6.2: Tree Diagrams and the Multiplication Axiom
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Learning Objectives
In this section you will learn to
- Use trees to count possible outcomes in a multi-step process
- Use the multiplication axiom to count possible outcomes in a multi-step process.
In this chapter, we are trying to develop counting techniques that will be used in the next chapter to study probability. One of the most fundamental of such techniques is called the Multiplication Axiom. Before we introduce the multiplication axiom, we first look at some examples.
Example \(\PageIndex{1}\)
If a woman has two blouses and three skirts, how many different outfits consisting of a blouse and a skirt can she wear?
Solution
Suppose we call the blouses \(b_1\) and \(b_2\), and skirts \(s_1\), \(s_2\), and \(s_3\).
We can have the following six outfits.
\[b_1s_1, b_1s_2, b_1s_3, b_2s_1, b_2s_2, b_2s_3 \nonumber \]
Alternatively, we can draw a tree diagram:
The tree diagram gives us all six possibilities. The method involves two steps. First the woman chooses a blouse. She has two choices: blouse one or blouse two. If she chooses blouse one, she has three skirts to match it with; skirt one, skirt two, or skirt three. Similarly if she chooses blouse two, she can match it with each of the three skirts, again. The tree diagram helps us visualize these possibilities.
The reader should note that the process involves two steps. For the first step of choosing a blouse, there are two choices, and for each choice of a blouse, there are three choices of choosing a skirt. So altogether there are \(2 \cdot 3 = 6\) possibilities.
If, in the previous example, we add the shoes to the outfit, we have the following problem.
Example \(\PageIndex{2}\)
If a woman has two blouses, three skirts, and two pumps, how many different outfits consisting of a blouse, a skirt, and a pair of pumps can she wear?
Solution
Suppose we call the blouses \(b_1\) and \(b_2\), the skirts \(s_1\), \(s_2\), and \(s_3\), and the pumps \(p_1\), and \(p_2\).
The following tree diagram results.
We count the number of branches in the tree, and see that there are 12 different possibilities.
This time the method involves three steps. First, the woman chooses a blouse. She has two choices: blouse one or blouse two. Now suppose she chooses blouse one. This takes us to step two of the process which consists of choosing a skirt. She has three choices for a skirt, and let us suppose she chooses skirt two. Now that she has chosen a blouse and a skirt, we have moved to the third step of choosing a pair of pumps. Since she has two pairs of pumps, she has two choices for the last step. Let us suppose she chooses pumps two. She has chosen the outfit consisting of blouse one, skirt two, and pumps two, or \(b_1s_2p_2\). By looking at the different branches on the tree, one can easily see the other possibilities.
The important thing to observe here, again, is that this is a three step process. There are two choices for the first step of choosing a blouse. For each choice of a blouse, there are three choices of choosing a skirt, and for each combination of a blouse and a skirt, there are two choices of selecting a pair of pumps.
All in all, we have \(2 \cdot 3 \cdot 2 = 12\) different possibilities.
Tree diagrams help us visualize the different possibilities, but they are not practical when the possibilities are numerous. Besides, we are mostly interested in finding the number of elements in the set and not the actual list of all possibilities; once the problem is envisioned, we can solve it without a tree diagram. The two examples we just solved may have given us a clue to do just that.
Let us now try to solve Example \(\PageIndex{2}\) without a tree diagram. The problem involves three steps: choosing a blouse, choosing a skirt, and choosing a pair of pumps. The number of ways of choosing each are listed below. By multiplying these three numbers we get 12, which is what we got when we did the problem using a tree diagram.
| The number of ways of choosing a blouse | The number of ways of choosing a skirt | The number of ways of choosing pumps |
| 2 | 3 | 2 |
The procedure we just employed is called the multiplication axiom.
The Multiplication Axiom
If a task can be done in \(m\) ways, and a second task can be done in \(n\) ways, then the operation involving the first task followed by the second can be performed in \(m \cdot n\) ways.
The general multiplication axiom is not limited to just two tasks and can be used for any number of tasks.
Example \(\PageIndex{3}\)
A truck license plate consists of a letter followed by four digits. How many such license plates are possible?
Solution
Since there are 26 letters and 10 digits, we have the following choices for each.
| Letter | Digit | Digit | Digit | Digit |
| 26 | 10 | 10 | 10 | 10 |
Therefore, the number of possible license plates is \(26 \cdot 10 \cdot 10 \cdot 10 \cdot 10=260,000\).
Example \(\PageIndex{4}\)
In how many different ways can a 3-question true-false test be answered?
Solution
Since there are two choices for each question, we have
| Question 1 | Question 2 | Question 3 |
| 2 | 2 | 2 |
Applying the multiplication axiom, we get \(2 \cdot 2 \cdot 2 = 8\) different ways.
We list all eight possibilities: TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF
The reader should note that the first letter in each possibility is the answer corresponding to the first question, the second letter corresponds to the answer to the second question, and so on. For example, TFF, says that the answer to the first question is given as true, and the answers to the second and third questions false.
Example \(\PageIndex{5}\)
In how many different ways can four people be seated in a row?
Solution
Suppose we put four chairs in a row, and proceed to put four people in these seats.
There are four choices for the first chair we choose. Once a person sits down in that chair, there are only three choices for the second chair, and so on. We list as shown below.
| 4 | 3 | 2 | 1 |
So there are altogether \(4 \cdot 3 \cdot 2 \cdot 1 = 24\) different ways.
Example \(\PageIndex{6}\)
How many three-letter word sequences can be formed using the letters { A, B, C } if no letter is to be repeated?
Solution
The problem is very similar to the previous example.
Imagine a child having three building blocks labeled A, B, and C. Suppose he puts these blocks on top of each other to make word sequences. For the first letter he has three choices, namely A, B, or C. Let us suppose he chooses the first letter to be a B, then for the second block which must go on top of the first, he has only two choices: A or C. And for the last letter he has only one choice. We list the choices below.
| 3 | 2 | 1 |
Therefore, 6 different word sequences can be formed.
Finally, we'd like to illustrate this with a tree diagram showing all six possibilities.


7.1: Introduction to Bacterial, Viral/COVID-19, and Fungal Infections
By the end of this section, you should be able to:
- 7.1.1 Describe the pathophysiology of infection and the body’s defense system.
- 7.1.2 Identify clinical manifestations related to infection.
- 7.1.3 Identify the etiology and diagnostic studies related to infection.
Infection and the Body’s Defense System
The body’s immune system has several different means of preventing infection from various microorganisms. These different barriers to infection can be broken down into two distinct types: innate immunity and adaptive immunity.
Innate immunity is the body’s first line of defense against many types of organisms. It includes physical barriers such as the skin, the acidic pH of the stomach, phagocytic cells (e.g., macrophages), and the inflammatory cascade (see Introduction to the Immune System and the Inflammatory Response). This type of immunity does not require prior exposure to an organism to provide protection and is not specific to any type of microorganism. Innate forms of immunity work quickly, within minutes to hours.
By comparison, adaptive immunity , also known as acquired immunity, is highly specific to individual microorganisms. Adaptive immunity works through various immune cells with the ability to recognize specific targets (antigens) as belonging to the self or as foreign. When the body detects foreign antigens, the adaptive immune system can mount a response against the pathogen to eliminate it (Figure 7.2). Adaptive immunity is initially slow upon the first exposure to an antigen, but it has a “memory” that allows it to mount fast responses upon repeated exposure to that same pathogen. This immune memory is durable and able to last the individual’s whole life in some cases. This same process is how some vaccines work to induce immunity to organisms.
When microorganisms evade the innate and adaptive immunities, infection can occur. The individual may develop signs and symptoms consistent with elevated levels of inflammation that are caused by the immune system attempting to fight the infection. These signs and symptoms can include fever, malaise, nausea, vomiting, and pain.
Various diagnostic tests are available to aid in diagnosis of an infection. These include laboratory blood tests (e.g., a complete blood count can reveal white blood cell abnormalities), imaging (e.g., a chest x-ray can reveal pneumonia), and various direct and indirect organism tests (e.g., an influenza swab can reveal the presence of influenza viral antigens). The health care provider’s choice of tests to order depends on the client.
People who are immunocompromised are more susceptible to, and have less immune defense against, infection. They may also be unable to mount “typical” signs and symptoms of infection, including fever, which can delay recognition of the infection. This can increase morbidity and mortality significantly in this client population because they are more prone to serious infections than clients with competent immune systems are. A client can be immunocompromised for a variety of reasons, such as by having human immunodeficiency virus (HIV) and the associated acquired immunodeficiency syndrome (AIDS), receiving chemotherapy for cancer treatment, or using corticosteroids (e.g., prednisone, dexamethasone). Nurses should take particular care to investigate any potential infections in clients who are immunocompromised.
