3.1: Introduction to Differential Geometry
General relativity is described mathematically in the language of differential geometry . Let’s take those two terms in reverse order.
The geometry of spacetime is non-Euclidean, not just in the sense that the 3+1-dimensional geometry of Lorentz frames is different than that of 4 interchangeable Euclidean dimensions, but also in the sense that parallels do not behave in the way described by E5 or A1-A3. In a Lorentz frame, which describes space without any gravitational fields, particles whose world-lines are initially parallel will continue along their parallel world-lines forever. But in the presence of gravitational fields, initially parallel world-lines of free-falling particles will in general diverge, approach, or even cross. Thus, neither the existence nor the uniqueness of parallels can be assumed. We can’t describe this lack of parallelism as arising from the curvature of the world-lines, because we’re using the world-lines of free-falling particles as our definition of a “straight” line. Instead, we describe the effect as coming from the curvature of spacetime itself. The Lorentzian geometry is a description of the case in which this curvature is negligible.
What about the word differential ? The equivalence principle states that even in the presence of gravitational fields, local Lorentz frames exist. How local is “local?” If we use a microscope to zoom in on smaller and smaller regions of spacetime, the Lorentzian approximation becomes better and better. Suppose we want to do experiments in a laboratory, and we want to ensure that when we compare some physically observable quantity against predictions made based on the Lorentz geometry, the resulting discrepancy will not be too large. If the acceptable error is \(\epsilon\), then we should be able to get the error down that low if we’re willing to make the size of our laboratory no bigger than \(\delta\). This is clearly very similar to the Weierstrass style of defining limits and derivatives in calculus. In calculus, the idea expressed by differentiation is that every smooth curve can be approximated locally by a line; in general relativity, the equivalence principle tells us that curved spacetime can be approximated locally by flat spacetime. But consider that no practitioner of calculus habitually solves problems by filling sheets of scratch paper with epsilons and deltas. Instead, she uses the Leibniz notation, in which dy and dx are interpreted as infinitesimally small numbers. You may be inclined, based on your previous training, to dismiss infinitesimals as neither rigorous nor necessary. In 1966, Abraham Robinson demonstrated that concerns about rigor had been unfounded; we’ll come back to this point in section 3.3 . Although it is true that any calculation written using infinitesimals can also be carried out using limits, the following example shows how much more well suited the infinitesimal language is to differential geometry.
Example 1: Areas on a sphere
The area of a region S in the Cartesian plane can be calculated as \(\int_{S}\) dA, where dA = dx dy is the area of an infinitesimal rectangle of width dx and height dy. A curved surface such as a sphere does not admit a global Cartesian coordinate system in which the constant coordinate curves are both uniformly spaced and perpendicular to one another. For example, lines of longitude on the earth’s surface grow closer together as one moves away from the equator. Letting \(\theta\) be the angle with respect to the pole, and \(\phi\) the azimuthal angle, the approximately rectangular patch bounded by \(\theta, \theta + d \theta, \phi\), and \(\phi +d \phi\) has width r sin \(\theta\) d\(\theta\) and height r d\(\phi\), giving dA = r 2 sin \(\theta\) d\(\theta\) d\(\phi\). If you look at the corresponding derivation in an elementary calculus textbook that strictly eschews infinitesimals, the technique is to start from scratch with Riemann sums. This is extremely laborious, and moreover must be carried out again for every new case. In differential geometry, the curvature of the space varies from one point to the next, and clearly we don’t want to reinvent the wheel with Riemann sums an infinite number of times, once at each point in space.


6.4B: Informal Structure
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The informal organization is the aggregate of behaviors, interactions, norms, and personal/professional connections.
Learning Objectives
- Explain the function of informal groups within a formal organizational structure
Key Points
- Keith Davis suggests that informal groups serve at least four major functions within the formal organizational structure.
- Informal groups perpetuate the cultural and social values that the group holds dear.
- Informal groups provide social status and satisfaction that may not be obtained from the formal organization.
- Informal groups develop a communication channel to keep its members informed about what management actions will affect them in various ways.
- Informal groups provide social control by influencing and regulating behavior inside and outside the group.
Key Terms
- Informal organizations : It consists of a dynamic set of personal relationships, social networks, communities of common interest, and emotional sources of motivation. The informal organization evolves organically and spontaneously in response to changes in the work environment, the flux of people through its porous boundaries, and the complex social dynamics of its members.
What is Informal Organization?
The informal organization is the interlocking social structure that governs how people work together in practice. It is the aggregate of behaviors, interactions, norms, and personal/professional connections through which work gets done and relationships are built among people. It consists of a dynamic set of personal relationships, social networks, communities of common interest, and emotional sources of motivation. The informal organization evolves organically in response to changes in the work environment, the flux of people through its porous boundaries, and the complex social dynamics of its members.
Key Characteristics of Informal Organizations
The nature of the informal organization becomes more distinct when its key characteristics are juxtaposed with those of the formal organization. The informal organization is characterized by constant evolution; grass roots; being dynamic and responsive; requiring insider knowledge to be seen; treating people as individuals; being flat and fluid; being cohered by trust and reciprocity; and being difficult to pin down.
Functions of Informal Organizations
Keith Davis suggests that informal groups serve at least four major functions within the formal organizational structure.
First, they perpetuate the cultural and social values that the group holds dear. Certain values are usually already commonly held among informal group members. Day-to-day interaction reinforces these values that perpetuate a particular lifestyle and preserve group unity and integrity. For example, a college management class of 50 students may contain several informal groups that constitute the informal organization within the formal structure of the class.
Second, they provide social status and satisfaction that may not be obtained from the formal organization. In a large organization, a worker may feel like an anonymous number rather than a unique individual. Members of informal groups share jokes and gripes, eat together, play and work together, and are friends—contributing to personal esteem, satisfaction, and a feeling of worth.
Third, the informal group develops a communication channel to keep its members informed about what management actions will affect them in various ways. Many astute managers use the grapevine to “informally” convey certain information about company actions and rumors.
Finally, they provide social control by influencing and regulating behavior inside and outside the group. Internal control persuades members of the group to conform to its lifestyle. For example, if a student starts to wear a coat and tie to class, informal group members may convince the student that such attire is not acceptable and therefore to return to sandals, jeans, and T-shirts.
Business Approaches
Under rapid growth business approach, Starbucks, which grew from 100 employees to over 100,000 in just over a decade, provides structures to support improvisation. Under the Learning Organization model, following a four-year study of the Toyota Production System, Steven J. Spear and H. Kent Bowen concluded in Harvard Business Review that the legendary flexibility of Toyota’s operations is due to the way the scientific method is ingrained in its workers—not through formal training or manuals but through unwritten principles that govern how workers work, interact, construct, and learn.


