4.2: Probability Distribution Function (PDF) for a Discrete Random Variable
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A discrete probability distribution function has two characteristics:
- Each probability is between zero and one, inclusive.
- The sum of the probabilities is one.
Example \(\PageIndex{1}\)
A child psychologist is interested in the number of times a newborn baby's crying wakes its mother after midnight. For a random sample of 50 mothers, the following information was obtained. Let \(X =\) the number of times per week a newborn baby's crying wakes its mother after midnight. For this example, \(x = 0, 1, 2, 3, 4, 5\).
\(P(x) =\) probability that \(X\) takes on a value \(x\).
| \(x\) | \(P(x)\) |
|---|---|
| 0 | \(P(x = 0) = \dfrac{2}{50}\) |
| 1 | \(P(x = 1) = \dfrac{11}{50}\) |
| 2 | \(P(x = 2) = \dfrac{23}{50}\) |
| 3 | \(P(x = 3) = \dfrac{9}{50}\) |
| 4 | \(P(x = 4) = \dfrac{4}{50}\) |
| 5 | \(P(x = 5) = \dfrac{1}{50}\) |
\(X\) takes on the values 0, 1, 2, 3, 4, 5. This is a discrete PDF because:
- Each \(P(x)\) is between zero and one, inclusive.
- The sum of the probabilities is one, that is,
\[\dfrac{2}{50} + \dfrac{11}{50} + \dfrac{23}{50} + \dfrac{9}{50} + \dfrac{4}{50} + \dfrac{1}{50} = 1\]
Exercise \(\PageIndex{1}\)
A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a 12-hour shift. For a random sample of 50 patients, the following information was obtained. Let \(X =\) the number of times a patient rings the nurse during a 12-hour shift. For this exercise, \(x = 0, 1, 2, 3, 4, 5\). \(P(x) =\) the probability that \(X\) takes on value \(x\). Why is this a discrete probability distribution function (two reasons)?
| \(X\) | \(P(x)\) |
|---|---|
| 0 | \(P(x = 0) = \dfrac{4}{50}\) |
| 1 | \(P(x = 1) = \dfrac{8}{50}\) |
| 2 | \(P(x = 2) = \dfrac{16}{50}\) |
| 3 | \(P(x = 3) = \dfrac{14}{50}\) |
| 4 | \(P(x = 4) = \dfrac{6}{50}\) |
| 5 | \(P(x = 5) = \dfrac{2}{50}\) |
Answer
Each \(P(x)\) is between 0 and 1, inclusive, and the sum of the probabilities is 1, that is:
\[\dfrac{4}{50} + \dfrac{8}{50} +\dfrac{16}{50} +\dfrac{14}{50} +\dfrac{6}{50} + \dfrac{2}{50} = 1\]
Example \(\PageIndex{2}\)
Suppose Nancy has classes three days a week. She attends classes three days a week 80% of the time, two days 15% of the time, one day 4% of the time, and no days 1% of the time. Suppose one week is randomly selected.
- Let \(X\) = the number of days Nancy ____________________.
- \(X\) takes on what values?
- Suppose one week is randomly chosen. Construct a probability distribution table (called a PDF table) like the one in Example . The table should have two columns labeled \(x\) and \(P(x)\). What does the \(P(x)\) column sum to?
Solutions
a. Let \(X\) = the number of days Nancy attends class per week.
b. 0, 1, 2, and 3
c
| \(x\) | \(P(x)\) |
|---|---|
| 0 | 0.01 |
| 1 | 0.04 |
| 2 | 0.15 |
| 3 | 0.80 |
Exercise \(\PageIndex{2}\)
Jeremiah has basketball practice two days a week. Ninety percent of the time, he attends both practices. Eight percent of the time, he attends one practice. Two percent of the time, he does not attend either practice. What is X and what values does it take on?
Answer
\(X\) is the number of days Jeremiah attends basketball practice per week. X takes on the values 0, 1, and 2.
Review
The characteristics of a probability distribution function (PDF) for a discrete random variable are as follows:
- Each probability is between zero and one, inclusive ( inclusive means to include zero and one).
- The sum of the probabilities is one.
Use the following information to answer the next five exercises: A company wants to evaluate its attrition rate, in other words, how long new hires stay with the company. Over the years, they have established the following probability distribution.
Let \(X =\) the number of years a new hire will stay with the company.
Let \(P(x) =\) the probability that a new hire will stay with the company x years.
Exercise 4.2.3
Complete Table using the data provided.
| \(x\) | \(P(x)\) |
|---|---|
| 0 | 0.12 |
| 1 | 0.18 |
| 2 | 0.30 |
| 3 | 0.15 |
| 4 | |
| 5 | 0.10 |
| 6 | 0.05 |
Answer
| \(x\) | \(P(x)\) |
|---|---|
| 0 | 0.12 |
| 1 | 0.18 |
| 2 | 0.30 |
| 3 | 0.15 |
| 4 | 0.10 |
| 5 | 0.10 |
| 6 | 0.05 |
Exercise 4.2.4
\(P(x = 4) =\) _______
Exercise 4.2.5
\(P(x \geq 5) =\) _______
Answer
0.10 + 0.05 = 0.15
Exercise 4.2.6
On average, how long would you expect a new hire to stay with the company?
Exercise 4.2.7
What does the column “ P ( x )” sum to?
Answer
1
Use the following information to answer the next six exercises: A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer. Through observation, the baker has established a probability distribution.
| \(x\) | \(P(x)\) |
|---|---|
| 1 | 0.15 |
| 2 | 0.35 |
| 3 | 0.40 |
| 4 | 0.10 |
Exercise 4.2.8
Define the random variable \(X\).
Exercise 4.2.9
What is the probability the baker will sell more than one batch? \(P(x > 1) =\) _______
Answer
0.35 + 0.40 + 0.10 = 0.85
Exercise 4.2.10
What is the probability the baker will sell exactly one batch? \(P(x = 1) =\) _______
Exercise 4.2.11
On average, how many batches should the baker make?
Answer
1(0.15) + 2(0.35) + 3(0.40) + 4(0.10) = 0.15 + 0.70 + 1.20 + 0.40 = 2.45
Use the following information to answer the next four exercises: Ellen has music practice three days a week. She practices for all of the three days 85% of the time, two days 8% of the time, one day 4% of the time, and no days 3% of the time. One week is selected at random.
Exercise 4.2.12
Define the random variable \(X\).
Exercise 4.2.13
Construct a probability distribution table for the data.
Answer
| \(x\) | \(P(x)\) |
|---|---|
| 0 | 0.03 |
| 1 | 0.04 |
| 2 | 0.08 |
| 3 | 0.85 |
Exercise 4.2.14
We know that for a probability distribution function to be discrete, it must have two characteristics. One is that the sum of the probabilities is one. What is the other characteristic?
Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35% of the time, four events 25% of the time, three events 20% of the time, two events 10% of the time, one event 5% of the time, and no events 5% of the time.
Exercise 4.2.15
Define the random variable \(X\).
Answer
Let \(X =\) the number of events Javier volunteers for each month.
Exercise 4.2.16
What values does \(x\) take on?
Exercise 4.2.17
Construct a PDF table.
Answer
| \(x\) | \(P(x)\) |
|---|---|
| 0 | 0.05 |
| 1 | 0.05 |
| 2 | 0.10 |
| 3 | 0.20 |
| 4 | 0.25 |
| 5 | 0.35 |
Exercise 4.2.18
Find the probability that Javier volunteers for less than three events each month. \(P(x < 3) =\) _______
Exercise 4.2.19
Find the probability that Javier volunteers for at least one event each month. \(P(x > 0) =\) _______
Answer
1 – 0.05 = 0.95
Glossary
- Probability Distribution Function (PDF)
- a mathematical description of a discrete random variable ( RV ), given either in the form of an equation (formula) or in the form of a table listing all the possible outcomes of an experiment and the probability associated with each outcome.


7.1.5B: Class, Crime, and the Criminal Justice System
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Learning Objectives
- Explain why white-collar crime is less likely to be tracked in the U.S.
Of the classical founders of social science, conflict theory is most commonly associated with Karl Marx. Based on a dialectical materialist account of history, Marxism posited that capitalism, like previous socioeconomic systems, would inevitably produce internal tensions leading to its own destruction. Marx ushered in radical change, advocating proletarian revolution and freedom from the ruling classes. In Marxist theory, the class structure of the capitalist mode of production is characterized by the conflict between two main classes: the bourgeoisie, the capitalists who own the means of production, and the much larger proletariat who must sell their own labor power.
Theory of Deviance
Clifford Shaw and Henry D. McKay theorized that social disorganization was a root cause of deviancy and crime, especially for minority youth. They discussed the fact that inner city kids tended to be more involved in a criminal lifestyle than kids who lived in the suburbs. Being able to afford to live in better parts of the city (and thus having wealth) afforded certain kids better opportunities in terms of lifestyle and education, leading to less crime and criminal involvement.
White-Collar Crime
White-collar crime is a financially motivated, nonviolent crime committed for illegal monetary gain. Within the field of criminology, white-collar crime initially was defined by sociologist Edwin Sutherland in 1939 as “a crime committed by a person of respectability and high social status in the course of his occupation. ” A clear example of how deviance reflects power imbalances is in the reporting and tracking of crimes. Indeed, white-collar crimes are typically committed by individuals in higher social classes. That white-collar crimes are less likely to be tracked, less likely to be reported, less likely to be prosecuted, and are more likely to be committed by people in higher social classes suggests that the way crimes are punished in the United States tends to favor the affluent while punitively punishing the less affluent. Additionally, men benefit more from white-collar crime than do women, as they are more likely to attempt these crimes when they are in more powerful positions, allowing them to reap greater rewards.
The Criminal Justice System
Criminal justice is the system of practices and institutions of governments directed at upholding social control, deterring and mitigating crime, or sanctioning those who violate laws with criminal penalties and rehabilitation efforts. Those accused of crime have protections against abuse of investigatory and prosecution powers.
Within the criminal justice system, there are three basic elements that constitute it: the police, the courts, and punishment. The police maintain public order by enforcing the law. Police use personal discretion in deciding whether and how to handle a situation. Research suggests that police are more likely to make an arrest if the offense is serious, if bystanders are present, or if the suspect is of a visible minority. Courts rely on an adversarial process in which attorneys-one representing the defendant and one representing the crown-present their cases in the presence of a judge who monitors legal procedures. In practice, courts resolve most cases through a plea bargain. Though efficient, this method puts less-powerful people at a disadvantage. There are four jurisdictions for punishment: retribution, deterrence, rehabilitation, and societal protection. Community-bases corrections include probation and parole. These programs lower the cost of supervising people convicted of crimes and reduce prison overcrowding but have not been shown to reduce recidivism.
Key Points
- In Marxist theory, the class structure of the capitalist mode of production is characterized by the conflict between two main classes.
- Bourgeoisie are the capitalists who own the means of production, while the much larger proletariat who must sell their own labor power.
- White-collar crime is a financially motivated, nonviolent crime committed for illegal monetary gain.
- Criminal justice is the system of practices and institutions of governments directed at upholding social control, deterring, and mitigating crime or sanctioning those who violate laws with criminal penalties and rehabilitation efforts.
- Criminal justice is the system of practices and institutions of governments directed at upholding social control, deterring and mitigating crime, or sanctioning those who violate laws with criminal penalties and rehabilitation efforts.
- There are four jurisdictions for punishment: retribution, deterrence, rehabilitation, and societal protection.
- Courts rely on an adversarial process in which attorneys-one representing the defendant and one representing the crown-present their cases in the presence of a judge who monitors legal procedures
- There are four jurisdictions for punishment: retribution, deterrence, rehabilitation, and societal protection.
Key Terms
- Marxist Theory : An economic and sociopolitical worldview and method of socioeconomic inquiry centered upon a materialist interpretation of history, a dialectical view of social change, and an analysis–critique of the development of capitalism.
- white-collar crime : A non-violent crime, generally for personal gain and often involving money.
- plea bargain : An agreement in which a defendant agrees to plead guilty to a lesser charge instead of not guilty to a greater one