Anti-infective Drugs
A variety of anti-infective medications are available today. Because many drugs are specific to the organisms that they treat, the first step in selection is to determine the most likely organism causing the infection. This allows for more judicious use of anti-infective medications and avoids the unnecessary use of drugs to treat organisms that are not present in the client. This consideration is commonly referred to as anti-infective stewardship .
Anti-infective drugs are grouped based on the organisms that they treat and include antibiotics, antivirals, antifungals, and antiparasitics. The following sections will detail the common types of anti-infectives, including their mechanisms of action, indications, adverse effects, and contraindications.


14.5: Report Conclusions in Non‐statistical Language
The hypothesis test has been conducted and we have reached a decision. We must now communicate these conclusions so they are complete, accurate, and understood by the targeted audience. How a conclusion is written is open to subjective analysis, but here are a few suggestions.
Be consistent with the results of the Hypothesis Test
Rejecting \(H_o\) requires a strong statement in support of \(H_a\), while failing to reject \(H_o\) does NOT support \(H_o\), but requires a weak statement of insufficient evidence to support \(H_a\).
Example
A researcher wants to support the claim that, on average, students send more than 1000 text messages per month, and the research hypotheses are \(H_o: \mu=1000\) vs. \(H_a: \mu>1000\)
Conclusion if \(H_o\) is rejected: The mean number of text messages sent by students exceeds 1000.
Conclusion if \(H_o\) is not rejected: There is insufficient evidence to support the claim that the mean number of text messages sent by students exceeds 1000.
Use language that is clearly understood in the context of the problem
Do not use technical language or jargon, but instead refer back to the language of the original general question or research hypotheses. Saying less is better than saying more.
Example
A test supported the Alternative Hypothesis that housing prices and size of homes in square feet were positively correlated. Compare these two conclusions and decide which is clearer:
Solution
Conclusion 1 : By rejecting the Null Hypothesis we are inferring that the Alterative Hypothesis is supported and that there exists a significant correlation between the independent and dependent variables in the original problem comparing home prices to square footage.
Conclusion 2 : Homes with more square footage generally have higher prices.
Limit the inference to the population that was sampled
Care must be taken to describe the population being sampled and understand that the any claim is limited to this sampled population. If a survey was taken of a subgroup of a population, then the inference applies only to the subgroup.
For example, studies by pharmaceutical companies will only test adult patients, making it difficult to determine effective dosage and side effects for children. “In the absence of data, doctors use their medical judgment to decide on a particular drug and dose for children. ‘Some doctors stay away from drugs, which could deny needed treatment,’ Blumer says. ‘Generally, we take our best guess based on what's been done before.’ The antibiotic chloramphenicol was widely used in adults to treat infections resistant to penicillin. But many newborn babies died after receiving the drug because their immature livers couldn't break down the antibiotic.” 73 We can see in this example that applying inference of the drug testing results on adults to the un‐sampled children led to tragic results.
Report sampling methods that could question the integrity of the random sample assumption
In practice it is nearly impossible to choose a random sample, and scientific sampling techniques that attempt to simulate a random sample need to be checked for bias caused by under‐sampling.
Telephone polling was found to under‐sample young people during the 2008 presidential campaign because of the increase in cell phone only households. Since young people were more likely to favor Obama, this caused bias in the polling numbers. Additionally, caller ID has dramatically reduced the percentage of successful connections to people being surveyed. The pollster Jay Leve of SurveyUSA said telephone polling was “doomed” and said his company was already developing new methods for polling. 74
Sampling that didn’t occur over the weekend may exclude many full time workers while self‐selected and unverified polls (such as ratemyprofessors.com) could contain immeasurable bias.
Conclusions should address the potential or necessity of further research, sending the process back to the first procedure
Answers often lead to new questions. If changes are recommended in a researcher’s conclusion, then further research is usually needed to analyze the impact and effectiveness of the implemented changes. There may have been limitations in the original research project (such as funding resources, sampling techniques, unavailability of data) that warrant more comprehensive studies.
For example, a math department modifies its curriculum based on the improved student success rates of an experimental course. The department would want to do further study of student outcomes to assess the effectiveness of the new program.


11.7: Sheep Brain Dissection
Background Information:
The sheep brain is remarkably similar to the human brain. One major difference, however, is in proportion. For example, the sheep brain has a proportionately smaller cerebrum. Another difference is in orientation of the spinal cord. The sheep spinal cord is orientated anterior to posterior, as in any four-legged animal. The human spinal cord is orientated superior to inferior. This orientation difference has a major affect on the location of the brain stem. The sheep brain stem is located more towards the rear (posteriorly). The sheep skull, in order to compensate for this, has the foramen magnum located more towards the rear of the skull. In humans, since we walk upright (bipedalism), the spinal cord is in a more vertical plane, thus the foramen magnum is located centrally on the bottom of the skull (inferiorly). By observing the movement of this major skull feature from the rear (as in very early human ancestors) to the current location (modern humans), scientists have been able to determine when the human species began to walk upright on two legs.
 Place this on your dissection tray.
- You will need the following dissection tools to properly perform this lab:
- scalpel
- scissors
- probes
3. The sheep brain is enclosed in a tough outer covering called the dura mater. You can still see some structures on the brain before you remove the dura mater. Take special note of the pituitary gland and the optic chiasma. These two structures will likely be pulled off when you remove the dura mater.
4. This image shows the ventral surface of the sheep's brain with most of the dura mater removed. The pituitary gland and the optic chiasma are still intact. (A = pituitary gland, B = optic chiasma, C = olfactory bulb)
5. On this image, the dura mater has been completely removed, you can still see the optic chiasma but the pituitary gland is missing. The infundibulum (pituitary stalk) is now visible in the center. Careful dissection also reveals two other large nerves: the oculomotor nerves (C.2). Often these two nerves are removed with the dura mater, but in this image they are still intact.
These two figures show the fissures located on the surface of the brain. Longitudinal fissure Transverse fissure
6. If you flip the brain over to the other side, you can see the cerebellum , it will be loosely attached to the cerebrum in most cases. If you did not carefully remove the dura mater you may have accidentally pulled the entire cerebellum away from the brain. The lobes of the brain are visible, as well as the transverse fissure , which separates the cerebrum from the cerebellum. The convolutions of the brain are also visible as bumps ( gyri ) and grooves ( sulci ).
7. The gap between the cerebrum and the cerebellum at the transverse fissure can reveal some internal parts of the brain. In this image, a student is bending the cerebellum down to show the superior and inferior colliculi . Just behind the colliculi, the pineal gland is just barely visible.
8. Using a scalpel and the longitudinal fissure as a guide, the brain is separated into the left and the right hemispheres. Sharp scalpels work best for this procedure. Always leave the specimen in the dissecting tray when cutting it, do NOT hold it in your hand!
If you are very careful, you will cleanly cut the brain into two halves and can see the internal structures, the most visible of them being the corpus callosum , which divides the left and right hemispheres. The cerebrum will still be visible as a wrinkled structrure, and you can even locate the "bumps" of the superior and inferior colliculi . Remember, you located those structures by pulling down the cerebellum.
The cerebellum, when cut will have a very distinct tree-like white area within it. This is called the arbor vitae , or the tree of life.
9. In the image below, a probe indicates the location of the lateral ventricle .
10. Once the brain is cut this way, the colliculi can also be seen from the inside and the pineal gland is revealed only if you made a very careful incision.
11. Other major structures are visible, here the probe indicates the
arbor vitae
(tree of life) found within the cerebellum. The fissure between the cerebrum and the cerebellum is called the
transverse fissure
. The cerebellum only loosely connects to the rest of the brain when the dura is removed.
12. This brain is pinned to show the pineal gland (blue pin), thalamus (red pin) and lateral ventricle (green pin).
13. The image below shows a cleanly separated brain with the major internal structures visible and labeled.
14. Finally, a section of the brain is cut to examine the difference between white matter and gray matter.
15. Once you have made the cut like in the above diagram, you should be able to see the difference between the white and gray matter, just like the diagram below.
16. The diagram below will help you understand the difference between the gyri and the sulcus.
Sheep brain dissection maybe downloaded at the following site: https://jb004.k12.sd.us/my%20website...P%20BRAIN%20DI SSECTION/SHEEP%20BRAIN%20LAB.htm
LAB 11 EXERCISES
\(\PageIndex{1}\):
Sheep Brain Dissection Analysis:
Match the structure to the description
| a | looks like a butt | Arbor Vitae | |
| b | leathery covering over the entire brain | Lateral Ventricle | |
| c | cauliflower, the area toward the back of the brain | Optic Chiasma | |
| d | behind the colliculi, looks like a little nub | Superior Colliculi | |
| e | looks like a "tree" | Dura Mater | |
| f | the rounded part of the brain stem | Cerebellum | |
| g | shaped like an X | Pineal Gland | |
| h | large area under the corpus callosum | Thalamus | |
| i | space for fluid between the corpus callosum and the fornix | Pons | |
| j | contains nerves, connects to the far front of the brain | Olfactory Bulb |


6.3A: Defining Boundaries
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Social groups are defined and separated by boundaries.