1.4: Excel Tutorials
Microsoft Excel
Microsoft Excel is an important tool for business leaders. It is especially helpful in making and communicating forecasting decisions.
To improve your proficiency in using Excel, access the following website: Free Excel Tutorials
The website contains a comprehensive collection of free tutorials that are designed to help users master Microsoft Excel. Tutorials include a wide range of topics, from basic spreadsheet navigation and data entry to more advanced features like formulas, functions, charts, and pivot tables. The tutorials are organized into easy-to-follow modules that include step-by-step instructions, visuals, and even interactive practice exercises. The tutorials are arranged in sections, as follows:
- Excel Basics: Sections 1 - 4
- Working with Cells and Sheets: Sections 5 - 12
- Formulas and Functions: Sections 13 - 16
- Working with Data: Sections 17 - 24
- Doing More with Excel: Sections 25 - 29
Practice replicating the forecasts from the End of Chapter Problems in section 1.3.8 of this textbook. Are you able to calculate the correct values?


29.6B: Evolution of Mammals
- Outline the evolution of mammals
Evolution of Mammals
The evolution of mammals passed through many stages since the first appearance of their synapsid ancestors in the late Carboniferous period. Mammals are synapsids: they have a single opening in the skull. They are the only living synapsids as earlier forms became extinct by the Jurassic period. The early, non-mammalian synapsids can be divided into two groups: the pelycosaurs and the therapsids. Within the therapsids, a group called the cynodonts are thought to be the ancestors of mammals. By the mid-Triassic, there were many synapsid species that looked like mammals. The lineage leading to today’s mammals split in the Jurassic. Synapsids from this period include Dryolestes (more closely related to extant placentals and marsupials than to monotremes) as well as Ambondro (more closely related to monotremes). Later, the eutherian and metatherian lineages separated. Metatherians are the animals more closely related to the marsupials, while eutherians are those more closely related to the placentals. Eutherians are distinguished from noneutherians by various features of the feet, ankles, jaws, and teeth. One of the major differences between placental and nonplacental eutherians is that placentals lack epipubic bones, which are present in all other fossil and living mammals (marsupials and monotremes).
Since Juramaia, the earliest-known eutherian, lived 160 million years ago in the Jurassic, this divergence must have occurred in the same period. After the Cretaceous–Paleogene extinction event wiped out the non-avian dinosaurs (birds are generally regarded as the surviving dinosaurs) and several other mammalian groups, placental and marsupial mammals diversified into many new forms and ecological niches throughout the Paleogene and Neogene, by the end of which all modern orders had appeared.
The synapsid lineage became distinct from the sauropsid lineage in the late Carboniferous period, between 320 and 315 million years ago. The sauropsids are today’s reptiles and birds, along with all the extinct animals more closely related to them than to mammals. This does not include the mammal-like reptiles, a group more closely related to the mammals. Throughout the Permian period, the synapsids included the dominant carnivores and several important herbivores. In the subsequent Triassic period, however, a previously-obscure group of sauropsids, the archosaurs, became the dominant vertebrates. The mammaliaforms appeared during this period; their superior sense of smell, backed up by a large brain, facilitated entry into nocturnal niches with less exposure to archosaur predation. The nocturnal lifestyle may have contributed greatly to the development of mammalian traits such as endothermy and hair. Later in the Mesozoic, after theropod dinosaurs replaced rauisuchians as the dominant carnivores, mammals spread into other ecological niches. For example, some became aquatic, some were gliders, and some even fed on juvenile dinosaurs. Most of the evidence consists of fossils. For many years, fossils of Mesozoic mammals and their immediate ancestors were very rare and fragmentary; however, since the mid-1990s, there have been many important new finds, especially in China. The relatively new techniques of molecular phylogenetics have also shed light on some aspects of mammalian evolution by estimating the timing of important divergence points for modern species. When used carefully, these techniques often, but not always, agree with the fossil record. Although mammary glands are a signature feature of modern mammals, little is known about the evolution of lactation. This is because these soft tissues are not often preserved in the fossil record. Most study of the evolution of mammals centers, rather, around the shapes of the teeth, the hardest parts of the tetrapod body. Other much-studied aspects include the evolution of the middle ear bones, erect limb posture, a bony secondary palate, fur and hair, and warm-bloodedness.
A key characteristic of synapsids is endothermy, rather than the ectothermy seen in most other vertebrates. The increased metabolic rate required to internally-modify body temperature went hand-in-hand with changes to certain skeletal structures. The later synapsids, which had more-evolved characteristics unique to mammals, possess cheeks for holding food and heterodont teeth (specialized for chewing by mechanically breaking down food to speed digestion and releasing the energy needed to produce heat). Chewing also requires the ability to chew and breathe at the same time, which is facilitated by the presence of a secondary palate. It separates the area of the mouth where chewing occurs from the area above where respiration occurs, allowing breathing to proceed uninterrupted during chewing. A secondary palate is not found in pelycosaurs, but is present in cynodonts and mammals. The jawbone also shows changes from early synapsids to later ones. The zygomatic arch, or cheekbone, is present in mammals and advanced therapsids such as cynodonts, but is not present in pelycosaurs. The presence of the zygomatic arch suggests the presence of the masseter muscle, which closes the jaw and functions in chewing.
Key Points
- Synapsids are defined by a single opening in the skull and the fact that they are endothermic.
- Mammals are the only living synapsids, derived from a lineage in the Jurassic period.
- Two groups of mammals include the eutherians, which are closely related to placentals and the metatherians, which are more closely related to the marsupials.
- Mammalian lineages from the Jurassic include Dryolestes, related to placentals and marsupials, and Ambondro, related to monotremes.
- Later synapsids had specialized structures for chewing, including teeth, cheeks that can hold food, and a secondary palate, which gave them the ability to chew and breathe at the same time.