1.7: Ethical Communication
Defining Communication Ethics
Communication has ethical implications. Ethics in the broadest sense asks questions about what we believe to be right and wrong. Communication ethics asks these questions when reflecting on our communication. Everyday we have to make communicative choices, and some of these choices will be more or less ethical than other options. It is because we have these different options that our ethics are tested. We can never really say that something is completely ethical or unethical, especially when it comes to communication. “Murdering someone is generally thought of as unethical and illegal, but many instances of hurtful speech, or even what some would consider hate speech, have been protected as free speech. This shows the complicated relationship between protected speech, ethical speech, and the law” (Communication in the Real World, 2013).
When we make communication choices, the question of whether they are ethical or not depends on a variety of situational, personal, and and/or contextual variables that can be difficult to navigate. Many professional organizations have created ethical codes to help guide this decision-making, and the field of Communication Studies is no different. In 1999, the National Communication Association officially adopted the Credo for Ethical Communication. The NCA Credo for Ethical Communication is a set of beliefs that Communication scholars have about the ethics of human communication (NCA Legislative Council, November 1999).
We should always strive for ethical communication, but it is particularly important in interpersonal interactions. We will talk more about climate, trust and honesty, and specific relationships in the coming chapters, but at the most basic level you should strive to make ethical choices in your communication. Communication is impactful. Our communication choices have lasting impacts on those with whom we engage. While ethics is a focus on what is right and wrong, it is not easy to navigate. What is right in one circumstance may not be in another. To help us make our way through difficult ethical choices we must be competent.
Communication Competence
Communication competence focuses on communicating effectively and appropriately in various contexts (Kiessling & Fabry, 2021). In order to be competent you must have knowledge, motivation, and skills. You have been communicating for most of your life, so you have observational knowledge about how communication works. You are also now a college student actively studying communication so your knowledge will continue to increase. As you learn more about communication, continue to observe these concepts around you and you will expand the information you have to draw on in any given context. In addition to having basic information you must also be motivated to better your own communication and you need to develop the skills necessary to do so. One way to improve your communication competence is to become a more mindful communicator. “A mindful communicator actively and fluidly processes information, is sensitive to communication contexts and multiple perspectives, and is able to adapt to novel communication situations” (Communication in the Real World), 2013. Your path to improving your interpersonal communication competence is just beginning. You will learn more about specific aspects of mindfulness, such as listening, conflict management, deception, etc., in the coming chapters. For now we hope you are motivated to improve your knowledge and grow your skills.


10: Descriptions
- Explain a technical description.
- Exemplify sample definitions and descriptions.
An important part of technical and report writing is the use of definitions, descriptions, and instructions. They often work together to clarify the purpose and information in a technical document for an audience. This series of three chapters look at these three elements: definitions, descriptions, and instructions, more closely, so that you understand how to incorporate them into your technical documents to achieve clarity and conciseness. For that reason, you will find this introduction at the beginning of each chapter to remind you of their interconnectedness.
Definitions
We grow up reading these from the time we are in elementary school. They are one of the more common patterns of organization used. Definitions can be simple short insertions in a sentence to clarify a term, or they can be an entire document known as an extended definition.
Descriptions
These are similar to a definition but can be longer, more detailed, and have a visual representation to further explain a concept. More than likely, you have read these as well. Descriptions explain objects, mechanisms, and processes – and they do so in a way that whatever is being described is easily understood. If a description of the greenhouse effect was created for specialists in the field, it might be more complex. If this same idea was written for a general audience, the language would be simplified.
Instructions
Instructions provide step-by-step explanations of how to do things: assemble something, operate something, repair something, or explain a personal process (enrolling in college, for example) so that readers may better understand it and possibly use it themselves.
What are Technical Descriptions?
Descriptions and definitions are closely connected. Often, a technical writer will include description in an extended definition or include definition while writing a description. We all use the word description loosely to refer to practically any discussion or explanation. But in a technical writing context, description means the detailed discussion of the physical aspects of a thing. That means discussing things like color, shape, size, weight, height, width, thickness, texture, density, contents, materials of construction, and so on.
The following sentence is not a description; it explains the function or purpose but provides little or no physical detail. It is a definition.
A computer diskette is a device used for storing electronic data.
However, this sentence is definitely description:
The common computer diskette is 3.5 inches by 3.5 inches and approximately 1/8 inch thick.
Contexts for Description
Descriptions are a common element in technical writing. Certain types of technical writing feature description:
- Accident reports requiring plenty of description.
- Product specifications—documents that describe the design and features of a new or changed product—have plenty of description.
- Instructions often require description to help readers visualize what they are doing and what they are working with.
Contents and Organization of Descriptions
Below is a review of the sections you will commonly find in descriptions:
Introduction. Plan the introduction to your description carefully. Make sure it does all of the following things (but not necessarily in this order) that apply to your particular description:
- Indicate the specific object about to be described.
- Indicate what the audience needs in terms of knowledge and background to understand the description.
- Provide a general description of the object.
- Include an overview of the contents of the description.
Background. If what you are describing is not likely to be familiar to most of your readers, add some background before you plunge into the description. If you are about to describe an SGO/3 density gauge to non-specialists, you should first discuss what it is, what it does, and where it is used.
Visual. The easiest way for someone to understand something is to see it. Visuals help with this. In addition, the parts of the object or process are often labeled (See Chapter ___ on Document Design) so the reader can better understand how each part works in conjunction with the entire item.
Discussion of the parts or characteristics. The main part of your description is the discussion of each part or characteristic. You must divide the item you are describing into parts, or characteristics, or both.
Parts are easy: for example, a wooden pencil has lead, a wooden barrel, an eraser, and a metal clip. Characteristics are describable aspects of a thing but are not parts: for example, the pencil has a certain weight, length, width, and so on.
If you were a budding real-estate tycoon and had to describe a vacant lot for company files, you'd probably describe it by its characteristics: its location, square footage, terrain, vegetation, access to utilities, and so on. (Check out the description of the Primitive Stone Scraper - part of it is arranged by characteristics and part by parts!) If the parts section follows a visual, it is often organized to match the visual. For example, it might list the parts in clockwise order or from top to bottom of how they are labeled on the visual. In addition, the descriptors labeled on the visual match what is listed in this section exactly. Consistency is key.
Once you've divided the item you are describing into parts, characteristics, or both, your next job is to describe each one. For mechanical items, it works well to start by defining the part, by explaining its function. After that, you describe the part from general to specific, using any of the sources of description that are appropriate.
Discussion of the related operation or process. At some point in a description, often at the end, it is useful to summarize the operation or process associated with the object you're describing. For example, if you've just described a mechanical pencil, you could briefly explain how it is used. If you've just described a snowflake, you could discuss the process by which it formed.
Note: In description, you can mix other types of writing. You'll find yourself explaining functions, defining terms, and discussing a bit of process as you describe. That's not a problem as long as the primary focus and the majority of the content is truly description.
Sources of Description
When you write a description, you need to think about the kinds of descriptive detail you can provide. Sometimes, descriptions are rather weak in this area. Use the following list to plan your description or to review a description you have written. Think of the categories of descriptive detail you could provide, or use the following list to identify categories you have not used:
|
color height width shape weight |
materials texture width location methods of attachment |
depth amount pattern, design ingredients age |
subparts length finish temperature moisture content smell |
Note : This is a typical or common model for the contents and organization—many others are possible.
Anatomy of a descriptive paragraph. Typically, it starts with some statement about the purpose or function of the part, with the descriptive detail following. Descriptive detail draws upon the "sources" of description—such things as color, shape, width, and height.
Graphics and Format in Descriptions
Visuals : In most descriptions, you'll need at least one illustration of the thing you are describing, with a caption , labels pointing to the parts, and callouts to highlight important information. See the chapter on Document Design Basics for more on captions , labels, and callouts
Headings. In descriptions, you'll want to use headings and subheadings to mark off the discussion of the individual parts or characteristics. Remember that, ideally, you want to describe each part in a separate paragraph or section—and flag that discussion with a heading. If you have a background section, use a heading for it too. See the chapter on Document Design Basics for the specific requirements of headings.
Lists. Lists are not nearly so important in descriptions as they are in instructions. However, if you itemize parts or subparts or list specifications, these are good situations for lists. See the Chapter on Document Design Basics for the specific requirements of lists .
Special notices. In descriptions, there is no role for special notices as there is in instructions. In a description, readers will not be following a procedure and therefore, will not be running any risks of damaging equipment, wasting supplies, or injuring themselves or others. However, you may find the N ote special notice to be useful to emphasize important points or exceptions . See the Chapter on Document Design Basics for a complete discussion of the proper use of these special notices as well as their format and placement within instructions.
GENERAL DESCRIPTION TIPS
- Descriptions are more detailed than definitions and typically include a visual representation.
- Technical descriptions discuss things like color, shape, size, weight, height, width, thickness, texture, density, contents, materials of construction, and so on.
- Descriptions should include an introduction, background, visual elements, and discussion of the parts and process.
- Description format should be accessible and include elements such as callouts, captions, labels, headings, lists, and special notices.
This work "Descriptions" is a derivative of Definitions, Descriptions, & Instructions CC-BY Amber Kinonen , Technical Definitions CC-BY Amber Kinonen , Technical Description: What Does it Look Like? CC-BY David McMurrey , and Technical Writing for Technicians CC-BY Will Fleming, used under a CC BY 4.0 license. "Descriptions" is licensed under CC BY 4.0 by Mary P. Richards.