Learning Objectives
- Explain what tends to happen to individuals when their group boundaries are impermeable, and also when they are permeable
Key Points
- Cultural sociologists define symbolic boundaries as “conceptual distinctions made by social actors…that separate people into groups and generate feelings of similarity and group membership. ” These boundaries are necessary for the existence of in-groups and out-groups.
- Where group boundaries are considered permeable (e.g., a group member may pass from a low status group into a high status group), individuals are more likely to engage in individual mobility strategies.
- Where group boundaries are considered impermeable, and where status relations are considered reasonably stable, individuals are predicted to engage in social creativity behaviors.
- One important factor in how symbolic boundaries function is how widely they are accepted as valid. Symbolic boundaries are a “necessary but insufficient” condition for social change.
- According to sociologists, it is “only when symbolic boundaries are widely agreed upon can they take on a constraining character… and become social boundaries. ” Thus, rituals and traditions to define boundaries are extremely influential in determining how groups interact.
- In the social sciences, the word “clique” is used to describe a group of 2 to 12 “persons who interact with each other more regularly and intensely than others in the same setting. “
Key Terms
- symbolic boundary : Conceptual distinctions made by social actors that separate people into groups and generate feelings of similarity and group membership.
- individual mobility : The ability of an individual to move from one social group to another.
Social groups are defined by boundaries. Cultural sociologists define symbolic boundaries as “conceptual distinctions made by social actors…that separate people into groups and generate feelings of similarity and group membership. ” In-groups, or social groups to which an individual feels he or she belongs as a member, and out-groups, or groups with which an individual does not identify, would be impossible without symbolic boundaries.
Permeability of Group Boundaries
The perceived permeability of group boundaries is important in determining how members define their identity. Where group boundaries are considered permeable (e.g., a group member may pass from a low status group into a high status group), individuals are more likely to engage in individual mobility strategies. That is, individuals “disassociate from the group and pursue individual goals designed to improve their personal lot rather than that of their in-group. ”
Where group boundaries are considered impermeable, and where status relations are considered reasonably stable, individuals are predicted to engage in social creativity behaviors. Here, without changing necessarily the objective resources of in the in-group or the out-group, low status in-group members are still able to increase their positive distinctiveness. This may be achieved by comparing the in-group to the out-group on some new dimension, changing the values assigned to the attributes of the group, and choosing an alternative out-group by which to compare the in-group.
Defining Boundaries
One important factor in how symbolic boundaries function is how widely they are accepted as valid. Symbolic boundaries are a “necessary but insufficient” condition for social change. According to sociologists, it is “only when symbolic boundaries are widely agreed upon can they take on a constraining character… and become social boundaries. ” Thus, rituals and traditions to define boundaries are extremely influential in determining how groups interact.
Emile Durkheim was interested in this idea. He saw the symbolic boundary between the sacred and the profane as the most profound of all social facts, and the one from which lesser symbolic boundaries were derived. Rituals, whether secular or religious, were for Durkheim the means by which groups maintained their symbolic and moral boundaries. Mary Douglas has subsequently emphasized the role of symbolic boundaries in organizing experience, private and public, even in a secular society.


1.4: Chapter Summary
Population health is a comprehensive approach that considers the impact of various factors on health and involves collaborative efforts among health care professionals, communities, industries, academia, and governmental organizations. Nurses across all settings and specialties are critical advocates for achieving population health goals. Population health takes a proactive approach to addressing the root causes of health problems as a means of prevention. Critical components of public health include outcomes, determinants, disparities, and risk factors. Concepts of upstream, midstream, and downstream approaches to intervention interact and evolve as population health evolves, with upstream interventions offering the widest-reaching health benefits to the population. Population health extends to the global stage, involving global nursing, which aims to promote sustainable planetary health and equity for all individuals.
Promoting health across a population is essential for improving health outcomes for individuals and society. Rather than treating individual illnesses as they arise, population health approaches aim to prevent conditions and manage diseases more effectively when they occur. National health indicators provide a macro-level perspective of a nation’s health status, contributing to a comprehensive understanding of population health. Population health approaches improve cost savings, actualize social benefits, promote equity, facilitate social justice, and improve health outcomes.
A shift toward population health and health promotion in nursing has encouraged collaboration and meaningful integration of systems across the health system. Emphasizing preventive care, early intervention, and chronic disease management is a positive change under a population health paradigm within the entire health care continuum, which includes preventive care, acute care, long-term care, and transitional care. Nurses are driving change in population health and have developed many innovative, proactive care programs. Today, student nurses are expected to achieve competencies relevant to providing care at the population level and leading change that ameliorates upstream factors contributing to poor health outcomes in the United States.


1.5: Asset, Liability and Stockholders’ Equity Accounts
The three other categories of accounts—assets, liabilities, and stockholders’ equity—are reported on another financial statement called the balance sheet. Unlike the temporary accounts on the income statement, these are permanent accounts because they are not closed out at the end of the accounting period. Instead, the account balances of the balance sheet accounts at the end of the period are carried forward and become the starting balances at the beginning of the next period.
1.5.1 Assets
Assets are anything of value to a business, including things a business owns so it can operate. Assets are recorded in the journal at what they cost the business, or what the business paid to acquire them. This is called the cost principle .
The first two asset accounts are those you are familiar with so far. These are current assets , which means they are either cash or are expected to be converted to cash within one year.
| Cash | Money a business possesses |
| Accounts Receivable | Amount customers owe to a business from being invoiced on account |
The following assets are fixed assets . They are relatively expensive and will last for more than one accounting year. Therefore, they are considered assets rather than expenses, which are costs related to a particular accounting period.
| Land | Real estate property a business owns |
| Building | Real estate property a business owns |
| Truck | Motor vehicle a business owns |
| Equipment | Electronic machinery a business owns |
| Furnishings | Furniture a business owns |
Debit Any ASSET when it increases
Credit Any ASSET when it decreases
1.5.2 Liabilities
Liabilities are debts a business has on the assets it possesses. They are claims on the assets by people and entities that are not owners of the business.
The following are liability accounts.
| Accounts Payable | Amount a business owes to vendors from being invoiced on account |
| Note Payable | Loan for cash or on any of the assets owned |
| ANY Payable | Debt owed for a specific reason |
Credit Any LIABILITY when it increases
Debit Any LIABILITY when it decreases
Both Accounts Payable and Note Payable are liability accounts, or debts. They are different, however. Accounts Payable is a payment agreement with a vendor who gives you time—usually thirty days—to pay for a product or service your business purchases. A note payable is a formal, signed loan contract that may include an interest rate and that spells out the terms and conditions of repayment over time.
1.5.3 Stockholders’ Equity
Stockholders’ equity is the stockholders’ share of ownership of the assets that the business possesses, or the claim on the business’s assets by its owners.
A corporation is a form of business that is a separate legal entity from its owners. The people and/or organizations who own a corporation are called stockholders . Stockholders (owners) receive shares of stock as receipts for theirinvestments in the business. This form of business offers limited liability to stockholders—the owners can only lose what they invested in the business. Their other assets cannot be taken to satisfy the obligations of the company they invest in.
Let’s say you start a lawn care business and invest $500 of your own cash and spend $1,500 for lawnmowers for a total investment of $2,000. If you do not incorporate, your business is a sole proprietorship. If you do incorporate, your business is a corporation. To form a corporation, a business needs to file paperwork called articles of incorporation (and pay a fee) with the state in which it will be operating. The state grants the business its corporate status.
If you damage the property of one of your customers and he submits a claim against you for $10,000, the most that you can be liable for as a corporation is the amount you have invested and earned in the business. As a sole proprietorship, however, it is possible the customer can be awarded more than the value of your ownership in the business. You would then have to pay out the difference using your personal money. If you don’t have enough, youcould even be forced to sell some of the things you own or make payments from your future wages to pay the claim off. If you are not organized as a corporation, your risk is not limited to the amount you invested and earned in the business.
The following are stockholders’ equity accounts:
| Common Stock | Account that shows the value of shares of stock issued to stockholders |
| Retained Earnings | Account where the corporation’s profits accumulate and are “stored” |
| Cash Dividends | Payouts of profits (retained earnings) to stockholders |
Stockholders’ equity is the amount of a business’s total assets that is owned by the stockholders. Only two accounts fall in this category: stockholders’ equity is the total of the balances in the Common Stock and Retained Earnings accounts.
Common stock is the ownership value in the business that comes from outside the company—investors who pay their own money into the business. Retained earnings is the ownership value in the business that comes from inside thecompany—the business makes a profit that is shared by the stockholders.
Cash dividends are payouts of profits from retained earnings to stockholders. Cash Dividends is a temporary account that substitutes for a debit to Retained Earnings and is classified as a contra (opposite) stockholders’ equity account. Cash dividends will reduce the Retained Earnings balance. This is ultimately accom- plished by closing the Cash Dividends balance into Retained Earnings at the end of the accounting period.