Key Terms
- eutherian : the mammals more closely related to animals like humans and rodents than to marsupials
- metatherian : belonging or pertaining to the infraclass Metatheria of marsupials


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accuracy standards
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benchmarks that substantiate that program evaluation methods, data, and documentation are appropriate and contain accurate information
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effectiveness
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ability of a community program to achieve the desired outcome in real-life settings
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efficacy
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maximum potential effect under ideal conditions
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efficiency
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occurs when the effect of program interventions, or outputs, are greater than the resources, or inputs, used to provide the interventions
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facilitator
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person or thing that makes implementation of program interventions and activities easier
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feasibility standards
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benchmarks involving the consideration of resources, which include money, time, and effort, that are available to conduct program evaluation
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formative evaluation
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an assessment that occurs during program development to confirm that program interventions are feasible and appropriate
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impact evaluation
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an assessment to determine the degree to which the community health program has achieved its primary goal
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outcome evaluation
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an assessment of the extent to which the program achieves its objectives within the target population and the effect the program has on the target populations’ knowledge, attitudes, and behaviors
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plain language
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communication that is visually appealing, logically organized, appropriate for the intended audience, and understandable at the first reading
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process evaluation
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an assessment focused on program implementation processes in order to determine if the program has been implemented as planned and in the most efficient way
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program evaluation
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ongoing, systematic collection, analysis, and use of data to examine program efficacy, effectiveness, and efficiency to make decisions about current and future health programs
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propriety standards
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criteria used to confirm that program evaluation is fair and ethical
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recruitment
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finding community members from the target population to participate in a community health program
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retention
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continued participation in a community health program until completion
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sustainability
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continuation of community health programs by decreasing dependence on one source of funding and shifting to a new funding stream because the program is valued, cost and resource efficient, effective, and supported by the community
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utility standards
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specifications for determining who needs evaluation information, what information is needed, the purpose of evaluation, and how the information will be used


15.3: Quick Tips for Evaluating Health Websites
Content on the Internet is unregulated; anyone can publish anything on the Internet. There is sound medical information on the Internet along with dangerous information. You need to be able to tell the difference.
Ask yourself the following:
- Why did the person create the page?
- What’s in it for them?
- Are they trying to sell me something?
Criteria for evaluating information from the web:
Accuracy
- Is the information based on sound medical research? Can the information on the web page be verified by another source?
- Are the sources cited reliable?
- Are there grammatical and spelling errors?
- Are there footnotes, bibliographies, or references so that you can verify the information? Are these reliable? (a citation to Parade magazine does not have the same weight as an article from JAMA )
Authority
- Who published the page? What are the person’s credentials? What do you know about them?
- Is the person backed by a known organization? (the American Association for Cancer Therapy may be a made-up name for something operating out of someone’s basement.)
- Is the person affiliated with a university? If so, is the person a student or a faculty member?
- Can you easily find contact information on the web page? Check the “about us” link, usually found at the beginning or the end of a webpage. What does the “About Us” section tell you about the purpose of the organization? Can you find a physical location for the organization? Or is the only way to contact the organization through a webform?
- What is the domain name? (.edu, .gov ) Is it a personal page or supported by the organization? The tilde (~) means that the site is a personal page (compare an address like med.harvard.edu/~jsmith/headache to med.harvard.edu/neurology/headache)
Bias/Objectivity
- Is the information showing just one point of view?
- What kind of institution sponsored the webpage? A pharmaceutical company? A non-profit organization?
- Is advertising clearly marked?
- Can you tell if the information you are reading is advertisement?
- Do the graphics, fonts, and verbiage play upon emotions? Beware of CAPITAL LETTERS, EXCLAMATION POINTS!!!!! Or words like MIRACLE CURE!!!
- Is the author using data improperly to promote a position or a product?
Currency/Timeliness
- Is there a date on the page?
- When was the page last updated?
- Do the links work?
- Has there been more recent research on the subject? Many medical treatments change with the publication of new studies. What was published a year ago may be outdated now.
Coverage
- Is the information complete?
- Are there sources given for additional information?
Additional Resources
- MedlinePlus Guide to Healthy Web Surfing: http://www.nlm.nih.gov/medlineplus/healthywebsurfing.html
- Trust It or Trash It? (evaluation tool)
- Trust It or Trash It? (printer-friendly handout)
Contributors and Attributions
Public Domain Content
- Evaluating Health Websites. Authored by: National Network of Libraries of Medicine (link is no longer active). Provided by: National Institutes of Health. Located at: nnlm.gov/outreach/consumer/evalsite.html. License: Public Domain: No Known Copyright


5.1E: Circles (Exercises)
Section 5.1 Exercise
1. Find the distance between the points (5,3) and (-1,-5).
2. Find the distance between the points (3,3) and (-3,-2).
3. Write an equation of the circle centered at (8 , -10) with radius 8.
4. Write an equation of the circle centered at (-9, 9) with radius 16.
5. Write an equation of the circle centered at (7, -2) that passes through (-10, 0).
6. Write an equation of the circle centered at (3, -7) that passes through (15, 13).
7. Write an equation for a circle where the points (2, 6) and (8, 10) lie along a diameter.
8. Write an equation for a circle where the points (-3, 3) and (5, 7) lie along a diameter.
9. Sketch a graph of \(\left(x-2\right)^{2} + \left(y+3\right)^{2} = 9\).
10. Sketch a graph of \(\left(x+1\right)^{2} + \left(y-2\right)^{2} = 16\).
11. Find the \(y\) intercept(s) of the circle with center (2, 3) with radius 3.
12. Find the \(x\) intercept(s) of the circle with center (2, 3) with radius 4.
13. At what point in the first quadrant does the line with equation \(y = 2x + 5\) intersect a circle with radius 3 and center (0, 5)?
14. At what point in the first quadrant does the line with equation \(y = x + 2\) intersect the circle with radius 6 and center (0, 2)?
15. At what point in the second quadrant does the line with equation \(y = 2x + 5\) intersect a circle with radius 3 and center (-2, 0)?
16. At what point in the first quadrant does the line with equation \(y = x + 2\) intersect the circle with radius 6 and center (-1,0)?
17. A small radio transmitter broadcasts in a 53 mile radius. If you drive along a straight line from a city 70 miles north of the transmitter to a second city 74 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?
18. A small radio transmitter broadcasts in a 44 mile radius. If you drive along a straight line from a city 56 miles south of the transmitter to a second city 53 miles west of the transmitter, during how much of the drive will you pick up a signal from the transmitter?
19. A tunnel connecting two portions of a space station has a circular cross-section of radius 15 feet. Two walkway decks are constructed in the tunnel. Deck A is along a horizontal diameter and another parallel Deck B is 2 feet below Deck A. Because the space station is in a weightless environment, you can walk vertically upright along Deck A, or vertically upside down along Deck B. You have been assigned to paint “safety stripes” on each deck level, so that a 6 foot person can safely walk upright along either deck. Determine the width of the “safe walk zone” on each deck. [UW]
20. A crawling tractor sprinkler is located as pictured here, 100 feet south of a sidewalk. Once the water is turned on, the sprinkler waters a circular disc of radius 20 feet and moves north along the hose at the rate of 1/2 inch/second. The hose is perpendicular to the 10 ft. wide sidewalk. Assume there is grass on both sides of the sidewalk. [UW]
a. Impose a coordinate system. Describe the initial coordinates of the sprinkler and find equations of the lines forming and find equations of the lines forming the north and south boundaries of the sidewalk.
b. When will the water first strike the sidewalk?
c. When will the water from the sprinkler fall completely north of the sidewalk?
d. Find the total amount of time water from the sprinkler falls on the sidewalk.
e. Sketch a picture of the situation after 33 minutes. Draw an accurate picture of the watered portion of the sidewalk.
f. Find the area of grass watered after one hour.
