21.2: Basic Concepts
Before discussing specific procedures related to facilitating bowel and bladder function, let’s review basic concepts related to urinary and bowel elimination. When facilitating alternative methods of elimination, it is important to understand the anatomy and physiology of the gastrointestinal and urinary systems, as well as the adverse effects of various conditions and medications on elimination. Use the hyperlinks below to review information about these topics.
For more information about the anatomy and physiology of the gastrointestinal system and medications used to treat diarrhea and constipation, visit the “ Gastrointestinal ” chapter of the Open RN Nursing Pharmacology textbook.
For more information about the anatomy and physiology of the kidneys and diuretic medications used to treat fluid overload, visit the “ Cardiovascular and Renal System ” chapter in Open RN Nursing Pharmacology textbook.
For more information about applying the nursing process to facilitate elimination, visit the “ Elimination ” chapter in Open RN Nursing Fundamentals .
Urinary Elimination Devices
This section will focus on the devices used to facilitate urinary elimination. Urinary catheterization is the insertion of a catheter tube into the urethral opening and placing it in the neck of the urinary bladder to drain urine. There are several types of urinary elimination devices, such as indwelling catheters, intermittent catheters, suprapubic catheters, and external devices. Each of these types of devices is described in the following subsections.
Indwelling Catheter
An indwelling catheter , often referred to as a “Foley catheter,” refers to a urinary catheter that remains in place after insertion into the bladder for the continual collection of urine. It has a balloon on the insertion tip to maintain placement in the neck of the bladder. The other end of the catheter is attached to a drainage bag for the collection of urine. See Figure \(\PageIndex{1}\) [1] for an illustration of the anatomical placement of an indwelling catheter in the bladder neck.
The distal end of an indwelling catheter has a urine drainage port that is connected to a drainage bag. The size of the catheter is marked at this end using the French catheter scale. A balloon port is also located at this end, where a syringe is inserted to inflate the balloon after it is inserted into the bladder. The balloon port is marked with the amount of fluid required to fill the balloon. See Figure \(\PageIndex{2}\) [2] for an image of the parts of an indwelling catheter.
Catheters have different sizes, with the larger the number indicating a larger diameter of the catheter. See Figure \(\PageIndex{3}\) [3] for an image of the French Catheter Scale.
There are two common types of bags that may be attached to an indwelling catheter. During inpatient or long-term care, larger collection bags that can hold up to 2 liters of fluid are used. See Figure \(\PageIndex{4}\) [4] for an image of a typical collection bag attached to an indwelling catheter. These bags should be emptied when they are half to two-thirds full to prevent traction on the urethra from the bag. Additionally, the collection bag should always be placed below the level of the patient’s bladder so that urine flows out of the bladder and urine does not inadvertently flow back into the bladder. Ensure the tubing is not coiled, kinked, or compressed so that urine can flow unobstructed into the bag. Slack should be maintained in the tubing to prevent injury to the patient’s urethra. To prevent the development of a urinary tract infection, the bag should not be permitted to touch the floor.
See Figure \(\PageIndex{5}\) [5] for an illustration of the placement of the urine collection bag when the patient is lying in bed.
 Leg bags provide discretion when the patient is in public because they can be worn under clothing. However, leg bags are small and must be emptied more frequently than those used during inpatient care. Figure \(\PageIndex{6}\) [6] for an image of leg bag and Figure \(\PageIndex{7}\) [7] for an illustration of an indwelling catheter attached to a leg bag.
Straight Catheter
A straight catheter is used for intermittent urinary catheterization. The catheter is inserted to allow for the flow of urine and then immediately removed, so a balloon is not required at the insertion tip. See Figure \(\PageIndex{8}\) [8] for an image of a straight catheter. Intermittent catheterization is used for the relief of urinary retention. It may be performed once, such as after surgery when a patient is experiencing urinary retention due to the effects of anesthesia, or performed several times a day to manage chronic urinary retention. Some patients may also independently perform self-catheterization at home to manage chronic urinary retention caused by various medical conditions. In some situations, a straight catheter is also used to obtain a sterile urine specimen for culture when a patient is unable to void into a sterile specimen cup. According to the Centers for Disease Control and Prevention (CDC), intermittent catheterization is preferred to indwelling urethral catheters whenever feasible because of decreased risk of developing a urinary tract infection. [9]
Other Types of Urinary Catheters
Coude Catheter Tip
Coude catheter tips are curved to follow the natural curve of the urethra during catheterization. They are often used when catheterizing male patients with enlarged prostate glands. See Figure \(\PageIndex{9}\) [10] for an example of a urinary catheter with a coude tip. During insertion, the tip of the Coude catheter must be pointed anteriorly or it can cause damage the urethra. A thin line embedded in the catheter provides information regarding orientation during the procedure; maintain the line upwards to keep it pointed anteriorly.
Irrigation Catheter
Irrigation catheters are typically used after prostate surgery to flush the surgical area. These catheters are larger in size to allow for larger amounts of fluid to flush. See Figure \(\PageIndex{10}\) [11] for an image comparing a larger 20 French catheter (typically used for irrigation) to a 14 French catheter (typically used for indwelling catheters).
Suprapubic Catheters
Suprapubic catheters are surgically inserted through the abdominal wall into the bladder. This type of catheter is typically inserted when there is a blockage within the urethra that does not allow the use of a straight or indwelling catheter. Suprapubic catheters may be used for a short period of time for acute medical conditions or may be used permanently for chronic conditions. See Figure \(\PageIndex{11}\) [12] for an image of a suprapubic catheter. The insertion site of a suprapubic catheter must be cleaned regularly according to agency policy with appropriate steps to prevent skin breakdown.
Male Condom Catheter
A condom catheter is a noninvasive device used for males with incontinence.  This device protects and promotes healing of the skin around the perineal area and inner legs and is used as an alternative to an indwelling urinary catheter. See Figure \(\PageIndex{12}\) [13] for an image of a condom catheter and Figure \(\PageIndex{13}\) [14] for an illustration of a condom catheter attached to a leg bag.
Female External Urinary Catheter
Female external urinary catheters (FEUC) have been recently introduced into practice to reduce the incidence of catheter-associated urinary tract infection (CAUTI) in women. [15] The external female catheter device is made of a purewick material that is placed externally over the female’s urinary meatus. The wicking material is attached to a tube that is hooked to a low-suction device. When the wick becomes saturated with urine, it is suctioned into a drainage canister. Preliminary studies have found that utilizing the FEUC device reduced the risk for CAUTI. [16] , [17]
- " Foley Catheter.png ” by BruceBlaus is licensed under CC BY-SA 4.0 ↵
- " Foley catheter inflated and deflated EN.svg ” by Olek Remesz (wiki-pl: Orem , commons: Orem ) is licensed under CC BY-SA 3.0 ↵
- " French catheter scale.gif ” by Glitzy queen00 is licensed under CC BY-SA 3.0 ↵
- "DSC_2104.jpg” by British Columbia Institute of Technology is licensed under CC BY 4.0 . Access for free at https://ecampusontario.pressbooks.pub/clinicalskills/chapter/2-2-head-to-toe-assessment-checklist/ ↵
- " Closed Urinary Drainage.png ” by BruceBlaus is licensed under CC BY-SA 4.0 ↵
- "Leg Bag_3I3A0667.jpg” by Deanna Hoyord, Chippewa Valley Technical College is licensed under CC BY 4.0 ↵
- " Foley Catheter Drainage (cropped).png ” by BruceBlaus is licensed under CC BY-SA 4.0 ↵
- " Urinary catheter.JPG ” by Bengt Oberger is licensed under CC BY-SA 3.0 ↵
- Centers for Disease Control and Prevention. (2015, November 15). Catheter-associated urinary tract infections (CAUTI) . https://www.cdc.gov/infectioncontrol/guidelines/cauti/ ↵
- "Self Cath_3I3A0676.jpg” by Deanna Hoyord, Chippewa Valley Technical College is licensed under CC BY 4.0 ↵
- "Irrigation Catheter and a 14 Fr. Catheter - 3I3A0753.jpg" by Deanna Hoyord, Chippewa Valley Technical College is licensed under CC BY 4.0 ↵
- "NewlyPlaceSubprapubic.jpg ” by James Heilman, MD is licensed under CC BY-SA 4.0 ↵
- "Male External Cath 1 .jpg” by Deanna Hoyord, Chippewa Valley Technical College is licensed under CC BY 4.0 ↵
- " Condom Cather Drainage.png ” by BruceBlaus is licensed under CC BY-SA 4.0 ↵
- Eckert, L., Mattia, L., Patel, S., Okumura, R., Reynolds, P., & Stuiver, I. Reducing the risk of indwelling catheter-associated urinary tract infection in female patients by implementing an alternative female external urinary collection device: A quality improvement project. Journal of Wound Ostomy Continence Nursing, 47 (1), 50-53. https://doi.org/10.1097/won.0000000000000601 ↵
- Eckert, L., Mattia, L., Patel, S., Okumura, R., Reynolds, P., & Stuiver, I. Reducing the risk of indwelling catheter-associated urinary tract infection in female patients by implementing an alternative female external urinary collection device: A quality improvement project. Journal of Wound Ostomy Continence Nursing, 47 (1), 50-53. https://doi.org/10.1097/won.0000000000000601 ↵
- Glover, E., Bleeker, E., Bauermeister, A., Koehlmoos, A., & Van Whye, M. (2018). E xternal catheters and reducing adverse effects in the female inpatient . Northwestern College Department of Nursing. https://nwcommons.nwciowa.edu/cgi/viewcontent.cgi?article=1026&context=celebrationofresearch ↵
- Madrid, S. (2019, June 19). Purewick. [Video]. YouTube. All rights reserved. https://youtu.be/1rnQaHvIMBc ↵
- Newton, C. (2016, August 4). PureWick User Instructions . [Video]. YouTube. All rights reserved. https://youtu.be/xSOuvcShikw ↵


8.4: Treatment, Jail, or Justice?