Credit
Common Stock
or
Retained Earnings
when it increases
Debit
Retained Earnings
when it decreases
Debit
Cash Dividends
when it increases
Credit
Cash Dividends
when it decreases
1.5.4 Balance Sheet Account Transactions
Six very typical business transactions that involve balance sheet accounts will be shown next.
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A company purchases equipment, paying $5,000 cash.
Date Account Debit Credit 6/1 Equipment 5,000 ▲ Equipment is an asset account that is increasing . Cash 5,000 ▼ Cash is an asset account that is decreasing . -
A company purchases equipment for $5,000 on account.
Date Account Debit Credit 6/1 Equipment 5,000 ▲ Equipment is an asset account that is increasing . Accounts Payable 5,000 ▲ Accounts Payable is a liability account that is increasing . -
A company purchases equipment that costs $5,000. The company pays a down payment of $1,000 and takes a loan for the remaining $4,000.
Date Account Debit Credit 6/1 Equipment 5,000 ▲ Equipment is an asset account that is increasing . Cash 1,000 ▼ Cash is an asset account that is decreasing . Note Payable 4,000 ▲ Note Payable is a liability account that is increasing . NOTE: Transaction #3 is called a compound transaction because there is more than one credit. (A compound transaction could also have more than one debit, if required.) The total of the debits must equal the total of the credits in each transaction. In this case one asset is being purchased, but there are two forms of payment—cash and the loan.
Also notice that in transactions #1, 2, and 3 above, the account debited is Equipment , an asset (and not Equipment Expense , which would be an expense account). The same holds true for the purchase of real estate: the assets Building and/or Land would be debited (not Building Expense or Land Expense ). This is because these assets will last more than one accounting period.
Usually one of the first steps in starting a business is opening the business’s bank account.
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An individual invests $10,000 of his own cash to open a new corporation’s checking account.
Date Account Debit Credit 6/1 Cash 10,000 ▲ Cash is an asset account that is increasing . Common Stock 10,000 ▲ Common Stock is an equity account that is increasing . Think of common stock as a receipt for an investor infusing money or other assets into the business. It recognizes that person’s ownership. A running total of all the investments that people make in a corporation is maintained in the Common Stock account.
Transaction #4 is recorded when an investor puts money or other assets into a corporation. There are also times when investors take money out of a business. This can only be done if the corporation has generated a profit over time, which is what the investors will draw from. The accumulated profit over time appears in the corporation’s Retained Earnings account.
The board of directors of large corporations or the owner(s) of small, closely- held corporations may decide to pay cash dividends to stockholders if there are sufficient retained earnings and sufficient cash to do so. Cash dividends are payouts of profit to stockholders; in other words, distributions of retained earnings. Cash dividends are not paid out of owner investments, or common stock.
-
The corporation pays $1,000 in dividends to its stockholders.
It might seem logical to debit Retained Earnings to reduce that stockholders’ equity account and credit Cash to reduce that asset account. That is not entirely wrong. However, we are going to reserve Retained Earnings for closing entries only, and payment of dividends is not a closing entry. Instead of a debit to Retained Earnings , therefore, we will substitute the Cash Dividends account in this transaction.
Date Account Debit Credit 6/15 Cash Dividends 1,000 ▲ Cash Dividends is a contra equity account that is increasing . Cash 1,000 ▼ Cash is an asset account that is decreasing . -
The corporation closes the Cash Dividends account at the end of the month.
Finally, at the end of the accounting period (in this case a month), there is one final closing entry in addition to the ones you already know for revenue and expense accounts. This closes the Cash Dividends account to Retained Earnings, so ultimately the Retained Earnings account is reduced by the profit paid out to stockholders. The Cash Dividends account balance is set back to zero as a result.
Date Account Debit Credit 6/30 Retained Earnings 1,000 Cash Dividends 1,000 ▼ Retained Earnings is an equity account that is decreasing .
▼ Cash Dividends is an equity account that is decreasing .
The following summarizes the two cash dividends transactions in #5 and #6— paying the dividends and closing the Cash Dividends account at the end of the month. If the debit and credit to Cash Dividends is struck through since the two combined would result in a balance of zero in the Cash Dividends account, you are ultimately left with a debit to Retained Earnings (reducing it) and a credit to Cash (reducing it) for the payment of a dividend.
| Date | Account | Debit | Credit | |
| 6/15 |
|
x | 1,000 | |
| 6/30 | Retained Earnings | x | 1,000 | |
|
|
x | 1,000 | ||
| Cash Dividends | |||||
| Date | Item | Debit | Credit | Debit | Credit |
| 6/15 | 1,000 | 1,000 | |||
| 6/30 | 1,000 | 0 | |||


8.4: The Real Numbers
We complete our construction of the real numbers (we have the irrational numbers remaining) with the objective of proving the ordercompleteness of the real numbers, and deriving some important consequences of completeness. Many of the most powerful and interesting results of calculus depend on this property of the real numbers. If you have been asked to accept some of these theorems on faith, now it is time to reward your trust.
There are a couple of different ways to construct the real numbers from the rational numbers. One approach is to define real numbers as convergent sequences of rational numbers. The other common approach is to characterize real numbers as subsets of rational numbers that satisfy certain conditions.
DEFINITION.  Let \(L\) be a Dedekind cut. Then there is some rational number \(a \in L\) , and therefore all rational numbers less then \(a\) are in \(L\) . Let \(R=\mathbb{Q} \backslash L\) . Since \(L \neq \mathbb{Q}\) , there is \(c \in R\) and every rational number greater than \(c\) is in \(R\) . It is clear that \(\{L, R\}\) is a partition of \(\mathbb{Q}\) and that every element of \(L\) is less than every element of \(R\) . So Dedekind cuts "split" the rational numbers. We shall associate each Dedekind cut with a real number located at the split on the real number line.
REMARK. To help our mental picture of what is going on, we think of \(L\) as all rational numbers to the left of some fixed real number \(\alpha\) , i.e. as \((-\infty, \alpha) \cap \mathbb{Q}\) , and \(R\) as the rational numbers to the right, \([\alpha, \infty) \cap \mathbb{Q}\) . Of course we don’t yet know what exactly we mean by "the real number \(\alpha\) ", but this is the idea to keep in mind. Note that \(R\) will have a least element iff \(\alpha\) is rational.
To understand how Dedekind cuts relate to numbers we construct an injection from the rational numbers to the Dedekind cuts. Let \(\mathcal{D}\) be the set of Dedekind cuts. We define an injection \(i: \mathbb{Q} \rightarrow \mathcal{D}\) by \[i(a)=\{b \in \mathbb{Q} \mid b<a\} .\] The function \(i\) is a well-defined injection that informs us of how \(\mathbb{Q}\) fits into \(\mathcal{D}\) .
We shall define order and operations on \(\mathcal{D}\) so that they agree with the usual linear ordering and operations on \(\mathbb{Q}\) that are inherited in \(i[\mathbb{Q}]\) . That is, we shall define the linear order, addition and multiplication on \(\mathcal{D}\) so that for \(a, b \in \mathbb{Q}\) , \[\begin{aligned} {[a \leq b] } & \Longleftrightarrow[i(a) \leq i(b)] \\ i(a+b) &=i(a)+i(b) \\ i(a \cdot b) &=i(a) \cdot i(b) \end{aligned}\] If we can do this, we can think of \(\mathcal{D}\) as an extension of \(\mathbb{Q}\) . How do we do it?
For \(L, K \in \mathcal{D}\) , we define the relation \(\leq\) in \(\mathcal{D}\) by \[[L \leq K] \Longleftrightarrow[L \subseteq K] .\] You should confirm that \(\leq\) is a linear ordering of \(\mathcal{D}\) and that the relation \(\leq\) on \(i[\mathbb{Q}]\) satisfies (1). If \(L \in \mathcal{D}\) and \(L<i(0)\) we say that \(L\) is negative. If \(L>i(0)\) , we say that \(L\) is positive. With a similar objective in mind we define addition and multiplication on \(\mathcal{D}\) . That is, we want the operations to satisfy certain properties of addition and multiplication and we want the operations defined on \(i[\mathbb{Q}]\) to agree with the operations on \(\mathbb{Q}\) .
If \(L, K \in \mathcal{D}\) , then \[L+K:=\{a+b \mid a \in L \text { and } b \in K\} .\] Verify that \(L+K\) is a Dedekind cut, and that (2) holds.
Multiplication takes a bit more effort to define. (Why can’t we let \(L \cdot K=\{a b \mid a \in L, b \in K\}\) ?) If \(L\) or \(K\) is \(i(0)\) , then \[L \cdot K:=i(0) .\] If \(L, K \in \mathcal{D}\) are both positive, then \[L \cdot K=\{a \cdot b \mid a \in L, b \in K, a>0 \text { and } b>0\} \cup\{c \in \mathbb{Q} \mid c \leq 0\} .\] Verify that \(L \cdot K\) is a Dedekind cut, and that (3) holds for \(a, b>0\) .