21. Erik’s disabled sailboat is floating anchored 3 miles East and 2 miles north of Kingsford. A ferry leaves Kingsford heading toward Eaglerock at 12 mph. Eaglerock is 6 miles due east of Kingsford. After 20 minutes the ferry turns, heading due south. Bander is 8 miles south and 1 mile west of Eaglerock. Impose coordinates with Bander as the origin. [UW]
a. Find equations for the lines along which the ferry is moving and draw in these lines.
b. The sailboat has a radar scope that will detect any object within 3 miles of the sailboat. Looking down from above, as in the picture, the radar region looks like a circular disk. The boundary is the “edge” or circle around this disk, the interior is everything inside of the circle, and the exterior is everything outside of the circle. Give the mathematical description (an equation or inequality) of the boundary, interior and exterior of the radar zone. Sketch an accurate picture of the radar zone by determining where the line connecting Kingsford and Eaglerock would cross the radar zone.
c. When does the ferry enter the radar zone?
d. Where and when does the ferry exit the radar zone?
e. How long does the ferry spend inside the radar zone?
22. Nora spends part of her summer driving a combine during the wheat harvest. Assume she starts at the indicated position heading east at 10 ft/sec toward a circular wheat field of radius 200 ft. The combine cuts a swath 20 feet wide and begins when the corner of the machine labeled “a” is 60 feet north and 60 feet west of the western-most edge of the field. [UW]
a. When does Nora’s combine first start cutting the wheat?
b. When does Nora’s combine first start cutting a swath 20 feet wide?
c. Find the total amount of time wheat is being cut during this pass across the field.
d. Estimate the area of the swath cut during this pass across the field.
23. The vertical cross-section of a drainage ditch is pictured to the right. Here, R indicates in each case the radius of a circle with R = 10 feet, where all of the indicated circle centers lie along a horizontal line 10 feet above and parallel to the ditch bottom. Assume that water is flowing into the ditch so that the level above the bottom is rising at a rate of 2 inches per minute. [UW]
a. When will the ditch be completely full?
b. Find a piecewise defined function that models the vertical cross-section of the ditch.
c. What is the width of the filled portion of the ditch after 1 hour and 18 minutes?
d. When will the filled portion of the ditch be 42 feet wide? 50 feet wide? 73 feet wide?
- Answer
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1. 10
3. \((x - 8)^2 + (y + 10)^2 = 8^2\)
5. \((x - 7)^2 + (y + 2)^2 = 293\)
7. \((x - 5)^2. + (y - 8)^2 = 13\)
9.
11. \((0, 3 + \sqrt{5})\) and \((0, 3 - \sqrt{5})\)
13. (1.3416407865, 7.683281573)
15. (-1.07335, 2.8533)
17. 29.87 miles


5.5: National Patient Safety Goals
Every year, national patient safety goals are published by The Joint Commission to improve patient safety. National Patient Safety Goals are goals and recommendations tailored to seven different types of health care agencies based on patient safety data from experts and stakeholders. The seven health care areas include ambulatory health care settings, behavioral health care settings, critical access hospitals, home care, hospital settings, laboratories, nursing care centers, and office-based surgery settings. These goals are updated annually based on safety data and include evidence-based interventions. It is important for nurses and nursing students to be aware of the current National Patient Safety Goals for the settings in which they provide patient care and use the associated recommendations.
The National Patient Safety Goals for nursing care settings (otherwise known as long-term care centers) are described in Table 5.5. (Note that the term “bedsore” is used in the last goal. This is a historic term for the current term “pressure injuries.”)
| Goal | Recommendations and Rationale |
|---|---|
| Identify residents correctly | Use at least two ways to identify patients or residents. For example, use the patient’s or resident’s name and date of birth. This is done to make sure that each patient or resident gets the correct medicine and treatment. |
| Use medicines safely |
Take extra care with patients and residents who take medications to thin their blood.
Record and pass along correct information about a patient’s or resident’s medications. Find out what medications the patient or resident is taking. Compare those medications to new medications given to the patient or resident. Give the patient or resident written information about the medications they need to take. Tell the patient or resident it is important to bring their up-to-date list of medications every time they visit a doctor. |
| Prevent infection | Use the hand hygiene guidelines from the Centers for Disease Control and Prevention or the World Health Organization. Set goals for improving hand cleaning. |
| Prevent residents from falling | Find out which patients and residents are most likely to fall. For example, is the patient or resident taking any medicines that might make them weak, dizzy, or sleepy? Take action to prevent falls for these patients and residents. |
| Prevent bed sores | Find out which patients and residents are most likely to have pressure injuries. Take action to prevent pressure injuries in these patients and residents. Per agency protocol, recheck patients and residents frequently for pressure injuries. |
Read more about National Patient Safety Goals established by The Joint Commission.
Read more details about how to identify patients correctly, administer medications safely, and prevent infection by visiting the following sections in Open RN Nursing Skills :
- Initiating Patient Interaction
- Aseptic Technique Basic Concepts
- Basic Concepts of Administering Medications
Read more about “ Pressure Injuries ” (the newest term used for bed sores) in the “Integumentary” chapter of this book.
- The Joint Commission. (n.d.). 2021 National patient safety goals . https://www.jointcommission.org/standards/national-patient-safety-goals/ ↵


15.8: Exercises
Complete the changes in concentrations for each of the following reactions:
(a)
(b)
(c)
(d)
(e)
Complete the changes in concentrations for each of the following reactions:
(a)
(b
(c)
(d)
(e)
How do the concentrations of Ag + and in a saturated solution above 1.0 g of solid Ag 2 CrO 4 change when 100 g of solid Ag 2 CrO 4 is added to the system? Explain.
How do the concentrations of Pb 2 + and S 2– change when K 2 S is added to a saturated solution of PbS?
What additional information do we need to answer the following question: How is the equilibrium of solid silver bromide with a saturated solution of its ions affected when the temperature is raised?