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- Christopher Byers
- Linn-Benton Community College via OpenOregon
Preface
I always thought of myself as someone who has been the underdog in life. From my sister’s death, to homelessness to drug addiction, I thought I had it worse than anyone in the world. My negative experiences played a role in shaping my belief systems about myself and the world around me and in a way, encapsulated my thinking by keeping me “unique.” But then, that all began to change for me. It all started with a class. A Human Development and Family Sciences class at LBCC, where I was introduced to a concept that altered my perception of the world and how I fit into it in a significant way. I began to learn how maybe after all these years of believing I had a hard life and everyone had it easy, this was not the case. I learned about concepts such as “Privilege,” Sociological Imagination, Equity, Social Constructivism, and many other concepts like those that helped me to slowly and sometimes painfully open my eyes to the reality of life. When I learned about Privilege, I had the hardest time wrapping my head around that one. Me, someone whose sister died, experienced homelessness and drug addiction be privileged? I thought at first that it didn’t apply to me. But through much internal reflection and writing and processing with people in my life and with my professor, I began to slowly understand what Privilege means. That concept alone was a catalyst for me to dive further into researching injustice in the U.S., and really inspired me to do a lot of deep reflective work on my own social identity and what it means to be who I am in the United States. This work has become some of the most important work I have ever done in my life, I see it as a path of healing not only for myself, but for many of us who choose to seek it.
Mental Health and Substance Abuse
To the extent that social inequality, social interaction, and drug culture contribute to substance use, it is a mistake to contribute substance use only to biological and psychological factors. While these factors do play a role, it would be a mistake to ignore the social environments in which people participate in substance use. The role that the family plays in substance abuse potential in members is vastly underlooked. Weak family bonds and school connections are often seen as a major role in the development of substance use in adolescents. Weak bonds to family members prompt adolescents to be less likely to conform to conventional norms and more likely to engage in using drugs and other deviant behavior. Healthy family bonds, coping with trauma, learning how to identify feelings, and open communication all play a positive role in the reduction and prevention of potential substance use of family members in the future.
Mental Health
So what exactly is Mental Health? And how is it defined today? Well, first, lets shine a brief but illuminating light on its history in the U.S. The mentally ill have been treated very poorly for hundreds of years. In the 1800s, It was believed that mental illness was caused by demonic possession, witchcraft, or an angry god. For example, in medieval times, abnormal behaviors were viewed as a sign that a person was possessed by demons. Actually, most people who displayed abnormal behavior were viewed as being possessed by demons. This was not an uncommon societal belief in the 1700s. The idea that mental illness was the result of demonic possession by either an evil spirit or an evil god incorrectly attributed all inexpiable phenomena to deities deemed either good or evil. [1] As a result of these prevailing theories of psychopathology, derived from folklore and inadequate scientific beliefs, these systems are still perpetuated today. Although science has shed a light to a better understanding of mental health, it is still common for stigma, discrimination , and ignorance to be the deciding factors in how people with mental health are cared for and treated.
Psychological disorders are very common in the U.S, yet there is still a great deal of inequity that encompasses mental health issues. Stigma, labeling, ignorance, discrimination and judgement are all still very prevalent and harsh realities in our society today. The biological, sociological, physiological and cultural determinants of mental health disorders vary from case to case and most mental health issues are still often difficult to understand, since the roots of mental illness are often misunderstood. We mentioned the DSM-5 earlier, which is another tool that psychologists and psychiatrists and CADC counselors (Drug and Alcohol Counselors) often use to diagnose mental health and substance use disorders. Our society has made a lot of progress in understanding how some operate, but we still have a ways to go until we as a society can see mental health through a collective, compassionate lens. It is important to remember that those who struggle with psychological disorders are not their disorder. It is something they have, through no fault of their own. As with cancer and diabetes, these people who have psychological disorders often suffer from debilitating, often painful conditions through no choice of their own. These individuals deserve to be treated and viewed with compassion, dignity, and understanding.
Substance Abuse
So what exactly is the relationship between substance abuse and mental health? Substance abuse and mental health are interconnecting and overlapping systems. There has been years and years of stigma, discrimination , and misunderstanding with people who both have mental health issues and substance abuse issues. Many people who suffer from substance abuse often have undealt-with trauma, depression, anxiety, and environmental and genetic factors that contribute to the use of substances as a way to self-medicate and cope with how they feel, regardless of the negative consequences that might happen as a result of using. For example, conduct disorder and antisocial personality disorder are strongly associated with the development of both substance use and serious mental illness such as major depression and bipolar disorder. Some substance abuse use can even mimic mental health issues, and are quite difficult to diagnose without dealing with the substance abuse issues first. Psychopharmacological reactions to withdrawal can also induce psychiatric symptoms and exacerbate underlying mental health issues for people as well.
Mental health issues are interconnected with substance abuse. Most people who practice counseling often deal with the substance abuse issues with a client and then proceed to determine if that individual is suffering from mental health issues after the substance abuse is addressed.
Mental Health, Substance Abuse, and the Criminalization of Drugs
Although slavery has been abolished, and the Jim Crow Laws are no longer legal, the systems that have oppressed people of color and marginalized communities from 300 years ago are perpetuated still to this day. It can be argued that the criminal justice system and the legislation and policies that were created to punish drug users and drug crimes were designed to perpetuate discrimination and oppression of people of color at disproportionate rates.
A significant aspect of the “War on Drugs,” which was a piece of legislation that disproportionately affected people of color in the United States. It imposed mandatory minimum sentencing laws that sent non-violent drug offenders to prison, rather than enrolling them in treatment programs. Seventy percent of inmates in the United States are non-White—a figure that surpasses the percentage of non-Whites in US society, which is approximately 23%, according to the 2015 US census. That means that non-White prisoners are far over-represented in the US criminal justice system. The United States has the highest incarceration rates for drug-related crimes. This figure based on the article “The Black/White marijuana arrest gap, in in nine charts ,” demonstrates the implicit bias our justice system still has for people of color in the United States.
It is a very sad and common societal view that addicts and substance abuse users as lesser human beings, a lower standard of individual than the rest of society. This is known as stigma. This overarching and negative view on people who struggle with substance abuse plays a role in the passing of policies and criminalization of millions of people every year in the United States, most importantly, affecting families of color. When people of color are targeted for nonviolent drug-related crimes, they are more likely to receive harsher punishments than White people. This has absolutely deleterious consequences for people of color and their families, with the head of households usually being the ones who receive these harsher punishments.
Women of color have been arrested at rates far higher than White women, even though they use drugs at a rate equal to or lower than White women. Furthermore, according to Bureau of Justice statistics from 2007, nearly two-thirds of US women prisoners had children under 18 years of age. Before incarceration, disproportionately, these women were the primary caregivers to their children and other family members so the impact on children, families, and communities is substantial when women are imprisoned. Finally, inmates often engage in prison labor for less than minimum wage. When these individuals are incarcerated, corporations contract prison labor that produces millions of dollars in profit. Therefore, the incarceration of millions of people artificially deflates the unemployment rate (something politicians benefit from) and creates a cheap labor force that generates millions of dollars in profit for private corporations. How do we make sense of this? What does this say about the state of democracy in the United States? When seen through an equity lens, we can establish some interesting points. One is that the rates that people of color and White people use drugs are about the same, but one important factor plays a role in the disproportionate rates that POC are incarcerated. This is implicit bias. People have subconscious ideas about who uses drugs in the United States. These ideas are based on false narratives derived from implicit biases that perpetuate the inequitable incarceration of people of color. And two, the war on drugs focused and funneled money into the punishment of and incarceration for drug-related offenses.
A question to ponder is this: What would society look like if instead of punishment and criminalizing drug use and drug users, we used that money to focus on treatment, recovery centers, and social services?
Licenses and Attributions
Open Content, Shared Previously
Figure 8.8. “One and Other-Mental Health” by Feggy Art. License: CC BY-NC-ND 2.0.
Open Content, Original
Figure 8.9. “Arrest Rates for Marijuana Possession by Race (2001-2010)”. License: CC BY 4.0 . Data source: FBI/Uniform Crime Reporting Program Data and U.S. Census Data compiled by the Washington Post .
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Lumen Learning. (n.d.). The state, law, and the prison system. Retrieved September 3, 2020, from
https://courses.lumenlearning.com/su...prison-system/
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2.1: Elements and Atoms - The Building Blocks of Matter
Learning Objectives
- Discuss the relationships between matter, mass, elements, compounds, atoms, and subatomic particles
- Distinguish between atomic number and mass number
- Identify the key distinction between isotopes of the same element
- Explain how electrons occupy electron shells and their contribution to an atom’s relative stability
The substance of the universe—from a grain of sand to a star—is called matter . Scientists define matter as anything that occupies space and has mass. An object’s mass and its weight are related concepts, but not quite the same. An object’s mass is the amount of matter contained in the object, and the object’s mass is the same whether that object is on Earth or in the zero-gravity environment of outer space. An object’s weight, on the other hand, is its mass as affected by the pull of gravity. Where gravity strongly pulls on an object’s mass its weight is greater than it is where gravity is less strong. An object of a certain mass weighs less on the moon, for example, than it does on Earth because the gravity of the moon is less than that of Earth. In other words, weight is variable, and is influenced by gravity. A piece of cheese that weighs a pound on Earth weighs only a few ounces on the moon.
Elements and Compounds
All matter in the natural world is composed of one or more of the 92 fundamental substances called elements. An element is a pure substance that is distinguished from all other matter by the fact that it cannot be created or broken down by ordinary chemical means. While your body can assemble many of the chemical compounds needed for life from their constituent elements, it cannot make elements. They must come from the environment. A familiar example of an element that you must take in is calcium (Ca ++ ). Calcium is essential to the human body; it is absorbed and used for a number of processes, including strengthening bones. When you consume dairy products your digestive system breaks down the food into components small enough to cross into the bloodstream. Among these is calcium, which, because it is an element, cannot be broken down further. The elemental calcium in cheese, therefore, is the same as the calcium that forms your bones. Some other elements you might be familiar with are oxygen, sodium, and iron. The elements in the human body are shown in Figure \(\PageIndex{1}\), beginning with the most abundant: oxygen (O), carbon (C), hydrogen (H), and nitrogen (N). Each element’s name can be replaced by a one- or two-letter symbol; you will become familiar with some of these during this course. All the elements in your body are derived from the foods you eat and the air you breathe.
In nature, elements rarely occur alone. Instead, they combine to form compounds. A compound is a substance composed of two or more elements joined by chemical bonds. For example, the compound glucose is an important body fuel. It is always composed of the same three elements: carbon, hydrogen, and oxygen. Moreover, the elements that make up any given compound always occur in the same relative amounts. In glucose, there are always six carbon and six oxygen units for every twelve hydrogen units. But what, exactly, are these “units” of elements?
Atoms and Subatomic Particles
An atom is the smallest quantity of an element that retains the unique properties of that element. In other words, an atom of hydrogen is a unit of hydrogen—the smallest amount of hydrogen that can exist. As you might guess, atoms are almost unfathomably small. The period at the end of this sentence is millions of atoms wide.
Atomic Structure and Energy
Atoms are made up of even smaller subatomic particles, three types of which are important: the proton , neutron , and electron . The number of positively-charged protons and non-charged (“neutral”) neutrons, gives mass to the atom, and the number of each in the nucleus of the atom determine the element. The number of negatively-charged electrons that “spin” around the nucleus at close to the speed of light equals the number of protons. An electron has about 1/2000 th the mass of a proton or neutron.
Figure \(\PageIndex{2}\) shows two models that can help you imagine the structure of an atom—in this case, helium (He). In the planetary model, helium’s two electrons are shown circling the nucleus in a fixed orbit depicted as a ring. Although this model is helpful in visualizing atomic structure, in reality, electrons do not travel in fixed orbits, but whiz around the nucleus erratically in a so-called electron cloud.
An atom’s protons and electrons carry electrical charges. Protons, with their positive charge, are designated p + . Electrons, which have a negative charge, are designated e – . An atom’s neutrons have no charge: they are electrically neutral. Just as a magnet sticks to a steel refrigerator because their opposite charges attract, the positively charged protons attract the negatively charged electrons. This mutual attraction gives the atom some structural stability. The attraction by the positively charged nucleus helps keep electrons from straying far. The number of protons and electrons within a neutral atom are equal, thus, the atom’s overall charge is balanced.
Atomic Number and Mass Number
An atom of carbon is unique to carbon, but a proton of carbon is not. One proton is the same as another, whether it is found in an atom of carbon, sodium (Na), or iron (Fe). The same is true for neutrons and electrons. So, what gives an element its distinctive properties—what makes carbon so different from sodium or iron? The answer is the unique quantity of protons each contains. Carbon by definition is an element whose atoms contain six protons. No other element has exactly six protons in its atoms. Moreover, all atoms of carbon, whether found in your liver or in a lump of coal, contain six protons. Thus, the atomic number , which is the number of protons in the nucleus of the atom, identifies the element. Because an atom usually has the same number of electrons as protons, the atomic number identifies the usual number of electrons as well.
In their most common form, many elements also contain the same number of neutrons as protons. The most common form of carbon, for example, has six neutrons as well as six protons, for a total of 12 subatomic particles in its nucleus. An element’s mass number is the sum of the number of protons and neutrons in its nucleus. So the most common form of carbon’s mass number is 12. (Electrons have so little mass that they do not appreciably contribute to the mass of an atom.) Carbon is a relatively light element. Uranium (U), in contrast, has a mass number of 238 and is referred to as a heavy metal. Its atomic number is 92 (it has 92 protons) but it contains 146 neutrons; it has the most mass of all the naturally occurring elements.
The periodic table of the elements , shown in Figure \(\PageIndex{3}\), is a chart identifying the 92 elements found in nature, as well as several larger, unstable elements discovered experimentally. The elements are arranged in order of their atomic number, with hydrogen and helium at the top of the table, and the more massive elements below. The periodic table is a useful device because for each element, it identifies the chemical symbol, the atomic number, and the mass number, while organizing elements according to their propensity to react with other elements. The number of protons and electrons in an element are equal. The number of protons and neutrons may be equal for some elements, but are not equal for all.
Isotopes
Although each element has a unique number of protons, it can exist as different isotopes. An isotope is one of the different forms of an element, distinguished from one another by different numbers of neutrons. The standard isotope of carbon is 12 C, commonly called carbon twelve. 12 C has six protons and six neutrons, for a mass number of twelve. All of the isotopes of carbon have the same number of protons; therefore, 13 C has seven neutrons, and 14 C has eight neutrons. The different isotopes of an element can also be indicated with the mass number hyphenated (for example, C-12 instead of 12 C). Hydrogen has three common isotopes, shown in Figure \(\PageIndex{4}\).
An isotope that contains more than the usual number of neutrons is referred to as a heavy isotope. An example is 14 C . Heavy isotopes tend to be unstable, and unstable isotopes are radioactive. A radioactive isotope is an isotope whose nucleus readily decays, giving off subatomic particles and electromagnetic energy. Different radioactive isotopes (also called radioisotopes) differ in their half-life, the time it takes for half of any size sample of an isotope to decay. For example, the half-life of tritium—a radioisotope of hydrogen—is about 12 years, indicating it takes 12 years for half of the tritium nuclei in a sample to decay. Excessive exposure to radioactive isotopes can damage human cells and even cause cancer and birth defects, but when exposure is controlled, some radioactive isotopes can be useful in medicine. For more information, see the Career Connections.
CAREER CONNECTION: Interventional Radiologist
The controlled use of radioisotopes has advanced medical diagnosis and treatment of disease. Interventional radiologists are physicians who treat disease by using minimally invasive techniques involving radiation. Many conditions that could once only be treated with a lengthy and traumatic operation can now be treated non-surgically, reducing the cost, pain, length of hospital stay, and recovery time for patients. For example, in the past, the only options for a patient with one or more tumors in the liver were surgery and chemotherapy (the administration of drugs to treat cancer). Some liver tumors, however, are difficult to access surgically, and others could require the surgeon to remove too much of the liver. Moreover, chemotherapy is highly toxic to the liver, and certain tumors do not respond well to it anyway. In some such cases, an interventional radiologist can treat the tumors by disrupting their blood supply, which they need if they are to continue to grow. In this procedure, called radioembolization, the radiologist accesses the liver with a fine needle, threaded through one of the patient’s blood vessels. The radiologist then inserts tiny radioactive “seeds” into the blood vessels that supply the tumors. In the days and weeks following the procedure, the radiation emitted from the seeds destroys the vessels and directly kills the tumor cells in the vicinity of the treatment.
Radioisotopes emit subatomic particles that can be detected and tracked by imaging technologies. One of the most advanced uses of radioisotopes in medicine is the positron emission tomography (PET) scanner, which detects the activity in the body of a very small injection of radioactive glucose, the simple sugar that cells use for energy. The PET camera reveals to the medical team which of the patient’s tissues are taking up the most glucose. Thus, the most metabolically active tissues show up as bright “hot spots” on the images (Figure \(\PageIndex{5}\)). PET can reveal some cancerous masses because cancer cells consume glucose at a high rate to fuel their rapid reproduction.
The Behavior of Electrons
In the human body, atoms do not exist as independent entities. Rather, they are constantly reacting with other atoms to form and to break down more complex substances. To fully understand anatomy and physiology you must grasp how atoms participate in such reactions. The key is understanding the behavior of electrons.
Although electrons do not follow rigid orbits a set distance away from the atom’s nucleus, they do tend to stay within certain regions of space called electron shells. An electron shell is a layer of electrons that encircle the nucleus at a distinct energy level.
The atoms of the elements found in the human body have from one to five electron shells, and all electron shells hold eight electrons except the first shell, which can only hold two. This configuration of electron shells is the same for all atoms. The precise number of shells depends on the number of electrons in the atom. Hydrogen and helium have just one and two electrons, respectively. If you take a look at the periodic table of the elements, you will notice that hydrogen and helium are placed alone on either sides of the top row; they are the only elements that have just one electron shell (Figure \(\PageIndex{6}\)). A second shell is necessary to hold the electrons in all elements larger than hydrogen and helium.
Lithium (Li), whose atomic number is 3, has three electrons. Two of these fill the first electron shell, and the third spills over into a second shell. The second electron shell can accommodate as many as eight electrons. Carbon, with its six electrons, entirely fills its first shell, and half-fills its second. With ten electrons, neon (Ne) entirely fills its two electron shells. Again, a look at the periodic table reveals that all of the elements in the second row, from lithium to neon, have just two electron shells. Atoms with more than ten electrons require more than two shells. These elements occupy the third and subsequent rows of the periodic table.
The factor that most strongly governs the tendency of an atom to participate in chemical reactions is the number of electrons in its valence shell. A valence shell is an atom’s outermost electron shell. If the valence shell is full, the atom is stable; meaning its electrons are unlikely to be pulled away from the nucleus by the electrical charge of other atoms. If the valence shell is not full, the atom is reactive; meaning it will tend to react with other atoms in ways that make the valence shell full. Consider hydrogen, with its one electron only half-filling its valence shell. This single electron is likely to be drawn into relationships with the atoms of other elements, so that hydrogen’s single valence shell can be stabilized.
All atoms (except hydrogen and helium with their single electron shells) are most stable when there are exactly eight electrons in their valence shell. This principle is referred to as the octet rule, and it states that an atom will give up, gain, or share electrons with another atom so that it ends up with eight electrons in its own valence shell. For example, oxygen, with six electrons in its valence shell, is likely to react with other atoms in a way that results in the addition of two electrons to oxygen’s valence shell, bringing the number to eight. When two hydrogen atoms each share their single electron with oxygen, covalent bonds are formed, resulting in a molecule of water, H 2 O.
In nature, atoms of one element tend to join with atoms of other elements in characteristic ways. For example, carbon commonly fills its valence shell by linking up with four atoms of hydrogen. In so doing, the two elements form the simplest of organic molecules, methane, which also is one of the most abundant and stable carbon-containing compounds on Earth. As stated above, another example is water; oxygen needs two electrons to fill its valence shell. It commonly interacts with two atoms of hydrogen, forming H 2 O. Incidentally, the name “hydrogen” reflects its contribution to water (hydro- = “water”; -gen = “maker”). Thus, hydrogen is the “water maker.”
Chapter Review
The human body is composed of elements, the most abundant of which are oxygen (O), carbon (C), hydrogen (H) and nitrogen (N).  The smallest unit of an element that retains all of the properties of that element is an atom. But, atoms themselves contain many subatomic particles, the three most important of which are protons, neutrons, and electrons. These particles do not vary in quality from one element to another; rather, what gives an element its distinctive identification is the quantity of its protons, called its atomic number. Protons and neutrons contribute nearly all of an atom’s mass; the number of protons and neutrons is an element’s mass number. Heavier and lighter versions of the same element can occur in nature because these versions have different numbers of neutrons. Different versions of an element are called isotopes.
The tendency of an atom to be stable or to react readily with other atoms is largely due to the behavior of the electrons within the atom’s outermost electron shell, called its valence shell. Helium, as well as larger atoms with eight electrons in their valence shell, is unlikely to participate in chemical reactions because they are stable. All other atoms tend to accept, donate, or share electrons in a process that brings the electrons in their valence shell to eight (or in the case of hydrogen, to two).
Interactive Link Questions
Visit this website to view the periodic table. In the periodic table of the elements, elements in a single column have the same number of electrons that can participate in a chemical reaction. These electrons are known as “valence electrons.” For example, the elements in the first column all have a single valence electron—an electron that can be “donated” in a chemical reaction with another atom. What is the meaning of a mass number shown in parentheses?
Answer: The mass number is the total number of protons and neutrons in the nucleus of an atom.
Review Questions
Q. Together, just four elements make up more than 95 percent of the body’s mass. These include ________.
A. calcium, magnesium, iron, and carbon
B. oxygen, calcium, iron, and nitrogen
C. sodium, chlorine, carbon, and hydrogen
D. oxygen, carbon, hydrogen, and nitrogen
Answer: D
Q. The smallest unit of an element that still retains the distinctive behavior of that element is an ________.
A. electron
B. atom
C. elemental particle
D. isotope
Q. The characteristic that gives an element its distinctive properties is its number of ________.
A. protons
B. neutrons
C. electrons
D. atoms
Answer: A
Q. On the periodic table of the elements, mercury (Hg) has an atomic number of 80 and a mass number of 200.59. It has seven stable isotopes. The most abundant of these probably have ________.
A. about 80 neutrons each
B. fewer than 80 neutrons each
C. more than 80 neutrons each
D. more electrons than neutrons
Answer: C
Q. Nitrogen has an atomic number of seven. How many electron shells does it likely have?
A. one
B. two
C. three
D. four
Answer: B
Critical Thinking Questions
Q. The most abundant elements in the foods and beverages you consume are oxygen, carbon, hydrogen, and nitrogen. Why might having these elements in consumables be useful?
A. These four elements—oxygen, carbon, hydrogen, and nitrogen—together make up more than 95 percent of the mass of the human body, and the body cannot make elements, so it is helpful to have them in consumables.
Q. Oxygen, whose atomic number is eight, has three stable isotopes: 16 O, 17 O, and 18 O. Explain what this means in terms of the number of protons and neutrons.
A. Oxygen has eight protons. In its most abundant stable form, it has eight neutrons, too, for a mass number of 16. In contrast, 17 O has nine neutrons, and 18 O has 10 neutrons.
Q. Magnesium is an important element in the human body, especially in bones. Magnesium’s atomic number is 12. Is it stable or reactive? Why? If it were to react with another atom, would it be more likely to accept or to donate one or more electrons?
A. Magnesium’s 12 electrons are distributed as follows: two in the first shell, eight in the second shell, and two in its valence shell. According to the octet rule, magnesium is unstable (reactive) because its valence shell has just two electrons. It is therefore likely to participate in chemical reactions in which it donates two electrons.
Glossary
- atom
- smallest unit of an element that retains the unique properties of that element
- atomic number
- number of protons in the nucleus of an atom
- compound
- substance composed of two or more different elements joined by chemical bonds
- electron
- subatomic particle having a negative charge and nearly no mass; found orbiting the atom’s nucleus
- electron shell
- area of space a given distance from an atom’s nucleus in which electrons are grouped
- element
- substance that cannot be created or broken down by ordinary chemical means
- isotope
- one of the variations of an element in which the number of neutrons differ from each other
- mass number
- sum of the number of protons and neutrons in the nucleus of an atom
- matter
- physical substance; that which occupies space and has mass
- neutron
- heavy subatomic particle having no electrical charge and found in the atom’s nucleus
- periodic table of the elements
- arrangement of the elements in a table according to their atomic number; elements having similar properties because of their electron arrangements compose columns in the table, while elements having the same number of valence shells compose rows in the table
- proton
- heavy subatomic particle having a positive charge and found in the atom’s nucleus
- radioactive isotope
- unstable, heavy isotope that gives off subatomic particles, or electromagnetic energy, as it decays; also called radioisotopes
- valence shell
- outermost electron shell of an atom


10.4: Node Voltage Method
Method for Node Voltage Calculation
We start with a circuit having conventional voltage sources. A common node E 0 is chosen as a reference point. The node voltages E 1 and E 2 are calculated with respect to this point.
A voltage source in series with a resistance must be replaced by an equivalent current source in parallel with the resistance. We will write KCL equations for each node. The right hand side of the equation is the value of the current source feeding the node.
Replacing voltage sources and associated series resistors with equivalent current sources and parallel resistors yields the modified circuit. Substitute resistor conductances in siemens for resistance in ohms.
The parallel conductances (resistors) may be combined by addition of the conductances. Though, we will not redraw the circuit. The circuit is ready for application of the node voltage method.
Deriving a general node voltage method, we write a pair of KCL equations in terms of unknown node voltages V 1 and V 2 this one time. We do this to illustrate a pattern for writing equations by inspection.
The coefficients of the last pair of equations above have been rearranged to show a pattern. The sum of conductances connected to the first node is the positive coefficient of the first voltage in equation (1). The sum of conductances connected to the second node is the positive coefficient of the second voltage in equation (2). The other coefficients are negative, representing conductances between nodes. For both equations, the right hand side is equal to the respective current source connected to the node. This pattern allows us to quickly write the equations by inspection. This leads to a set of rules for the node voltage method of analysis.
Node Voltage Rules:
- Convert voltage sources in series with a resistor to an equivalent current source with the resistor in parallel.
- Change resistor values to conductances.
- Select a reference node(E 0 )
- Assign unknown voltages (E 1 )(E 2 ) ... (E N )to remaining nodes.
- Write a KCL equation for each node 1,2, ... N. The positive coefficient of the first voltage in the first equation is the sum of conductances connected to the node. The coefficient for the second voltage in the second equation is the sum of conductances connected to that node. Repeat for coefficient of third voltage, third equation, and other equations. These coefficients fall on a diagonal.
- All other coefficients for all equations are negative, representing conductances between nodes. The first equation, second coefficient is the conductance from node 1 to node 2, the third coefficient is the conductance from node 1 to node 3. Fill in negative coefficients for other equations.
- The right hand side of the equations is the current source connected to the respective nodes.
- Solve system of equations for unknown node voltages.
Example: Set up the equations and solve for the node voltages using the numerical values in the above figure.
Solution:
The solution of two equations can be performed with a calculator, or with octave (not shown).[octav] The solution is verified with SPICE based on the original schematic diagram with voltage sources. [spi] .Though, the circuit with the current sources could have been simulated.
One more example. This one has three nodes. We do not list the conductances on the schematic diagram. However, G 1 = 1/R 1 , etc.
There are three nodes to write equations for by inspection. Note that the coefficients are positive for equation (1) E 1 , equation (2) E 2 , and equation (3) E 3 . These are the sums of all conductances connected to the nodes. All other coefficients are negative, representing a conductance between nodes. The right hand side of the equations is the associated current source, 0.136092 A for the only current source at node 1. The other equations are zero on the right hand side for lack of current sources. We are too lazy to calculate the conductances for the resistors on the diagram. Thus, the subscripted G’s are the coefficients.
We are so lazy that we enter reciprocal resistances and sums of reciprocal resistances into the octave “A” matrix, letting octave compute the matrix of conductances after “A=”. [octav] The initial entry line was so long that it was split into three rows. This is different than previous examples. The entered “A” matrix is delineated by starting and ending square brackets. Column elements are space separated. Rows are “new line” separated. Commas and semicolons are not need as separators. Though, the current vector at “b” is semicolon separated to yield a column vector of currents.
Note that the “A” matrix diagonal coefficients are positive, That all other coefficients are negative.
The solution as a voltage vector is at “x”. E 1 = 24.000 V, E 2 = 17.655 V, E 3 = 19.310 V. These three voltages compare to the previous mesh current and SPICE solutions to the unbalanced bridge problem. This is no coincidence, for the 0.13609 A current source was purposely chosen to yield the 24 V used as a voltage source in that problem.
Summary
- Given a network of conductances and current sources, the node voltage method of circuit analysis solves for unknown node voltages from KCL equations.
- See rules above for details in writing the equations by inspection.


4.2.5E: Peer Groups
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Learning Objectives
- Analyze the importance of the peer group in terms of childhood and adolescent socialization
A peer group is a social group whose members have interests, social positions, and age in common. This is where children can escape supervision and learn to form relationships on their own. The influence of the peer group typically peaks during adolescence. However, peer groups generally only affect short term interests unlike the family, which has long term influence.
Unlike the family and the school, the peer group lets children escape the direct supervision of adults. Among peers, children learn to form relationships on their own. Peer groups also offer the chance to discuss interests that adults may not share with their children (such as clothing and popular music) or permit (such as drugs and sex ).
Peer groups have a significant influence on psychological and social adjustments for group individuals. They provide perspective outside of individual’s viewpoints. Members inside peer groups also learn to develop relationships with others in the social system. Peers, particularly group members, become important social referents for teaching members’ customs, social norms, and different ideologies.
Peer groups can also serve as a venue for teaching members gender roles. Through gender-role socialization group members learn about sex differences, social and cultural expectations. While boys and girls differ greatly there is not a one to one link between sex and gender role with males always being masculine and female always being feminine. Both genders can contain different levels of masculinity and femininity.
Adolescent peer groups provide support for children and teens as they assimilate into the adult society decreasing dependence on parents, increasing feeling of self-sufficiency, and connecting with a much larger social network. Peer groups cohesion is determined and maintained by such factors as group communication, group consensus, and group conformity concerning attitude and behavior. As members of peer groups interconnect, and agree, a normative code arises. This normative code can become very rigid deciding group behavior and dress. Peer group individuality is increased by normative codes, and intergroup conflict. Member deviation from the strict normative code can lead to rejection from the group. The term “peer pressure” is often used to describe instances where an individual feels indirectly pressured into changing their behavior to match that of their peers. Taking up smoking and underage drinking are two of the best known examples. In spite of the often negative connotations of the term, peer pressure can be used positively.
Key Points
- This is where children can escape supervision and learn to form relationships on their own.
- The influence of the peer group typically peaks during adolescence.
- However, peer groups generally only affect short term interests unlike the family, which has long term influence.
- Peer groups can also serve as a venue for teaching members gender roles.
- Adolescent peer groups provide support for children and teens as they assimilate into the adult society decreasing dependence on parents, increasing feeling of self-sufficiency, and connecting with a much larger social network.
- The term ” peer pressure ” is often used to describe instances where an individual feels indirectly pressured into changing their behavior to match that of their peers.
Key Terms
- peer pressure : Peer pressure is the influence exerted by a peer group, encouraging individuals to change their attitudes, values, or behaviors in order to conform to group norms.
- gender roles : Sets of social and behavioral norms that are generally considered appropriate for either a man or a woman in a social or interpersonal relationship.
- Peer group : A peer group is a social group whose members have interests, social positions, and age in common.


5: Geostrophic Balance
Now that we have an understanding of the main forces acting on fluid parcels moving through the ocean and the atmosphere, we focus on how these forces impact the flows in the atmosphere and the ocean. In the horizontal momentum balance equations \((1.2a)\) and \((1.2b)\) in Section 1 , the pressure gradient and Coriolis terms are almost always much larger than the other terms. Therefore, to a good approximation, there exists a balance between the pressure gradient and Coriolis forces named geostrophic balance :
\[\dfrac{\left(\dfrac{dp}{dx}\right)}{\rho}=f * v \label{5.1a}\]
\[\dfrac{\left(\dfrac{dp}{dy}\right)}{\rho}=-f * u \label{5.1b}\]
Geostrophic balance is arguably the most central concept in physical oceanography and dynamical meteorology. Although higher-order processes are responsible for all the interesting dynamics, it is important to realize that almost all large-scale flows in the atmosphere and ocean are in geostrophic balance to leading order. The only place where geostrophic balance never holds is at the Equator where there is no Coriolis force. A key feature of geostrophic balance is that rather than flowing from high to low pressure, the fluid actually moves parallel to lines of equal pressure (isobars). Viewing in the direction of the flow, low pressure is to the left and high pressure is to the right on the Northern Hemisphere (and vice versa in the Southern Hemisphere). This is illustrated in Figure \(\PageIndex{1}\).
To understand conceptually how geostrophic balance emerges, imagine a fluid parcel that starts accelerating due to a pressure gradient. As the parcel moves, it is deflected by the Coriolis force (see left panel below), until it flows parallel to the isobars and the pressure gradient and Coriolis force cancel each other (right panel below).
From the Equations \ref{5.1a} and \ref{5.1b}, it can easily be seen that
\[u_g = -\dfrac{\left(\dfrac{dp}{dy}\right)}{f * \rho}\]
and
\[v_g = \dfrac{\left(\dfrac{dp}{dx}\right)}{f * \rho},\]
with \(u_g\) and \(v_g\) the geostrophic velocities in the zonal and meridional directions, respectively. Thus, the geostrophic velocities increase with an increasing pressure gradient. Somewhat counterintuitively, geostrophic velocities decrease with increasing latitude, although the Coriolis force is stronger at higher latitudes. This is because with a weaker Coriolis force, the fluid is able to accelerate more under the influence of the pressure gradient force, before geostrophic equilibrium is reached.


1.5.1: Logging in
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The lock screen displays the current date, time, and usually and image background. You will learn how to change this later. The lock screen is displayed when the computer first boots or starts up and if you leave your computer unattended for a while. Windows 10 can also be set up to allow multiple users to have their own profile and files on the PC and utilize the lock screen to switch to their account.
To begin the login process, press the Spacebar on the keyboard to continue the login process. If the computer requires a username and/or password, you will be prompted to enter it here. You will learn how to create a password for your account later in Chapter 6 Windows Security. Depending on your computer hardware, windows may permit login passwords based upon a username and password, personal identification numbers (PIN) biometrics like your facial features or fingerprint, or some other password identification (such as a personal password) When you have completed entering your password, please click on the right arrow to complete the login into Windows.


2.3: Free Press- Simon and Schuster, Inc. v. Members of N. Y. State Crime Victims Bd. 502 U.S. 105 (1991)
OCTOBER TERM, 1991
SIMON & SCHUSTER, INC. v. MEMBERS OF THE NEW YORK STATE CRIME VICTIMS BOARD ET AL.
CERTIORARI TO THE UNITED STATES COURT OF APPEALS FOR THE SECOND CIRCUIT
No. 90-1059. Argued October 15, 1991-Decided December 10,1991
Among other things, New York’s “Son of Sam” law provides that an “entity” contracting with a person “accused or convicted of a crime” for the production of a book or other work describing the crime must pay to respondent Crime Victims Board any moneys owed to that person under the contract; requires the Board to deposit such funds in an escrow account for payment to any victim who, within five years, obtains a civil judgment against the accused or convicted person and to the criminal’s other creditors; and defines “person convicted of a crime” to include “any person who has voluntarily and intelligently admitted the commission of a crime for which such person is not prosecuted.” After it discovered that petitioner publisher had signed an agreement with an author who had contracted with admitted organized crime figure Henry Hill for the production of a book about Hill’s life, the Board, inter alia, determined that petitioner had violated the Son of Sam law and ordered it to turn over all money payable to Hill. Petitioner then brought suit under 42 U. S. C. § 1983, seeking a declaration that the law violates the First Amendment and an injunction barring the law’s enforcement. The District Court found the law to be consistent with the Amendment, and the Court of Appeals affirmed.
Held: The Son of Sam law is inconsistent with the First Amendment.
Pp. 115-123.
(a) Whether the First Amendment “speaker” is considered to be Hill, whose income the New York law places in escrow because of the story he has told, or petitioner, which can publish books about crime with the assistance of only those criminals willing to forgo remuneration for at least five years, the law singles out speech on a particular subject for a financial burden that it places on no other speech and no other income and, thus, is presumptively inconsistent with the Amendment. Leathers v. Medlock, 499 U. S. 439 , 447; Arkansas Writers’ Project, Inc. v. Ragland, 481 U. S. 221 , 230. The fact that the law escrows speechderived income, rather than taxing a percentage of it outright as did the law invalidated in Arkansas Writers’ Project, cannot serve as the basis for disparate treatment under the Amendment, since both forms of financial burden operate as disincentives to speak. Moreover, the
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Board’s assertion that discriminatory financial treatment is suspect only when the legislature intends to suppress certain ideas is incorrect, since this Court has long recognized that even regulations aimed at proper governmental concerns can restrict unduly the exercise of rights under the Amendment. Furthermore, the Board’s claim that the law is permissible under the Amendment because it focuses generally on an “entity” rather than specifically on the media falters, first, on semantic grounds, since any entity that enters into a contract with a convicted person to transmit that person’s speech becomes by definition a medium of communication, and, second, on constitutional grounds, since the governmental power to impose content-based financial disincentives on speech does not vary with the identity of the speaker. Accordingly, in order to justify the differential treatment imposed by the law, the State must show that its regulation is necessary to serve a compelling state interest and is narrowly drawn to achieve that end. Id., at 231. Pp. 115-118.
(b) The State has a compelling interest in compensating victims from the fruits of crime. Cf. Caplin & Drysdale, Chartered v. United States, 491 U. S. 617 , 629. However, contrary to the Board’s assertion, the State has little if any interest in limiting such compensation to the proceeds of the wrongdoer’s speech about the crime. The Board cannot explain why the State should have any greater interest in compensating victims from the proceeds of criminals’ “storytelling” than from any of their other assets, nor offer any justification for a distinction between this expressive activity and any other activity in connection with its interest in transferring the fruits of crime from criminals to their victims. Cf., e. g., Arkansas Writers’ Project, supra, at 231. Like the governmental entities in the latter and similar cases, the Board has taken the effect of the statute and posited that effect as the State’s interest. Pp. 118-121.
(c) The New York law is not narrowly tailored to achieve the State’s objective of compensating victims from the profits of crime. The law is significantly overinclusive, since it applies to works on any subject provided that they express the author’s thoughts or recollections about his crime, however tangentially or incidentally, and since its broad definition of “person convicted of a crime” enables the Board to escrow the income of an author who admits in his work to having committed a crime, whether or not he was ever actually accused or convicted. These two provisions combine to encompass a wide range of existing and potential works that do not enable a criminal to profit from his crime while a victim remains uncompensated. Pp. 121-123.
Contributors and Attributions
- Located at : https://supreme.justia.com/cases/federal/us/502/105/ . License : Public Domain: No Known Copyright


12.3: Arithmetic Sequences
By the end of this section, you will be able to:
- Determine if a sequence is arithmetic
- Find the general term (\(n\)th term) of an arithmetic sequence
- Find the sum of the first \(n\) terms of an arithmetic sequence
Before you get started, take this readiness quiz.
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Evaluate \(4n−1\) for the integers \(1, 2, 3\), and \(4\).
If you missed this problem, review Example 1.6. -
Solve the system of equations: \(\left\{\begin{array}{l}{x+y=7} \\ {3 x+4 y=23}\end{array}\right.\).
If you missed this problem, review Example 4.9. -
If \(f(n)=\frac{n}{2}(3 n+5)\), find \(f(1)+f(20)\).
If you missed this problem, review Example 3.49.
Determine if a Sequence is Arithmetic
The last section introduced sequences and now we will look at two specific types of sequences that each have special properties. In this section we will look at arithmetic sequences and in the next section, geometric sequences.
An arithmetic sequence is a sequence where the difference between consecutive terms is constant. The difference between consecutive terms in an arithmetic sequence, a_{n}-a_{n-1}, is \(d\), the common difference , for \(n\) greater than or equal to two.
An arithmetic sequence is a sequence where the difference between consecutive terms is always the same.
The difference between consecutive terms, a_{n}-a_{n-1}, is \(d\), the common difference , for \(n\) greater than or equal to two.
Determine if each sequence is arithmetic. If so, indicate the common difference.
- \(5,9,13,17,21,25, \dots\)
- \(4,9,12,17,20,25, \dots\)
- \(10,3,-4,-11,-18,-25, \dots\)
Solution :
To determine if the sequence is arithmetic, we find the difference of the consecutive terms shown.
a. \(\begin{array}{cccccc}{5,} & {9,} & {13,} & {17} & {21,} & {25, \ldots} \\ {\text { Find the difference of the consecutive terms.}} & {9-5} & {13-9} & {17-13} & {21-17} & {25-21} \\ & {4} & {4} & {4} & {4}&{4}\end{array}\)
The sequence is arithmetic. The common difference is \(d=4\).
b. \(\begin{array}{cccccc}{4,} & {9,} & {12,} & {17} & {20,} & {25, \ldots} \\ {\text { Find the difference of the consecutive terms.}} & {9-4} & {12-9} & {17-12} & {20-17} & {25-20} \\ & {2} & {3} & {5} & {3}&{5}\end{array}\)
The sequence is not arithmetic as all the differences between the consecutive terms are not the same. There is no common difference.
c. \(\begin{array}{cccccc}{10,} & {3,} & {-4,} & {-11} & {-18,} & {-25, \ldots} \\ {\text { Find the difference of the consecutive terms.}} & {3-10} & {-4-3} & {-11-(-4)} & {-18-(-11)} & {-25-(-18)} \\ & {-7} & {-7} & {-7} & {-7}&{-7}\end{array}\)
Answer :
The sequence is arithmetic. The common difference is \(d=-7\).
Determine if each sequence is arithmetic. If so, indicate the common difference.
- \(9,20,31,42,53,64, \dots\)
- \(12,6,0,-6,-12,-18, \dots\)
- \(7,1,10,4,13,7, \dots\)
- Answer
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- The sequence is arithmetic with common difference \(d=11\).
- The sequence is arithmetic with common difference \(d=-6\).
- The sequence is not arithmetic as all the differences between the consecutive terms are not the same.
Determine if each sequence is arithmetic. If so, indicate the common difference.
- \(-4,4,2,10,8,16, \dots\)
- \(-3,-1,1,3,5,7, \dots\)
- \(7,2,-3,-8,-13,-18, \dots\)
- Answer
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- The sequence is not arithmetic as all the differences between the consecutive terms are not the same.
- The sequence is arithmetic with common difference \(d=2\).
- The sequence is arithmetic with common difference \(d=−5\).
If we know the first term, \(a_{1}\), and the common difference, \(d\), we can list a finite number of terms of the sequence.
Write the first five terms of the sequence where the first term is \(5\) and the common difference is \(d=−6\).
Solution :
We start with the first term and add the common difference. Then we add the common difference to that result to get the next term, and so on.
\(\begin{array}{cccc}{a_{1}} & {a_{2}} & {a_{3}} & {a_{4}} & {a_{5}} \\ {5} & {5+(-6)} & {-1+(-6)} & {-7+(-6)} & {-13+(-6)} \\ {}&{-1} & {-7} & {-13} & {-19}\end{array}\)
Answer :
The sequence is \(5,-1,-7,-13,-19, \dots\)
Write the first five terms of the sequence where the first term is \(7\) and the common difference is \(d=−4\).
- Answer
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\(7,3,-1,-5,-9, \dots\)
Write the first five terms of the sequence where the first term is \(11\) and the common difference is \(d=−8\).
- Answer
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\(11,3,-5,-13,-21, \dots\)
Find the General Term (\(n\)th Term) of an Arithmetic Sequence
Just as we found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence.
Let’s write the first few terms of a sequence where the first term is \(a_{1}\) and the common difference is \(d\). We will then look for a pattern.
As we look for a pattern we see that each term starts with \(a_{1}\).
The first term adds \(0d\) to the \(a_{1}\), the second term adds \(1d\), the third term adds \(2d\), the fourth term adds \(3d\), and the fifth term adds \(4d\). The number of \(ds\) that were added to \(a_{1}\) is one less than the number of the term. This leads us to the following
\(a_{n}=a_{1}+(n-1) d\)
The general term of an arithmetic sequence with first term \(a_{1}\) and the common difference \(d\) is
\(a_{n}=a_{1}+(n-1) d\)
We will use this formula in the next example to find the 15 th term of a sequence.
Find the fifteenth term of a sequence where the first term is \(3\) and the common difference is \(6\).
Solution :
\(\begin{array}{cc}{\text{To find the fifteenth term, }a_{15}\text{, use the formula with } a_{1}=3 \:\text{and} \:d=6.}&{a_{n}=a_{1}+(n-1) d} \\ {\text{Substitute in the values.}}&{a_{15}=3+(15-1) 6} \\{\text{Simplify.}}& {a_{15}=3+(14) 6} \\ {}&{a_{15}=87}\end{array}\)
Find the twenty-seventh term of a sequence where the first term is \(7\) and the common difference is \(9\).
- Answer
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\(241\)
Find the eighteenth term of a sequence where the first term is \(13\) and the common difference is \(−7\).
- Answer
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\(-106\)
Sometimes we do not know the first term and we must use other given information to find it before we find the requested term.
Find the twelfth term of a sequence where the seventh term is \(10\) and the common difference is \(−2\). Give the formula for the general term.
Solution :
To first find the first term, \(a_{1}\), use the formula with \(a_{7}=10\),\(n=7\), and \(d=−2\). Substitute in the values. Simplify.
\(a_{n}=a_{1}+(n-1) d\)
\(10=a_{1}+(7-1)(-2)\)
\(10=a_{1}+(6)(-2)\)
\(10=a_{1}-12\)
\(a_{1}=22\)
Find the twelfth term, \(a_{12}\),using the formula with \(a_{1}=22\), \(n=12\), and \(d=-2\). Substitute in the values. Simplify.
\(a_{n}=a_{1}+(n-1) d\)
\(a_{12}=22+(12-1)(-2)\)
\(a_{12}=22+(11)(-2)\)
\(a_{12}=0\)
The twelfth term of the sequence is \(0, a_{12}=0\)
To find the general term, substitute the values into the formula.
\(a_{n}=a_{1}+(n-1) d\)
\(a_{n}=22+(n-1)(-2)\)
\(a_{n}=22-2 n+2\)
Answer
:
The general term is \(a_{n}=-2 n+24\)
Find the eleventh term of a sequence where the ninth term is \(8\) and the common difference is \(−3\). Give the formula for the general term.
- Answer
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\(a_{11}=2 .\) The general term is \(a_{n}=-3 n+35\)
Find the nineteenth term of a sequence where the fifth term is \(1\) and the common difference is \(−4\).Give the formula for the general term.
- Answer
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\(a_{19}=-55 .\) The general term is \(a_{n}=-4 n+21\)
Sometimes the information given leads us to two equations in two unknowns. We then use our methods for solving systems of equations to find the values needed.
Find the first term and common difference of a sequence where the fifth term is \(19\) and the eleventh term is \(37\). Give the formula for the general term.
Solution :
Since we know two terms, we can make a system of equations using the formula for the general term.
| We know the value of \(a_{5}\) and \(a_{11}\), so we will use \(n=5\) and \(n=11\). | |
|
Substitute in the values, \(a_{5}=19\) and \(a_{11}=37\). |
|
| Simplify. | |
|
Prepare to eliminate the \(a_{1}\) term by multiplying the top equation by \(−1\).
Add the equations. |
|
| Substituting \(d=3\) back into the first equation. | |
| Solve for \(a_{1}\). | |
| Use the formula with \(a_{1}=7\) and \(d=3\). | |
| Substitute in the values. | |
| Simplify. | |
|
The first term is \(a_{1}=7\).
The common difference is \(d=3\). |
|
| The general term of the sequence is \(a_{n}=3n+4\). |
Answer :
The general term of the sequence is \(a_{n}=3n+4\).
Find the first term and common difference of a sequence where the fourth term is \(17\) and the thirteenth term is \(53\). Give the formula for the general term.
- Answer
-
\(a_{1}=5, d=4 .\) The general term is \(a_{n}=4 n+1\).
Find the first term and common difference of a sequence where the third term is \(2\) and the twelfth term is \(−25\). Give the formula for the general term.
- Answer
-
\(a_{1}=8, d=-3 .\) The general term is \(a_{n}=-3 n+11\).
Find the Sum of the First \(n\) Terms of an Arithmetic Sequence
As with the general sequences, it is often useful to find the sum of an arithmetic sequence. The sum, \(S_{n}\), of the first \(n\) terms of any arithmetic sequence is written as \(S_{n} =a_{1} +a_{2} +a_{3} +\ldots +a_{n}\). To find the sum by merely adding all the terms can be tedious. So we can also develop a formula to find the sum of a sequence using the first and last term of the sequence.
We can develop this new formula by first writing the sum by starting with the first term, \(a_{1}\), and keep adding a \(d\) to get the next term as:
\(S_{n}=a_{1}+\left(a_{1}+d\right)+\left(a_{1}+2 d\right)+\ldots+a_{n}\).
We can also reverse the order of the terms and write the sum by starting with \(a_{n}\) and keep subtracting \(d\) to get the next term as
\(S_{n}=a_{n}+\left(a_{n}-d\right)+\left(a_{n}-2 d\right)+\ldots+a_{1}\).
If we add these two expressions for the sum of the first \(n\) terms of an arithmetic sequence, we can derive a formula for the sum of the first \(n\) terms of any arithmetic series.
\(\begin{aligned} &S_{n}= a_{1} \quad+\left(a_{1}+d\right)+\left(a_{1}+2 d\right)+\ldots+a_{n} \\+&S_{n} =a_{n} \quad+\left(a_{n}-d\right)+\left(a_{n}-2 d\right)+\ldots+a_{1} \\ \hline \\ &2S_{n}=(a_{1}+a_{n})+(a_{1}+a_{n})+(a_{1}+a_{n})+\dots+(a_{1}+a_{n}) \end{aligned}\)
Because there are \(n\) sums of \((a_{1}+a_{n})\) on the right side of the equation, we rewrite the right side as \(n(a_{1}+a_{n})\).
\(2 S_{n}=n\left(a_{1}+a_{n}\right)\)
We divide by two to solve for \(S_{n}\).
\(S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right)\)
This give us a general formula for the sum of the first \(n\) terms of an arithmetic sequence.
The sum, \(S_{n}\), of the first \(n\) terms of an arithmetic sequence is
\(S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right)\)
where \(a_{1}\) is the first term and \(a_{n}\) is the \(n\)th term.
We apply this formula in the next example where the first few terms of the sequence are given.
Find the sum of the first \(30\) terms of the arithmetic sequence: \(8, 13, 18, 23, 28, …\)
Solution :
To find the sum, we will use the formula \(S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right)\). We know \(a_{1}=8, d=5\) and \(n=30\), but we need to find \(a_{n}\) in order to use the sum formula.
Find \(a_{n}\) where \(a_{1}=8, d=5\) and \(n=30\). Simplify.
\(\begin{aligned} a_{n} &=a_{1}+(n-1) d \\ a_{30} &=8+(30-1) 5 \\ a_{30} &=8+(29) 5 \\ a_{30} &=153 \end{aligned}\)
Knowing \(a_{1}=8, n=30\), and \(a_{30}=153\), use the sum formula. Substitute in the values. Simplify. Simplify.
\(\begin{aligned} S_{n} &=\frac{n}{2}\left(a_{1}+a_{n}\right) \\ S_{30} &=\frac{30}{2}(8+153) \\ S_{30} &=15(161) \\ S_{30} &=2,415 \end{aligned}\)
Find the sum of the first \(30\) terms of the arithmetic sequence: \(5, 9, 13, 17, 21, …\)
- Answer
-
\(1,890\)
Find the sum of the first \(30\) terms of the arithmetic sequence: \(7, 10, 13, 16, 19, …\)
- Answer
-
\(1,515\)
In the next example, we are given the general term for the sequence and are asked to find the sum of the first \(50\) terms.
Find the sum of the first \(50\) terms of the arithmetic sequence whose general term is \(a_{n}=3n−4\).
Solution :
To find the sum, we will use the formula \(S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right)\). We know \(n=50\), but we need to find \(a_{1}\) and \(a_{n}\) in order to use the sum formula.
| Find \(a_{1}\), by substituting \(n=1\). | |
| Find \(a_{n}\) by substituting \(n=50\). | |
| Simplify. | |
| Knowing \(n=50, a_{1}=−1,\) and \(a_{50}=146\) use the sum formula. | |
| Substitute in the values. | |
| Simplify. | |
| Simplify. |
Find the sum of the first \(50\) terms of the arithmetic sequence whose general term is \(a_{n}=2n−5\).
- Answer
-
\(2,300\)
Find the sum of the first \(50\) terms of the arithmetic sequence whose general term is \(a_{n}=4n+3\).
- Answer
-
\(5,250\)
In the next example we are given the sum in summation notation. To add all the terms would be tedious, so we extract the information needed to use the formula to find the sum of the first \(n\) terms.
Find the sum: \(\sum_{i=1}^{25}(4 i+7)\).
Solution :
To find the sum, we will use the formula \(S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right)\). We know \(n=25\), but we need to find \(a_{1}\) and \(a_{n}\) in order to use the sum formula.
| Expand the summation notation. | |
| Simplify. | |
| Identify \(a_{1}\). | |
| Identify \(a_{25}\). | |
| Knowing \(n=25, a_{1}=11\), and \(a_{25} = 107\) use the sum formula. | |
| Substitute in the values. | |
| Simplify. | |
| Simplify. |
Find the sum: \(\sum_{i=1}^{30}(6 i-4)\).
- Answer
-
\(2,670\)
Find the sum: \(\sum_{i=1}^{35}(5 i-3)\).
- Answer
-
\(3,045\)
Access these online resources for additional instruction and practice with arithmetic sequences
Key Concepts
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General Term (\(n\)th term) of an Arithmetic Sequence
The general term of an arithmetic sequence with first term \(a_{1}\) and the common difference \(d\) is\(a_{n}=a_{1}+(n-1) d\)
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Sum of the First \(n\) Terms of an Arithmetic Sequence
The sum, \(S_{n}\), of the first \\(n\) terms of an arithmetic sequence, where \(a_{1}\) is the first term and \(a_{n}\) is the \(n\)th term is\(S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right)\)
Glossary
- arithmetic sequence
- An arithmetic sequence is a sequence where the difference between consecutive terms is constant.
- common difference
- The difference between consecutive terms in an arithmetic sequence, \(a_{n}−a_{n−1}\), is \(d\), the common difference, for \(n\) greater than or equal to two.


8.7: Titrations - Neutralization and Stoichiometry
One of the standard laboratory exercises in General Chemistry is an acid-base titration . In order to perform an acid-base titration, you must have a solution of acid or base with a known concentration. You then slowly add a known volume of this solution, using a volumetric burette , to an acid or base solution with an unknown concentration until neutrality has been achieved. At that point, you know the volume and concentration of the reactant you have added, which means that you can calculate the number of moles that you added. Based on the stoichiometry of your neutralization reaction, you then know how many moles of acid or base were in the unknown sample. How do you know when you have reached neutrality? Generally an indicator or a p H meter is used (as described in Section 8.5). For example, if we had a solution of NaOH that was exactly 0.100 M and we had a beaker containing an unknown concentration of HCl. To perform the titration we would add a few drops of a stock phenolphthalein solution to our HCl, and then slowly add a measured amount of the NaOH solution until all of the acid had been consumed and the indicator changed from colorless to pink.
HCl (aq) + NaOH (aq) → NaCl (aq) + H 2 O
Working with the above example, if the volume of base that we added (as measured on the buret) was 12.6 mL, we could calculate the number of moles present in the unknown acid solution. This is equal to the known concentration of our NaOH (0.100 M) multiplied by the volume required for neutrality (0.0126 L), or:
\[(0.100moles/L)(0.0126L)=1.26\times 10^{-3}moles \nonumber \]
If we had used exactly 100.0 mL of our unknown acid in our titration, the concentration of our acid would be:
\[\frac{(1.26\times 10^{-3}moles)}{(0.1000L)}=1.26\times 10^{-2}M \nonumber \]
- You are given a solution containing an unknown concentration of HCl. You carefully measure 50.0 mL of this solution into a flask and then add a few drops of phenolphthalein solution. You prepare a buret containing 0.055 M NaOH and note that the initial level of the solution in the buret is 12.6 mL. You slowly add the NaOH solution to the acid until the color change just occurs (as evidence of the color change becomes visible, you carefully stir the solution after each drop has been added). When the acid solution turns (and remains) pink, you note that the volume in the buret is now 28.9 mL. What is the concentration of the unknown acid solution?
- If 25.00 mL of HCl solution with a concentration of 0.1234 M is neutralized by 23.45 mL of NaOH, what is the concentration of the base?