How do we define multiplication by "negative" Dedekind cuts? Let’s start with defining multiplication by \(-1\) . Let \(L \in \mathcal{D}\) and \(R=\) \(\mathbb{Q} \backslash L\) . We define \(-L\) by \[-L:=\{c \in \mathbb{Q} \mid(\exists r \in R)-c>r\} .\] Now we can define multiplication on arbitrary elements of \(\mathcal{D}\) to satisfy the properties we desire. If \(L, K \in \mathcal{D}\) and both are negative, then \[L \cdot K:=(-L \cdot-K) .\] If exactly one of \(L\) and \(K\) is negative, then \[L \cdot K:=-(-L \cdot K) .\] DEFINITION. Real numbers, \(\mathbb{R}\) The real numbers are the Dedekind cuts, with addition, multiplication and \(\leq\) defined as above. We denote the real numbers by \(\mathbb{R}\) when we do not need to think of them explicitly as Dedekind cuts.
We have defined the real numbers as sets of rational numbers. Since the rational numbers were defined using basic ideas about sets, functions and relations, so are the real numbers. The properties of the real numbers that we discussed at the beginning of this section are satisfied by the Dedekind cuts. For every rational number \(a\) , we identify \(a\) with the Dedekind cut \(i(a)\) .
THEOREM 8.2. The real numbers as defined above satisfy:
(i) Addition and multiplication are both commutative and associative.
(ii) \((\forall L \in \mathcal{D}) L+0=L, L \cdot 1=L\) .
(iii) \((\forall L \in \mathcal{D}) L+(-L)=0\) .
(iv) \((\forall L \in \mathcal{D} \backslash\{0\})(\exists K \in \mathcal{D}) L \cdot K=1\) .
(v) \((\forall L, K, J \in \mathcal{D}) L \cdot(K+J)=L \cdot K+L \cdot J\) .
Proof. Exercise.


4.2: Fertility and Conception
By the end of this section, you will be able to:
- Describe the components of fertility, both physical and psychosocial
- Describe the stages of conception, including ovulation and fertilization
- Describe the different causes of infertility
Human reproduction and conception have been celebrated in mythology, with symbols of female fertility evident in ancient civilizations all around the world. In addition, world mythology reflects the historical importance and need for fertility and reproduction in different cultures (Behjati-Ardakani et al., 2016). People as far back as 3500 BCE to 500 CE used the principles of insemination and “magic potions” in attempts to remedy childless royal families (Sharma et al., 2018). Though the stigma of infertility has largely changed and is still changing in some countries, the desire for a child remains strong for many people (Sharma et al., 2018). Infertility can be a life-changing experience for people. It affects both the body and mind and can completely alter a person’s relationship with their partner, family, and friends. It is important for nurses to consider both the physical and psychosocial components of fertility when working with people who are having difficulty in conceiving.
This section will address the concepts of fertility and infertility, including when patients should be advised to seek help, as well as the physical causes of and psychosocial factors involved with fertility. It will also delve into the diagnostic testing used to assess the fertility of both partners and the measures that can be used to improve fertility.
Components of Fertility
A person’s ability to conceive a pregnancy is called fertility . The inability to become pregnant after 1 year of regular sexual intercourse without using any form of birth control is called infertility (American College of Obstetricians and Gynecologists [ACOG], 2022b). It is a global health problem, affecting approximately 1 in 6 adults worldwide (World Health Organization, 2023b). This rate is fairly consistent across different regions and countries, and there is no significant difference between resource-poor and resource-rich countries. Global access to affordable reproductive care is lacking. In most cases, care is provided as an out-of-pocket expense, which can be financially devastating—or completely inaccessible—for people with limited resources.
Physical
Three factors are essential for pregnancy to occur: the release of an egg, the presence of sperm, and a healthy environment for implantation . Many factors can influence the health of the egg, sperm, and uterus. Environmental factors such as pesticides, radiation, and unclean water can cause difficulty with conception . Genetics, infections, and substance use can also affect fertility.
Egg
The egg, or ovum, is the female gamete, or reproductive cell, and contains one-half the number of chromosomes found in the body’s cells. Eggs are formed very early during the fetal development of female embryos, with nearly one million immature eggs (known as oogonia) present in the ovarian follicles at birth. That number decreases over the course of a person’s lifespan, with only 400,000 remaining by puberty (Krajnik et al., 2023).
When stimulated by follicle-stimulating hormone (FSH), the oogonium in the immature follicle undergoes mitosis to form a primary oocyte and a stem cell. Both new cells are genetically identical diploid cells, meaning that each contains the full 46 chromosomes (Hutter, 2023). The primary oocyte then undergoes the first meiotic division to form a secondary oocyte and a polar body. The second meiotic division causes the polar body to form two daughter polar bodies. The secondary oocyte forms a mature ovum and a third polar body. The result is three polar bodies and one mature ovum, all of which contain the haploid number of chromosomes (Figure 4.2).
 Unlike the female gamete, sperm cells do not develop in the male body until after puberty in a process known as spermatogenesis . Each seminiferous tubule in the testes is lined with diploid cells, called spermatogonia, which contain the full 46 chromosomes and constantly undergo mitotic division. Some spermatogonia move away from the lining of the seminiferous tubule to mature and become primary spermatocytes. The primary spermatocytes undergo meiotic division to become two haploid (containing 23 chromosomes, instead of the usual 46) secondary spermatocytes. These secondary spermatocytes undergo a second round of meiosis to become four haploid spermatids, which eventually develop into four sperm cells (Figure 4.3). This entire process takes around 74 days (Komeya et al., 2018).
These developing sperm cells move from the seminiferous tubules in the testicles into the head of the epididymis. They travel through the epididymis, from the head to the tail, over the course of about 12 days, where they absorb nutrients and testosterone (Figure 4.4). By the time they reach the tail of the epididymis, the sperm cells are mature and fully able to fertilize an ovum. These mature sperm cells are then stored in the epididymal tail until just before ejaculation. When this occurs, sperm cells leave the epididymis and rapidly travel through the vas deferens and internal duct system. They combine with fluid released from the seminal vesicles, the Cowper (bulbourethral) glands, and the prostate gland to create semen, a nutrient-rich fluid that nourishes the sperm cells. The semen enters the urethra near the base of the penis, where strong muscular contractions push the semen out of the body (Soni et al., 2022).
Each ejaculation releases up to 200 million sperm cells. If sperm cells are ejaculated into the vagina, they propel themselves through the cervix and into the female reproductive tract by rapid movements of the flagella, a long tail that protrudes from the end of the sperm cell. It takes sperm cells approximately 4 to 6 hours to travel to the fallopian tubes. Unlike egg cells, sperm cells can live up to about 5 days in the female reproductive tract ( Conception: How it works , n.d.).
Without the presence of both viable sperm and eggs in the female reproductive tract at the same time, conception cannot occur.
Uterus
The final component needed for conception is the presence of a hormonally receptive uterus for implantation . During the menstrual cycle , the uterus undergoes several important changes in anticipation of a pregnancy. As the ovarian follicles enlarge, they begin to secrete estrogen. That estrogen signals the lining of the uterus, the endometrium, to proliferate and thicken. This is the proliferative phase of the uterine cycle.
After ovulation occurs and progesterone levels rise, the endometrium enters the secretory phase. The lining becomes even thicker and more vascular, which helps to prepare for the implantation of a fertilized egg, or zygote (Figure 4.5). If pregnancy does not occur, the endometrium begins to break down and slough off during menstruation, triggered by declining progesterone levels in the corpus luteum (Cable & Grider, 2023).
Even if ovulation perfectly coincides with sexual activity to result in a fertilized egg, pregnancy does not occur unless the zygote can properly implant in the uterine lining, triggering the hormonal cascade that occurs with pregnancy.
Psychosocial
Though conception and pregnancy are largely physiologic events, the nurse must not neglect the important psychosocial factors that are associated with fertility and conception. Stress and other societal pressure can impact both a person’s ability and their desire to conceive.
Societal Pressures
Social implications of fertility and reproduction extend beyond the individual and their family to the whole society, just as cultural and social change can affect the individual and their family makeup (Behjati-Ardakani et al., 2016). Likewise, changes in society have also impacted family-building practices in the United States and globally. People, for many different reasons, have opted to wait longer to start a family. “These changing fertility rates by age shifted the median age at which women gave birth in the United States from age 27 in 1990 to age 30 in 2019” (Morse, 2022). Several reasons for the delay in parenthood have been identified, including:
- a belief that assisted reproductive technologies could correct any fertility problems
- recognition that families could be formed through adoption
- not having a partner with whom to build a family
- not having the social support or financial stability in place to have children
- ambivalence toward wanting to have children
- more women focusing on building their career rather than starting a family
- lack of knowledge about how fertility declines with age (Delbaere et al., 2020).