Which of the following slightly soluble compounds has a solubility greater than that calculated from its solubility product because of hydrolysis of the anion present: CoSO 3 , CuI, PbCO 3 , PbCl 2 , Tl 2 S, KClO 4 ?
Which of the following slightly soluble compounds has a solubility greater than that calculated from its solubility product because of hydrolysis of the anion present: AgCl, BaSO 4 , CaF 2 , Hg 2 I 2 , MnCO 3 , and ZnS?
Write the ionic equation for dissolution and the solubility product ( K sp ) expression for each of the following slightly soluble ionic compounds:
(a) PbCl 2
(b) Ag 2 S
(c) Sr 3 (PO 4 ) 2
(d) SrSO 4
Write the ionic equation for the dissolution and the K sp expression for each of the following slightly soluble ionic compounds:
(a) LaF 3
(b) CaCO 3
(c) Ag 2 SO 4
(d) Pb(OH) 2
The Handbook of Chemistry and Physics gives solubilities of the following compounds in grams per 100 mL of water. Because these compounds are only slightly soluble, assume that the volume does not change on dissolution and calculate the solubility product for each.
(a) BaSiF 6 , 0.026 g/100 mL (contains ions)
(b) Ce(IO 3 ) 4 , 1.5 10 –2 g/100 mL
(c) Gd 2 (SO 4 ) 3 , 3.98 g/100 mL
(d) (NH 4 ) 2 PtBr 6 , 0.59 g/100 mL (contains ions)
The Handbook of Chemistry and Physics gives solubilities of the following compounds in grams per 100 mL of water. Because these compounds are only slightly soluble, assume that the volume does not change on dissolution and calculate the solubility product for each.
(a) BaSeO 4 , 0.0118 g/100 mL
(b) Ba(BrO 3 ) 2 ·H 2 O, 0.30 g/100 mL
(c) NH 4 MgAsO 4 ·6H 2 O, 0.038 g/100 mL
(d) La 2 (MoO 4 ) 3 , 0.00179 g/100 mL
Use solubility products and predict which of the following salts is the most soluble, in terms of moles per liter, in pure water: CaF 2 , Hg 2 Cl 2 , PbI 2 , or Sn(OH) 2 .
Assuming that no equilibria other than dissolution are involved, calculate the molar solubility of each of the following from its solubility product:
(a) KHC 4 H 4 O 6
(b) PbI 2
(c) Ag 4 [Fe(CN) 6 ], a salt containing the ion
(d) Hg 2 I 2
Assuming that no equilibria other than dissolution are involved, calculate the molar solubility of each of the following from its solubility product:
(a) Ag 2 SO 4
(b) PbBr 2
(c) AgI
(d) CaC 2 O 4 ·H 2 O
Assuming that no equilibria other than dissolution are involved, calculate the concentration of all solute species in each of the following solutions of salts in contact with a solution containing a common ion. Show that changes in the initial concentrations of the common ions can be neglected.
(a) AgCl( s ) in 0.025 M NaCl
(b) CaF 2 ( s ) in 0.00133 M KF
(c) Ag 2 SO 4 ( s ) in 0.500 L of a solution containing 19.50 g of K 2 SO 4
(d) Zn(OH) 2 ( s ) in a solution buffered at a pH of 11.45
Assuming that no equilibria other than dissolution are involved, calculate the concentration of all solute species in each of the following solutions of salts in contact with a solution containing a common ion. Show that changes in the initial concentrations of the common ions can be neglected.
(a) TlCl( s ) in 1.250 M HCl
(b) PbI 2 ( s ) in 0.0355 M CaI 2
(c) Ag 2 CrO 4 ( s ) in 0.225 L of a solution containing 0.856 g of K 2 CrO 4
(d) Cd(OH) 2 ( s ) in a solution buffered at a pH of 10.995
Assuming that no equilibria other than dissolution are involved, calculate the concentration of all solute species in each of the following solutions of salts in contact with a solution containing a common ion. Show that it is not appropriate to neglect the changes in the initial concentrations of the common ions.
(a) TlCl( s ) in 0.025 M TlNO 3
(b) BaF 2 ( s ) in 0.0313 M KF
(c) MgC 2 O 4 in 2.250 L of a solution containing 8.156 g of Mg(NO 3 ) 2
(d) Ca(OH) 2 ( s ) in an unbuffered solution initially with a pH of 12.700
Explain why the changes in concentrations of the common ions in Exercise 15.17 can be neglected.
Explain why the changes in concentrations of the common ions in Exercise 15.18 cannot be neglected.
Calculate the solubility of aluminum hydroxide, Al(OH) 3 , in a solution buffered at pH 11.00.
Refer to Appendix J for solubility products for calcium salts. Determine which of the calcium salts listed is most soluble in moles per liter and which is most soluble in grams per liter.
Most barium compounds are very poisonous; however, barium sulfate is often administered internally as an aid in the X-ray examination of the lower intestinal tract (Figure 15.4). This use of BaSO 4 is possible because of its low solubility. Calculate the molar solubility of BaSO 4 and the mass of barium present in 1.00 L of water saturated with BaSO 4 .
Public Health Service standards for drinking water set a maximum of 250 mg/L (2.60 10 –3 M ) of because of its cathartic action (it is a laxative). Does natural water that is saturated with CaSO 4 (“gyp” water) as a result or passing through soil containing gypsum, CaSO 4 ·2H 2 O, meet these standards? What is the concentration of in such water?
Perform the following calculations:
(a) Calculate [Ag + ] in a saturated aqueous solution of AgBr.
(b) What will [Ag + ] be when enough KBr has been added to make [Br – ] = 0.050 M ?
(c) What will [Br – ] be when enough AgNO 3 has been added to make [Ag + ] = 0.020 M ?
The solubility product of CaSO 4 ·2H 2 O is 2.4 10 –5 . What mass of this salt will dissolve in 1.0 L of 0.010 M
Assuming that no equilibria other than dissolution are involved, calculate the concentrations of ions in a saturated solution of each of the following (see Appendix J for solubility products).