15.9: Assignment- Additional Assignments for Job Candidates
- Discuss the usefulness of creating additional assignments for potential candidates to complete
In Module 15: Recruiting and Selecting New Employees, we cover additional methods to determine a job applicant’s skills. Have you ever had to complete a test or demonstrate your skills as part of the hiring process? Have you been a manager who required one of these skill assessments of your job applicants?
Using your cell phone or any other recording device, create a short video advising a fellow student who shares your time management style. You don’t have to edit or create a professional-grade film. You’ve most likely have done this type of recording already on social media, so feel free to use the same informal conversational tone.
Do an internet search for a job posting website, such as Indeed, and read about their skills tests. Think of your audience as fellow students who are interested to learn about these skills assessments as fellow managers. In your video, you can address the following:
- How would these tests help you determine a candidate’s knowledge, skills, and abilities?
- How might these tests inhibit your ability as a manager to find the best candidate?
- If you have had experience with these tests, and you feel comfortable sharing your experience, tell your audience what worked or didn’t work for you.
A Note To Teachers: For this assignment, the first term students will be creating the videos, and then the next term’s students can respond to the videos. After you have two terms of examples, use the best three from the batch as examples and start the process over again. Using the videos as starting points for OL discussion boards may work as well.
Contributors and Attributions
- Open Pedagogy Assignment 3: Additional Assignments for Job Candidates . Authored by : Lumen Learning . License : CC BY: Attribution


3.6: Culturally Sensitive Care
Providing culturally responsive care integrates an individual’s cultural beliefs into their health care. Begin by conveying cultural sensitivity to patients and their family members with these suggestions: [1]
- Set the stage by introducing yourself by name and role when meeting the patient and their family for the first time. Until you know differently, address the patient formally by using their title and last name. Ask the patient how they wish to be addressed and record this in the patient’s chart. Respectfully acknowledge any family members and visitors at the patient’s bedside.
- Begin by standing or sitting at least arm’s length from the patient.
- Observe the patient and family members in regards to eye contact, space orientation, touch, and other nonverbal communication behaviors and follow their lead.
- Make note of the language the patient prefers to use and record this in the patient’s chart. If English is not the patient’s primary language, determine if a medical interpreter is required before proceeding with interview questions. See the box below for guidelines in using a medical interpreter.
- Use inclusive language that is culturally sensitive and appropriate. For example, do not refer to someone as “wheelchair bound”; instead say “a person who uses a wheelchair.” [2]
- Be open and honest about the extent of your knowledge of their culture. It is acceptable to politely ask questions about their beliefs and seek clarification to avoid misunderstandings.
- Adopt a nonjudgmental approach and show respect for the patient’s cultural beliefs, values, and practices. It is possible that you may not agree with a patient’s cultural expressions, but it is imperative that the patient’s rights are upheld. As long as the expressions are not unsafe for the patient or others, the nurse should attempt to integrate them into their care.
- Assure the patient that their cultural considerations are a priority in their care.
When caring for a patient whose primary language is not English and they have a limited ability to speak, read, write, or understand the English language, seek the services of a trained medical interpreter. Health care facilities are mandated by The Joint Commission to provide qualified medical interpreters. Use of a trained medical interpreter is linked to fewer communication errors, shorter hospital stays, reduced 30-day readmission rates, and improved patient satisfaction.
Refrain from asking a family member to act as an interpreter. The patient may withhold sensitive information from them, or family members may possibly edit or change the information provided. Unfamiliarity with medical terminology can also cause misunderstanding and errors.
Medical interpreters may be on-site or available by videoconferencing or telephone. The nurse should also consider coordinating patient and family member conversations with other health care team members to streamline communication, while being aware of cultural implications such who can discuss what health care topics and who makes the decisions. When possible, obtain a medical interpreter of the same gender as the patient to prevent potential embarrassment if a sensitive matter is being discussed.
Guidelines for working with a medical interpreter:
- Allow extra time for the interview or conversation with the patient.
- Whenever possible, meet with the interpreter beforehand to provide background.
- Document the name of the medical interpreter in the progress note.
- Always face and address the patient directly, using a normal tone of voice. Do not direct questions or conversation to the interpreter.
- Speak in the first person (using “I”).
- Avoid using idioms, such as, “Are you feeling under the weather today?” Avoid abbreviations, slang, jokes, and jargon.
- Speak in short paragraphs or sentences. Ask only one question at a time. Allow sufficient time for the interpreter to finish interpreting before beginning another statement or topic.
- Ask the patient to repeat any instructions and explanations given to verify that they understood.
- Brooks, L., Manias, E., & Bloomer, M. (2019). Culturally sensitive communication in healthcare: A concept analysis. Collegian, 26 (3), 383-391. https://doi.org/10.1016/j.colegn.2018.09.007 ↵
- UK Office for Disability Issues. (2018, December 13). Inclusive language: Words to use and avoid when writing about disability . https://www.gov.uk/government/publications/inclusive-communication/inclusive-language-words-to-use-and-avoid-when-writing-about-disability. ↵
- Juckett, G., & Unger, K. (2014). Appropriate use of medical interpreters. American Family Physician, 90 (7), 476-80. https://pubmed.ncbi.nlm.nih.gov/25369625 / ↵


11.5: The Solid State of Matter
- Define and describe the bonding and properties of ionic, molecular, metallic, and covalent network crystalline solids
- Describe the main types of crystalline solids: ionic solids, metallic solids, covalent network solids, and molecular solids
- Explain the ways in which crystal defects can occur in a solid
When most liquids are cooled, they eventually freeze and form crystalline solids , solids in which the atoms, ions, or molecules are arranged in a definite repeating pattern. It is also possible for a liquid to freeze before its molecules become arranged in an orderly pattern. The resulting materials are called amorphous solids or noncrystalline solids (or, sometimes, glasses). The particles of such solids lack an ordered internal structure and are randomly arranged (Figure \(\PageIndex{1}\)).
Metals and ionic compounds typically form ordered, crystalline solids. Substances that consist of large molecules, or a mixture of molecules whose movements are more restricted, often form amorphous solids. For examples, candle waxes are amorphous solids composed of large hydrocarbon molecules. Some substances, such as boron oxide (Figure \(\PageIndex{2}\)), can form either crystalline or amorphous solids, depending on the conditions under which it is produced. Also, amorphous solids may undergo a transition to the crystalline state under appropriate conditions.
Crystalline solids are generally classified according the nature of the forces that hold its particles together. These forces are primarily responsible for the physical properties exhibited by the bulk solids. The following sections provide descriptions of the major types of crystalline solids: ionic, metallic, covalent network, and molecular.
Ionic Solids
Ionic solids , such as sodium chloride and nickel oxide, are composed of positive and negative ions that are held together by electrostatic attractions, which can be quite strong (Figure \(\PageIndex{3}\)). Many ionic crystals also have high melting points. This is due to the very strong attractions between the ions—in ionic compounds, the attractions between full charges are (much) larger than those between the partial charges in polar molecular compounds. This will be looked at in more detail in a later discussion of lattice energies. Although they are hard, they also tend to be brittle, and they shatter rather than bend. Ionic solids do not conduct electricity; however, they do conduct when molten or dissolved because their ions are free to move. Many simple compounds formed by the reaction of a metallic element with a nonmetallic element are ionic.
Metallic Solids
Metallic solids such as crystals of copper, aluminum, and iron are formed by metal atoms Figure \(\PageIndex{4}\). The structure of metallic crystals is often described as a uniform distribution of atomic nuclei within a “sea” of delocalized electrons. The atoms within such a metallic solid are held together by a unique force known as metallic bonding that gives rise to many useful and varied bulk properties. All exhibit high thermal and electrical conductivity, metallic luster, and malleability. Many are very hard and quite strong. Because of their malleability (the ability to deform under pressure or hammering), they do not shatter and, therefore, make useful construction materials. The melting points of the metals vary widely. Mercury is a liquid at room temperature, and the alkali metals melt below 200 °C. Several post-transition metals also have low melting points, whereas the transition metals melt at temperatures above 1000 °C. These differences reflect differences in strengths of metallic bonding among the metals.
Covalent Network Solids
Covalent network solids include crystals of diamond, silicon, some other nonmetals, and some covalent compounds such as silicon dioxide (sand) and silicon carbide (carborundum, the abrasive on sandpaper). Many minerals have networks of covalent bonds. The atoms in these solids are held together by a network of covalent bonds, as shown in Figure \(\PageIndex{5}\). To break or to melt a covalent network solid, covalent bonds must be broken. Because covalent bonds are relatively strong, covalent network solids are typically characterized by hardness, strength, and high melting points. For example, diamond is one of the hardest substances known and melts above 3500 °C.
Molecular Solids
Molecular solids , such as ice, sucrose (table sugar), and iodine, as shown in Figure \(\PageIndex{6}\), are composed of neutral molecules. The strengths of the attractive forces between the units present in different crystals vary widely, as indicated by the melting points of the crystals. Small symmetrical molecules (nonpolar molecules), such as H 2 , N 2 , O 2 , and F 2 , have weak attractive forces and form molecular solids with very low melting points (below −200 °C). Substances consisting of larger, nonpolar molecules have larger attractive forces and melt at higher temperatures. Molecular solids composed of molecules with permanent dipole moments (polar molecules) melt at still higher temperatures. Examples include ice (melting point, 0 °C) and table sugar (melting point, 185 °C).
Properties of Solids
A crystalline solid, like those listed in Table \(\PageIndex{1}\) has a precise melting temperature because each atom or molecule of the same type is held in place with the same forces or energy. Thus, the attractions between the units that make up the crystal all have the same strength and all require the same amount of energy to be broken. The gradual softening of an amorphous material differs dramatically from the distinct melting of a crystalline solid. This results from the structural nonequivalence of the molecules in the amorphous solid. Some forces are weaker than others, and when an amorphous material is heated, the weakest intermolecular attractions break first. As the temperature is increased further, the stronger attractions are broken. Thus amorphous materials soften over a range of temperatures.
| Type of Solid | Type of Particles | Type of Attractions | Properties | Examples |
|---|---|---|---|---|
| ionic | ions | ionic bonds | hard, brittle, conducts electricity as a liquid but not as a solid, high to very high melting points | NaCl, Al 2 O 3 |
| metallic | atoms of electropositive elements | metallic bonds | shiny, malleable, ductile, conducts heat and electricity well, variable hardness and melting temperature | Cu, Fe, Ti, Pb, U |
| covalent network | atoms of electronegative elements | covalent bonds | very hard, not conductive, very high melting points | C (diamond), SiO 2 , SiC |
| molecular | molecules (or atoms) | IMFs | variable hardness, variable brittleness, not conductive, low melting points | H 2 O, CO 2 , I 2 , C 12 H 22 O 11 |
Carbon is an essential element in our world. The unique properties of carbon atoms allow the existence of carbon-based life forms such as ourselves. Carbon forms a huge variety of substances that we use on a daily basis, including those shown in Figure \(\PageIndex{7}\). You may be familiar with diamond and graphite, the two most common allotropes of carbon. (Allotropes are different structural forms of the same element.) Diamond is one of the hardest-known substances, whereas graphite is soft enough to be used as pencil lead. These very different properties stem from the different arrangements of the carbon atoms in the different allotropes.
You may be less familiar with a recently discovered form of carbon: graphene. Graphene was first isolated in 2004 by using tape to peel off thinner and thinner layers from graphite. It is essentially a single sheet (one atom thick) of graphite. Graphene, illustrated in Figure \(\PageIndex{8}\), is not only strong and lightweight, but it is also an excellent conductor of electricity and heat. These properties may prove very useful in a wide range of applications, such as vastly improved computer chips and circuits, better batteries and solar cells, and stronger and lighter structural materials. The 2010 Nobel Prize in Physics was awarded to Andre Geim and Konstantin Novoselov for their pioneering work with graphene.
Crystal Defects
In a crystalline solid, the atoms, ions, or molecules are arranged in a definite repeating pattern, but occasional defects may occur in the pattern. Several types of defects are known, as illustrated in Figure \(\PageIndex{9}\). Vacancies are defects that occur when positions that should contain atoms or ions are vacant. Less commonly, some atoms or ions in a crystal may occupy positions, called interstitial sites , located between the regular positions for atoms. Other distortions are found in impure crystals, as, for example, when the cations, anions, or molecules of the impurity are too large to fit into the regular positions without distorting the structure. Trace amounts of impurities are sometimes added to a crystal (a process known as doping) in order to create defects in the structure that yield desirable changes in its properties. For example, silicon crystals are doped with varying amounts of different elements to yield suitable electrical properties for their use in the manufacture of semiconductors and computer chips.
Summary
Some substances form crystalline solids consisting of particles in a very organized structure; others form amorphous (noncrystalline) solids with an internal structure that is not ordered. The main types of crystalline solids are ionic solids, metallic solids, covalent network solids, and molecular solids. The properties of the different kinds of crystalline solids are due to the types of particles of which they consist, the arrangements of the particles, and the strengths of the attractions between them. Because their particles experience identical attractions, crystalline solids have distinct melting temperatures; the particles in amorphous solids experience a range of interactions, so they soften gradually and melt over a range of temperatures. Some crystalline solids have defects in the definite repeating pattern of their particles. These defects (which include vacancies, atoms or ions not in the regular positions, and impurities) change physical properties such as electrical conductivity, which is exploited in the silicon crystals used to manufacture computer chips.
Glossary
- amorphous solid
- (also, noncrystalline solid) solid in which the particles lack an ordered internal structure
- covalent network solid
- solid whose particles are held together by covalent bonds
- crystalline solid
- solid in which the particles are arranged in a definite repeating pattern
- interstitial sites
- spaces between the regular particle positions in any array of atoms or ions
- ionic solid
- solid composed of positive and negative ions held together by strong electrostatic attractions
- metallic solid
- solid composed of metal atoms
- molecular solid
- solid composed of neutral molecules held together by intermolecular forces of attraction
- vacancy
- defect that occurs when a position that should contain an atom or ion is vacant


5.3: Ethics Bowl - Cases and Score Sheets
Module Introduction
This module is designed to give you a brief orientation in the Ethics Bowl competition. It is designed to complement and complete other modules concerning the ethics bowl that you will find in the Corporate Governance course.
Ethics Bowl Rules (briefly)
- The moderator will begin the competition by flipping a coin to determine which team will present first. If the team that calls wins the toss, they choose whether they or the other team go first.
- Monday: (1) Team 1 will have one minute to consult and seven minutes to give its initial presentation. The presentation must be tied to the question/task given to it by the moderator. (2) Team 2 has a minute to consult and seven minutes to make its Commentary on Team 1's presentation. Team 2 can close its commentary by posing a question to Team 1. (3) Team 1 then has a minute to consult and five minutes to respond to Team 2's Commentary. (4) Team 1 will then answer questions posed by the two peer-review teams. Each peer review team will ask a question. A quick follow-up is allowed. The peer-review question and answer session will go for 15 minutes. (5) The peer-review teams will score the first half of the competition but not announce the results.
- Wednesday: The same procedure will occur while reversing the roles between Teams 1 and 2. Thus, team 2 will present, team 1 comment, team 2 respond, and then team 2 will answer questions from the peer review panels. The peer-review panels will add the scores for the second part of the competition but will hold off on announcing the results until Friday's class.
- Friday: The two peer-review teams will present and explain their scores. Peer Review teams will take note: you're objective is not to criticize or evaluate the debating teams but to provide them feedback in terms of the four categories.
- Debating teams may trade minutes from consulting to presenting. For example, Team 1 may decide to take two minutes to consult when given their case and task. This means that they will have 6 minutes, instead of 7, to present.
- Nota Bene: Debating teams and Peer Review teams are not allowed to bring notes into the competition. You will be provided with paper to take notes once the competition starts.
- Even though the national Ethics Bowl competition allows only one presenter, debating teams will be allowed to "pass the baton." When one person finishes speaking, another can step in his or her place. It is absolutely forbidden that more than one person speaks at a time. Also, the competing team's speaking time is limited to its commentary. Once that is over, they are instructed to quietly listen. Infractions will be followed first by a warning. Second infractions will result in points being taken away.
Competition Timeline
- Team 1 Presentation: One minute to consult, seven minutes to present.
- Team 2 Commentary: One minute to consult, seven minutes to present.
- Team 1 Response to Commentary: One minute to consult, five minutes to respond.
- The question and answer session between Team 1 and the Peer Review teams will last 15 minutes (running clock). The first peer review team will have 7 minutes 30 seconds for its questions and the second will have roughly the same time.
- In the second round, the timeline is the same while the debating teams change roles.
Advice to Debating Teams
- Tell us what you are going to do, do it, and then tell us what you have done. In other words, start your presentation with a summary, then launch into the main body of your presentation, and then conclude with another summary. This will help the listening audience understand what you are trying to do.
- Be professional, formal, and courteous. Address yourself to the other team and the peer review team. It is a good idea to stand when you are giving your initial presentation.
- Be sure to communicate your understanding of the scoring criteria. What do you and your team understand by intelligibility, ethical integration, feasibility, and moral imagination/creativity? Take time to listen to the other team and the peer review teams to gain insights into their understanding. During the commentary and the question and answer session, you will get crucial clues into what others think you have achieved and where you need further work. Use this feedback.
- Be sure to thank the peer review teams, moderators, and your opponents during and after the competition. Such formalities make it possible to penetrate to the deeper practices that underlie the virtue of reasonableness.
- Relax and have fun! You may not have the opportunity to say everything you want to say. One of the purposes behind this competition is to help you see just how hard it is to advocate for ethical positions. We almost always have to do so under serious constraints such as time limits. Also, remember that you have other forums for "getting it said," namely, your group self-evaluation and your in-depth case analysis. In these places, you will be able to discuss these issues in the kind of depth you think necessary.
Advice to the Peer Review Teams on Scoring
- Remember that all three scoring events of the competition are worth 20 points. The initial presentation, the response to the commentary and questions, and the commentary on the other team's presentation all count for the same 20 points.
- Although you have the complete rubric only for the initial presentation, you will score the other parts of the presentation based on the four criteria: intelligibility, ethical integration, feasibility, and moral imagination/creativity. You will score 1 to 5 on each criterion for a total of 20.
- Three is the middle of the road score. In other words, three is a good, average score. It is not a C--don't think of scoring as grading. Start each team off from a default of three. Then move off that default only when something exceptionally good or not so good happens. If your scores deviate much from straight twelves (36), then you are scoring too high or too low.
Ethics Bowl Scoring Criteria
- Intelligibility includes three skills or abilities: (1) the ability to construct and compare multiple arguments representing multiple viewpoints; (2) the ability to construct arguments and provide reasons that are clear, coherent, and factually correct; (3) evidence of realizing the virtue of reasonableness by formulating and presenting value integrative solutions?
- Integrating Ethical Concerns includes three skills: (1) presenting positions that are clearly reversible between stakeholders; (2) identifying and weighing key consequences of positions considered; (3) developing positions that integrate values like integrity, responsibility, reasonableness, honesty, humility, and justice.
- Feasibility implies that the positions taken and the arguments formulated demonstrate full recognition and integration of interest, resource, and technical constraints. While solutions are designed with constraints in mind, these do not serve to trump ethical considerations.
- Moral Imagination and Creativity demonstrate four skill sets: (1) ability to clearly formulate and frame ethical issues and problems; (2) ability to provide multiple framings of a given situation; (3) ability to identify and integrate conflicting stakeholders and stakes; (4) ability to generate solutions and positions that are non-obvious, i.e., go beyond what is given in the situation.
Peer Review Team Responsibilities
- Attend the debate sessions and the feedback session on Friday after the competition. Remember this is the capstone event of the course. It looks bad if you do not bother to attend.
- Your team will ask questions during the debate. This will constitute, at a minimum, one question and a quick follow up if necessary. You are not to debate with the presenting team. So your questions should not be designed to trap them. Rather, seek through your questions to explore seeming weak points, unclear statements, and incomplete thoughts. Use your questions to help you line up the debating team against the four criteria.
- Fill out the score sheet and assess the debating teams in terms of intelligibility, integrating ethics, feasibility and moral imagination/creativity.
- Lead, with the other Peer Review team, the feedback sessions. This requires that you prepare a short, informal presentation that shows your scoring and then explains it.
- Always, always, always be courteous in your feedback comments. Try to present things positively and proactively. This is difficult but practice now will serve you well later when you are trying to explain to a supervisor how he or she has made a mistake.
Media Files with Cases and Score Sheets
Score Sheet Team One: Revised_ScoreSheet_T1_V2.doc
Score Sheet Team Two: Revised_ScoreSheet_T2_V2.doc
Word file containing the 12 Ethics Bowl classes for Business Ethics Spring 2007: Ethics Bowl Cases for Spring 2007.doc
These are the cases for the Ethics Bowl Competition for the Fall Semester in the year 2007. These scenarios or decision points are taken from Incident at Morales, Hughes Aircraft Case, Biomatrix Case, and Toysmart Case: EB_ Fall07_W97.doc
This presentation was given Friday, April 27 to the Ethics Bowl teams that debated on the Therac-25 case and the Inkjet case: Debriefing_Round_2.ppt


8.17: Discussion- Early Adulthood
Contributors and Attributions
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DISCUSSION : In this discussion, reflect upon and discuss BOTH of the following questions:
STEP 1 : First, write a response with at least EIGHT substantial sentences, integrating concepts you learned from the reading and other materials (include links with necessary). Show that you can think critically on the topic by integrating your own thoughts, analysis, or experiences.
STEP 2 : Then return to the discussion to comment on at least TWO classmates’ posts (in at least FIVE sentences). Expand on a classmate’s comments in a value-adding, topic-related way. Promote a collaborative, supportive community, and advance the dialogue through follow-up questions. Reply posts cannot be one-liners, off-topic posts, vague statements, unsupported opinions, inadequate explanations or simply say, “I agree” or “good job.”


3.2: The Main Phase Transition
The membranes that provide structure and definition to cellular compartments are composed of dynamic and heterogeneous lipids whose varying chemical properties allow for their function. The transition between the two main phases of a lipid bilayer, the liquid crystalline phase ( L α ) and the gel phase ( L β ) determines how the membrane will behave in terms of fluidity at a particular temperature and the formation of biologically useful membrane microdomains.
Relevant Properties of Lipid Phases
Membranes composed of a lipid bilayer can exist in either the liquid or gel phase, depending on the strength of interaction between the molecules. The two phases are quite different in their physical properties and biological relevance.
A.
L
α
The L
α
, or “liquid” phase, is characterized by more thermal motion than the gel phase. Head groups in the liquid phase are loosely packed with rotational freedom, giving rise to a larger area per molecule at room temperature than observed in the gel phase
[
1
]
. The acyl chain tails are not rigid or efficiently packed due to the presence of “kinks” caused by gauche orientation of the alkane chain.
In the liquid phase, phospholipid molecules can move freely in the X and Y axis, but distance between the bilayers (the Z axis direction) is shorter in than in the gel phase, as would be expected due to the kinks in the hydrophobic tails
[
2
]
. The presence of gel phase domains within the bilayer can inhibit the lateral diffusion of liquid phase molecules
[
3
]
.
B.
L
β
In the
L
β
phase, or “gel” phase, acyl chain tails are fully extended and a uniform trans orientation is present for the length of the alkyl chain. This gives the gel phase a longer Z axis, or length, because the acyl chains are more rigid and straight.
Tightly packed head groups result in a lower area per molecule at room temperature than in the liquid phase
[
1
]
.
Ehrenfest’s Definition of Phase Transition
The Austrian physicist Paul Ehrenfest provided a definition for phase transitions, where phase behavior is described in terms of free energy ( Δ G ) as a function of different thermodynamic variables. The “order” is the derivative of Δ G (as a function of some physical or chemical variable) and the point at which it is discontinuous is the point of phase change [ 4 ] .
Physical state phase changes, such as the transition between a solid, liquid, gas (or liquid to gel state of a lipid bilayer) are first order phase transitions because they are discontinuous in density at their T m , which is the first derivative of Δ G with respect to chemical potential.
In a second order phase transition, the first derivative of
Δ
G is continuous but the second derivative is discontinuous for the second derivative of
Δ
G
. Magnetization is an example of a second order phase transition process.
The phase transition of gel to liquid is a first order process and involves a latent heat, in which the thermodynamic system (membrane) absorbs or emits a fixed amount of enthalpy ΔS.
During phase transition, the temperature stays constant while heat is added to the system
[
4
]
.
A simplified diagram of phase change
The heat flow required for a first order phase change can be modeled as TΔS. Plotting heat flow versus temperature shows intuitively the energy required for phase transition, with an integral of the curved area representing the free energy of the process:
Hysteresis is the phenomenon where there is a difference in T m depending on which direction the phase transition is going (a difference between T liquid→gel and T gel→liquid ).This phenomenon occurs only in first and not second order phase transitions [ 5 ] .
Characteristics of Phase Transitions
The close packing of the gel phase is accomplished in part by the long and straight acyl chains. These chains are kept straight and rigid by the trans orientation of the alkane chain. During phase transitions, the acyl chain portion of the phospholipid hydrophobic tail undergoes a trans-gauche isomerization produces kinks, a shorter and skewed tail, and leads to less efficient packing [ 6 ] .
Biologically relevant membranes exist as complex mixtures of different lipids with varying melting temperatures. As a fraction of the membrane reaches its phase transition temperature (T m ) a mixed phase will exist with a heterogenous mixture of liquid and gel phases. Should a sufficient amount of mixed phase exist, a substantial amount of lipid molecules may become exposed to an aqueous environment as the liquid phase molecules will be shorter in the Z axis than the gel phase, making the membrane thinner and exposing a portion of the hydrophobic tails of adjacent phospholipid molecules. This phenomenon is known as “Hydrophobic mismatch” and is the driving force for phase separation. Thus, the disparity of phase transition temperatures in complex lipid mixtures of a lipid bilayer can be thought of as a driving force for phase separation and microdomain formation [ 7 ] .
Forces Driving Phase Transition
Lipids undergo temperature specific phase transitions from liquid crystalline to gel phase. The specific temperature at which this transition occurs is referred to as T m and varies depending on the specific molecule. Lipid molecules with longer acyl chains tend to have higher T m than shorter tailed phospholipids because the longer tail provides more opportunity for Van der Waals interactions to occur between two adjacent molecules, lowering over all energy and requiring more thermal energy to reach the point of phase transition and making the phospholipid more likely to exist in the gel phase [ 5 ] .
Double bonding within the acyl chain also affects T
m
[
8
]
.
Unsaturated lipids have a lower T
m
whereas saturated lipids have a higher T
m.
In addition to the degree of unsaturation or number of double bonds, the position of the double bond in the acyl chain also affects phase behavior
[
7
]
.
In addition to the acyl tail, the lipid headgroup also influences phase behavior. The size of the headgroup is thought to have a strong influence on T
m.
Glycocylated head groups have been shown to exhibit less hydrogen bonding than normal phosphatidylcholine resulting in a more fluid and flexible membrane
[
9
]
. Somewhat related, the surrounding pH and the hydration of the membrane can also affect T
m
, with lower pH and higher hydration state resulting in a higher T
m
for the same lipid composition by affecting the hydrogen bonding between the lipid components
[
10
]
.
Change in melting temperature for lipids in pollen grains containing different water contents [ 11 ]
Techniques for Detecting and Measuring Phase Transitions
Differential scanning calorimetry (DSC) is a technique that can be used to measure phase transitions in lipid samples and even in simple cells like intact bacteria [ 12 ] . DSC works by measuring the amount of heat energy required to raise the temperature of the sample. By slowly ramping the temperature of the sample and accurately measuring temperature change, DSC can determine phase transition phases. By conducting a DSC scan in both directions, hysteresis can be identified.
Fourier transform infrared spectroscopy (FTIR) is a powerful technique that can provide a wealth of information about lipid structure, membrane organization and phase behavior [ 13 ] . FTIR detects the absorbance of IR type light emission, which at certain wavelengths is absorbed by different carbon-carbon bonds. From FTIR absorbance spectra, information about lipid conformation can be extrapolated. For example, the phase transition from gel to liquid phase represent change from a more ordered to a more disordered state; the disordered state provides for more rotational and vibrational freedom of the lipid molecules, which is accompanied by a broadening of the IR absorbance of the higher energy molecule [ 14 ] . It has been used to find T m of intact cells, including human platelets [ 15 ] .
FTIR spectra showing the C-H stretch at 6° of DMPS in an aqeous buffer [ 14 ]
Biological Consequences
Packing defects of the lipid bilayer can lead to leakage during phase transition the formation of microdomains that occurs as consequence of phase separation can alter the curvature and fluidity of the membrane, which can impart new biological characteristics to the plasma membrane, tonoplast or a vesicle. In addition, protein movement through the membrane can be impaired, or a protein could be restricted to a lipid environment in which it does not optimally function. [ 16 ]
Myelin is a notable exception to the general rule that lipids are fluid during physiological temperatures. In persons with multiple sclerosis, the T m of myelin is lowered by 20°C [ 17 ] , altering the physical properties of neuronal myelin sheath and possibly causing increased susceptibility to degradation [ 18 ] .
References
- Nagle, J.F., Theory of lipid monolayer and bilayer phase transitions: Effect of headgroup interactions. The Journal of Membrane Biology, 1976. 27 (1): p. 233-250.
- John, K., et al., Transbilayer Movement of Phospholipids at the Main Phase Transition of Lipid Membranes: Implications for Rapid Flip-Flop in Biological Membranes. Biophysical Journal, 2002. 83 (6): p. 3315-3323.
- Ratto, T.V. and M.L. Longo, Obstructed diffusion in phase-separated supported lipid bilayers: a combined atomic force microscopy and fluorescence recovery after photobleaching approach. Biophys J, 2002. 83 (6): p. 3380-92.
- Jaeger, G., The Ehrenfest Classification of Phase Transitions: Introduction and Evolution. Archive for History of Exact Sciences, 1998. 53 (1): p. 51-81.
- Faller, R., MCB/PBH 241 Course Lectures . 2014.
- Hjort Ipsen, J., et al., Phase equilibria in the phosphatidylcholine-cholesterol system. Biochimica et Biophysica Acta (BBA) - Biomembranes, 1987. 905 (1): p. 162-172.
- Wallace, E.J., N.M. Hooper, and P.D. Olmsted, Effect of hydrophobic mismatch on phase behavior of lipid membranes. Biophys J, 2006. 90 (11): p. 4104-18.
- Applegate, K.R. and J.A. Glomset, Effect of acyl chain unsaturation on the packing of model diacylglycerols in simulated monolayers. J Lipid Res, 1991. 32 (10): p. 1645-55.
- Popova, A.V. and D.K. Hincha, Effects of the sugar headgroup of a glycoglycerolipid on the phase behavior of phospholipid model membranes in the dry state. Glycobiology, 2005. 15 (11): p. 1150-5.
- Seddon, J.M., et al., Phosphatidylcholine–fatty acid membranes: effects of headgroup hydration on the phase behaviour and structural parameters of the gel and inverse hexagonal (HII) phases. Biochimica et Biophysica Acta (BBA) - Biomembranes, 1997. 1327 (1): p. 131-147.
- Crowe, J.H., et al., Lipid phase transitions measured in intact cells with fourier transform infrared spectroscopy. Cryobiology, 1989. 26 (1): p. 76-84.
- Lepock, J.R., H.E. Frey, and W.E. Inniss, Thermal analysis of bacteria by differential scanning calorimetry: relationship of protein denaturation in situ to maximum growth temperature. Biochim Biophys Acta, 1990. 1055 (1): p. 19-26.
- Lewis, R.N. and R.N. McElhaney, Membrane lipid phase transitions and phase organization studied by Fourier transform infrared spectroscopy. Biochim Biophys Acta, 2013. 1828 (10): p. 2347-58.
- Lewis, R.N.A.H. and R.N. McElhaney, Membrane lipid phase transitions and phase organization studied by Fourier transform infrared spectroscopy. Biochimica et Biophysica Acta (BBA) - Biomembranes, 2013. 1828 (10): p. 2347-2358.
- Tablin, F., et al., Membrane phase transition of intact human platelets: Correlation with cold-induced activation. Journal of Cellular Physiology, 1996. 168 (2): p. 305-313.
- van Meer, G., D.R. Voelker, and G.W. Feigenson, Membrane lipids: where they are and how they behave. Nat Rev Mol Cell Biol, 2008. 9 (2): p. 112-24.
- Chia, L.S., J.E. Thompson, and M.A. Moscarello, Alteration of lipid-phase behavior in multiple sclerosis myelin revealed by wide-angle x-ray diffraction. Proc Natl Acad Sci U S A, 1984. 81 (6): p. 1871-4.
- Moscarello, M.A., et al., Myelin in multiple sclerosis is developmentally immature. J Clin Invest, 1994. 94 (1): p. 146-54.


21.1: Introduction
Ideally, the postpartum period is a time for parents to bond with newborns. It is also a time for the laboring person to rest and reflect on the labor and birth. In some cases, however, complications occur and interrupt the routine postpartum course. Some postpartum complications, such as breast-feeding complications, perineal discomforts, and recovery from cesarean birth, are common. Other complications, such as infections, hemorrhage, and depression, can be more serious. The nurse plays a vital role in identifying and assisting persons with postpartum complications and collaborates with other health-care providers to promote the best possible outcomes for both the birthing person and the newborn.


10.5: Multiply Polynomials (Part 2)
Using the FOIL Method
Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes there are no like terms to combine. Let's look at the last example again and pay particular attention to how we got the four terms.
\[\begin{split} (x + 2)&(x - y) \\ x^{2} - xy &+ 2x - 2y \end{split}\]
Where did the first term, x 2 , come from?
It is the product of x and x, the first terms in (x + 2) and (x − y).
The next term, −xy, is the product of x and − y, the two outer terms.
The third term, +2x, is the product of 2 and x, the two inner terms.
And the last term, −2y, came from multiplying the two last terms.
We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. The word FOIL is easy to remember and ensures we find all four products. We might say we use the FOIL method to multiply two binomials.
Let's look at (x + 3)(x + 7) again. Now we will work through an example where we use the FOIL pattern to multiply two binomials.
Multiply using the FOIL method: (x + 6)(x + 9).
Solution
| Step 1 : Multiply the First terms. | |
| Step 2 : Multiply the Outer terms. | |
| Step 3 : Multiply the Inner terms. | |
| Step 4 : Multiply the Last terms. | |
| Step 5 : Combine like terms, when possible. | x 2 + 15x + 54 |
Multiply using the FOIL method: (x + 7)(x + 8).
- Answer
-
\(x^2+15x+56 \)
Multiply using the FOIL method: (y + 14)(y + 2).
- Answer
-
\(y^2+16y+28\)
We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!
Step 1. Multiply the First terms.
Step 2. Multiply the Outer terms.
Step 3. Multiply the Inner terms.
Step 4. Multiply the Last terms.
Step 5. Combine like terms, when possible.
Multiply: (y − 8)(y + 6).
Solution
| Step 1 : Multiply the First terms. | |
| Step 2 : Multiply the Outer terms. | |
| Step 3 : Multiply the Inner terms. | |
| Step 4 : Multiply the Last terms. | |
| Step 5 : Combine like terms, when possible. | y 2 - 2y - 48 |
Multiply: (y − 3)(y + 8).
- Answer
-
\(y^2+5y-24 \)
Multiply: (q − 4)(q + 5).
- Answer
-
\(q^2+q-20 \)
Multiply: (2a + 3)(3a − 1).
Solution
| Multiply the First terms. | |
| Multiply the Outer terms. | |
| Multiply the Inner terms. | |
| Multiply the Last terms. | |
| Combine like terms. | 6a 2 + 7a - 3 |
Multiply: (4a + 9)(5a − 2).
- Answer
-
\(20a^2+37a-18 \)
Multiply: (7x + 4)(7x − 8).
- Answer
-
\( 49x^2-28x-32\)
Multiply: (5x − y)(2x − 7).
Solution
| Multiply the First terms. | |
| Multiply the Outer terms. | |
| Multiply the Inner terms. | |
| Multiply the Last terms. | |
| Combine like terms. There are none. | 10x 2 - 35x - 2xy + 7y |
Multiply: (12x − y)(x − 5).
- Answer
-
\(12 x^{2}-60 x-x y+5 y \)
Multiply: (6a − b)(2a − 9).
- Answer
-
\(12 a^{2}-54 a-2 a b+9 b \)
Using the Vertical Method
The FOIL method is usually the quickest method for multiplying two binomials, but it works only for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.
You start by multiplying 23 by 6 to get 138. Then you multiply 23 by 4, lining up the partial product in the correct columns. Last, you add the partial products. Now we'll apply this same method to multiply two binomials.
Multiply using the vertical method: (5x − 1)(2x − 7).
Solution
It does not matter which binomial goes on the top. Line up the columns when you multiply as we did when we multiplied 23(46).
| Multiply 2x − 7 by −1. | |
| Multiply 2x − 7 by 5x. | |
| Add like terms. |
Notice the partial products are the same as the terms in the FOIL method.
Multiply using the vertical method: (4m − 9)(3m − 7).
- Answer
-
\( 12 m^{2}-55 m+63\)
Multiply using the vertical method: (6n − 5)(7n − 2).
- Answer
-
\(42 n^{2}-47 n+10 \)
We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The three methods are listed here to help you remember them.
To multiply binomials, use the:
- Distributive Property
- FOIL Method
- Vertical Method
Remember, FOIL only works when multiplying two binomials.
Multiply a Trinomial by a Binomial
We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we're ready to multiply a trinomial by a binomial. Remember, the FOIL method will not work in this case, but we can use either the Distributive Property or the Vertical Method. We first look at an example using the Distributive Property.
Multiply using the Distributive Property: (x + 3)(2x 2 − 5x + 8).
Solution
| Distribute. | |
| Multiply. | 2x 3 − 5x 2 + 8x + 6x 2 − 15x + 24 |
| Combine like terms. | 2x 3 + x 2 − 7x + 24 |
Multiply using the Distributive Property: (y − 1)(y 2 − 7y + 2).
- Answer
-
\( y^{3}-8 y^{2}+9 y-2\)
Multiply using the Distributive Property: (x + 2)(3x 2 − 4x + 5).
- Answer
-
\( 3 x^{3}+2 x^{2}-3 x+10\)
Now let's do this same multiplication using the Vertical Method.
Multiply using the Vertical Method: (x + 3)(2x 2 − 5x + 8).
Solution
It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.
| Multiply (2x 2 − 5x + 8) by 3. | |
| Multiply (2x 2 − 5x + 8) by x. | |
| Add like terms. |
Multiply using the Vertical Method: (y − 1)(y 2 − 7y + 2).
- Answer
-
\(y^{3}-8 y^{2}+9 y-2 \)
Multiply using the Vertical Method: (x + 2)(3x 2 − 4x + 5).
- Answer
-
\( 3 x^{3}+2 x^{2}-3 x+10\)
Multiply Monomials
Multiply Polynomials
Multiply Polynomials 2
Multiply Polynomials Review
Multiply Polynomials Using the Distributive Property
Multiply Binomials
Practice Makes Perfect
Multiply a Polynomial by a Monomial
In the following exercises, multiply.
- 4(x + 10)
- 6(y + 8)
- 15(r − 24)
- 12(v − 30)
- −3(m + 11)
- −4(p + 15)
- −8(z − 5)
- −3(x − 9)
- u(u + 5)
- q(q + 7)
- n(n 2 − 3n)
- s(s 2 − 6s)
- 12x(x − 10)
- 9m(m − 11)
- −9a(3a + 5)
- −4p(2p + 7)
- 6x(4x + y)
- 5a(9a + b)
- 5p(11p − 5q)
- 12u(3u − 4v)
- 3(v 2 + 10v + 25)
- 6(x 2 + 8x + 16)
- 2n(4n 2 − 4n + 1)
- 3r(2r 2 − 6r + 2)
- −8y(y 2 + 2y − 15)
- −5m(m 2 + 3m − 18)
- 5q 3 (q 2 − 2q + 6)
- 9r 3 (r 2 − 3r + 5)
- −4z 2 (3z 2 + 12z − 1)
- −3x 2 (7x 2 + 10x − 1)
- (2y − 9)y
- (8b − 1)b
- (w − 6) • 8
- (k − 4) • 5
Multiply a Binomial by a Binomial
In the following exercises, multiply the following binomials using: (a) the Distributive Property (b) the FOIL method (c) the Vertical method
- (x + 4)(x + 6)
- (u + 8)(u + 2)
- (n + 12)(n − 3)
- (y + 3)(y − 9)
In the following exercises, multiply the following binomials. Use any method.
- (y + 8)(y + 3)
- (x + 5)(x + 9)
- (a + 6)(a + 16)
- (q + 8)(q + 12)
- (u − 5)(u − 9)
- (r − 6)(r − 2)
- (z − 10)(z − 22)
- (b − 5)(b − 24)
- (x − 4)(x + 7)
- (s − 3)(s + 8)
- (v + 12)(v − 5)
- (d + 15)(d − 4)
- (6n + 5)(n + 1)
- (7y + 1)(y + 3)
- (2m − 9)(10m + 1)
- (5r − 4)(12r + 1)
- (4c − 1)(4c + 1)
- (8n − 1)(8n + 1)
- (3u − 8)(5u − 14)
- (2q − 5)(7q − 11)
- (a + b)(2a + 3b)
- (r + s)(3r + 2s)
- (5x − y)(x − 4)
- (4z − y)(z − 6)
Multiply a Trinomial by a Binomial
In the following exercises, multiply using (a) the Distributive Property and (b) the Vertical Method.
- (u + 4)(u 2 + 3u + 2)
- (x + 5)(x 2 + 8x + 3)
- (a + 10)(3a 2 + a − 5)
- (n + 8)(4n 2 + n − 7)
In the following exercises, multiply. Use either method.
- (y − 6)(y 2 − 10y + 9)
- (k − 3)(k 2 − 8k + 7)
- (2x + 1)(x 2 − 5x − 6)
- (5v + 1)(v 2 − 6v − 10)
Everyday Math
-
Mental math
You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 13 times 15. Think of 13 as 10 + 3 and 15 as 10 + 5.
- Multiply (10 + 3)(10 + 5) by the FOIL method.
- Multiply 13 • 15 without using a calculator.
- Which way is easier for you? Why?
-
Mental math
You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 18 times 17. Think of 18 as 20 − 2 and 17 as 20 − 3.
- Multiply (20 − 2)(20 − 3) by the FOIL method.
- Multiply 18 • 17 without using a calculator.
- Which way is easier for you? Why?
Writing Exercises
- Which method do you prefer to use when multiplying two binomials—the Distributive Property, the FOIL method, or the Vertical Method? Why?
- Which method do you prefer to use when multiplying a trinomial by a binomial—the Distributive Property or the Vertical Method? Why?
Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) What does this checklist tell you about your mastery of this section? What steps will you take to improve?
Contributors and Attributions
-
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at<EMAIL_ADDRESS>."


5.4: Spectroscopy in Astronomy
By the end of this section, you will be able to:
- Describe the properties of light
- Explain how astronomers learn the composition of a gas by examining its spectral lines
- Discuss the various types of spectra
Electromagnetic radiation carries a lot of information about the nature of stars and other astronomical objects. To extract this information, however, astronomers must be able to study the amounts of energy we receive at different wavelengths of light in fine detail. Let’s examine how we can do this and what we can learn.
Properties of Light
Light exhibits certain behaviors that are important to the design of telescopes and other instruments. For example, light can be reflected from a surface. If the surface is smooth and shiny, as with a mirror, the direction of the reflected light beam can be calculated accurately from knowledge of the shape of the reflecting surface. Light is also bent, or refracted , when it passes from one kind of transparent material into another—say, from the air into a glass lens.
Reflection and refraction of light are the basic properties that make possible all optical instruments (devices that help us to see things better)—from eyeglasses to giant astronomical telescopes. Such instruments are generally combinations of glass lenses, which bend light according to the principles of refraction, and curved mirrors, which depend on the properties of reflection. Small optical devices, such as eyeglasses or binoculars, generally use lenses, whereas large telescopes depend almost entirely on mirrors for their main optical elements. We will discuss astronomical instruments and their uses more fully in Astronomical Instruments. For now, we turn to another behavior of light, one that is essential for the decoding of light.
In 1672, in the first paper that he submitted to the Royal Society, Sir Isaac Newton described an experiment in which he permitted sunlight to pass through a small hole and then through a prism. Newton found that sunlight, which looks white to us, is actually made up of a mixture of all the colors of the rainbow (Figure 5.9).
Figure 5.9 shows how light is separated into different colors with a prism—a piece of glass in the shape of a triangle with refracting surfaces. Upon entering one face of the prism, the path of the light is refracted (bent), but not all of the colors are bent by the same amount. The bending of the beam depends on the wavelength of the light as well as the properties of the material, and as a result, different wavelengths (or colors of light) are bent by different amounts and therefore follow slightly different paths through the prism. The violet light is bent more than the red. This phenomenon is called dispersion and explains Newton’s rainbow experiment.
Upon leaving the opposite face of the prism, the light is bent again and further dispersed. If the light leaving the prism is focused on a screen, the different wavelengths or colors that make up white light are lined up side by side just like a rainbow (Figure 5.10). (In fact, a rainbow is formed by the dispersion of light though raindrops; see The Rainbow feature box.) Because this array of colors is a spectrum of light, the instrument used to disperse the light and form the spectrum is called a spectrometer .
The Value of Stellar Spectra
When Newton described the laws of refraction and dispersion in optics, and observed the solar spectrum, all he could see was a continuous band of colors. If the spectrum of the white light from the Sun and stars were simply a continuous rainbow of colors, astronomers would have little interest in the detailed study of a star’s spectrum once they had learned its average surface temperature. In 1802, however, William Wollaston built an improved spectrometer that included a lens to focus the Sun’s spectrum on a screen. With this device, Wollaston saw that the colors were not spread out uniformly, but instead, some ranges of color were missing, appearing as dark bands in the solar spectrum. He mistakenly attributed these lines to natural boundaries between the colors. In 1815, German physicist Joseph Fraunhofer , upon a more careful examination of the solar spectrum, found about 600 such dark lines (missing colors), which led scientists to rule out the boundary hypothesis (Figure 5.11).
Later, researchers found that similar dark lines could be produced in the spectra (“spectra” is the plural of “spectrum”) of artificial light sources. They did this by passing their light through various apparently transparent substances—usually containers with just a bit of thin gas in them.
These gases turned out not to be transparent at all colors: they were quite opaque at a few sharply defined wavelengths. Something in each gas had to be absorbing just a few colors of light and no others. All gases did this, but each different element absorbed a different set of colors and thus showed different dark lines. If the gas in a container consisted of two elements, then light passing through it was missing the colors (showing dark lines) for both of the elements. So it became clear that certain lines in the spectrum “go with” certain elements. This discovery was one of the most important steps forward in the history of astronomy.
What would happen if there were no continuous spectrum for our gases to remove light from? What if, instead, we heated the same thin gases until they were hot enough to glow with their own light? When the gases were heated, a spectrometer revealed no continuous spectrum, but several separate bright lines. That is, these hot gases emitted light only at certain specific wavelengths or colors.
When the gas was pure hydrogen, it would emit one pattern of colors; when it was pure sodium, it would emit a different pattern. A mixture of hydrogen and sodium emitted both sets of spectral lines. The colors the gases emitted when they were heated were the very same colors as those they had absorbed when a continuous source of light was behind them. From such experiments, scientists began to see that different substances showed distinctive spectral signatures by which their presence could be detected (Figure 5.12). Just as your signature allows the bank to identify you, the unique pattern of colors for each type of atom (its spectrum) can help us identify which element or elements are in a gas.
Types of Spectra
In these experiments, then, there were three different types of spectra . A continuous spectrum (formed when a solid or very dense gas gives off radiation) is an array of all wavelengths or colors of the rainbow. A continuous spectrum can serve as a backdrop from which the atoms of much less dense gas can absorb light. A dark line, or absorption spectrum , consists of a series or pattern of dark lines—missing colors—superimposed upon the continuous spectrum of a source. A bright line, or emission spectrum , appears as a pattern or series of bright lines; it consists of light in which only certain discrete wavelengths are present. (Figure 5.11 shows an absorption spectrum, whereas Figure 5.12 shows the emission spectrum of a number of common elements along with an example of a continuous spectrum.)
When we have a hot, thin gas, each particular chemical element or compound produces its own characteristic pattern of spectral lines—its spectral signature. No two types of atoms or molecules give the same patterns. In other words, each particular gas can absorb or emit only certain wavelengths of the light peculiar to that gas. In contrast, absorption spectra occur when passing white light through a cool, thin gas. The temperature and other conditions determine whether the lines are bright or dark (whether light is absorbed or emitted), but the wavelengths of the lines for any element are the same in either case. It is the precise pattern of wavelengths that makes the signature of each element unique. Liquids and solids can also generate spectral lines or bands, but they are broader and less well defined—and hence, more difficult to interpret. Spectral analysis, however, can be quite useful. It can, for example, be applied to light reflected off the surface of a nearby asteroid as well as to light from a distant galaxy.
The dark lines in the solar spectrum thus give evidence of certain chemical elements between us and the Sun absorbing those wavelengths of sunlight. Because the space between us and the Sun is pretty empty, astronomers realized that the atoms doing the absorbing must be in a thin atmosphere of cooler gas around the Sun. This outer atmosphere is not all that different from the rest of the Sun, just thinner and cooler. Thus, we can use what we learn about its composition as an indicator of what the whole Sun is made of. Similarly, we can use the presence of absorption and emission lines to analyze the composition of other stars and clouds of gas in space.
Such analysis of spectra is the key to modern astronomy. Only in this way can we “sample” the stars, which are too far away for us to visit. Encoded in the electromagnetic radiation from celestial objects is clear information about the chemical makeup of these objects. Only by understanding what the stars were made of could astronomers begin to form theories about what made them shine and how they evolved.
In 1860, German physicist Gustav Kirchhoff became the first person to use spectroscopy to identify an element in the Sun when he found the spectral signature of sodium gas. In the years that followed, astronomers found many other chemical elements in the Sun and stars. In fact, the element helium was found first in the Sun from its spectrum and only later identified on Earth. (The word “helium” comes from helios , the Greek name for the Sun.)
Why are there specific lines for each element? The answer to that question was not found until the twentieth century; it required the development of a model for the atom. We therefore turn next to a closer examination of the atoms that make up all matter.
You can use the Spectrum Constructor to explore the three types of spectra and how they appear through a diffraction grating when the brightness of each wavelength is measured with a spectrometer.
Rainbows are an excellent illustration of the dispersion of sunlight. You have a good chance of seeing a rainbow any time you are between the Sun and a rain shower, as illustrated in Figure 5.13. The raindrops act like little prisms and break white light into the spectrum of colors. Suppose a ray of sunlight encounters a raindrop and passes into it. The light changes direction—is refracted—when it passes from air to water; the blue and violet light are refracted more than the red. Some of the light is then reflected at the backside of the drop and reemerges from the front, where it is again refracted. As a result, the white light is spread out into a rainbow of colors.
Note that violet light lies above the red light after it emerges from the raindrop. When you look at a rainbow, however, the red light is higher in the sky. Why? Look again at Figure 5.13. If the observer looks at a raindrop that is high in the sky, the violet light passes over her head and the red light enters her eye. Similarly, if the observer looks at a raindrop that is low in the sky, the violet light reaches her eye and the drop appears violet, whereas the red light from that same drop strikes the ground and is not seen. Colors of intermediate wavelengths are refracted to the eye by drops that are intermediate in altitude between the drops that appear violet and the ones that appear red. Thus, a single rainbow always has red on the outside and violet on the inside.


2.3: Colorado River Aqueduct
Metropolitan Water District was formed in 1928 with the explicit purpose of picking up on where the City of Los Angeles left off with the planning for the use of the Colorado River. The Colorado River Aqueduct was funded by voters three years later in 1931, begun in 1933 and completed in 1935. Water first began to flow in 1939. Even though this time frame is relatively short, the Colorado River Aqueduct was the result of a series of lengthy and heated negotiations, which we will now explore. Collectively the agreements that govern the Colorado River are known as The Law of the River.
Learning Objectives
After reading this section, you should be able to:
- Differentiate among key agreements that are part of The Law of the River
- Conjecture as to possible sources of future disagreement
The primary tension with all Law of the River negotiations is similar to the tension in California: spatial. The water doesn’t exist where the population centers are. But the geography of the Western United States is much more vast than just California. There are seven states considered to be within the Colorado River Basin: Wyoming, Colorado, Utah, New Mexico, Arizona, Nevada, and California. These are all primary stakeholders in the use of the Colorado River. The upper basin states include Wyoming, Colorado, Utah, and New Mexico. Upper basin states were especially sparse in population and anticipated that they might need the Colorado River water at some point as their populations increased or industries like natural gas extraction grew, but they didn’t have an immediate need for the water. The lower basin states included California, Arizona, and Nevada, most of whom had burgeoning populations. The upper basin states were originally concerned that if the lower basin states constructed dams, including Hoover Dam, and began to use the Colorado River that under the doctrine of prior appropriation, the upper basin states would not be able to use the Colorado River’s water again.
The first important agreement in The Law of the River is the Colorado River Compact of 1922, which divided the basin in half. Each basin had the right to use 7.5 MAF of water based on an average flow of the Colorado River of 18 million acre-feet per year, plus a small allocation to Mexico. While most states ratified this agreement, it took Arizona 22 years to ratify it, which is a window into the contentiousness. Arizona was greatly concerned about not receiving enough water eventually and of California taking water that Arizona was entitled to.
The second important agreement in The Law of the River was the Boulder Canyon Project Act of 1928. Negotiations for this agreement lasted seven years, and including four bills being introduced and a filibuster. In this agreement, the lower basin was apportioned among the states:
- Arizona - 2.8 MAF
- California - 4.4 MAF
- Nevada - 0.3 MAF
The Secretary of the Interior was directed to function as the authority for water use in the lower basin, including the ability to commission studies on feasibility for dams and storage. It is interesting to note how small the apportionment for Nevada is.
As a slightly less important agreement in the Law of the River, the California Seven Party Agreement of 1931 is an agreement for division of water among seven entities. The parties included Palo Verde Irrigation District, Yuma Project, Imperial Irrigation District, Coachella Valley Irrigation District, Metropolitan Water District, and the City and County of San Diego. This agreement temporarily settled longstanding disagreements between agricultural and urban users.
The third important act in The Law of The River is an international treaty. In the Mexican Water Treaty of 1944, 1.5 MAF of the Colorado River flow were committed to be received in Mexico. While the amount was agreed upon, the quality of the water remained undetermined. Due to agricultural and urban use, the quality of the Colorado River becomes worse the further south the river flows. A desalting plant for removing salt was eventually built in Yuma, Arizona to increase the quality of the water released to Mexico.
In the fourth important agreement in The Law of the River, the Upper Colorado River Basin Act of 1948, the upper Colorado River Commission was created and the Upper Basin was apportioned:
- Colorado - 51.75%
- New Mexico - 11.25%
- Utah - 23%
- Wyoming - 14%
During the negotiations, it was noted that the originally agreed upon amount of 18 million acre-feet of normal flow for the Colorado River might be an overestimation and that agreements would be better done by percent rather than flat allocations of water.
The Law of the River is a complicated set of agreements, and the previous pages only covered the major agreements. It should be clear that there will be issues of allocations, water rights, and urban v. agricultural differences for many years to come. Several recent modifications to the Law of the River were made because of droughts. In 2007, interim guidelines for allocating the Colorado River during water shortages through 2026 were signed by the Secretary of the Interior. Water shortages were determined based on the surface elevation of Lake Mead. In 2012, the United States and Mexico signed Minute 319 that addressed how Mexico’s allocation of 1.5 MAF would change in drought conditions also based on the surface elevation of Lake Mead.
Try It!
- What is the difference between the Upper Colorado Basin Act and the Boulder Canyon Project Act?
- What is a likely source of disagreement in the future?
Key Terms
Boulder Canyon Project Act of 1928—Apportioned the lower Colorado River basin among three states:
- Arizona - 2.8 MAF
- California - 4.4 MAF
- Nevada - 0.3 MAF
Colorado River Compact of 1922—Which divided the basin in half; each basin had the right to use 7.5 MAF of water based on an average flow of the Colorado River of 18 million acre-feet per year, plus a small allocation to Mexico
The Law of the River—Collectively the laws and regulations that govern the Colorado River
Mexican Water Treaty of 1944—Committed 1.5 MAF of the Colorado River flow to Mexico; did not describe the quality of water, just quantity
Minute 319—An international agreement between the United States and Mexico regarding the Colorado River that addressed how Mexico’s allocation of 1.5 MAF would change in drought conditions also based on the surface elevation of Lake Mead
Upper Colorado River Basin Act of 1948—Apportioned the Upper Colorado River Basin by percent:
- Colorado - 51.75%
- New Mexico - 11.25%
- Utah - 23%
- Wyoming - 14%


16.6: Review Questions
The nurse is watching a client’s cardiac monitor. The nurse recognizes that the QRS complex represents which phase of the cardiac cycle?
- Answer
-
Ventricular depolarization
Which of the following cardiac chambers pumps blood to the pulmonary artery and thus is prone to failure in clients with pulmonary arterial hypertension?
- Answer
-
Right ventricle
The nurse is caring for a client following a myocardial infarction. Which blood vessel does the nurse explain is blocked?
- Answer
-
Coronary artery
A client is being sent to the cardiac catheterization laboratory after an acute myocardial infarction. During the procedure, a physician will puncture and enter the femoral artery to gain access to various heart structures. Which of the following differentiates arteries like the femoral artery from veins?
- Answer
-
Arteries carry oxygenated blood to the body for use by the tissues.
The left ventricle pumps blood from the heart to the aorta for use by the body. Which of the following processes occurs nearly simultaneously with left ventricular systole?
- Answer
-
Right ventricular systole
Which of the following carries an electrical signal from the atria to the ventricles in the healthy heart?
- Answer
-
Atrioventricular node
Which of the following terms describes the ability of cardiac tissue to act as the pacemaker of the heart?
- Answer
-
Automaticity
A client is admitted to the hospital with dizziness and shortness of breath. The provider orders an ECG. Which of the following can be diagnosed based on the results of the ECG?
- Answer
-
Dysrhythmia
The nurse is teaching a group of cardiac rehab clients about the electrical conduction of the heart. What statement made by a client best indicates the teaching is effective?
- Answer
-
“The sinoatrial node is a specialized bundle of nerve tissue in the right atrium called the pacemaker of the heart.”
A client is admitted with bradycardia detected by an ECG. Which statement describes bradycardia?
- Answer
-
Heart rate less than 60 beats per minute


29.5: Practice Exercises
29.5 Practice Exercises
Day One
1.
Analyze the figured bass to provide Roman numerals in the blanks below the staff, then voice lead the progression. Include the key signature.
Answer.
2.
Voice lead the following Roman numeral progression. Include the key signature. Add the following non-chord tones: 1 suspension, 1 passing tone, 1 anticipation, and 3 neighbor tones.
Answer.
Day Two
3.
Analyze the figured bass to provide Roman numerals in the blanks below the staff, then voice lead the progressions in chorale style. Include the key signatures. Include one suspension in each measure.
Answer.
Day Three
4.
Analyze the figured bass to provide Roman numerals in the blanks below the staff. Voice lead the progressions in chorale style. Include the key signatures.
Answer.
Day Four
5.
Voice lead the following Roman numeral progression. Include the key signature. Add the following non-chord tones: an escape tone, a 4-3 suspension, an anticipation, and a passing tone.
Answer.
Click here to download the first homework assignment for this chapter. Click here to download the second homework assignment for this chapter. Click here to download the third homework assignment for this chapter. Click here to download the fourth homework assignment for this chapter. Click here to download the Unit 9 Practice Test.