Whatever the reason for delay, the effect of delayed childbearing is generally the same: Rates of fecundity decrease as the person assigned female at birth ages (Table 4.1). It is important to recognize that while the data about the effects of age on fertility are fairly consistent, they are an average. This means that one person may have significantly decreased fertility by the time they are 45, while another may have a similar decline by only 35 or 40 years of age.
| Age of Patient in Years | Chance of Conception within 1 Year |
|---|---|
| <30 | 85% |
| 30 | 75% |
| 35 | 66% |
| 40 | 44% |
In 2018, a large systematic review of 71 studies found that many study participants reported low to moderate knowledge about age-related fertility decline (Pedro et al., 2018). Researchers found that interventions to increase this knowledge would be helpful, especially for men, persons with lower education, and family planning clinics (Pedro et al., 2018). The studies analyzed in this review found that these populations in particular were found to have lower knowledge about fertility and family planning (Pedro et al., 2018). Nurses working in family planning or women’s health settings should be aware of this knowledge gap and intervene as necessary.
Falling Birth Rates
Many countries all over the world have been dealing with low birth rates for the past several years. Countries need, on average, a birth rate of at least 2.1 children per woman to maintain their population, yet the average birth rate in Europe is only 1.59 (BBC News, 2020), and it is as low as 0.78 in South Korea (Young & Bae, 2023). There are many reasons why people are choosing to not have as many children or to not have children at all:
- Strenuous work culture
- Poor wages
- Inflation and high cost of living
- Changing opinions about marriage (Young & Bae, 2023).
Scandinavia and France have seen success with increasing birth rates by increasing social benefits for families, including tax breaks, subsidized childcare and generous family leave. However, Scandinavia also began to see falling birth rates in recent years (BBC News, 2020), and similar policies were also not successful in South Korea (Young & Bae, 2023). More study is needed to determine what policy changes and social support systems are needed to improve birth rates moving forward.
(Young & Bae, 2023; BBC News, 2020)
Conception
Conception is a complex process, requiring several different precise processes. Ovulation, fertilization, and implantation must all occur safely. Factors can interrupt the process of conception , such as hormonal imbalances, structural abnormalities within the fallopian tube or uterus, or an inhospitable endometrium not allowing implantation. The quality of the egg and sperm can also affect conception. Timing of these processes must also be precise. When infertility occurs, health-care providers can examine these processes to potentially identify the cause of infertility.
Ovulation
Each month after menarche (the first menstrual period), a cyclic pattern of hormonal changes and menstrual bleeding occurs until menopause , which is the cessation of those cycles. This cycle is initiated by rising gonadotropin (follicle-stimulating hormone [FSH]) levels, causing a cohort of ovarian follicles to grow, develop, and secrete estrogen in the form of estradiol (E2). Eventually, most of these egg follicles atrophy, leaving one egg follicle to become dominant and secrete larger and larger amounts of estradiol (see Figure 4.2). Once the egg is mature, rising estradiol levels trigger the surge of another hormone, luteinizing hormone (LH), which causes the release of the mature egg through the process of ovulation . Estrogen levels fall, but the empty egg follicle, now known as the corpus luteum (“yellow body”), secretes the hormone progesterone (and small amounts of estradiol) to prepare the body for pregnancy (Cable & Grider, 2023). The corpus luteum is hormonally active for about 14 days before it degenerates and triggers the next menstrual period if pregnancy has not occurred. If conception does occur, the corpus luteum will continue to secrete progesterone until the placenta is mature enough to support the growing pregnancy.
The mature egg is released into the abdomen and is gently swept by the fimbriae, finger-like structures, into the infundibulum of the fallopian tube , the small tube that allows the passage of the egg from the ovary to the uterus. Small cilia and gentle muscular contractions, called peristalsis, move the ovum down toward the uterus, which it enters just below the fundus. In most cases, fertilization with a single sperm cell occurs in the distal third, or ampulla, of the fallopian tube. The egg is viable for only about 12 to 24 hours after ovulation ( Conception: How it works , n.d.). After that time, it begins to disintegrate if fertilization has not occurred.
Fertilization
The penetration of an ovum by a single sperm cell is called fertilization . When that occurs, the membrane of the ovum, the zona pellucida, becomes impenetrable to other sperm cells ( Conception: How it works , n.d.). This process triggers a series of other reactions:
- completion of the second meiotic division of the secondary oocyte
- formation of the female pronucleus from the nucleus of the secondary oocyte
- formation of the male pronucleus from the head of the sperm
- degeneration of the tail of the sperm
- fusion of the two pronuclei and combination of the two sets of chromosomes to form the diploid number
All these reactions occur in the fallopian tube, and once they are complete, the fertilized egg is called a zygote. As the zygote begins to travel down the fallopian tube toward the uterus, a series of mitotic divisions occur that cause the single-celled zygote to become two cells, then 4 cells, then 8, and finally 16 cells, a solid ball of cells called a morula . The zygote does not increase in size during these cellular divisions, so the daughter cells become progressively smaller (Figure 4.6). It takes about 4 days for a fertilized egg to form a morula after fertilization (Khan & Ackerman, 2023).
As the morula floats in the uterus, fluid begins to enter the intracellular space and separate the morula into two distinct parts. Eventually, the morula develops into a blastocyst on about day 5 after fertilization. The inner cell mass contains stem cells, while the outer layer of cells is the trophoblast (Rehman & Muzio, 2023).
Implantation
Implantation occurs most often in the fundus of the uterus around 5 to 6 days after fertilization ( Conception: How it works , n.d.). As the zygote reaches the uterine wall, the trophoblast cells surrounding the blastocyst begin to push away the endometrial cells, allowing the blastocyst to burrow into the lining of the uterus. The trophoblast now begins to secrete enzymes that allow it to become completely buried in the endometrial lining, a process known as implantation . This can cause the blood vessels at the site of implantation to break down slightly, sometimes causing spotting known as implantation bleeding.
Chorionic villi are long, finger-like projections that extend from the trophoblast into the endometrium. They facilitate the exchange of oxygen, nutrients, and waste products through the placenta. The zygote officially becomes an embryo after implantation, from about day 15 until week 8 of pregnancy (Oliver & Basit, 2023).


9.8: Bibliography
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17.4: Preparation of Alcohols- A Review
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Objectives
After completing this section, you should be able to describe in detail the methods of preparing alcohols and diols, which you encountered in previous chapters.
Study Notes
If necessary, you should review Sections 8.4 and 8.5 describing direct hydration of alkenes, and Section 8.7 describing preparation of cis and trans diols from alkenes. Section 8.7 also gives more details on the formation of the respective osmate and epoxide intermediates of these reactions.
Introduction
Alcohols are considered on of the more important functional groups in organic chemistry. They can be prepared from compounds containing a wide assortment of functional groups. Also, the can be used to create compounds with a wide variety of functional groups such as: alkenes, ketones, carboxylic acids, and others. Many functional group conversions can be accomplished through the preparation of an alcohol giving them an important central position in organic synthesis. ex. Alkene → Alcohol → Ketone.
Many methods for the preparation of alcohols have been discussed in previous chapter of this textbook and will be review in this section.
Alcohols from Substitution Reactions
Methyl and primary alkyl halides can be converted to alcohols by using an S N 2 reaction with OH - as a nucleophile ( Section 11.5 ) . Also, secondary and tertiary alkyl halides can be converted to alcohols by an S N 1 reaction using water as the nucleophile (and it can even be the solvent). Recall that S N 1 reactions are promoted in polar, protic solvents ( Section 11.7 ).
The Synthesis of Methanol Using an S N 2 Reaction
Alcohols from Alkenes
Oxymercuration - demercurationis a special electrophilic addition ( Section 8.5 ). It is anti-stereospecific and regioselective. This reaction involves mercury undergoing electrophilic addition to the alkene double bond forming a Mercurinium Ion Bridge . A water molecule then attacks the most substituted carbon to open the mercurium ion bridge, followed by proton transfer to form a hydroxyl group (-OH). The organomercury intermediate is then reduced by sodium borohydride. Notice that overall, the oxymercuration - demercuration mechanism follows Markovnikov's Regioselectivity with the OH group attached to the most substituted carbon and the H is attached to the least substituted carbon. Also, the H and OH species will be anti to each other in the product.
Hydroboration-Oxidation is a two step pathway used to produce alcohols (Section 8.6). It is syn-stereospecific and regioselective. The reaction proceeds in an Anti-Markovnikov manner, where the hydrogen (from BH 3 or BHR 2 ) attaches to the more substituted carbon and the boron attaches to the least substituted carbon in the alkene double bond. The organoborane intermediate is then converted to an alcohol by reaction with hydrogen peroxide (H 2 O 2 ) and sodium hydroxide (NaOH). The hydroboration mechanism has the elements of both hydrogenation and electrophilic addition and it is a stereospecific ( syn addition) , meaning that the addition of the H and OH species takes place on the same face of the double bond leading to their cis configuration in the product. Because the BH 3 can attack either face of the alkene, this reaction can product a racemic mixture of enantiomers in the product.