(a) TlCl
(b) BaF 2
(c) Ag 2 CrO 4
(d) CaC 2 O 4 ·H 2 O
(e) the mineral anglesite, PbSO 4
Assuming that no equilibria other than dissolution are involved, calculate the concentrations of ions in a saturated solution of each of the following (see Appendix J for solubility products):
(a) AgI
(b) Ag 2 SO 4
(c) Mn(OH) 2
(d) Sr(OH) 2 ·8H 2 O
(e) the mineral brucite, Mg(OH) 2
The following concentrations are found in mixtures of ions in equilibrium with slightly soluble solids. From the concentrations given, calculate K sp for each of the slightly soluble solids indicated:
(a) AgBr: [Ag + ] = 5.7 10 –7 M , [Br – ] = 5.7 10 –7 M
(b) CaCO 3 : [Ca 2+ ] = 5.3 10 –3 M , = 9.0 10 –7 M
(c) PbF 2 : [Pb 2+ ] = 2.1 10 –3 M , [F – ] = 4.2 10 –3 M
(d) Ag 2 CrO 4 : [Ag + ] = 5.3 10 –5 M , 3.2 10 –3 M
(e) InF 3 : [In 3+ ] = 2.3 10 –3 M , [F – ] = 7.0 10 –3 M
The following concentrations are found in mixtures of ions in equilibrium with slightly soluble solids. From the concentrations given, calculate K sp for each of the slightly soluble solids indicated:
(a) TlCl: [Tl + ] = 1.21 10 –2 M , [Cl – ] = 1.2 10 –2 M
(b) Ce(IO 3 ) 4 : [Ce 4+ ] = 1.8 10 –4 M , = 2.6 10 –13 M
(c) Gd 2 (SO 4 ) 3 : [Gd 3+ ] = 0.132 M , = 0.198 M
(d) Ag 2 SO 4 : [Ag + ] = 2.40 10 –2 M , = 2.05 10 –2 M
(e) BaSO 4 : [Ba 2+ ] = 0.500 M , = 4.6 10 −8 M
Which of the following compounds precipitates from a solution that has the concentrations indicated? (See Appendix J for K sp values.)
(a) KClO 4 : [K + ] = 0.01 M , = 0.01 M
(b) K 2 PtCl 6 : [K + ] = 0.01 M , = 0.01 M
(c) PbI 2 : [Pb 2+ ] = 0.003 M , [I – ] = 1.3 10 –3 M
(d) Ag 2 S: [Ag + ] = 1 10 –10 M , [S 2– ] = 1 10 –13 M
Which of the following compounds precipitates from a solution that has the concentrations indicated? (See Appendix J for K sp values.)
(a) CaCO 3 : [Ca 2+ ] = 0.003 M , = 0.003 M
(b) Co(OH) 2 : [Co 2+ ] = 0.01 M , [OH – ] = 1 10 –7 M
(c) CaHPO 4 : [Ca 2+ ] = 0.01 M , = 2 10 –6 M
(d) Pb 3 (PO 4 ) 2 : [Pb 2+ ] = 0.01 M , = 1 10 –13 M
Calculate the concentration of Tl + when TlCl just begins to precipitate from a solution that is 0.0250 M in Cl – .
Calculate the concentration of sulfate ion when BaSO 4 just begins to precipitate from a solution that is 0.0758 M in Ba 2+ .
Calculate the concentration of Sr 2+ when SrCrO 4 starts to precipitate from a solution that is 0.0025 M in CrO 4 2– .
Calculate the concentration of when Ag 3 PO 4 starts to precipitate from a solution that is 0.0125 M in Ag + .
Calculate the concentration of F – required to begin precipitation of CaF 2 in a solution that is 0.010 M in Ca 2+ .
Calculate the concentration of Ag + required to begin precipitation of Ag 2 CO 3 in a solution that is 2.50 10 –6 M in
What [Ag + ] is required to reduce to 8.2 10 –4 M by precipitation of Ag 2 CO 3 ?
What [F – ] is required to reduce [Ca 2+ ] to 1.0 10 –4 M by precipitation of CaF 2 ?
A volume of 0.800 L of a 2 10 –4 - M Ba(NO 3 ) 2 solution is added to 0.200 L of 5 10 –4 M Li 2 SO 4 . Does BaSO 4 precipitate? Explain your answer.
Perform these calculations for nickel(II) carbonate. (a) With what volume of water must a precipitate containing NiCO 3 be washed to dissolve 0.100 g of this compound? Assume that the wash water becomes saturated with NiCO 3 ( K sp = 1.36 10 –7 ).
(b) If the NiCO 3 were a contaminant in a sample of CoCO 3 ( K sp = 1.0 10 –12 ), what mass of CoCO 3 would have been lost? Keep in mind that both NiCO 3 and CoCO 3 dissolve in the same solution.
Iron concentrations greater than 5.4 10 –6 M in water used for laundry purposes can cause staining. What [OH – ] is required to reduce [Fe 2+ ] to this level by precipitation of Fe(OH) 2 ?
A solution is 0.010 M in both Cu 2+ and Cd 2+ . What percentage of Cd 2+ remains in the solution when 99.9% of the Cu 2+ has been precipitated as CuS by adding sulfide?
A solution is 0.15 M in both Pb 2+ and Ag + . If Cl – is added to this solution, what is [Ag + ] when PbCl 2 begins to precipitate?
What reagent might be used to separate the ions in each of the following mixtures, which are 0.1 M with respect to each ion? In some cases it may be necessary to control the pH. (Hint: Consider the K sp values given in Appendix J.)
(a) and Cu 2+
(b) and Cl –
(c) Hg 2+ and Co 2+
(d) Zn 2+ and Sr 2+
(e) Ba 2+ and Mg 2+
(f) and OH –
A solution contains 1.0 10 –5 mol of KBr and 0.10 mol of KCl per liter. AgNO 3 is gradually added to this solution. Which forms first, solid AgBr or solid AgCl?
A solution contains 1.0 10 –2 mol of KI and 0.10 mol of KCl per liter. AgNO 3 is gradually added to this solution. Which forms first, solid AgI or solid AgCl?
The calcium ions in human blood serum are necessary for coagulation (Figure 15.5). Potassium oxalate, K 2 C 2 O 4 , is used as an anticoagulant when a blood sample is drawn for laboratory tests because it removes the calcium as a precipitate of CaC 2 O 4 ·H 2 O. It is necessary to remove all but 1.0% of the Ca 2+ in serum in order to prevent coagulation. If normal blood serum with a buffered pH of 7.40 contains 9.5 mg of Ca 2+ per 100 mL of serum, what mass of K 2 C 2 O 4 is required to prevent the coagulation of a 10 mL blood sample that is 55% serum by volume? (All volumes are accurate to two significant figures. Note that the volume of serum in a 10-mL blood sample is 5.5 mL. Assume that the K sp value for CaC 2 O 4 in serum is the same as in water.)
About 50% of urinary calculi (kidney stones) consist of calcium phosphate, Ca 3 (PO 4 ) 2 . The normal mid range calcium content excreted in the urine is 0.10 g of Ca 2+ per day. The normal mid range amount of urine passed may be taken as 1.4 L per day. What is the maximum concentration of phosphate ion that urine can contain before a calculus begins to form?
