attainment. With the believed correlation between educational attainment and occupational attainment (and with it, so many other opportunities for social mobility), connections between other statuses that produce differential social capital, like sex and race, and education have inspired sociological research. The spread of mass education in the 19th and 20th centuries offered opportunities for social mobility that were previously unavailable to many groups. However, especially in its earliest years, the spread of mass education also presented the spread of unequal access to adequate education. Historically and today, race, class, and gender remain influential factors on access to education and educational attainment.


3.22: Linear Relationships (3 of 4)
Learning Objectives
- Use a correlation coefficient to describe the direction and strength of a linear relationship. Recognize its limitations as a measure of the relationship between two quantitative variables.
Now we interpret the value of r in the context of some familiar examples.
Example
Highway Sign
In a previous example, we looked at this scatterplot to investigate the relationship between the age of a driver and the maximum distance at which the driver can read a highway sign. Because the form of the relationship is linear, we can use the correlation coefficient as a measure of direction and strength of the linear relationship.
The r -value is −0.793. The r -value is negative ( r < 0), which means that the linear relationship has a negative direction. We can see this in the scatterplot. Because r is somewhat close to −1, the relationship is moderately strong.
In the context of the data, the negative correlation confirms that the maximum reading distance decreases with age. Because r indicates a moderately strong linear relationship, we expect that drivers of similar age will have some (but not a lot) of variability in their maximum reading distance.
Example
Biology Courses
A biology department is interested in tracking the progress of its students from entry until graduation. As part of the study, the department tabulates the performance of 10 students in an introductory course and later in an upper-level course required for graduation. What is the relationship between the students’ course grades in the two courses? Here are two scatterplots of the same data.
Both scatterplots show a relationship that is positive in direction and linear in form. The strength appears different in the two scatterplots because of the difference in scales. This illustrates why we support our visual assessment of strength with a measurement of strength. We can use the correlation coefficient as a measure of the strength of the linear relationship. The correlation coefficient is r = 0.91, which is close to 1. The correlation coefficient confirms that the linear relationship is very strong.
Comment
Note that in both examples, we supplemented the scatterplot with the correlation ( r ). Now that we have the correlation, why do we still need to look at a scatterplot when examining the relationship between two quantitative variables?
The correlation coefficient can be interpreted only as the measure of the strength of a linear relationship , so we need the scatterplot to verify that the relationship indeed looks linear. This point and its importance will be clearer after we examine a few properties of r .
Contributors and Attributions
- Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution


15.21G: West Nile Virus
- Discuss the causes, symptoms and diseases (West Nile encephalitis, meningitis, meningoencephalitis and poliomyelitis) caused by the West Nile virus (WNV)
West Nile virus (WNV) is a mosquito-borne zoonotic arbovirus belonging to the genus Flavivirus, and is found in temperate and tropical regions of the world.
WNV was first identified in the West Nile subregion in the East African nation of Uganda in 1937. Prior to the mid 1990s, WNV disease occurred only sporadically and was considered a minor risk for humans. This was until an outbreak in Algeria in 1994, with cases of WNV-caused encephalitis, and the first large outbreak in Romania in 1996, with a high number of cases with neuroinvasive disease. WNV has now spread globally, with the first case in the Western Hemisphere being identified in New York City in 1999. The virus has now spread across the continental United States, north into Canada, and southward into the Caribbean Islands and Latin America. The US experienced one of its worst epidemics to date in 2012. WNV also spread to Europe, beyond the Mediterranean Basin, with a new strain of the virus recently identified in Italy (2012). WNV is now considered to be an endemic pathogen in Africa, Asia, Australia, the Middle East, Europe and in the United States.
The main mode of WNV transmission is by mosquitoes, the prime vector. WNV has been found in various species of ticks. However, current research suggests they are not important vectors of the virus. WNV infects various mammal species, including humans. It has also been identified in reptilians (including alligators and crocodiles) and amphibians. Birds, especially passerines, are the most commonly infected animal, and serve as the prime reservoir host. Many species – including humans – do not develop viral levels sufficient to transmit the disease to uninfected mosquitoes, and are thus not considered major factors in WNV transmission.
Approximately 80% of West Nile virus infections in humans cause no symptoms. West Nile fever is the manifestation of symptoms, with an incubation period typically between 2 and 15 days. Symptoms may include fever, headaches, fatigue, muscle pain or aches, malaise, nausea, anorexia, vomiting, myalgias and rash. Less than 1% of the cases are severe and result in neurological disease when the central nervous system is affected. People of advanced age, the very young, or those with immunosuppression are most susceptible.
The specific neurological diseases which may occur are:
- West Nile encephalitis, which causes inflammation of the brain
- West Nile meningitis, which causes inflammation of the meninges (the protective membranes that cover the brain and spinal cord)
- West Nile meningoencephalitis, which causes inflammation of the brain and surrounding meninges
- West Nile poliomyelitis (spinal cord inflammation, which results in a syndrome similar to polio that may cause acute flaccid paralysis).
Currently, no vaccine against WNV infection is available. The best method to reduce the rates of WNV infection is by public mosquito control (particularly the elimination of standing water) and by personal protection (mosquito nets and repellent).
Key Points
- WNV is considered to be an endemic pathogen in Africa, Asia, Australia, the Middle East, Europe and in the United States, with one of the worst epidemics occurring in 2012.
- Birds are the most commonly infected animal, and serve as the prime reservoir host.
- The specific neurological diseases which may occur are encephalitis, meningitis, meningoencephalitis, and poliomyelitis.
Key Terms
- zoonotic : of or relating to zoonosis, the transmission of an infectious disease between species.