Diols from alkenes
Epoxides may be cleaved by aqueous acid to give anti-1,2-diols which are also called glycols. Proton transfer from the acid catalyst generates the conjugate acid of the epoxide, which is attacked by a water nucleophile. Because the nucleophilic attach utilizes an S N 2 mechanism the result is an anti-stereospecific configuration of the diol product. The water nucleophile prefers to attack the more substituted carbon on the epoxide which allows for regioselectivity in the reaction.
Osmium tetroxide oxidizes alkenes to give 1,2-diols through syn addition. The reaction with OsO 4 is a concerted process that creates a cyclic intermediate with no rearrangements. The intermediate osmium compound is reduced to the diol product by reduction with H 2 S or NaHSO 4 with H 2 O.
This syn -dihydroxylation complements the epoxide-hydrolysis sequence which creates an anti- dihydroxylation product. This reaction lack regioselectivity so an alkene reacts with osmium tetroxide there is a possibility of a reaction producing a racemic mixture as the product. In general for this reaction, cis alkenes produce a meso product while trans alkenes produce a racemic mixture as the product.
Exercise \(\PageIndex{1}\)
1) Predict the products of the following reactions:
a)
b)
c)
- Answer
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1)
a)
b)
c)


1.6: The Building Blocks of Molecules
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At its most fundamental level, life is made up of matter. Matter occupies space and has mass. All matter is composed of elements, substances that cannot be broken down or transformed chemically into other substances. Each element is made of atoms, each with a constant number of protons and unique properties. A total of 118 elements have been defined; however, only 92 occur naturally, and fewer than 30 are found in living cells. The remaining 26 elements are unstable and, therefore, do not exist for very long or are theoretical and have yet to be detected.
Each element is designated by its chemical symbol (such as H, N, O, C, and Na), and possesses unique properties. These unique properties allow elements to combine and to bond with each other in specific ways.
Atoms
An atom is the smallest component of an element that retains all of the chemical properties of that element. For example, one hydrogen atom has all of the properties of the element hydrogen, such as it exists as a gas at room temperature, and it bonds with oxygen to create a water molecule. Hydrogen atoms cannot be broken down into anything smaller while still retaining the properties of hydrogen. If a hydrogen atom were broken down into subatomic particles, it would no longer have the properties of hydrogen.
At the most basic level, all organisms are made of a combination of elements. They contain atoms that combine together to form molecules. In multicellular organisms, such as animals, molecules can interact to form cells that combine to form tissues, which make up organs. These combinations continue until entire multicellular organisms are formed.
All atoms contain protons, electrons, and neutrons (Figure \(\PageIndex{1}\)). The only exception is hydrogen (H), which is made of one proton and one electron. A proton is a positively charged particle that resides in the nucleus (the core of the atom) of an atom and has a mass of 1 and a charge of +1. An electron is a negatively charged particle that travels in the space around the nucleus. In other words, it resides outside of the nucleus. It has a negligible mass and has a charge of –1.
Figure \(\PageIndex{1}\): Atoms are made up of protons and neutrons located within the nucleus, and electrons surrounding the nucleus.
Neutrons, like protons, reside in the nucleus of an atom. They have a mass of 1 and no charge. The positive (protons) and negative (electrons) charges balance each other in a neutral atom, which has a net zero charge.
Because protons and neutrons each have a mass of 1, the mass of an atom is equal to the number of protons and neutrons of that atom. The number of electrons does not factor into the overall mass, because their mass is so small.
As stated earlier, each element has its own unique properties. Each contains a different number of protons and neutrons, giving it its own atomic number and mass number. The atomic number of an element is equal to the number of protons that element contains. The mass number, or atomic mass, is the number of protons plus the number of neutrons of that element. Therefore, it is possible to determine the number of neutrons by subtracting the atomic number from the mass number.
These numbers provide information about the elements and how they will react when combined. Different elements have different melting and boiling points, and are in different states (liquid, solid, or gas) at room temperature. They also combine in different ways. Some form specific types of bonds, whereas others do not. How they combine is based on the number of electrons present. Because of these characteristics, the elements are arranged into the periodic table of elements, a chart of the elements that includes the atomic number and relative atomic mass of each element. The periodic table also provides key information about the properties of elements ( Figure \(\PageIndex{2}\) )—often indicated by color-coding. The arrangement of the table also shows how the electrons in each element are organized and provides important details about how atoms will react with each other to form molecules.
Isotopes are different forms of the same element that have the same number of protons, but a different number of neutrons. Some elements, such as carbon, potassium, and uranium, have naturally occurring isotopes. Carbon-12, the most common isotope of carbon, contains six protons and six neutrons. Therefore, it has a mass number of 12 (six protons and six neutrons) and an atomic number of 6 (which makes it carbon). Carbon-14 contains six protons and eight neutrons. Therefore, it has a mass number of 14 (six protons and eight neutrons) and an atomic number of 6, meaning it is still the element carbon. These two alternate forms of carbon are isotopes. Some isotopes are unstable and will lose protons, other subatomic particles, or energy to form more stable elements. These are called radioactive isotopes or radioisotopes.
ART CONNECTION
Chemical Bonds
How elements interact with one another depends on how their electrons are arranged and how many openings for electrons exist at the outermost region where electrons are present in an atom. Electrons exist at energy levels that form shells around the nucleus. The closest shell can hold up to two electrons. The closest shell to the nucleus is always filled first, before any other shell can be filled. Hydrogen has one electron; therefore, it has only one spot occupied within the lowest shell. Helium has two electrons; therefore, it can completely fill the lowest shell with its two electrons. If you look at the periodic table, you will see that hydrogen and helium are the only two elements in the first row. This is because they only have electrons in their first shell. Hydrogen and helium are the only two elements that have the lowest shell and no other shells.
The second and third energy levels can hold up to eight electrons. The eight electrons are arranged in four pairs and one position in each pair is filled with an electron before any pairs are completed.
Looking at the periodic table again (Figure \(\PageIndex{2}\)), you will notice that there are seven rows. These rows correspond to the number of shells that the elements within that row have. The elements within a particular row have increasing numbers of electrons as the columns proceed from left to right. Although each element has the same number of shells, not all of the shells are completely filled with electrons. If you look at the second row of the periodic table, you will find lithium (Li), beryllium (Be), boron (B), carbon (C), nitrogen (N), oxygen (O), fluorine (F), and neon (Ne). These all have electrons that occupy only the first and second shells. Lithium has only one electron in its outermost shell, beryllium has two electrons, boron has three, and so on, until the entire shell is filled with eight electrons, as is the case with neon.
Not all elements have enough electrons to fill their outermost shells, but an atom is at its most stable when all of the electron positions in the outermost shell are filled. Because of these vacancies in the outermost shells, we see the formation of chemical bonds, or interactions between two or more of the same or different elements that result in the formation of molecules. To achieve greater stability, atoms will tend to completely fill their outer shells and will bond with other elements to accomplish this goal by sharing electrons, accepting electrons from another atom, or donating electrons to another atom. Because the outermost shells of the elements with low atomic numbers (up to calcium, with atomic number 20) can hold eight electrons, this is referred to as the octet rule. An element can donate, accept, or share electrons with other elements to fill its outer shell and satisfy the octet rule.
When an atom does not contain equal numbers of protons and electrons, it is called an ion. Because the number of electrons does not equal the number of protons, each ion has a net charge. Positive ions are formed by losing electrons and are called cations. Negative ions are formed by gaining electrons and are called anions.
For example, sodium only has one electron in its outermost shell. It takes less energy for sodium to donate that one electron than it does to accept seven more electrons to fill the outer shell. If sodium loses an electron, it now has 11 protons and only 10 electrons, leaving it with an overall charge of +1. It is now called a sodium ion.
The chlorine atom has seven electrons in its outer shell. Again, it is more energy-efficient for chlorine to gain one electron than to lose seven. Therefore, it tends to gain an electron to create an ion with 17 protons and 18 electrons, giving it a net negative (–1) charge. It is now called a chloride ion. This movement of electrons from one element to another is referred to as electron transfer. As Figure \(\PageIndex{4}\) illustrates, a sodium atom (Na) only has one electron in its outermost shell, whereas a chlorine atom (Cl) has seven electrons in its outermost shell. A sodium atom will donate its one electron to empty its shell, and a chlorine atom will accept that electron to fill its shell, becoming chloride. Both ions now satisfy the octet rule and have complete outermost shells. Because the number of electrons is no longer equal to the number of protons, each is now an ion and has a +1 (sodium) or –1 (chloride) charge.