The pH of normal urine is 6.30, and the total phosphate concentration
Magnesium metal (a component of alloys used in aircraft and a reducing agent used in the production of uranium, titanium, and other active metals) is isolated from sea water by the following sequence of reactions:
Sea water has a density of 1.026 g/cm 3 and contains 1272 parts per million of magnesium as Mg 2+ ( aq ) by mass. What mass, in kilograms, of Ca(OH) 2 is required to precipitate 99.9% of the magnesium in 1.00 10 3 L of sea water?
Hydrogen sulfide is bubbled into a solution that is 0.10 M in both Pb 2+ and Fe 2+ and 0.30 M in HCl. After the solution has come to equilibrium it is saturated with H 2 S ([H 2 S] = 0.10 M ). What concentrations of Pb 2+ and Fe 2+ remain in the solution? For a saturated solution of H 2 S we can use the equilibrium:
(Hint: The changes as metal sulfides precipitate.)
Perform the following calculations involving concentrations of iodate ions:
(a) The iodate ion concentration of a saturated solution of La(IO 3 ) 3 was found to be 3.1 10 –3 mol/L. Find the K sp .
(b) Find the concentration of iodate ions in a saturated solution of Cu(IO 3 ) 2 ( K sp = 7.4 10 –8 ).
Calculate the molar solubility of AgBr in 0.035 M NaBr ( K sp = 5 10 –13 ).
How many grams of Pb(OH) 2 will dissolve in 500 mL of a 0.050- M PbCl 2 solution ( K sp = 1.2 10 –15 )?
Use the simulation from the earlier Link to Learning to complete the following exercise. Using 0.01 g CaF 2 , give the K sp values found in a 0.2- M solution of each of the salts. Discuss why the values change as you change soluble salts.
How many grams of Milk of Magnesia, Mg(OH) 2 ( s ) (58.3 g/mol), would be soluble in 200 mL of water. K sp = 7.1 10 –12 . Include the ionic reaction and the expression for K sp in your answer. ( K w = 1 10 –14 = [H 3 O + ][OH – ])
Two hypothetical salts, LM 2 and LQ, have the same molar solubility in H 2 O. If K sp for LM 2 is 3.20 10 –5 , what is the K sp value for LQ?
The carbonate ion concentration is gradually increased in a solution containing equal concentrations of the divalent cations of magnesium, calcium, strontium, barium, and manganese. Which of the following carbonates will precipitate first? Which will precipitate last? Explain.
(a)
(b)
(c)
(d)
(e)
How many grams of Zn(CN) 2 ( s ) (117.44 g/mol) would be soluble in 100 mL of H 2 O? Include the balanced reaction and the expression for K sp in your answer. The K sp value for Zn(CN) 2 ( s ) is 3.0 10 –16 .
Even though Ca(OH) 2 is an inexpensive base, its limited solubility restricts its use. What is the pH of a saturated solution of Ca(OH) 2 ?
Under what circumstances, if any, does a sample of solid AgCl completely dissolve in pure water?
Explain why the addition of NH 3 or HNO 3 to a saturated solution of Ag 2 CO 3 in contact with solid Ag 2 CO 3 increases the solubility of the solid.
Calculate the cadmium ion concentration, [Cd 2+ ], in a solution prepared by mixing 0.100 L of 0.0100 M Cd(NO 3 ) 2 with 0.150 L of 0.100 NH 3 ( aq ).
Explain why addition of NH 3 or HNO 3 to a saturated solution of Cu(OH) 2 in contact with solid Cu(OH) 2 increases the solubility of the solid.
Sometimes equilibria for complex ions are described in terms of dissociation constants, K d . For the complex ion the dissociation reaction is:
and
Calculate the value of the formation constant, K f , for
Using the value of the formation constant for the complex ion calculate the dissociation constant.
Using the dissociation constant, K d = 7.8 10 –18 , calculate the equilibrium concentrations of Cd 2+ and CN – in a 0.250- M solution of
Using the dissociation constant, K d = 3.4 10 –15 , calculate the equilibrium concentrations of Zn 2+ and OH – in a 0.0465- M solution of
Using the dissociation constant, K d = 2.2 10 –34 , calculate the equilibrium concentrations of Co 3+ and NH 3 in a 0.500- M solution of
Using the dissociation constant, K d = 1 10 –44 , calculate the equilibrium concentrations of Fe 3+ and CN – in a 0.333 M solution of
Calculate the mass of potassium cyanide ion that must be added to 100 mL of solution to dissolve 2.0 10 –2 mol of silver cyanide, AgCN.
Calculate the minimum concentration of ammonia needed in 1.0 L of solution to dissolve 3.0 10 –3 mol of silver bromide.
A roll of 35-mm black and white photographic film contains about 0.27 g of unexposed AgBr before developing. What mass of Na 2 S 2 O 3 ·5H 2 O (sodium thiosulfate pentahydrate or hypo) in 1.0 L of developer is required to dissolve the AgBr as ( K f = 4.7 10 13 )?
We have seen an introductory definition of an acid: An acid is a compound that reacts with water and increases the amount of hydronium ion present. In the chapter on acids and bases, we saw two more definitions of acids: a compound that donates a proton (a hydrogen ion, H + ) to another compound is called a Brønsted-Lowry acid, and a Lewis acid is any species that can accept a pair of electrons. Explain why the introductory definition is a macroscopic definition, while the Brønsted-Lowry definition and the Lewis definition are microscopic definitions.
Write the Lewis structures of the reactants and product of each of the following equations, and identify the Lewis acid and the Lewis base in each:
(a)
(b)
(c)
(d) (use Al-Cl single bonds)
(e)
Write the Lewis structures of the reactants and product of each of the following equations, and identify the Lewis acid and the Lewis base in each:
(a)
(b)
(c)
(d)
(e)
Using Lewis structures, write balanced equations for the following reactions:
(a)
(b)
(c)
(d)
Calculate in a solution prepared by adding 0.0200 mol of NaCl to 0.250 L of a 0.100- M HgCl 2 solution.
In a titration of cyanide ion, 28.72 mL of 0.0100 M AgNO 3 is added before precipitation begins. [The reaction of Ag + with CN – goes to completion, producing the complex.] Precipitation of solid AgCN takes place when excess Ag + is added to the solution, above the amount needed to complete the formation of How many grams of NaCN were in the original sample?
What are the concentrations of Ag + , CN – , and in a saturated solution of AgCN?