5.7: Copyright infringement versus plagiarism
Because plagiarism and copyright infringement are both related to the ethical reuse of ideas and content, it can be confusing to understand where one ends and the other begins. One way to differentiate between the two is to examine the consequences of each.
When someone commits copyright infringement by sharing materials without permission from the copyright holder, this is a legal offense and can incur financial penalties. The consequences of these actions could include having something taken down from the internet, fines, or even jail time.
In contrast, when someone commits plagiarism, they haven’t broken any state or federal law and can’t be thrown in jail for that offense. However, because plagiarism involves stealing someone else’s ideas, it’s still unethical. For students, this can lead to failing a course or being kicked out of college. After graduation, plagiarism could lead to the loss of professional credibility and even get them fired.
Copyright infringement in the context of research
Now that you understand some basics, let’s compare plagiarism and copyright infringement in the context of being a college student. Building off of the works of other researchers is a core component of scholarship. You don’t need anyone’s permission to quote or cite their work, but you do need to carefully cite your sources to avoid committing plagiarism. Avoiding copyright infringement is more complicated.
Let’s say you want to include some copyrighted photographs (not your own) in a paper you’re writing for a class assignment. Your professor will read your paper, and perhaps your paper will be discussed in class, but it will not be shared outside of the classroom. You carefully cite the photographs in your paper, so you know you are not plagiarizing, but are you infringing on copyright? No, this is not considered copyright infringement because the photos will only be shared in your classroom.
The situation is very different if you want to reuse those photos outside of your class. Not only would you need to cite the images properly, but you would also need to contact the copyright holder and ask for permission to reuse their photographs. The copyright holder would get to decide whether to grant that permission and determine any fees or conditions for reuse. Failing to comply with those conditions would be copyright infringement and put you at risk to be sued.


1.6: An Electric Field Computation
We have established that vectors may be used to code complex numbers. Conversely, complex numbers may be used to code or represent the orthogonal components of any two-dimensional vector. This makes them invaluable in electromagnetic field theory, where they are used to represent the components of electric and magnetic fields.
The basic problem in electromagnetic field theory is to determine the electric or magnetic field that is generated by a static or dynamic distribution of charge. The key idea is to isolate an infinitesimal charge, determine the field set up by this charge, and then to sum the fields contributed by all such infinitesimal charges. This idea is illustrated in Figure, where the charge \(λ\), uniformly distributed over a line segment of length \(dx\) at point \(−x\), produces a field \(dE(x)\) at the test point \((0,h)\). The field \(dE(x)\) is a “vector” field (as opposed to a “scalar” field), with components \(E1(x)\) and \(E2(x)\). The intensity or field strength of the field \(dE(x)\) is
\[|dE(x)|=\frac {λdx} {4πϵ_0(h^2+x^2)} \nonumber \]
But the field strength is directed at angle \(θ(x)\), as illustrated in Figure. The field \(dE(x)\) is real with components \(dE_1(x)\) and \(dE_2(x)\), but we code it as a complex field. We say that the “complex” field at test point \((0,h)\) is
\[dE(x)=\frac {λdx} {4πϵ_0(h^2+x^2)} e^{jθ(x)} \nonumber \]
with components \(dE_1(x)\) and \(dE_2(x)\). That is,
\[dE(x)=dE_1(x)+jdE_2(x) \nonumber \]
\[dE_1(x)=\frac {λdx} {4πϵ_0(h^2+x^2)} \cosθ(x) \nonumber \]
\[dE_2(x)=\frac {λdx} {4πϵ_0(h^2+x^2)} \sinθ(x) \nonumber \]
For charge uniformally distributed with density \(λ\) along the x-axis, the total field at the test point \((0,h)\) is obtained by integrating \(dE\)
\[∫^∞_{−∞}dE(x)=∫^∞_{−∞} \frac {λ} {4πϵ_0(h^2+x^2)} [\cosθ(x)+j\sinθ(x)]dx \nonumber \]
The functions \(\cosθ(x)\) and \(\sinθ(x)\) are
\[\cosθ(x)=\frac {x} {(x^2+h^2)^{1/2}} ; \sinθ(x)=\frac {h} {(x^2+h^2)^{1/2}} \nonumber \]
We leave it as a problem to show that the real component \(E_1\) of the field is zero. The imaginary component \(E_2\) is
\[E=jE_2=j∫^∞_{−∞} \frac {λh} {4πϵ_0} \frac {dx} {(x^2+h^2)^{3/2}} \nonumber \]
\[=j\frac {λh} {4πϵ_0} \frac {x} {h^2(x^2+h^2)^{1/2}}∣^∞_{−∞} \nonumber \]
\[=j\frac {λh} {4πϵ_0} [\frac 1 {h^2} + \frac 1 {h_2}] = j\frac {λ} {2πϵ_0h} \nonumber \]
\[E_2=\frac λ {2πϵ_0h} \nonumber \]
We emphasize that the field at (0,h) is a real field. Our imaginary answer simply says that the real field is oriented in the vertical direction because we have used the imaginary part of the complex field to code the vertical component of the real field.
Show that the horizontal component of the field \(E\) is zero. Interpret this finding physically.
From the symmetry of this problem, we conclude that the field around the infinitely long wire of the Figure is radially symmetric. So, in polar coordinates, we could say
\[E(r,θ)=\frac λ {2πϵ_0r} \nonumber \]
which is independent of θ. If we integrated the field along a radial line perpendicular to the wire, we would measure the voltage difference
\[V(r_1)−V(r_0)=∫^{r_1}_{r_0}\frac λ {2πϵ_0r}dr=\frac λ {2πϵ_0}[\mathrm{log}r_1−\mathrm{log}r_0] \nonumber \]
An electric field has units of volts/meter, a charge density λ has units of coulombs/meter, and \(ϵ_0\) has units of coulombs/volt-meter; voltage has units of volts (of course).


8.14: Sources of Magnetic Fields (Answers)
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Check Your Understanding
12.1. 1.41 meters
12.2. \(\displaystyle \frac{μ_0I}{2R}\)
12.3. 4 amps flowing out of the page
12.4. Both have a force per unit length of \(\displaystyle 9.23×10^{−12}N/m\)
12.5. 0.608 meters
12.6. In these cases the integrals around the Ampèrian loop are very difficult because there is no symmetry, so this method would not be useful.
12.7. a. 1.00382;
b. 1.00015
12.8. a. \(\displaystyle 1.0×10^{−4}T\);
b. 0.60 T;
c. \(\displaystyle 6.0×10^3\)
Conceptual Questions
1. Biot-Savart law’s advantage is that it works with any magnetic field produced by a current loop. The disadvantage is that it can take a long time.
3. If you were to go to the start of a line segment and calculate the angle \(\displaystyle θ\) to be approximately \(\displaystyle 0°\), the wire can be considered infinite. This judgment is based also on the precision you need in the result.
5. You would make sure the currents flow perpendicular to one another.
7. A magnetic field line gives the direction of the magnetic field at any point in space. The density of magnetic field lines indicates the strength of the magnetic field.
9. The spring reduces in length since each coil with have a north pole-produced magnetic field next to a south pole of the next coil.
11. Ampère’s law is valid for all closed paths, but it is not useful for calculating fields when the magnetic field produced lacks symmetry that can be exploited by a suitable choice of path.
13. If there is no current inside the loop, there is no magnetic field (see Ampère’s law). Outside the pipe, there may be an enclosed current through the copper pipe, so the magnetic field may not be zero outside the pipe.
15. The bar magnet will then become two magnets, each with their own north and south poles. There are no magnetic monopoles or single pole magnets.
Problems
17. \(\displaystyle 5.66×10^{−5}T\)
19. \(\displaystyle B=\frac{μ_oI}{8}(\frac{1}{a}−\frac{1}{b})\) out of the page
21. \(\displaystyle a=\frac{2R}{π}\); the current in the wire to the right must flow up the page.
23. 20 A
25. Both answers have the magnitude of magnetic field of \(\displaystyle 4.5×10^{−5}T\).
27. At P1, the net magnetic field is zero. At P2, \(\displaystyle B=\frac{3μ_oI}{8πa}\) into the page.
29. The magnetic field is at a minimum at distance a from the top wire, or half-way between the wires.
31. a. \(\displaystyle F/l=8×10^{−6}\) N/m away from the other wire;
b. \(\displaystyle F/l=8×10^{−6}\) N/m toward the other wire
33. \(\displaystyle B=\frac{μ_oIa}{2πb^2}\) into the page
35. 0.019 m
37. \(\displaystyle 6.28×10^{−5}T\)
39. \(\displaystyle B=\frac{\mu_{o} I R^{2}}{\left(\left(\frac{d}{2}\right)^{2}+R^{2}\right)^{3 / 2}}\)
41. a. \(\displaystyle μ0I;\)
b. 0;
c. \(\displaystyle μ0I\);
d. 0
43. a. \(\displaystyle 3μ_0I\);
b. 0;
c. \(\displaystyle 7μ_0I\);
d. \(\displaystyle −2μ_0I\)
45. at the radius R
47.
49. \(\displaystyle B=1.3×10^{−2}T\)
51. roughly eight turns per cm
53. \(\displaystyle B=\frac{1}{2}μ_0nI\)
55. 0.0181 A
57. 0.0008 T
59. 317.31
61. \(\displaystyle 2.1×10^{−4}A⋅m^2\) \(\displaystyle 2.7A\)
63. 0.18 T
Additional Problems
65. \(\displaystyle B=6.93×10^{−5}T\)
67. \(\displaystyle 3.2×10^{−19}N\) in an arc away from the wire
69. a. above and below \(\displaystyle B=μ_0j\), in the middle \(\displaystyle B=0\);
b. above and below \(\displaystyle B=0\), in the middle \(\displaystyle B=μ_0j\)
71. \(\displaystyle \frac{dB}{B}=−\frac{dr}{r}\)
73. a. 52778 turns;
b. 0.10 T
75. \(\displaystyle B_1(x)=\frac{μ_0IR^2}{2(R^2+z^2)^{3/2}}\)
77. \(\displaystyle B=\frac{μ_0σω}{2}R\)
79. derivation
81. derivation
83. As the radial distance goes to infinity, the magnetic fields of each of these formulae go to zero.
85. a. \(\displaystyle B=\frac{μ_0I}{2πr}\);
b. \(\displaystyle B=\frac{μ_0J_0r^2}{3R}\)
87. \(\displaystyle B(r)=μ_0NI/2πr\)
Challenge Problems
89. \(\displaystyle B=\frac{μ_0I}{2πx}\).
91. a. \(\displaystyle B=\frac{μ_0σω}{2}[\frac{2h^2+R^2}{\sqrt{R^2+h^0}}]\);
b. \(\displaystyle B=4.09×10^{−5}T\), 82% of Earth’s magnetic field


16.1: Introduction
Urban geography is a subfield of geography that studies cities and urban areas. Urban geographers examine the physical, social, and economic dimensions of urban environments, and how they interact with each other. They also study the processes that shape urban development and evolution of them, and the impact of urbanization on the environment and society, (which includes ethnically and economically).
Urban geography is a broad field, and there are many different subtopics that urban geographer’s study. Some of the most common subtopics include:
- Urban Form: This refers to the physical layout of cities, including the size and shape of streets, blocks, and buildings. Urban geographers study how urban form is influenced by factors such as history, culture, and economic development.
- Urban Land Use: This refers to the way that land is used in cities, such as for residential, commercial, or industrial purposes. Urban geographers study how urban land use is influenced by factors such as transportation, accessibility, and environmental concerns.
- Urban Transportation: This refers to the movement of people and goods within and between cities. Urban geographers study how urban transportation systems are designed and operated, and how they impact the environment and society.
- Urban Social Geography: This refers to the social characteristics of cities, such as population density, ethnicity, and income inequality. Urban geographers study how these social characteristics are distributed across urban space, and how they shape people's lives.
- Urban Environmental Geography: This refers to the environmental impacts of cities, such as air pollution, water pollution, and climate change. Urban geographers study how cities interact with their natural environment, and how they can be made more sustainable.


12.3: Reading- Ore Genesis
The various theories of ore genesis explain how the various types of mineral deposits form within the Earth’s crust. Ore genesis theories are dependent on the mineral or commodity.
Ore genesis theories generally involve three components: source, transport or conduit, and trap. This also applies to the petroleum industry, which was first to use this methodology.
- Source is required because metal must come from somewhere, and be liberated by some process
- Transport is required first to move the metal bearing fluids or solid minerals into the right position, and refers to the act of physically moving the metal, as well as chemical or physical phenomenon which encourage movement
- Trapping is required to concentrate the metal via some physical, chemical or geological mechanism into a concentration which forms mineable ore
The biggest deposits are formed when the source is large, the transport mechanism is efficient, and the trap is active and ready at the right time.
Ore Genesis Processes
Endogenous
Magmatic Processes
- Fractional crystallization: separates ore and non-ore minerals according to their crystallization temperature. As early crystallizing minerals form, they incorporate certain elements, some of which are metals. These crystals may settle onto the bottom of the intrusion, concentrating ore minerals there. Chromite and magnetite are ore minerals that form in this way.
- Liquid immiscibility: sulfide ores containing copper, nickel or platinum may form from this process. As a magma changes, parts of it may separate from the main body of magma. Two liquids that will not mix are called immiscible; oil and water are an example. In magmas, sulfides may separate and sink below the silicate-rich part of the intrusion or be injected into the rock surrounding it. These deposits are found in mafic and ultramafic rocks.
Hydrothermal Processes
These processes are the physicochemical phenomena and reactions caused by movement of hydrothermal water within the crust, often as a consequence of magmatic intrusion or tectonic upheavals. The foundations of hydrothermal processes are the source-transport-trap mechanism.
Sources of hydrothermal solutions include seawater and meteoric water circulating through fractured rock, formational brines (water trapped within sediments at deposition) and metamorphic fluids created by dehydration of hydrous minerals during metamorphism.
Metal sources may include a plethora of rocks. However most metals of economic importance are carried as trace elements within rock-forming minerals, and so may be liberated by hydrothermal processes. This happens because of:
- incompatibility of the metal with its host mineral, for example zinc in calcite, which favours aqueous fluids in contact with the host mineral during diagenesis.
- solubility of the host mineral within nascent hydrothermal solutions in the source rocks, for example mineral salts (halite), carbonates (cerussite), phosphates (monazite and thorianite) and sulfates (barite)
- elevated temperatures causing decomposition reactions of minerals
Transport by hydrothermal solutions usually requires a salt or other soluble species which can form a metal-bearing complex. These metal-bearing complexes facilitate transport of metals within aqueous solutions, generally as hydroxides, but also by processes similar to chelation.
This process is especially well understood in gold metallogeny where various thiosulfate, chloride and other gold-carrying chemical complexes (notably tellurium-chloride/sulfate or antimony-chloride/sulfate). The majority of metal deposits formed by hydrothermal processes include sulfide minerals, indicating sulfur is an important metal-carrying complex.
Sulfide deposition: Sulfide deposition within the trap zone occurs when metal-carrying sulfate, sulfide or other complexes become chemically unstable due to one or more of the following processes;
- falling temperature, which renders the complex unstable or metal insoluble
- loss of pressure, which has the same effect
- reaction with chemically reactive wall rocks, usually of reduced oxidation state, such as iron bearing rocks, mafic or ultramafic rocks or carbonate rocks
- degassing of the hydrothermal fluid into a gas and water system, or boiling, which alters the metal carrying capacity of the solution and even destroys metal-carrying chemical complexes
Metal can also become precipitated when temperature and pressure or oxidation state favour different ionic complexes in the water, for instance the change from sulfide to sulfate, oxygen fugacity, exchange of metals between sulfide and chloride complexes, et cetera.
Metamorphic Processes
Lateral secretion: Ore deposits formed by lateral secretion are formed by metamorphic reactions during shearing, which liberate mineral constituents such as quartz, sulfides, gold, carbonates and oxides from deforming rocks and focus these constituents into zones of reduced pressure or dilation such as faults. This may occur without much hydrothermal fluid flow, and this is typical of podiform chromite deposits.
Metamorphic processes also control many physical processes which form the source of hydrothermal fluids, outlined above.
Surficial Processes (Exogenous)
Surficial processes are the physical and chemical phenomena which cause concentration of ore material within the regolith, generally by the action of the environment. This includes placerdeposits, laterite deposits and residual or eluvial deposits. The physical processes of ore deposit formation in the surficial realm include;
- erosion
- deposition by sedimentary processes, including winnowing, density separation (e.g.; gold placers)
- weathering via oxidation or chemical attack of a rock, either liberating rock fragments or creating chemically deposited clays, laterites or supergene enrichment
- Deposition in low-energy environments in beach environments
Classification of Ore Deposits
Ore deposits are usually classified by ore formation processes and geological setting. For example, SEDEX deposits, literally meaning “sedimentary exhalative” are a class of ore deposit formed on the sea floor (sedimentary) by exhalation of brines into seawater (exhalative), causing chemical precipitation of ore minerals when the brine cools, mixes with sea water and loses its metal carrying capacity.
Ore deposits rarely fit snugly into the boxes in which geologists wish to place them. Many may be formed by one or more of the basic genesis processes above, creating ambiguous classifications and much argument and conjecture. Often ore deposits are classified after examples of their type, for instance Broken Hill type lead-zinc-silver deposits or Carlin–type gold deposits.
Classification of hydrothermal ore deposits is also achieved by classifying according to the temperature of formation, which roughly also correlates with particular mineralising fluids, mineral associations and structural styles. This scheme, proposed by Waldemar Lindgren (1933) classified hydrothermal deposits as hypothermal , mesothermal , epithermal and telethermal .
Genesis of Common Ores
As they require the conjunction of specific environmental conditions to form, particular mineral deposit types tend to occupy specific geodynamic niches, [1] therefore, this page has been organised by metal commodity. It is also possible to organise theories the other way, namely according to geological criteria of formation. Often ores of the same metal can be formed by multiple processes, and this is described here under each metal or metal complex.
Iron
Iron ores are overwhelmingly derived from ancient sediments known as banded iron formations (BIFs). These sediments are composed of iron oxide minerals deposited on the sea floor. Particular environmental conditions are needed to transport enough iron in sea water to form these deposits, such as acidic and oxygen-poor atmospheres within the Proterozoic Era.
Often, more recent weathering is required to convert the usual magnetite minerals into more easily processed hematite. Some iron deposits within the Pilbara of West Australia are placer deposits , formed by accumulation of hematite gravels called pisolites which form channel-iron deposits. These are preferred because they are cheap to mine.
Lead Zinc Silver
Lead-zinc deposits are generally accompanied by silver, hosted within the lead sulfide mineral galena or within the zinc sulfide mineral sphalerite.
Lead and zinc deposits are formed by discharge of deep sedimentary brine onto the sea floor (termed sedimentary exhalative or SEDEX), or by replacement of limestone, in skarn deposits, some associated with submarine volcanoes (called volcanogenic massive sulfide ore deposits or VMS) or in the aureole of subvolcanic intrusions of granite. The vast majority of SEDEX lead and zinc deposits are Proterozoic in age, although there are significant Jurassic examples in Canada and Alaska.
The carbonate replacement type deposit is exemplified by the Mississippi valley type (MVT) ore deposits. MVT and similar styles occur by replacement and degradation of carbonate sequences byhydrocarbons, which are thought important for transporting lead.
Gold
Gold deposits are formed via a very wide variety of geological processes. Deposits are classified as primary, alluvial or placerdeposits, or residual or laterite deposits. Often a deposit will contain a mixture of all three types of ore.
Plate tectonics is the underlying mechanism for generating gold deposits. The majority of primary gold deposits fall into two main categories: lode gold deposits or intrusion-related deposits.
Lode gold deposits are generally high-grade, thin, vein and fault hosted. They are primarily made up of quartz veins also known as lodes or reefs , which contain either native gold or gold sulfides and tellurides. Lode gold deposits are usually hosted in basalt or in sediments known as turbidite, although when in faults, they may occupy intrusive igneous rocks such as granite.
Lode-gold deposits are intimately associated with orogeny and other plate collision events within geologic history. Most lode gold deposits sourced from metamorphic rocks because it is thought that the majority are formed by dehydration of basalt during metamorphism. The gold is transported up faults by hydrothermal waters and deposited when the water cools too much to retain gold in solution.
Intrusive related gold is generally hosted in granites, porphyry or rarely dikes. Intrusive related gold usually also contains copper, and is often associated with tin and tungsten, and rarely molybdenum, antimony and uranium. Intrusive-related gold deposits rely on gold existing in the fluids associated with the magma (White, 2001), and the inevitable discharge of thesehydrothermal fluids into the wall-rocks (Lowenstern, 2001). Skarn deposits are another manifestation of intrusive-related deposits.
Placer deposits are sourced from pre-existing gold deposits and are secondary deposits. Placer deposits are formed by alluvialprocesses within rivers, streams and on beaches. Placer gold deposits form via gravity, with the density of gold causing it to sink into trap sites within the river bed, or where water velocity drops, such as bends in rivers and behind boulders. Often placer deposits are found within sedimentary rocks and can be billions of years old, for instance the Witwatersrand deposits in South Africa. Sedimentary placer deposits are known as “leads” or “deep leads.”
Placer deposits are often worked by fossicking, and panning for gold is a popular pastime.
Laterite gold deposits are formed from pre-existing gold deposits (including some placer deposits) during prolonged weathering of the bedrock. Gold is deposited within iron oxides in the weathered rock or regolith, and may be further enriched by reworking by erosion. Some laterite deposits are formed by wind erosion of the bedrock leaving a residuum of native gold metal at surface.
A bacterium, Cupriavidus metallidurans plays a vital role in the formation of gold nuggets, by precipitating metallic gold from a solution of gold (III) tetrachloride, a compound highly toxic to most other microorganisms. Similarly, Delftia acidovorans can form gold nuggets.
Platinum
Platinum and palladium are precious metals generally found in ultramafic rocks. The source of platinum and palladium deposits is ultramafic rocks which have enough sulfur to form a sulfide mineral while the magma is still liquid. This sulfide mineral (usually pentlandite, pyrite, chalcopyrite or pyrrhotite) gains platinum by mixing with the bulk of the magma because platinum is chalcophile and is concentrated in sulfides. Alternatively, platinum occurs in association with chromite either within the chromite mineral itself or within sulfides associated with it.
Sulfide phases only form in ultramafic magmas when the magma reaches sulfur saturation. This is generally thought to be nearly impossible by pure fractional crystallisation, so other processes are usually required in ore genesis models to explain sulfur saturation. These include contamination of the magma with crustal material, especially sulfur-rich wall-rocks or sediments; magma mixing; volatile gain or loss.
Often platinum is associated with nickel, copper, chromium, and cobalt deposits.
Nickel
Nickel deposits are generally found in two forms, either as sulfide or laterite.
Sulfide type nickel deposits are formed in essentially the same manner as platinum deposits. Nickel is a chalcophile element which prefers sulfides, so an ultramafic or mafic rock which has a sulfide phase in the magma may form nickel sulfides. The best nickel deposits are formed where sulfide accumulates in the base of lava tubes or volcanic flows—especially komatiite lavas.
Komatiitic nickel-copper sulfide deposits are considered to be formed by a mixture of sulfide segregation, immiscibility, and thermal erosion of sulfidic sediments. The sediments are considered to be necessary to promote sulfur saturation.
Some subvolcanic sills in the Thompson Belt of Canada host nickel sulfide deposits formed by deposition of sulfides near the feeder vent. Sulfide was accumulated near the vent due to the loss of magma velocity at the vent interface. The massive Voisey’s Bay nickel deposit is considered to have formed via a similar process.
The process of forming nickel laterite deposits is essentially similar to the formation of gold laterite deposits, except that ultramafic or mafic rocks are required. Generally nickel laterites require very large olivine-bearing ultramafic intrusions. Minerals formed in laterite nickel deposits include gibbsite.
Copper
Copper is found in association with many other metals and deposit styles. Commonly, copper is either formed within sedimentary rocks, or associated with igneous rocks.
The world’s major copper deposits are formed within the granitic porphyry copper style. Copper is enriched by processes during crystallisation of the granite and forms as chalcopyrite—a sulfide mineral, which is carried up with the granite.
Sometimes granites erupt to surface as volcanoes, and copper mineralisation forms during this phase when the granite and volcanic rocks cool via hydrothermal circulation.
Sedimentary copper forms within ocean basins in sedimentary rocks. Generally this forms by brine from deeply buried sediments discharging into the deep sea, and precipitating copper and oftenlead and zinc sulfides directly onto the sea floor. This is then buried by further sediment. This is a process similar to SEDEX zinc and lead, although some carbonate-hosted examples exist.
Often copper is associated with gold, lead, zinc and nickel deposits.
Uranium
Uranium deposits are usually sourced from radioactive granites, where certain minerals such as monazite are leached during hydrothermal activity or during circulation of groundwater. The uranium is brought into solution by acidic conditions and is deposited when this acidity is neutralised. Generally this occurs in certain carbon-bearing sediments, within an unconformity in sedimentary strata. The majority of the world’s nuclear power is sourced from uranium in such deposits.
Uranium is also found in nearly all coal at several parts per million, and in all granites. Radon is a common problem during mining of uranium as it is a radioactive gas.
Uranium is also found associated with certain igneous rocks, such as granite and porphyry. The Olympic Dam deposit in Australia is an example of this type of uranium deposit. It contains 70% of Australia’s share of 40% of the known global low-cost recoverable uranium inventory.
Titanium and Zirconium
Mineral sands are the predominant type of titanium, zirconium and thorium deposit. They are formed by accumulation of such heavy minerals within beach systems, and are a type of placer deposits . The minerals which contain titanium are ilmenite, rutile and leucoxene, zirconium is contained within zircon, and thorium is generally contained within monazite. These minerals are sourced from primarily granite bedrock by erosion and transported to the sea by rivers where they accumulate within beach sands. Rarely, but importantly, gold, tin and platinum deposits can form in beach placer deposits.
Tin, Tungsten, and Molybdenum
These three metals generally form in a certain type of granite, via a similar mechanism to intrusive-related gold and copper. They are considered together because the process of forming these deposits is essentially the same. Skarn type mineralisation related to these granites is a very important type of tin, tungsten and molybdenum deposit. Skarn deposits form by reaction of mineralised fluids from the granite reacting with wall rocks such as limestone. Skarn mineralisation is also important in lead, zinc, copper, gold and occasionally uranium mineralisation.
Greisen granite is another related tin-molybdenum and topaz mineralisation style.
Rare Earth Elements, Niobium, Tantalum, Lithium
The overwhelming majority of rare earth elements, tantalum and lithium are found within pegmatite. Ore genesis theories for these ores are wide and varied, but most involve metamorphism and igneous activity. Lithium is present as spodumene or lepidolite within pegmatite.
Carbonatite intrusions are an important source of these elements. Ore minerals are essentially part of the unusual mineralogy of carbonatite.
Phosphate
Phosphate is used in fertilisers. Immense quantities of phosphate rock or phosphorite occur in sedimentary shelf deposits, ranging in age from the Proterozoic to currently forming environments. Phosphate deposits are thought to be sourced from the skeletons of dead sea creatures which accumulated on the seafloor. Similar to iron ore deposits and oil, particular conditions in the ocean and environment are thought to have contributed to these deposits within the geological past.
Phosphate deposits are also formed from alkaline igneous rocks such as nepheline syenites, carbonatites and associated rock types. The phosphate is, in this case, contained within magmaticapatite, monazite or other rare-earth phosphates.
Vanadium
Due to the presence of vanabins, concentration of vanadium found in the blood cells of Ascidia gemmata belonging to the suborder Phlebobranchia is 10,000,000 times higher than that in the surrounding seawater. A similar biological process might have played a role in the formation of vanadium ores. Vanadium is also present in fossil fuel deposits such as crude oil, coal, oil shale and oil sands. In crude oil, concentrations up to 1200 ppm have been reported.
Contributors and Attributions
- Ore genesis. Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/Ore_genesis . License : CC BY-SA: Attribution-ShareAlike


22.3: Applications
Distributing/choosing indistinguishable objects
How many ways are there to distribute seven coins amongst three children? (Assume the coins are indistinguishable. But children are obviously distinguishable.)
Solution
Here is one scheme by which we can decide how many coins each child will get. Line the children up in some order. (There is no need to count the number of ways to do this — see the end of the solution.) Also lay out the coins in a line:
\begin{equation*} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \text{.} \end{equation*}
Now grab two Hickory Sticks™ from the snack table to act as dividers to split the coins up into three groups. For example,
\begin{equation*} \mathord{\circ} \mathord{\mid} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\mid} \mathord{\circ} \mathord{\circ} \end{equation*}
means that the first child will receive one coin, the second will receive four, and the third child will receive two, whereas
\begin{equation*} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\mid} \mathord{\mid} \end{equation*}
means that the first child gets all seven coins.
We are now back to the red bottle, blue bottle problem (see Worked Example 21.3.4 and Worked Example 22.2.2 ): how many different symbol patterns can we obtain by arranging two indistinguishable \(\mathord{\mid}\) symbols and seven indistinguishable \(\mathord{\circ}\) symbols? Just choose two of the nine available positions in the pattern to place the \(\mathord{\mid}\) symbols. And so we have arrived at the answer \(C(9, 2) = 36\text{.}\)
Now, why do we not have to take into account the ordering of the children at the beginning? Let \(c_1, c_2, c_3\) represent the three children. Relative to that ordering of children, the symbol pattern
\begin{equation*} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\mid} \mathord{\mid} \end{equation*}
means that the child \(c_1\) gets all seven coins, as above. But relative to the ordering \(c_3,c_2,c_1\text{,}\) the different symbol pattern
\begin{equation*} \mathord{\mid} \mathord{\mid} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \mathord{\circ} \end{equation*}
also
means that child \(c_1\) gets all seven coins, which is the same result. So if we allow both symbol patterns and orderings of children to vary, we will end up over-counting.
There are \(C(n+k-1, k-1)\) ways to distribute \(n\) indistinguishable objects amongst \(k\) distinguishable containers.
- Proof.
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Just as in the last example, use \(n\) \(\mathord{\circ}\) symbols to represent the indistinguishable objects and \(k-1\) indistinguishable \(\mathord{\mid}\) symbols to represent the division into \(k\) containers. So each word from the alphabet \(\Sigma=\{\mathord{\circ},\mathord{\mid}\}\) that contains exactly \(n\) \(\mathord{\circ}\) symbols and \(k-1\) \(\mathord{\mid}\) symbols represents a unique way to divide the objects into the containers. The length of such a word is \(n+k-1\text{,}\) and every such word can be constructed by choosing \(k-1\) positions for the \(\mathord{\mid}\) symbols from the \(n+k-1\) available letter positions.
Your professor throws a discrete math party, but only nine students show up (sad face). The professor sends one of the students to the corner store to get cans of soda pop for everyone. The student decides to get a mix of four different varieties. How many possible mixes of soda varieties can the student come back with? (Assume that the cans are indistinguishable except by variety, and that the store has more than ten cans of each variety available.)
Solution
Here is one scheme by which the student can decide how to choose ten cans in some combination of soda varieties. Make four boxes labelled by soda variety. Have the student choose the soda cans while blindfolded, but has the store clerk place each can in the appropriate box as the cans are chosen (a permissible assumption, since it has no bearing on the outcome). In this way, we may assume that the cans are initially indistinguishable and remain so until they are placed in the appropriate box, at which time they magically become the variety specified by the box's label. The previous theorem now tells us that there are
\begin{equation*} C(10+4-1, 4-1) = C(13, 3) = 286 \end{equation*}
ways to do this.
There are \(C(n+k-1, k-1)\) ways to choose \(n\) objects from amongst \(k\) types of object, where objects are indistinguishable except by type, and there are at least \(n\) objects of each type available.
- Proof Idea.
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Appeal to Theorem \(\PageIndex{1}\) exactly as in Worked Example \(\PageIndex{2}\).
Counting edges in connected graphs
For \(n \ge 2\text{,}\) the complete graph with \(n\) vertices has \(C(n, 2)\) edges.
- Proof.
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A complete graph has no loops and exactly one edge between each pair of vertices. So to count the edges we can just count the number of pairs of vertices, which is \(C(n, 2)\) for \(n \ge 2\text{.}\)
Let's summarize what we know about the number of edges in an arbitrary connected graph.
- A connected graph with \(n\) vertices has at least \(n - 1\) edges ( Theorem 15.3.1 ).
- A connected graph with \(n\) vertices is a tree if and only if it has exactly \(n - 1\) edges ( Theorem 16.3.1 ).
- A simple graph with \(n \ge 2\) vertices is complete if and only if it has exactly \(C(n, 2)\) edges (Proposition \(\PageIndex{1}\) in the forward direction, Statement 4 of Proposition 14.2.1 in the reverse direction).
The first fact tells us the minimum number of edges a connected graph must have, but it does not guarantee that a graph with that many edges must be connected, even if the graph is simple. The following is something of a converse to this fact, as it does provide such a guarantee: it tells use how many edges a (simple) graph must have before we can be certain that it is connected.
If \(G = (V,E)\) is a simple graph such that \(\vert V \vert = n\) and \(\vert E \vert \gt C(n-1, 2)\text{,}\) then \(G\) is connected.
- Proof Idea.
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Considering the contrapositive, assume that \(G\) is simple but not connected. In Activity 15.5.6 , we discovered that such a \(G\) will be maximal when it has exactly two connected components, each of which is a complete graph. Among graphs with those two characteristics (and still \(n\) vertices), the largest possible value for \(\vert E \vert\) occurs when the connected components of \(G\) are an isolated vertex \(v\) and the complete graph \(K_{n-1}\text{,}\) in which case the number of edges is \(C(n-1, 2)\) (Proposition \(\PageIndex{1}\) for \(K_{n-1}\)). All other nonconnected, simple graphs will then have \(\vert E \vert \le C(n-1, 2)\text{,}\) as required to complete the proof by contrapositive.


2.15: Problems
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1. Three non-degenerate energy levels are available to a set of five distinguishable molecules, \(\{A,\ B,\ C,\ D,\ E\}\). The energies of these levels are \(1\), \(2\), and \(3\), in arbitrary units. Find all of the population sets that are possible in this system. For each population set, find the system energy, \(E\), and the number of microstates, \(W\). For each system energy, \(E\), list the associated population sets and the total number of microstates. How many population sets are there? What is \(W_{max}\)? If this system is isolated with \(E=10\), how many population sets are possible? What is \({\mathit{\Omega}}_E\) for \(E=10\)?
2. For the particle in a box, the allowed energies are proportional to the squares of the successive integers. What population sets are possible for the distinguishable molecules, \(\{A,\ B,\ C,\ D,\ E\}\), if they can occupy three quantum states whose energies are \(1\), \(4\), and \(9\)? For each population set, find the system energy, \(E\), and the number of microstates. For each system energy, \(E\), list the associated population sets and the total number of microstates. How many population sets are there? What is \(W_{max}\)? If this system is isolated with \(E=24\), how many population sets are possible? What is \({\mathit{\Omega}}_E\) for \(E=24\)?
3. Consider the results you obtained in problem 2. In general, when the allowed energies are proportional to the squares of successive integers, how many population sets do you think will be associated with each system energy?
4.
(a) Compare \(W\) for the population set \(\{3,3,3\}\) to \(W\) for the population set \(\{2,5,2\}\). The energy levels are non-degenerate.
(b) Consider an \(N\)-molecule system that has a finite number, \(M\), of quantum states. Show that \(W\) is (at least locally) a maximum when \(N_1=N_2=\dots =N_M={N}/{M}\). (Hint: Let \(U={N}/{M}\), and assume that \(N\) can be chosen so that \(U\) is an integer. Let \[W_U={N!}/{\left[U!U!\prod^{i=M-2}_{i=1}{U!}\right]} \nonumber \]
and let \[W_O={N!}/{\left[\left(U+1\right)!\left(U-1\right)!\prod^{i=M-2}_{i=1}{U!}\right]} \nonumber \]
Show that \({W_O}/{W_U}<1\).)
5. The energy levels available to isomer \(A\) are \({\epsilon }_0=1\), \({\epsilon }_2=2\), and \({\epsilon }_4=3\), in arbitrary units. The energy levels available to isomer B are \({\epsilon }_1=2\), \({\epsilon }_3=3\), and \({\epsilon }_5=4\). The energy levels are non-degenerate.
(a) A system contains five molecules. The energy of the system is \(10\). List the population sets that are consistent with \(N=5\) and \(E=10\). Find \(W\) for each of these population sets. What are \(W^{max}_{A,B}\), \(W^{max}_A\), and \(W^{max}_B\)? What is the total number of microstates, \(=\mathit{\Omega}_{A,B}\), available to the system in all of the cases in which \(A\) and \(B\) molecules are present? What is the ratio \(\mathit{\Omega}_{A,B}/W^{max}_{A,B}\)?
(b) Repeat this analysis for a system that contains six molecules and whose energy is \(12\).
(c) Would the ratio \(\mathit{\Omega}_{A,B}/W^{max}_{A,B}\) be larger or smaller for a system with \(N=50\) and \(E=100\)?
(d) What would happen to this ratio if the number of molecules became very large, while the average energy per molecule remained the same?
6. In Section 20.11 , we assume that all of the energy levels available to an isomeric pair of molecules have the same degeneracy. We then argue that the thermodynamic probabilities of a mixture of the isomers must be greater than the thermodynamic probability of either pure isomer: \(W^{max}_{A,B}>W^{max}_A\) and \(W^{max}_{A,B}>W^{max}_B\). Implicitly, we assume that many energy levels are multiply occupied: \(N_i>1\) for many energy levels \({\epsilon }_i\). Now consider the case that \(g_i>1\) for most \({\epsilon }_i\), but that nearly all energy levels are either unoccupied or contain only one molecule: \(N_i=0\) or \(N_i=1\). Show that under this assumption also, we must have \(W^{max}_{A,B}>W^{max}_A\) and \(W^{max}_{A,B}>W^{max}_B\).
Notes
\({}^{1}\)The statistical-mechanical procedures that have been developed for finding the energy levels available to a molecule express molecular energies as the difference between the molecule energy and the energy that its constituent atoms have when they are motionless. This is usually effected in two steps. The molecular energy levels are first expressed relative to the energy of the molecule’s own lowest energy state. The energy released when the molecules is formed in its lowest energy state from the isolated constituent atoms is then added. The energy of each level is then equal to the work done on the component atoms when they are brought together from infinite separation to form the molecule in that energy level. (Since energy is released in the formation of a stable molecule, the work done on the atoms and the energy of the resulting molecule are less than zero.) In our present discussion, we suppose that we can solve the Schrödinger equation to find the energies of the allowed quantum states. This corresponds to choosing the isolated constituent electrons and nuclei as the zero of energy for both isomers.