Ionic Bonds
There are four types of bonds or interactions: ionic, covalent, hydrogen bonds, and van der Waals interactions. Ionic and covalent bonds are strong interactions that require a larger energy input to break apart. When an element donates an electron from its outer shell, as in the sodium atom example above, a positive ion is formed. The element accepting the electron is now negatively charged. Because positive and negative charges attract, these ions stay together and form an ionic bond, or a bond between ions. The elements bond together with the electron from one element staying predominantly with the other element. When Na + and Cl – ions combine to produce NaCl, an electron from a sodium atom stays with the other seven from the chlorine atom, and the sodium and chloride ions attract each other in a lattice of ions with a net zero charge.
Covalent Bonds
Another type of strong chemical bond between two or more atoms is a covalent bond. These bonds form when an electron is shared between two elements and are the strongest and most common form of chemical bond in living organisms. Covalent bonds form between the elements that make up the biological molecules in our cells. Unlike ionic bonds, covalent bonds do not dissociate in water.
The hydrogen and oxygen atoms that combine to form water molecules are bound together by covalent bonds. The electron from the hydrogen atom divides its time between the outer shell of the hydrogen atom and the incomplete outer shell of the oxygen atom. To completely fill the outer shell of an oxygen atom, two electrons from two hydrogen atoms are needed, hence the subscript “2” in H 2 O. The electrons are shared between the atoms, dividing their time between them to “fill” the outer shell of each. This sharing is a lower energy state for all of the atoms involved than if they existed without their outer shells filled.
There are two types of covalent bonds: polar and nonpolar. Nonpolar covalent bonds form between two atoms of the same element or between different elements that share the electrons equally. For example, an oxygen atom can bond with another oxygen atom to fill their outer shells. This association is nonpolar because the electrons will be equally distributed between each oxygen atom. Two covalent bonds form between the two oxygen atoms because oxygen requires two shared electrons to fill its outermost shell. Nitrogen atoms will form three covalent bonds (also called triple covalent) between two atoms of nitrogen because each nitrogen atom needs three electrons to fill its outermost shell. Another example of a nonpolar covalent bond is found in the methane (CH 4 ) molecule. The carbon atom has four electrons in its outermost shell and needs four more to fill it. It gets these four from four hydrogen atoms, each atom providing one. These elements all share the electrons equally, creating four nonpolar covalent bonds (Figure \(\PageIndex{5}\)).
In a polar covalent bond, the electrons shared by the atoms spend more time closer to one nucleus than to the other nucleus. Because of the unequal distribution of electrons between the different nuclei, a slightly positive (δ+) or slightly negative (δ–) charge develops. The covalent bonds between hydrogen and oxygen atoms in water are polar covalent bonds. The shared electrons spend more time near the oxygen nucleus, giving it a small negative charge, than they spend near the hydrogen nuclei, giving these molecules a small positive charge.
Hydrogen Bonds
Ionic and covalent bonds are strong bonds that require considerable energy to break. However, not all bonds between elements are ionic or covalent bonds. Weaker bonds can also form. These are attractions that occur between positive and negative charges that do not require much energy to break. Two weak bonds that occur frequently are hydrogen bonds and van der Waals interactions. These bonds give rise to the unique properties of water and the unique structures of DNA and proteins.
When polar covalent bonds containing a hydrogen atom form, the hydrogen atom in that bond has a slightly positive charge. This is because the shared electron is pulled more strongly toward the other element and away from the hydrogen nucleus. Because the hydrogen atom is slightly positive (δ+), it will be attracted to neighboring negative partial charges (δ–). When this happens, a weak interaction occurs between the δ+ charge of the hydrogen atom of one molecule and the δ– charge of the other molecule. This interaction is called a hydrogen bond. This type of bond is common; for example, the liquid nature of water is caused by the hydrogen bonds between water molecules (Figure \(\PageIndex{6}\)). Hydrogen bonds give water the unique properties that sustain life. If it were not for hydrogen bonding, water would be a gas rather than a liquid at room temperature.
Hydrogen bonds can form between different molecules and they do not always have to include a water molecule. Hydrogen atoms in polar bonds within any molecule can form bonds with other adjacent molecules. For example, hydrogen bonds hold together two long strands of DNA to give the DNA molecule its characteristic double-stranded structure. Hydrogen bonds are also responsible for some of the three-dimensional structure of proteins.
van der Waals Interactions
Like hydrogen bonds, van der Waals interactions are weak attractions or interactions between molecules. They occur between polar, covalently bound, atoms in different molecules. Some of these weak attractions are caused by temporary partial charges formed when electrons move around a nucleus. These weak interactions between molecules are important in biological systems.
Matter is anything that occupies space and has mass. It is made up of atoms of different elements. All of the 92 elements that occur naturally have unique qualities that allow them to combine in various ways to create compounds or molecules. Atoms, which consist of protons, neutrons, and electrons, are the smallest units of an element that retain all of the properties of that element. Electrons can be donated or shared between atoms to create bonds, including ionic, covalent, and hydrogen bonds, as well as van der Waals interactions.
Glossary
- anion
- a negative ion formed by gaining electrons
- atomic number
- the number of protons in an atom
- cation
- a positive ion formed by losing electrons
- chemical bond
- an interaction between two or more of the same or different elements that results in the formation of molecules
- covalent bond
- a type of strong bond between two or more of the same or different elements; forms when electrons are shared between elements
- electron
- a negatively charged particle that resides outside of the nucleus in the electron orbital; lacks functional mass and has a charge of –1
- electron transfer
- the movement of electrons from one element to another
- element
- one of 118 unique substances that cannot be broken down into smaller substances and retain the characteristic of that substance; each element has a specified number of protons and unique properties
- hydrogen bond
- a weak bond between partially positively charged hydrogen atoms and partially negatively charged elements or molecules
- ion
- an atom or compound that does not contain equal numbers of protons and electrons, and therefore has a net charge
- ionic bond
- a chemical bond that forms between ions of opposite charges
- isotope
- one or more forms of an element that have different numbers of neutrons
- mass number
- the number of protons plus neutrons in an atom
- matter
- anything that has mass and occupies space
- neutron
- a particle with no charge that resides in the nucleus of an atom; has a mass of 1
- nonpolar covalent bond
- a type of covalent bond that forms between atoms when electrons are shared equally between atoms, resulting in no regions with partial charges as in polar covalent bonds
- nucleus
- (chemistry) the dense center of an atom made up of protons and (except in the case of a hydrogen atom) neutrons
- octet rule
- states that the outermost shell of an element with a low atomic number can hold eight electrons
- periodic table of elements
- an organizational chart of elements, indicating the atomic number and mass number of each element; also provides key information about the properties of elements
- polar covalent bond
- a type of covalent bond in which electrons are pulled toward one atom and away from another, resulting in slightly positive and slightly negative charged regions of the molecule
- proton
- a positively charged particle that resides in the nucleus of an atom; has a mass of 1 and a charge of +1
- radioactive isotope
- an isotope that spontaneously emits particles or energy to form a more stable element
- van der Waals interaction
- a weak attraction or interaction between molecules caused by slightly positively charged or slightly negatively charged atoms
Contributors and Attributions
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Samantha Fowler (Clayton State University), Rebecca Roush (Sandhills Community College), James Wise (Hampton University). Original content by OpenStax (CC BY 4.0; Access for free at https://cnx.org/contents/b3c1e1d2-83...4-e119a8aafbdd ).


26.1: Introduction
Planning a pregnancy and growing a family is typically a joyful, exciting time. Unfortunately, pregnancy can also be a time of sadness, anxiety, and grief. When a loss occurs during the pregnancy, labor, birth, or newborn period, families often deal with feelings of anger, guilt, distress, and emptiness. Families want answers to how and why this could happen. Obstetric and newborn nurses can help by providing emotional and physical support, referrals to counseling and support groups, and assistance with contacting funeral homes. The nurse plays a vital role in helping families to cope and heal during this time of loss. This chapter will cover perinatal loss and bereavement.


9.1: The First Remove
Now away we must go with those barbarous creatures, with our bodies wounded and bleeding, and our hearts no less than our bodies. About a mile we went that night, up upon a hill within sight of the town, where they intended to lodge. There was hard by a vacant house (deserted by the English before, for fear of the Indians). I asked them whether I might not lodge in the house that night, to which they answered, “What, will you love English men still?” This was the dolefulest night that ever my eyes saw.   To add to the dolefulness of the former day, and the dismalness of the present night, my thoughts ran upon my losses and sad bereaved condition.  There remained nothing to me but one poor wounded babe, and it seemed at present worse than death that it was in such a pitiful condition, bespeaking compassion, and I had no refreshing for it, nor suitable things to revive it. Little do many think what is the savageness and brutishness of this barbarous enemy, Ay, even those that seem to profess more than others among them, when the English have fallen into their hands.