In dilute aqueous solution HF acts as a weak acid. However, pure liquid HF (boiling point = 19.5 °C) is a strong acid. In liquid HF, HNO 3 acts like a base and accepts protons. The acidity of liquid HF can be increased by adding one of several inorganic fluorides that are Lewis acids and accept F – ion (for example, BF 3 or SbF 5 ). Write balanced chemical equations for the reaction of pure HNO 3 with pure HF and of pure HF with BF 3 .
The simplest amino acid is glycine, H 2 NCH 2 CO 2 H. The common feature of amino acids is that they contain the functional groups: an amine group, –NH 2 , and a carboxylic acid group, –CO 2 H. An amino acid can function as either an acid or a base. For glycine, the acid strength of the carboxyl group is about the same as that of acetic acid, CH 3 CO 2 H, and the base strength of the amino group is slightly greater than that of ammonia, NH 3 .
(a) Write the Lewis structures of the ions that form when glycine is dissolved in 1 M HCl and in 1 M KOH.
(b) Write the Lewis structure of glycine when this amino acid is dissolved in water. (Hint: Consider the relative base strengths of the –NH 2 and groups.)
Boric acid, H 3 BO 3 , is not a Brønsted-Lowry acid but a Lewis acid.
(a) Write an equation for its reaction with water.
(b) Predict the shape of the anion thus formed.
(c) What is the hybridization on the boron consistent with the shape you have predicted?
A saturated solution of a slightly soluble electrolyte in contact with some of the solid electrolyte is said to be a system in equilibrium. Explain. Why is such a system called a heterogeneous equilibrium?
Calculate the equilibrium concentration of Ni 2+ in a 1.0- M solution [Ni(NH 3 ) 6 ](NO 3 ) 2 .
Calculate the equilibrium concentration of Zn 2+ in a 0.30- M solution of
Calculate the equilibrium concentration of Cu 2+ in a solution initially with 0.050 M Cu 2+ and 1.00 M NH 3 .
Calculate the equilibrium concentration of Zn 2+ in a solution initially with 0.150 M Zn 2+ and 2.50 M CN – .
Calculate the Fe 3+ equilibrium concentration when 0.0888 mole of K 3 [Fe(CN) 6 ] is added to a solution with 0.0.00010 M CN – .
Calculate the Co 2+ equilibrium concentration when 0.010 mole of [Co(NH 3 ) 6 ](NO 3 ) 2 is added to a solution with 0.25 M NH 3 . Assume the volume is 1.00 L.
Calculate the molar solubility of Sn(OH) 2 in a buffer solution containing equal concentrations of NH 3 and
Calculate the molar solubility of Al(OH) 3 in a buffer solution with 0.100 M NH 3 and 0.400 M
What is the molar solubility of CaF 2 in a 0.100- M solution of HF? K a for HF = 6.4 10 –4 .
What is the molar solubility of BaSO 4 in a 0.250- M solution of NaHSO 4 ? K a for = 1.2 10 –2 .
What is the molar solubility of Tl(OH) 3 in a 0.10- M solution of NH 3 ?
What is the molar solubility of Pb(OH) 2 in a 0.138- M solution of CH 3 NH 2 ?
A solution of 0.075 M CoBr 2 is saturated with H 2 S ([H 2 S] = 0.10 M ). What is the minimum pH at which CoS begins to precipitate?
A 0.125- M solution of Mn(NO 3 ) 2 is saturated with H 2 S ([H 2 S] = 0.10 M ). At what pH does MnS begin to precipitate?
Both AgCl and AgI dissolve in NH 3 .
(a) What mass of AgI dissolves in 1.0 L of 1.0 M NH 3 ?
(b) What mass of AgCl dissolves in 1.0 L of 1.0 M NH 3 ?
The following question is taken from a Chemistry Advanced Placement Examination and is used with the permission of the Educational Testing Service.
Solve the following problem:
In a saturated solution of MgF 2 at 18 °C, the concentration of Mg 2+ is 1.21 10 –3 M . The equilibrium is represented by the preceding equation.
(a) Write the expression for the solubility-product constant, K sp , and calculate its value at 18 °C.
(b) Calculate the equilibrium concentration of Mg 2+ in 1.000 L of saturated MgF 2 solution at 18 °C to which 0.100 mol of solid KF has been added. The KF dissolves completely. Assume the volume change is negligible.
(c) Predict whether a precipitate of MgF 2 will form when 100.0 mL of a 3.00 10 –3 - M solution of Mg(NO 3 ) 2 is mixed with 200.0 mL of a 2.00 10 –3 - M solution of NaF at 18 °C. Show the calculations to support your prediction.
(d) At 27 °C the concentration of Mg 2+ in a saturated solution of MgF 2 is 1.17 10 –3 M . Is the dissolving of MgF 2 in water an endothermic or an exothermic process? Give an explanation to support your conclusion.
Which of the following compounds, when dissolved in a 0.01- M solution of HClO 4 , has a solubility greater than in pure water: CuCl, CaCO 3 , MnS, PbBr 2 , CaF 2 ? Explain your answer.
Which of the following compounds, when dissolved in a 0.01- M solution of HClO 4 , has a solubility greater than in pure water: AgBr, BaF 2 , Ca 3 (PO 4 ) 2 , ZnS, PbI 2 ? Explain your answer.
What is the effect on the amount of solid Mg(OH) 2 that dissolves and the concentrations of Mg 2+ and OH – when each of the following are added to a mixture of solid Mg(OH) 2 and water at equilibrium?
(a) MgCl 2
(b) KOH
(c) HClO 4
(d) NaNO 3
(e) Mg(OH) 2
What is the effect on the amount of CaHPO 4 that dissolves and the concentrations of Ca 2+ and when each of the following are added to a mixture of solid CaHPO 4 and water at equilibrium?
(a) CaCl 2
(b) HCl
(c) KClO 4
(d) NaOH
(e) CaHPO 4
Identify all chemical species present in an aqueous solution of Ca 3 (PO 4 ) 2 and list these species in decreasing order of their concentrations. (Hint: Remember that the ion is a weak base.)


7.8: Direct person-to-person transmission of pathogens
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A new infection begins when pathogens leave the body of their host – the infected individual in which the pathogens are multiplying – and enter a new host. They may be repelled by defense mechanisms (i.e. skin) in the new host, or they may survive and reproduce in sufficient numbers to cause an infectious disease.
Transmission of pathogens can occur directly between people, or indirectly in the air, water or food, or via other animals to humans, or from sources in the environment. In this section we explore direct transmission.
Figure \(\PageIndex{1}\). Direct person-to-person transmission of infection. (a) Contagion and sexual transmission. (b) Mother-to-child transmission.
Figure \(\PageIndex{1}\) represents the three ways in which pathogens can be transmitted by direct person-to-person contact. They are: