21.3.3: Order of Operations
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Learning Outcomes
- Use the order of operations to correctly perform multi-step arithmetic
- Apply the order of operations to statistics related complex questions.
When we are given multiple arithmetic operations within a calculation, there is a, established order that we must do them in based on how the expression is written. Understanding these rules is especially important when using a calculator, since calculators are programmed to strictly follow the order of operations. This comes up in every topic in statistics, so knowing the order of operations is an essential skill for all successful statistics students to have.
PEMDAS
The order of operations are as follows:
- P arentheses
- E xponents
- M ultiplication and D ivision
- A ddition and S ubtraction
When there is a tie, the rule is to go from left to right.
Notice that Multiplication and division are listed together as item 3. If you see multiplication and division in the same expression the rule is to go from left to right. Similarly, if you see addition and subtraction in the same expression the rule is to from go left to right. The same goes for two of the same arithmetic operators.
Example \(\PageIndex{1}\)
Evaluate: \(20-6\div3+\left(2\times3^2\right)\)
Solution
We start with what is inside the parentheses: \(2+3^2\). Since exponents comes before addition, we find \(3^2=9\) first. We now have
\[20-6\div3+\left(2\times9\right) \nonumber\]
We continue inside the parentheses and perform the multiplication: \(2\times9=18\).
This gives
\[20-6\div3+18 \nonumber\]
Since division comes before addition and subtraction, we next calculate \(6\div3=2\) to get
\[20-2+18 \nonumber\]
Since subtraction and addition are tied, we go from left to right. We calculate: \(20-2=18\) to get
\[18+18\:=36 \nonumber\]
The key to arriving at the correct answer is to go slow and write down each step in the arithmetic.
Hidden Parentheses
You may think that since you always have a calculator or computer at hand, that you don't need to worry about order of operations. Unfortunately, the way that expressions are written is not the same as the way that they are entered into a computer or calculator. In particular, exponents need to be treated with care as do fractions bars.
Example \(\PageIndex{3}\)
Evaluate \(2.1^{6-2}\)
Solution
First, note that we use the symbol "^" to tell a computer or calculator to exponentiate. If you were to enter 2.1^6-2 into a computer, it would give you the answer of 83.766121 which is not correct, since the computer will first expontiate and then subtract. Since the subtraction is within the exponent, it must be performed first. To tell a calculator or computer to perform the subtraction first, we use parentheses:
2.1^(6 - 2) = 19.4481
Example \(\PageIndex{4}\): z-scores
The "z-score" is defined by:
\[z=\frac{x-\mu}{\sigma} \nonumber\]
Find the z-score rounded to one decimal place if:
\[x=2.323,\:\mu=1.297,\:\sigma=0.241 \nonumber\]
Solution
Once again, if we put these numbers into the z-score formula and use a computer or calculator by entering \(3.323\:-\:1.297\:\div\:0.241\) we will get -0.259 which is the wrong answer. Instead, we need to know that the fraction bar separates the numerator and the denominator, so the subtraction must be done first. We compute
\[\frac{2.323-1.297}{0.241}\:=\left(2.323-1.297\right)\div0.241=\:4.25726141 \nonumber\]
Now round to one decimal place to get 4.3. Notice that if you rounded before you did the arithmetic, you would get exactly 5 which is very different. 4.3 is more accurate.
Exercise
Suppose the equation of the regression line for the number of pairs of socks a person owns, \(y\), based on the number of pairs of shoes, \(x\), the person owns is
\[\hat y=6+2x \nonumber\]
Use this regression line to predict the number of pairs of socks a person owns for a person who owns 4 pairs of shoes.


10.1: Introduction to Brønsted-Lowry Acids and Bases
Introduction
Liquid water is essential to life on our planet, and chemistry involving the characteristic ions of water, H + and OH – , is widely encountered in nature and society. As introduced in another chapter of this text, acid-base chemistry involves the transfer of hydrogen ions from donors (acids) to acceptors (bases). These H + transfer reactions are reversible, and the equilibria established by acid-base systems are essential aspects of phenomena ranging from sinkhole formation ( Figure \(\PageIndex{1}\) ) to oxygen transport in the human body. This chapter will further explore acid-base chemistry with an emphasis on the equilibrium aspects of this important reaction class.
Introduction to Equilibrium Reactions
Acid base reactions can go in both the forward and the reverse directions. The rate that the reaction goes in either direction depends upon the activation energy - discussed in the chapter on kinteics. For some acid base reactions the forward reaction is much faster, so the reaction ends up with more product. For other acid base reactions the reverse reaction is much faster so the reaction produces little product. Because acid/base reactions are very fast this equlibrium between forward and reverse reactions is reached quickly. For a reaction where:
reactant ⇌ product
The ratio of concentrations of the reactants and products is a constant at equilibrium. This equilibrium constant takes the form of this equation:
\[ K =\dfrac{\left[\mathrm{Products}\right]}{\left[\mathrm{Reactant}\right]} \]
If K is large, the reaction favors the product, and if K is small the reaction favors the reactant.
Brønsted-Lowry acids and Bases
By the end of this section, you will be able to:
- Identify acids, bases, and conjugate acid-base pairs according to the Brønsted-Lowry definition
- Write equations for acid and base ionization reactions
- Use the ion-product constant for water to calculate hydronium and hydroxide ion concentrations
- Describe the acid-base behavior of amphiprotic substances
The acid-base reaction class has been studied for quite some time. In 1680, Robert Boyle reported traits of acid solutions that included their ability to dissolve many substances, to change the colors of certain natural dyes, and to lose these traits after coming in contact with alkali (base) solutions. In the eighteenth century, it was recognized that acids have a sour taste, react with limestone to liberate a gaseous substance (now known to be CO 2 ), and interact with alkalis to form neutral substances. In 1815, Humphry Davy contributed greatly to the development of the modern acid-base concept by demonstrating that hydrogen is the essential constituent of acids. Around that same time, Joseph Louis Gay-Lussac concluded that acids are substances that can neutralize bases and that these two classes of substances can be defined only in terms of each other. The significance of hydrogen was reemphasized in 1884 when Svante Arrhenius defined an acid as a compound that dissolves in water to yield hydrogen cations (now recognized to be hydronium ions) and a base as a compound that dissolves in water to yield hydroxide anions.
Johannes Brønsted and Thomas Lowry proposed a more general description in 1923 in which acids and bases were defined in terms of the transfer of hydrogen ions, H + . (Note that these hydrogen ions are often referred to simply as protons , since that subatomic particle is the only component of cations derived from the most abundant hydrogen isotope, 1 H.) A compound that donates a proton to another compound is called a Brønsted-Lowry acid , and a compound that accepts a proton is called a Brønsted-Lowry base . An acid-base reaction is, thus, the transfer of a proton from a donor (acid) to an acceptor (base).
The concept of conjugate pairs is useful in describing Brønsted-Lowry acid-base reactions (and other reversible reactions, as well). When an acid donates H + , the species that remains is called the conjugate base of the acid because it reacts as a proton acceptor in the reverse reaction. Likewise, when a base accepts H + , it is converted to its conjugate acid . The reaction between water and ammonia illustrates this idea. In the forward direction, water acts as an acid by donating a proton to ammonia and subsequently becoming a hydroxide ion, OH − , the conjugate base of water. The ammonia acts as a base in accepting this proton, becoming an ammonium ion, the conjugate acid of ammonia. In the reverse direction, a hydroxide ion acts as a base in accepting a proton from ammonium ion, which acts as an acid.
The reaction between a Brønsted-Lowry acid and water is called acid ionization . For example, when hydrogen fluoride dissolves in water and ionizes, protons are transferred from hydrogen fluoride molecules to water molecules, yielding hydronium ions, H 3 O + , and fluoride ions:
Base ionization of a species occurs when it accepts protons from water molecules. In the example below, pyridine molecules, C 5 NH 5 , undergo base ionization when dissolved in water, yielding hydroxide and pyridinium ions:
The preceding ionization reactions suggest that water may function as both a base (as in its reaction with hydrogen fluoride) and an acid (as in its reaction with ammonia or pyridine). Species capable of either donating or accepting protons are called amphiprotic , or more generally, amphoteric , a term that may be used for acids and bases per definitions other than the Brønsted-Lowry one.
In the liquid state, molecules of an amphiprotic substance can react with one another as illustrated for water in the equations below:
The process in which like molecules react to yield ions is called autoionization . Liquid water undergoes autoionization to a very slight extent; at 25 °C, approximately two out of every billion water molecules are ionized. The extent of the water autoionization process is reflected in the value of its equilibrium constant, the ion-product constant for water, K w :
Notice that the concentration of water is not included in the equlibrium constant, since the concentration of water is so great that it does not change. The slight ionization of pure water is reflected in the small value of the equilibrium constant; at 25 °C, K w has a value of 1.0 10 −14 . The process is endothermic, and so the extent of ionization and the resulting concentrations of hydronium ion and hydroxide ion increase with temperature. For example, at 100 °C, the value for K w is about 5.6 10 −13 , roughly 50 times larger than the value at 25 °C.
Example 14.1
Ion Concentrations in Pure Water
What are the hydronium ion concentration and the hydroxide ion concentration in pure water at 25 °C?Solution
The autoionization of water yields the same number of hydronium and hydroxide ions. Therefore, in pure water, [H 3 O + ] = [OH − ] = x . At 25 °C:So:
The hydronium ion concentration and the hydroxide ion concentration are the same, 1.0 10 −7 M .
Check Your Learning
The ion product of water at 80 °C is 2.4 10 −13 . What are the concentrations of hydronium and hydroxide ions in pure water at 80 °C?Answer:
[H 3 O + ] = [OH − ] = 4.9 10 −7 M
Example 14.2
The Inverse Relation between [H 3 O + ] and [OH − ]
A solution of an acid in water has a hydronium ion concentration of 2.0 10 −6 M . What is the concentration of hydroxide ion at 25 °C?Solution
Use the value of the ion-product constant for water at 25 °Cto calculate the missing equilibrium concentration.
Rearrangement of the K w expression shows that [OH − ] is inversely proportional to [H 3 O + ]:
Compared with pure water, a solution of acid exhibits a higher concentration of hydronium ions (due to ionization of the acid) and a proportionally lower concentration of hydroxide ions. This may be explained via Le Chatelier’s principle as a left shift in the water autoionization equilibrium resulting from the stress of increased hydronium ion concentration.
Substituting the ion concentrations into the K w expression confirms this calculation, resulting in the expected value:
Check Your Learning
What is the hydronium ion concentration in an aqueous solution with a hydroxide ion concentration of 0.001 M at 25 °C?Answer:
[H 3 O + ] = 1 10 −11 M


11.1: Introduction to Apply Knowledge of Body Systems to Client Care
Learning Objectives
- Apply basic knowledge of body systems to specialize care for chronic conditions
- Promote healthy outcomes for individuals with compromised bodily functions
- Identify signs and symptoms that should be reported to the supervising nurse
- Provide interventions for chronic conditions within the scope of practice for nursing assistants
In this chapter you will learn the basic structure and functions of body systems and apply this knowledge to the individuals for whom you provide care. Age-related changes and common chronic conditions in each body system are outlined with related nursing assistant interventions to promote optimal patient outcomes. Observations related to each condition are discussed, as well as what should be immediately reported to the nurse.


4.3: Connective Tissue Supports and Protects
Learning Objectives
- Identify and distinguish between the types of connective tissue: proper, supportive, and fluid
- Explain the functions of connective tissues
As may be obvious from its name, one of the major functions of connective tissue is to connect tissues and organs. Unlike epithelial tissue, which is composed of cells closely packed with little or no extracellular space in between, connective tissue cells are dispersed in a matrix . The matrix usually includes a large amount of extracellular material produced by the connective tissue cells that are embedded within it. The matrix plays a major role in the functioning of this tissue. The major component of the matrix is a ground substance often crisscrossed by protein fibers. This ground substance is usually a fluid, but it can also be mineralized and solid, as in bones. Connective tissues come in a vast variety of forms, yet they typically have in common three characteristic components: cells, large amounts of amorphous ground substance, and protein fibers. The amount and structure of each component correlates with the function of the tissue, from the rigid ground substance in bones supporting the body to the inclusion of specialized cells; for example, a phagocytic cell that engulfs pathogens and also rids tissue of cellular debris.
Functions of Connective Tissues
Connective tissues perform many functions in the body, but most importantly, they support and connect other tissues; from the connective tissue sheath that surrounds muscle cells, to the tendons that attach muscles to bones, and to the skeleton that supports the positions of the body. Protection is another major function of connective tissue, in the form of fibrous capsules and bones that protect delicate organs and, of course, the skeletal system. Specialized cells in connective tissue defend the body from microorganisms that enter the body. Transport of fluid, nutrients, waste, and chemical messengers is ensured by specialized fluid connective tissues, such as blood and lymph. Adipose cells store surplus energy in the form of fat and contribute to the thermal insulation of the body.
Embryonic Connective Tissue
All connective tissues derive from the mesodermal layer of the embryo (see [link]). The first connective tissue to develop in the embryo is mesenchyme , the stem cell line from which all connective tissues are later derived. Clusters of mesenchymal cells are scattered throughout adult tissue and supply the cells needed for replacement and repair after a connective tissue injury. A second type of embryonic connective tissue forms in the umbilical cord, called mucous connective tissue or Wharton’s jelly. This tissue is no longer present after birth, leaving only scattered mesenchymal cells throughout the body.
Classification of Connective Tissues
The three broad categories of connective tissue are classified according to the characteristics of their ground substance and the types of fibers found within the matrix (Table \(\PageIndex{1}\)). Connective tissue proper includes loose connective tissue and dense connective tissue . Both tissues have a variety of cell types and protein fibers suspended in a viscous ground substance. Dense connective tissue is reinforced by bundles of fibers that provide tensile strength, elasticity, and protection. In loose connective tissue, the fibers are loosely organized, leaving large spaces in between. Supportive connective tissue —bone and cartilage—provide structure and strength to the body and protect soft tissues. A few distinct cell types and densely packed fibers in a matrix characterize these tissues. In bone, the matrix is rigid and described as calcified because of the deposited calcium salts. In fluid connective tissue , in other words, lymph and blood, various specialized cells circulate in a watery fluid containing salts, nutrients, and dissolved proteins.
Table \(\PageIndex{1}\)
| Connective Tissue Examples | ||
|---|---|---|
| Connective tissue proper | Supportive connective tissue | Fluid connective tissue |
|
Loose connective tissue
|
Cartilage
|
Blood |
|
Dense connective tissue
|
Bones
|
Lymph |
Connective Tissue Proper
Fibroblasts are present in all connective tissue proper (Figure \(\PageIndex{1}\)). Fibrocytes, adipocytes, and mesenchymal cells are fixed cells, which means they remain within the connective tissue. Other cells move in and out of the connective tissue in response to chemical signals. Macrophages, mast cells, lymphocytes, plasma cells, and phagocytic cells are found in connective tissue proper but are actually part of the immune system protecting the body.
Cell Types
The most abundant cell in connective tissue proper is the fibroblast . Polysaccharides and proteins secreted by fibroblasts combine with extra-cellular fluids to produce a viscous ground substance that, with embedded fibrous proteins, forms the extra-cellular matrix. As you might expect, a fibrocyte , a less active form of fibroblast, is the second most common cell type in connective tissue proper.
Adipocytes are cells that store lipids as droplets that fill most of the cytoplasm. There are two basic types of adipocytes: white and brown. The brown adipocytes store lipids as many droplets, and have high metabolic activity.  Their effectiveness at storing large amounts of fat is witnessed in obese individuals. The number and type of adipocytes depends on the tissue and location, and vary among individuals in the population.
The mesenchymal cell is a multipotent adult stem cell. These cells can differentiate into any type of connective tissue cells needed for repair and healing of damaged tissue.
The macrophage cell is a large cell derived from a monocyte, a type of blood cell, which enters the connective tissue matrix from the blood vessels. The macrophage cells are an essential component of the immune system, which is the body’s defense against potential pathogens and degraded host cells. When stimulated, macrophages release cytokines, small proteins that act as chemical messengers. Cytokines recruit other cells of the immune system to infected sites and stimulate their activities. Roaming, or free, macrophages move rapidly by amoeboid movement, engulfing infectious agents and cellular debris. In contrast, fixed macrophages are permanent residents of their tissues.
The mast cell, found in connective tissue proper, has many cytoplasmic granules. These granules contain the chemical signals histamine and heparin. When irritated or damaged, mast cells release histamine, an inflammatory mediator, which causes vasodilation and increased blood flow at a site of injury or infection, along with itching, swelling, and redness you recognize as an allergic response. Like blood cells, mast cells are derived from hematopoietic stem cells and are part of the immune system.
Connective Tissue Fibers and Ground Substance
Three main types of fibers are secreted by fibroblasts: collagen fibers, elastic fibers, and reticular fibers. Collagen fiber is made from fibrous protein subunits linked together to form a long and straight fiber. Collagen fibers, while flexible, have great tensile strength, resist stretching, and give ligaments and tendons their characteristic resilience and strength. These fibers hold connective tissues together, even during the movement of the body.
Elastic fiber contains the protein elastin along with lesser amounts of other proteins and glycoproteins. The main property of elastin is that after being stretched or compressed, it will return to its original shape. Elastic fibers are prominent in elastic tissues found in skin and the elastic ligaments of the vertebral column.
Reticular fiber is also formed from the same protein subunits as collagen fibers; however, these fibers remain narrow and are arrayed in a branching network. They are found throughout the body, but are most abundant in the reticular tissue of soft organs, such as liver and spleen, where they anchor and provide structural support to the parenchyma (the functional cells, blood vessels, and nerves of the organ).
All of these fiber types are embedded in ground substance. Secreted by fibroblasts, ground substance is made of polysaccharides, specifically hyaluronic acid, and proteins. These combine to form a proteoglycan with a protein core and polysaccharide branches. The proteoglycan attracts and traps available moisture forming the clear, viscous, colorless matrix you now know as ground substance.
Loose Connective Tissue
Loose connective tissue is found between many organs where it acts both to absorb shock and bind tissues together. It allows water, salts, and various nutrients to diffuse through to adjacent or imbedded cells and tissues.
Adipose tissue consists mostly of fat storage cells, with little extracellular matrix (Figure \(\PageIndex{2}\)). A large number of capillaries allow rapid storage and mobilization of lipid molecules. White adipose tissue is most abundant. It can appear yellow and owes its color to carotene and related pigments from plant food. White fat contributes mostly to lipid storage and can serve as insulation from cold temperatures and mechanical injuries. White adipose tissue can be found protecting the kidneys and cushioning the back of the eye. Brown adipose tissue is more common in infants, hence the term “baby fat.” In adults, there is a reduced amount of brown fat and it is found mainly in the neck and clavicular regions of the body. The many mitochondria in the cytoplasm of brown adipose tissue help explain its efficiency at metabolizing stored fat. Brown adipose tissue is thermogenic, meaning that as it breaks down fats, it releases metabolic heat, rather than producing adenosine triphosphate (ATP), a key molecule used in metabolism.
Areolar tissue shows little specialization. It contains all the cell types and fibers previously described and is distributed in a random, web-like fashion. It fills the spaces between muscle fibers, surrounds blood and lymph vessels, and supports organs in the abdominal cavity. Areolar tissue underlies most epithelia and represents the connective tissue component of epithelial membranes, which are described further in a later section.
Reticular tissue is a mesh-like, supportive framework for soft organs such as lymphatic tissue, the spleen, and the liver (Figure \(\PageIndex{3}\)). Reticular cells produce the reticular fibers that form the network onto which other cells attach. It derives its name from the Latin reticulus , which means “little net.”
Dense Connective Tissue
Dense connective tissue contains more collagen fibers than does loose connective tissue. As a consequence, it displays greater resistance to stretching. There are two major categories of dense connective tissue: regular and irregular. Dense regular connective tissue fibers are parallel to each other, enhancing tensile strength and resistance to stretching in the direction of the fiber orientations. Ligaments and tendons are made of dense regular connective tissue, but in ligaments not all fibers are parallel. Dense regular elastic tissue contains elastin fibers in addition to collagen fibers, which allows the ligament to return to its original length after stretching. The ligaments in the vocal folds and between the vertebrae in the vertebral column are elastic.
In dense irregular connective tissue, the direction of fibers is random. This arrangement gives the tissue greater strength in all directions and less strength in one particular direction. In some tissues, fibers crisscross and form a mesh. In other tissues, stretching in several directions is achieved by alternating layers where fibers run in the same orientation in each layer, and it is the layers themselves that are stacked at an angle. The dermis of the skin is an example of dense irregular connective tissue rich in collagen fibers. Dense irregular elastic tissues give arterial walls the strength and the ability to regain original shape after stretching (Figure \(\PageIndex{4}\)).
DISORDERS OF THE...
  That dull ache in the wrist that you ignored through the summer is now an unbearable pain. The game is over for now.
After examining your swollen wrist, the doctor in the emergency room announces that you have developed wrist tendinitis. She recommends icing the tender area, taking non-steroidal anti-inflammatory medication to ease the pain and to reduce swelling, and complete rest for a few weeks. She interrupts your protests that you cannot stop playing. She issues a stern warning about the risk of aggravating the condition and the possibility of surgery. She consoles you by mentioning that well known tennis players such as Venus and Serena Williams and Rafael Nadal have also suffered from tendinitis related injuries.
What is tendinitis and how did it happen? Tendinitis is the inflammation of a tendon, the thick band of fibrous connective tissue that attaches a muscle to a bone. The condition causes pain and tenderness in the area around a joint. On rare occasions, a sudden serious injury will cause tendinitis. Most often, the condition results from repetitive motions over time that strain the tendons needed to perform the tasks.
Persons whose jobs and hobbies involve performing the same movements over and over again are often at the greatest risk of tendinitis. You hear of tennis and golfer’s elbow, jumper's knee, and swimmer’s shoulder. In all cases, overuse of the joint causes a microtrauma that initiates the inflammatory response. Tendinitis is routinely diagnosed through a clinical examination. In case of severe pain, X-rays can be examined to rule out the possibility of a bone injury. Severe cases of tendinitis can even tear loose a tendon. Surgical repair of a tendon is painful. Connective tissue in the tendon does not have abundant blood supply and heals slowly.
While older adults are at risk for tendinitis because the elasticity of tendon tissue decreases with age, active people of all ages can develop tendinitis. Young athletes, dancers, and computer operators; anyone who performs the same movements constantly is at risk for tendinitis. Although repetitive motions are unavoidable in many activities and may lead to tendinitis, precautions can be taken that can lessen the probability of developing tendinitis. For active individuals, stretches before exercising and cross training or changing exercises are recommended. For the passionate athlete, it may be time to take some lessons to improve technique. All of the preventive measures aim to increase the strength of the tendon and decrease the stress put on it. With proper rest and managed care, you will be back on the court to hit that slice-spin serve over the net.
Watch this animation to learn more about tendonitis, a painful condition caused by swollen or injured tendons.
Supportive Connective Tissues
Two major forms of supportive connective tissue, cartilage and bone, allow the body to maintain its posture and protect internal organs.
Cartilage
The distinctive appearance of cartilage is due to polysaccharides called chondroitin sulfates, which bind with ground substance proteins to form proteoglycans. Embedded within the cartilage matrix are chondrocytes , or cartilage cells, and the space they occupy are called lacunae (singular = lacuna). A layer of dense irregular connective tissue, the perichondrium, encapsulates the cartilage. Cartilaginous tissue is avascular, thus all nutrients need to diffuse through the matrix to reach the chondrocytes. This is a factor contributing to the very slow healing of cartilaginous tissues.
The three main types of cartilage tissue are hyaline cartilage, fibrocartilage, and elastic cartilage (Figure \(\PageIndex{5}\)). Hyaline cartilage , the most common type of cartilage in the body, consists of short and dispersed collagen fibers and contains large amounts of proteoglycans. Under the microscope, tissue samples appear clear. The surface of hyaline cartilage is smooth. Both strong and flexible, it is found in the rib cage and nose and covers bones where they meet to form moveable joints. It makes up a template of the embryonic skeleton before bone formation. A plate of hyaline cartilage at the ends of bone allows continued growth until adulthood. Fibrocartilage is tough because it has thick bundles of collagen fibers dispersed through its matrix. The knee and jaw joints and the the intervertebral discs are examples of fibrocartilage. Elastic cartilage contains elastic fibers as well as collagen and proteoglycans. This tissue gives rigid support as well as elasticity. Tug gently at your ear lobes, and notice that the lobes return to their initial shape. The external ear contains elastic cartilage.
Bone
Bone is the hardest connective tissue. It provides protection to internal organs and supports the body. Bone’s rigid extracellular matrix contains mostly collagen fibers embedded in a mineralized ground substance containing hydroxyapatite, a form of calcium phosphate. Both components of the matrix, organic and inorganic, contribute to the unusual properties of bone. Without collagen, bones would be brittle and shatter easily. Without mineral crystals, bones would flex and provide little support. Osteocytes, bone cells like chondrocytes, are located within lacunae. The histology of transverse tissue from long bone shows a typical arrangement of osteocytes in concentric circles around a central canal. Bone is a highly vascularized tissue. Unlike cartilage, bone tissue can recover from injuries in a relatively short time.
Cancellous bone looks like a sponge under the microscope and contains empty spaces between trabeculae, or arches of bone proper. It is lighter than compact bone and found in the interior of some bones and at the end of long bones. Compact bone is solid and has greater structural strength.
Fluid Connective Tissue
Blood and lymph are fluid connective tissues. Cells circulate in a liquid extracellular matrix. The formed elements circulating in blood are all derived from hematopoietic stem cells located in bone marrow (Figure \(\PageIndex{6}\)). Erythrocytes, red blood cells, transport oxygen and some carbon dioxide. Leukocytes, white blood cells, are responsible for defending against potentially harmful microorganisms or molecules. Platelets are cell fragments involved in blood clotting. Some white blood cells have the ability to cross the endothelial layer that lines blood vessels and enter adjacent tissues. Nutrients, salts, and wastes are dissolved in the liquid matrix and transported through the body.
Lymph contains a liquid matrix and white blood cells. Lymphatic capillaries are extremely permeable, allowing larger molecules and excess fluid from interstitial spaces to enter the lymphatic vessels. Lymph drains into blood vessels, delivering molecules to the blood that could not otherwise directly enter the bloodstream. In this way, specialized lymphatic capillaries transport absorbed fats away from the intestine and deliver these molecules to the blood.
View the University of Michigan Webscope at virtualslides.med.umich.edu/H....svs/view.apml to explore the tissue sample in greater detail.
Visit this link to test your connective tissue knowledge with this 10-question quiz. Can you name the 10 tissue typesshown in the histology slides?
Chapter Review
Connective tissue is a heterogeneous tissue with many cell shapes and tissue architecture. Structurally, all connective tissues contain cells that are embedded in an extracellular matrix stabilized by proteins. The chemical nature and physical layout of the extracellular matrix and proteins vary enormously among tissues, reflecting the variety of functions that connective tissue fulfills in the body. Connective tissues separate and cushion organs, protecting them from shifting or traumatic injury. Connect tissues provide support and assist movement, store and transport energy molecules, protect against infections, and contribute to temperature homeostasis.
Many different cells contribute to the formation of connective tissues. They originate in the mesodermal germ layer and differentiate from mesenchyme and hematopoietic tissue in the bone marrow.  The extracellular matrix contains fluid, proteins, polysaccharide derivatives, and, in the case of bone, mineral crystals. Protein fibers fall into three major groups: collagen fibers that are thick, strong, flexible, and resist stretch; reticular fibers that are thin and form a supportive mesh; and elastin fibers that are thin and elastic.
The major types of connective tissue are connective tissue proper, supportive tissue, and fluid tissue. Loose connective tissue proper includes adipose tissue, areolar tissue, and reticular tissue. These serve to hold organs and other tissues in place and, in the case of adipose tissue, isolate and store energy reserves. The matrix is the most abundant feature for loose tissue although adipose tissue does not have much extracellular matrix. Dense connective tissue proper is richer in fibers and may be regular, with fibers oriented in parallel as in ligaments and tendons, or irregular, with fibers oriented in several directions. Organ capsules (collagenous type) and walls of arteries (elastic type) contain dense irregular connective tissue. Cartilage and bone are supportive tissue. Cartilage contains chondrocytes and is somewhat flexible. Hyaline cartilage is smooth and clear, covers joints, and is found in the growing portion of bones. Fibrocartilage is tough because of extra collagen fibers and forms, among other things, the intervertebral discs. Elastic cartilage can stretch and recoil to its original shape because of its high content of elastic fibers. The matrix contains very few blood vessels. Bones are made of a rigid, mineralized matrix containing calcium salts, crystals, and osteocytes lodged in lacunae. Bone tissue is highly vascularized. Cancellous bone is spongy and less solid than compact bone. Fluid tissue, for example blood and lymph, is characterized by a liquid matrix and no supporting fibers.
Interactive Link Questions
Visit this link to test your connective tissue knowledge with this 10-question quiz. Can you name the 10 tissue types shown in the histology slides?
Answer: Click at the bottom of the quiz for the answers.
Review Questions
Q. Connective tissue is made of which three essential components?
A. cells, ground substance, and carbohydrate fibers
B. cells, ground substance, and protein fibers
C. collagen, ground substance, and protein fibers
D. matrix, ground substance, and fluid
Answer: B
Q. Under the microscope, a tissue specimen shows cells located in spaces scattered in a transparent background. This is probably ________.
A. loose connective tissue
B. a tendon
C. bone
D. hyaline cartilage
Answer: D
Q. Which connective tissue specializes in storage of fat?
A. tendon
B. adipose tissue
C. reticular tissue
D. dense connective tissue
Answer: B
Q. Ligaments connect bones together and withstand a lot of stress. What type of connective tissue should you expect ligaments to contain?
A. areolar tissue
B. adipose tissue
C. dense regular connective tissue
D. dense irregular connective tissue
Answer: C
Q. In adults, new connective tissue cells originate from the ________.
A. mesoderm
B. mesenchyme
C. ectoderm
D. endoderm
Answer: B
Q. In bone, the main cells are ________.
A. fibroblasts
B. chondrocytes
C. lymphocytes
D. osteocytes
Answer: D
Critical Thinking Questions
Q. One of the main functions of connective tissue is to integrate organs and organ systems in the body. Discuss how blood fulfills this role.
A. Blood is a fluid connective tissue, a variety of specialized cells that circulate in a watery fluid containing salts, nutrients, and dissolved proteins in a liquid extracellular matrix. Blood contains formed elements derived from bone marrow. Erythrocytes, or red blood cells, transport the gases oxygen and carbon dioxide. Leukocytes, or white blood cells, are responsible for the defense of the organism against potentially harmful microorganisms or molecules. Platelets are cell fragments involved in blood clotting. Some cells have the ability to cross the endothelial layer that lines vessels and enter adjacent tissues. Nutrients, salts, and waste are dissolved in the liquid matrix and transported through the body.
Q. Why does an injury to cartilage, especially hyaline cartilage, heal much more slowly than a bone fracture?
A. A layer of dense irregular connective tissue covers cartilage. No blood vessels supply cartilage tissue. Injuries to cartilage heal very slowly because cells and nutrients needed for repair diffuse slowly to the injury site.
Glossary
- adipocytes
- lipid storage cells
- adipose tissue
- specialized areolar tissue rich in stored fat
- areolar tissue
- (also, loose connective tissue) a type of connective tissue proper that shows little specialization with cells dispersed in the matrix
- chondrocytes
- cells of the cartilage
- collagen fiber
- flexible fibrous proteins that give connective tissue tensile strength
- connective tissue proper
- connective tissue containing a viscous matrix, fibers, and cells.
- dense connective tissue
- connective tissue proper that contains many fibers that provide both elasticity and protection
- elastic cartilage
- type of cartilage, with elastin as the major protein, characterized by rigid support as well as elasticity
- elastic fiber
- fibrous protein within connective tissue that contains a high percentage of the protein elastin that allows the fibers to stretch and return to original size
- fibroblast
- most abundant cell type in connective tissue, secretes protein fibers and matrix into the extracellular space
- fibrocartilage
- tough form of cartilage, made of thick bundles of collagen fibers embedded in chondroitin sulfate ground substance
- fibrocyte
- less active form of fibroblast
- fluid connective tissue
- specialized cells that circulate in a watery fluid containing salts, nutrients, and dissolved proteins
- ground substance
- fluid or semi-fluid portion of the matrix
- hyaline cartilage
- most common type of cartilage, smooth and made of short collagen fibers embedded in a chondroitin sulfate ground substance
- lacunae
- (singular = lacuna) small spaces in bone or cartilage tissue that cells occupy
- loose connective tissue
- (also, areolar tissue) type of connective tissue proper that shows little specialization with cells dispersed in the matrix
- matrix
- extracellular material which is produced by the cells embedded in it, containing ground substance and fibers
- mesenchymal cell
- adult stem cell from which most connective tissue cells are derived
- mesenchyme
- embryonic tissue from which connective tissue cells derive
- mucous connective tissue
- specialized loose connective tissue present in the umbilical cord
- parenchyma
- functional cells of a gland or organ, in contrast with the supportive or connective tissue of a gland or organ
- reticular fiber
- fine fibrous protein, made of collagen subunits, which cross-link to form supporting “nets” within connective tissue
- reticular tissue
- type of loose connective tissue that provides a supportive framework to soft organs, such as lymphatic tissue, spleen, and the liver
- supportive connective tissue
- type of connective tissue that provides strength to the body and protects soft tissue


2.6: Examples of Groups/Nongroups, Part II
Example \(\PageIndex{1}\)
Let \(n\in \mathbb{Z}^+\text{.}\) We define \(n\mathbb{Z}\) by
\begin{equation*} n\mathbb{Z}=\{nx: x\in \mathbb{Z}\}: \end{equation*}
that is, \(n\mathbb{Z}\) is the set of all (integer) multiples of \(n\text{.}\)
Theorem \(\PageIndex{1}\)
\(n\mathbb{Z}\) is a group under \(+\) (the usual addition of integers).
- Proof
-
Let \(x,y∈n\mathbb{Z}\). Then there exist \(a,b∈\mathbb{Z}\) such that \(x=na\) and \(y=nb\). Then \(x+y=na+nb=n(a+b)∈n\mathbb{Z}\). So \(\langle n\mathbb{Z},+ \rangle\) \(\langle n\mathbb{Z},+ \rangle\) is a binary structure. The remainder of the proof is left as an exercise for the reader.
Remark
When we are discussing a group \(n\mathbb{Z}\text{,}\) assume that \(n\in \mathbb{Z}^+\text{,}\) unless otherwise noted.
We use an example from our next class of groups all the time; in fact, most six-year-olds do as well, since it is used when telling time! Before we get to the example, we need some more definitions and some notation. Throughout the following discussion, assume \(n\) is a fixed positive integer.
Definition: Congruent modulo (mod)
We say integers \(a\) and \(b\) are congruent modulo [or mod ] \(n\) if \(n\) divides \(a-b\text{.}\) If \(a\) and \(b\) are congruent mod \(n\text{,}\) we write \(a \equiv_n b\text{.}\)
Example \(\PageIndex{2}\)
\(1, 7, 13,\) and \(-5\) are all congruent mod \(6\text{.}\)
The following is a profoundly useful theorem; it's so important, it has a special name. We omit the proof of this theorem, but direct interested readers to for, instance, p. 5 in [3].
Theorem \(\PageIndex{2}\): Division Algorithm
Let \(n\in \mathbb{Z}^+\) and let \(a\) be any integer. Then there exist unique integers \(q\) and \(r\text{,}\) with \(0\leq r \lt n\text{,}\) such that \(a=qn+r\text{.}\)
(This is actually a special case of a more general theorem, which states that given any integers \(n\) and \(a\text{,}\) there exist unique integers \(q\) and \(r\text{,}\) with \(0\leq r\lt |n|\text{,}\) such that \(a=qn+r\text{.}\))
It follows that for each positive integer \(n\) and integer \(a\text{,}\) there exists a unique element \(R_n(a)\) (the \(r\) in the above theorem) of the set \(\{0,1,2,\ldots, n-1\}\) such that \(a\) is congruent to \(R_n(a)\) modulo \(n\text{.}\) For example, \(R_3(4)=1\text{,}\) \(R_3(0)=0\text{,}\) \(R_3(17)=2\text{,}\) and \(R_3(-5)=1\text{.}\)
Definition: Remainder
\(R_n(a)\) is the remainder when we divide \(a\) by \(n\text{.}\)
(Note: You were probably already familiar with the remainder when you divide a positive integer by \(n\text{.}\))
Definition: Addition modulo
We define addition modulo \(n\), \(+_n\text{,}\) on \(\mathbb{Z}\) by, for all \(a,b\in \mathbb{Z}\text{,}\)
\begin{equation*} a+_n b=R_n(a+b), \end{equation*}
that is, the unique element of \(\{0,1,\ldots, n-1\}\) that's congruent to the integer \(a+b\) modulo \(n\text{.}\)
Remark
Addition mod \(24\) is what we use to tell time!
The set \(\{0,1,2,\ldots, n-1\}\) of remainders when dividing by \(n\) is so important we give it a special notation.
Definition
We define \(\mathbb{Z}_n\) to be the set \(\{0,1,2,\ldots,n-1\}\text{.}\)
Note
Note that by our definition of \(\mathbb{Z}_n\text{,}\) the integer \(n\) itself is not in \(\mathbb{Z}_n\text{!}\)
We are now ready to consider our next type of group.
Example \(\PageIndex{3}\)
For each \(n\in \mathbb{Z}^+\text{,}\) \(\langle \mathbb{Z}_n,+_n\rangle\) is a group, called the cyclic group of order \(n\) (we will see later why we use the word “cyclic” here). This group is abelian and of order \(n\text{.}\)
Proof
We first check that \(\langle \mathbb{Z}n,+_n\rangle\) is a binary structure. Note that by the definition of \(+_n\), \(a+_nb∈\mathbb{Z}_n\) for each \(a,b∈Z\). Thus, \(a+_nb∈\mathbb{Z}_n\) for each \(a,b∈\mathbb{Z}_n\).
We next check that \(\mathbb{Z}_n\) under \(+_n\) satisfies the three group axioms. Note that since addition is commutative on \(\mathbb{Z}\),
\(a+_nb=R_n(a+b)=R_n(b+a)=b+_na\)
for all \(a,b∈\mathbb{Z}_n\). Again, a simpler way of stating this is that commutativity of \(+_n\) on \(\mathbb{Z}_n\) is inherited from the commutativity of addition on \(\mathbb{Z}\). One nice result of this is that since \(+_n\) is commutative on \(\mathbb{Z}_n\), we have less to check when verifying group axioms \(\mathcal{G}_2\) and \(\mathcal{G}_3\).
Now, let \(a,b,c∈\mathbb{Z}_n\). We want to show that \((a+_nb)+_nc=a+_n(b+_nc)\). Now,
\(\begin{array}& (a+_nb)+_nc &=R_n(a+b)+_nc \\ &≡_nR_n(a+b)+c \\ &≡_n(a+b)+c \\ &≡_na+(b+c)\\ &≡_na+R_n(b+c)\\ &≡_na+_n(b+_nc).\end{array} \)
So \((a+_nb)+_nc\) and \(a+_n(b+_nc)\) are congruent mod \(n\). Since both of these values are in \(\{0,1,…,n−1\}\), this implies that they are equal, as desired.
Next, clearly, \(0∈\mathbb{Z}_n\) acts as an identity element under \(+_n\), since
\(0+_na=R_n(0+a)=R_n(a)\)
for each \(a∈\mathbb{Z}_n\).
Finally, let \(a∈\mathbb{Z}_n\). If \(a=0\), then clearly aa has inverse \(0∈\mathbb{Z}_n\) since \(0+n_0=0\). If \(a≠0\), then the element \(n−a∈\mathbb{Z}_n\) is an inverse for aa since
\(a+_n(n−a)=R_n(a+(n−a))=R_n(n)=0.\)
Since \(+_n\) is commutative on \(\mathbb{Z}_n\), \(\mathbb{Z}_n\) is an abelian group under \(+_n\).
Finally, we already know that \(|\mathbb{Z}_n|=|\{0,1,2,…,n−1\}|=n\).
Remark
In practice, we often omit the subscript \(n\) and just write \(+\) when discussing addition modulo \(n\) on \(\mathbb{Z}_n\text{.}\)
Note
Do not confuse \(n\mathbb{Z}\) and \(\mathbb{Z}_n\text{!}\) They are very different as sets and as groups.
Example \(\PageIndex{4}\)
In the group \(\langle \mathbb{Z}_8,+\rangle\) (where, as indicated by our above remark, \(+\) means addition modulo \(8\)), we have, for instance, \(3+7=2\) and \(7+7=6\text{.}\) The numbers 2 and 6 are each other's inverse, and \(7^{-1}=1\text{.}\) The number \(0\) has inverse \(0\) (it can't be \(8\text{,}\) since \(8\not\in \mathbb{Z}_8\text{!}\)).
Definition: Multiplication modulo
For \(n\in \mathbb{Z}^+\text{,}\) we define multiplication modulo \(n\), denoted \(\cdot_n\text{,}\) on \(\mathbb{Z}_n\) by \(a\cdot_n b = R_n(ab)\text{,}\) the remainder when \(ab\) is divided by \(n\text{.}\)
Remark
\(\mathbb{Z}_n\) is never a group under \(\cdot_n\) (do you see why?).
But we can consider the following
Definition
For \(n\in \mathbb{Z}^+\text{,}\) we define \(\mathbb{Z}_n^{\times}\) to be the set
\begin{equation*} \{a\in \mathbb{Z}_n\,:\,\gcd(a,n)=1\}\text{.} \end{equation*}
Example \(\PageIndex{5}\)
\(\langle \mathbb{Z}_n^{\times},\,\cdot_n\, \rangle\) is a group under multiplication. We omit the proof.
We end this section by considering a few more examples.
Example \(\PageIndex{6}\)
Let \(F\) be the set of all functions from \(\mathbb{R}\) to \(\mathbb{R}\text{,}\) and define pointwise addition \(+\) on \(F\) by
\begin{equation*} (f+g)(x)=f(x)+g(x) \end{equation*}
for all \(f,g\in F\) and \(x\in \mathbb{R}\text{.}\) We claim that \(F\) is a group under pointwise addition. (For variety, in this proof we don't explicitly refer to \(\mathcal{G}_1\)–\(\mathcal{G}_3\text{,}\) though we certainly do verify they hold.)
Proof
If \(f,g∈F\) then clearly \(f+g\) is also a function from \(\mathbb{R}\) to \(\mathbb{R}\), so \(\langle F,+ \rangle\) is a binary structure.
Next, let \(f,g,h∈F\). Then for all \(x∈\mathbb{R}\),
\( \begin{array} &((f+g)+h)(x) &=(f+g)(x)+h(x)& \\&=((f(x)+g(x))+h(x)&\\ &=f(x)+(g(x)+h(x)) &\text{(since addition is associative on \(\mathbb{R}\))} \\ &=f(x)+(g+h)(x)& \\&=(f+(g+h))(x).& \end{array}\)
Note that the key fact used in this argument is that \(f(x)\), \(g(x)\) and \(h(x)\) all lie in \(\mathbb{R}\), and addition is associative on \(\mathbb{R}\). When you get used to such arguments, it is sufficient to say that associativity of \(+\) on \(F\) is inherited from the associativity of addition on \(\mathbb{R}\).
Next, let \(z:\mathbb{R}→\mathbb{R}\) be the function \(z(x)=0\) for all \(x\). Then for all \(f∈F\) and \(x∈R\),
\(\begin{array}& (f+z)(x) &=f(x)+z(x)\\ &=f(x)+0\\ &=f(x)\\ &=0+f(x)\\ &=z(x)+f(x)\\ &=(z+f)(x). \end{array}\)
So \(z\) is an identity element of \(\langle F,+ \rangle\).
Finally, let \(f∈F\), and define \(g∈F\) by \(g(x)=−f(x)\) for all \(x∈\mathbb{R}\). It is easy then to see that \(g\) is an inverse for \(f\) in \(F\).
Hence, \(F\) is a group under pointwise addition. Note that it is uncountably infinite and abelian.
Example \(\PageIndex{7}\)
The set \(F\) is not a group under function composition (do you see why?). But if we define \(B\) to be the set of all bijections from \(\mathbb{R}\) to \(\mathbb{R}\text{,}\) then \(B\) is a group under function composition. (Prove it!) \(B\) is uncountably infinite and nonabelian.
Example \(\PageIndex{8}\)
Let \(\langle G_1,*_1\rangle\text{,}\) \(\langle G_2,*_2\rangle\) , \(\ldots\text{,}\) \(\langle G_n,*_n\rangle\) be groups (\(n\in \mathbb{Z}^+\)). Then the group product
\begin{equation*} G=G_1\times G_2\times \cdots \times G_n \end{equation*}
is a group under the componentwise operation \(*\) defined by
\begin{equation*} (g_1,g_2,\ldots, g_n)*(h_1,h_2,\ldots,h_n)=(g_1*_1h_1, g_2*_2h_2,\ldots, g_n*_nh_n) \end{equation*}
for all \((g_1,g_2,\ldots, g_n),(h_1,h_2,\ldots,h_n)\in G\text{.}\)
For instance, considering multiplication on \(\mathbb{R}^*\text{,}\) matrix multiplication on \(GL(2,\mathbb{R})\text{,}\) and addition modulo \(6\) on \(\mathbb{Z}_6\text{,}\) we have that \(\langle \mathbb{R}^*\times GL(2,\mathbb{R}) \times \mathbb{Z}_6,*\rangle\) is a group in which, for instance,
\begin{equation*} \left(-1, \begin{bmatrix} 1 &3 \\ 0 &-1 \end{bmatrix}, 3\right) *\left(\pi, \begin{bmatrix} 2 &1 \\ 1 &1 \end{bmatrix}, 4\right)=\left(-\pi, \begin{bmatrix} 5 &4 \\ -1 &-1 \end{bmatrix} ,1\right). \end{equation*}
Example \(\PageIndex{9}\)
A common example of a group product is the group \(\mathbb{Z}_2^2\text{,}\) equipped with componentwise addition modulo \(2\).
Definition: Klein-4-group
The group \(\mathbb{Z}_2^2\) is known as the Klein 4-group . (Felix Klein was a German mathematician; you may have heard of him in relation to the Klein Bottle.) The group \(\mathbb{Z}_2^2\) is sometimes denoted by \(V\text{,}\) which stands for “Vierergruppe,” the German word for “four-group”.


10.1.2: Blue Bar
Below the image there is a Blue Bar with the entries Catalog, Libraries, Collections, Homework and Under Development
Under Development
Under development lists all projects for LibreTexts which are, of course, under development. If you know of an open resource that LibreTexts should include you can submit a request at https://harvest.libretexts.org . Doing so will start a project for integration of the resource into LibreTexts. This will be discussed in the next section about the LibreTexts Conductor Project Management. Selecting the Under Development Button brings up a listing.
The three numbers are first, the percentage of the import that has been completed, peer review, and accessibility review. One of the advantages of your institution being a member of the LibreTexts network is that your submissions would be dealt with on a priority basis.
Homework
Moving from right to left, the next button lists books for which homework is available.
If you click on View Assignments you will see
and if you click on View Courses
You can see the number of questions available in the ADAPT homework system, but without registering for an instructor's account in ADAPT you will not be able to see the questions.
Collections
Collections are LibreTexts books from a single source.
Libraries
LibreTexts has 14 libraries
and each library has a number of sections
A detailed map can be found on the next page. You can go to the library directly or to any of the sections by clicking on the links


2.1: The Independent-Molecule Approximation
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In Chapter 18 , our survey of quantum mechanics introduces the idea that a molecule can have any of an infinite number of discrete energies, which we can put in order starting with the smallest. We now turn our attention to the properties of a system composed of a large number of molecules. This multi-molecule system must obey the laws of quantum mechanics. Therefore, there exists a Schrödinger equation, whose variables include all of the inter-nucleus, inter-electron, and electron-nucleus distance and potential terms in the entire multi-molecule system. The relevant boundary conditions apply at the physical boundaries of the macroscopic system. The solutions of this equation include a set of infinitely many wavefunctions, \({\mathit{\Psi}}_{i,j}\), each describing a quantum mechanical state of the entire multi-molecule system. In general, the collection of elementary particles that can be assembled into a particular multi-molecule system can also be assembled into many other multi-molecule systems. For example, an equimolar mixture of \(CO\) and \(H_2O\) can be reassembled into a system comprised of equimolar \(CO_2\) and \(H_2\), or into many other systems containing mixtures of \(CO\), \(H_2O\), \(CO_2\), and \(H_2\). Infinitely many quantum-mechanical states are available to each of these multi-molecule systems.
For every such multi-molecule wavefunction, \({\psi }_{i,j}\), there is a corresponding system energy, \(E_i\). In general, the system energy, \(E_i\), is \({\mathit{\Omega}}_i\)-fold degenerate; there are \({\mathit{\Omega}}_i\) wavefunctions, \({\mathit{\Psi}}_{i,1}\), \({\mathit{\Psi}}_{i,2}\), … , \({\mathit{\Psi}}_{i,{\mathit{\Omega}}_i}\), whose energy is \(E_i\). The wavefunctions include all of the interactions among the molecules of the system, and the energy levels of the system reflect all of these interactions. While generating and solving this multi-molecule Schrödinger equation is straightforward in principle, it is completely impossible in practice.
Fortunately, we can model multi-molecule systems in another way. The primary focus of chemistry is the study of the properties and reactions of molecules. Indeed, the science of chemistry exists, as we know it, only because the atoms comprising a molecule stick together more tenaciously than molecules stick to one another. (Where this is not true, we get macromolecular materials like metals, crystalline salts, etc .) This occurs because the energies that characterize the interactions of atoms within a molecule are much greater than the energies that characterize the interaction of one molecule with another. Consequently, the energy of the system can be viewed as the sum of two terms. One term is a sum of the energies that the component molecules would have if they were all infinitely far apart. The other term is a sum of the energies of all of the intermolecular interactions, which is the energy change that would occur if the molecules were brought from a state of infinite separation to the state of interest.
In principle, we can describe a multi-molecule system in this way with complete accuracy. This description has the advantage that it breaks a very large and complex problem into two smaller problems, one of which we have already solved: In Chapter 18 , we see that we can approximate the quantum-mechanical description of a molecule and its energy levels by factoring molecular motions into translational, rotational, vibrational, and electronic components. It remains only to describe the intermolecular interactions. When intramolecular energies are much greater than intermolecular-interaction energies, it may be a good approximation to ignore the intermolecular interactions altogether. This occurs when we describe ideal gas molecules; in the limit that a gas behaves ideally, the force between any two of its molecules is nil.
In Chapter 23 , we return to the idea of multi-molecule wavefunctions and energy levels. Meanwhile we assume that intermolecular interactions can be ignored. This is a poor approximation for many systems. However, it is a good approximation for many others, and it enables us to keep our description of the system simple while we use molecular properties in our development of the essential ideas of statistical thermodynamics.
We focus on developing a theory that gives the macroscopic thermodynamic properties of a pure substance in terms of the energy levels available to its individual molecules. To begin, we suppose that we solve the Schrödinger equation for an isolated molecule. In this Schrödinger equation, the variables include the inter-nucleus, inter-electron, and electron-nucleus distance and potential terms that are necessary to describe the molecule. The solutions are a set of infinitely many wavefunctions, \({\psi }_{i,j}\), each describing a different quantum-mechanical state of an isolated molecule. We refer to each of the possible wavefunctions as quantum state of the molecule. For every such wavefunction, there is a corresponding molecular energy, \({\epsilon }_i\). Every unique molecular energy, \({\epsilon }_i\), is called an energy level . Several quantum states can have the same energy. When two or more quantum states have the same energy, we say that they belong to the same energy level, and the energy level is said to be degenerate . In general, there are \(g_i\) quantum states that we can represent by the \(g_i\) wavefunctions, \({\psi }_{i,1}\), , \({\psi }_{i,2}\), ..., , \({\psi }_{i,g_i}\), each of whose energy is \({\epsilon }_i\). The number of quantum states that have the same energy is called the degeneracy of the energy level. Figure 1 illustrates the terms we use to describe the quantum states and energy levels available to a molecule.
In our development of classical thermodynamics, we find it convenient to express the value of a thermodynamic property of a pure substance as the change that occurs during a formal process that forms one mole of the substance, in its standard state, from its unmixed constituent elements, in their standard states. In developing statistical thermodynamics, we find it convenient to express the value of a molecular energy, \({\epsilon }_i\), as the change that occurs during a formal process that forms a molecule of the substance, in one of its quantum states, \({\psi }_{i,j}\), from its infinitely separated, stationary, constituent atoms. That is, we let the isolated constituent atoms be the reference state for the thermodynamic properties of a pure substance.


2.4: Ethical Considerations in Sexuality Research
As discussed in the previous chapter, research done by the Father of Modern Gynecology , Dr. Marion Simms was tainted by his unethical approach to patient care, based on perceived racism and beliefs about the pain threshold of Black enslaved women. As research became something that aided in the pursuit of knowledge, many researchers who were laser focused on wanting to know chose to overlook ethical considerations, and their actions led to great harm in numerous cases. Ethics are taken very seriously in institutions of higher learning because of this shameful past, and we as present day practitioners are held to high standards when it comes to research on human subjects.
At a broader societal level, members of some groups have historically faced more than their fair share of the risks of scientific research, including people who are institutionalized, are disabled, or belong to a racial or ethnic minority or otherwise disadvantaged group. A particularly tragic example of this is the Tuskegee (Box 2.3 above) and Guatemala Syphilis studies both led by Dr. John Cutler. In Tuskegee, the study conducted for the Public Health Service took place from 1932 to 1972. These participants were poor African American men in the vicinity of Tuskegee, Alabama, who were told that they were being treated for bad blood . Although they were given free medical care, they were not treated for their syphilis. Instead, they were observed to see how the disease developed in untreated patients. Even after the use of penicillin became the standard treatment for syphilis in the 1940s, these men continued to be denied treatment without being given an opportunity to leave the study. The study was eventually discontinued only after details were made known to the general public by journalists and activists. It is now widely recognized that researchers need to consider issues of justice and fairness at the societal level.
Beginning in 1946, the United States government, under Dr. Cutler, immorally and unethically engaged in research experiments, in which more than 5000 uninformed and unconsenting Guatemalan people were intentionally infected with bacteria that cause sexually transmitted diseases. Many remain untreated to this day (Rodriguez, & García, 2013).
In 1997, 65 years after the Tuskegee Syphilis Study began, and 25 years after it ended, President Bill Clinton formally apologized on behalf of the government to those who were affected. The United States eventually provided treatment and compensation for victims, families, and heirs in Tuskegee, including funding to locate the victims and pay attorneys’ fees. The ethical principle of equal justice strongly suggests that similar relief should be provided for the Guatemalan victims. While the U.S. now acknowledges this was an ethical failing, efforts to remediate the damage as in the case of Tuskegee, have not been undertaken, In a 2013 article entitled, First, do no harm: the US sexually transmitted disease experiments in Guatemala , The authors write, “Although US President Barack Obama apologized in 2010, and although the US Presidential Commission for the Study of Bioethical Issues found the Guatemalan experiments morally wrong, little if anything has been done to compensate the victims and their families.” (Rodriguez, & García, 2013).
Researchers have an obligation to practice their research in an ethical manner, but what does that mean exactly? They must respect their participants' dignity and their autonomy, giving them the opportunity to act without coercion. Researchers must obtain informed consent, which essentially involves the participants' agreement and documentation of their agreement to participate in a study after having been informed of everything that might reasonably be expected of them as participants. Included in this is also a right to privacy so that those being studied have their identity kept confidential as well as the right to opt out. None of this was done in Guatemala nor in Tuskegee, and had they been informed, they most likely would not have agreed to participate in the first place.
Because of these and other heinous examples of ethical violations, measures have been put in place to offset this type of exploitation. One of the earliest ethics codes was the Nuremberg Code, a set of 10 principles written in 1947, in conjunction with the trials of Nazi physicians accused of shockingly cruel research on concentration camp prisoners during World War II. It provided a standard against which to compare the behavior of the men on trial, many of whom were eventually convicted and either imprisoned or sentenced to death. The Nuremberg Code was particularly clear about the importance of carefully weighing risks against benefits and the need for informed consent. The Declaration of Helsinki is a similar ethics code that was created by the World Medical Council in 1964. Among the standards that it added to the Nuremberg Code was that research with human participants should be based on a written protocol, and it must include a detailed description of the research that is reviewed by an independent committee. The Declaration of Helsinki has been revised several times, most recently in 2004. In the U.S., concerns about the Tuskegee experiment and others led to the publication of federal guidelines known as the Belmont Report. The Belmont Report explicitly recognized the principle of seeking justice, including the importance of conducting research in a way that distributes risks and benefits fairly across different groups at the societal level. The Belmont Report became the basis of a set of laws the Federal Policy for the Protection of Human Subjects that apply to research conducted, supported, or regulated by the federal government. An extremely important part of these regulations is that universities, hospitals, and other institutions that receive support from the federal government must establish an Institutional Review Board (IRB), a committee that is responsible for reviewing research protocols for potential ethical problems. An IRB must consist of at least five people with varying backgrounds, including members of different professions, scientists and nonscientists, men and women, and at least one person not otherwise affiliated with the institution. The IRB helps to make sure that the risks of the proposed research are minimized, the benefits outweigh the risks, the research is carried out in a fair manner, and the informed consent procedure is adequate. The federal regulations also distinguish research that poses three levels of risk. Exempt research includes research on the effectiveness of normal educational activities, the use of standard psychological measures and surveys of a nonsensitive nature that are administered in a way that maintains confidentiality, and research using existing data from public sources. It is called exempt because the regulations do not apply to it. Minimal risk research exposes participants to risks that are no greater than those encountered by healthy people in daily life or during routine physical or psychological examinations. Minimal risk research can receive an expedited review by one member of the IRB, or by a separate committee under the authority of the IRB that can only approve minimal risk research. (Many departments of psychology have such separate committees.) Finally, at-risk research poses greater than minimal risk and must be reviewed by the entire IRB.
Specific to human sexuality research, The American Psychological Association (APA) is the governing body for much of the sex research undertaken. Ethical Principles of Psychologists and Code of Conduct (also known as the APA Ethics Code ) was first published in 1953 and has been revised several times since then, most recently in 2002. It includes about 150 specific ethical standards that psychologists and their students are expected to follow. Much of the APA Ethics Code concerns the clinical practice of psychology advertising purposes, the most relevant part, is linked here: American Psychological Association Ethics Code .
Just as the medical field grew in the West out of predominately white, heteronormative, cisgender persons, so did the study of sexuality. Much of the body of research in the field is done on white, heterosexual cisgender people. Despite the world of human sexuality, being as vast and diverse as it is, research is slow to catch up. New spaces and opportunities for different research are opening up as the field of educators become more diverse but as we will learn throughout this book, there is much still to be done. One space that facilitates engaging with sexuality beyond those borders is afrosexology.com . Their mission to educate, explore and help people reclaim their sexual agency which then give them agency in their larger lives is one way in which current sex research is moving the needle towards equity.
Putting Ethics Into Practice
In this section, we look at some practical advice for conducting ethical research. Who knows, some of you may be undertaking sex research one day! Again, it is important to remember that ethical issues arise well before you begin to collect data and continue to arise through publication and beyond.
As the APA Ethics code notes in its introduction, “Lack of awareness or misunderstanding of an ethical standard is not itself a defense to a charge of unethical ethical responsibilities” (Ethical Principles of Psychologists and Code of Conduct, 2017). At a minimum, this means reading and understanding the relevant standards of the APA Ethics Code, distinguishing minimal risk from at-risk research, and knowing the specific policies and procedures of your institution including how to prepare and submit a research protocol for institutional review board (IRB) review. If you are conducting research as a course requirement, there may be specific course standards, policies, and procedures. If any standard, policy, or procedure is unclear, or you are unsure what to do about an ethical issue that arises, you must seek clarification. You can do this by reviewing the relevant ethics codes, reading about how similar issues have been resolved by others, or consulting with more experienced researchers, your IRB, or your course instructor. Ultimately, you as the researcher must take responsibility for the ethics of the research you conduct.
As you design your study, you must identify and minimize risks to participants via the informed consent procedure, which in parts states:
A researcher must inform participants about (1) the purpose of the research, expected duration, and procedures; (2) their right to decline to participate and to withdraw from the research once participation has begun; (3) the foreseeable consequences of declining or withdrawing; (4) reasonably foreseeable factors that may be expected to influence their willingness to participate such as potential risks, discomfort, or adverse effects; (5) any prospective research benefits; (6) limits of confidentiality; (7) incentives for participation; and (8) whom to contact for questions about the research and research participants' rights. They provide opportunity for the prospective participants to ask questions and receive answers. (Ethical Principles of Psychologists and Code of Conduct, 2017).
Start by listing all the risks, including risks of physical and psychological harm and violations of confidentiality. Remember that some risks might apply only to some participants. For example, while many people would have no problem completing a survey about their fear of various sex crimes, this may be triggering for those who have been a victim of one of those crimes. This is why you should seek input from a variety of people, including your research collaborators, more experienced researchers, and even from nonresearchers who might be better able to take the perspective of a participant.
Once you have identified the risks, you can often reduce or eliminate many of them. One way is to modify the research design. For example, you might be able to shorten or simplify the procedure to prevent boredom and frustration. You might be able to replace upsetting or offensive stimulus materials (e.g., graphic photos) with less upsetting or offensive ones (e.g., milder photos of the sort people are likely to see in the newspaper).
A second way to minimize risks is to use a pre-screening procedure to identify and eliminate participants who are at high risk. You can do this in part through the informed consent process. For example, you can warn participants that a survey includes questions about their fear of sex crimes, and remind them that they are free to withdraw if they think this might upset them. Prescreening can also involve collecting data to identify and eliminate participants.
A third way to minimize risks is to take active steps to maintain confidentiality. You should keep signed consent forms separate from any data that you collect, and in such a way that no individual's name can be linked to their data. You should only collect information that you actually need to answer your research question. If a person’s sexual orientation or ethnicity is not clearly relevant to your research question, for example, then do not ask them about it. Be aware also that certain data collection procedures can lead to unintentional violations of confidentiality. When participants respond to an oral survey in a shopping mall or complete a questionnaire in a classroom setting, it is possible that their responses will be overheard or seen by others. If the responses are personal, it is better to administer the survey or questionnaire individually in private or to use other techniques to prevent the unintentional sharing of personal information.
Be sure to identify and minimize deception. The APA code of ethics standard 8.07 states that a researcher may:
(a) not conduct a study involving deception unless they have determined that the use of deceptive techniques is justified by the study's significant prospective scientific, educational, or applied value and that effective non deceptive alternative procedures are not feasible.
(b) not deceive prospective participants about research that is reasonably expected to cause physical pain or severe emotional distress.
(c) must explain any deception that is an integral feature of the design and conduct of an experiment to participants as early as is feasible, preferably at the conclusion of their participation, but no later than at the conclusion of the data collection, and permit participants to withdraw their data. (Ethical Principles of Psychologists and Code of Conduct, 2017).
Remember that deception can take a variety of forms, not all of which involve actively misleading participants. It is also deceptive to allow participants to make incorrect assumptions or simply withhold information about the full design or purpose of the study. It is best to identify and minimize all forms of deception.
Remember that according to the APA Ethics Code, deception is ethically acceptable only if there is no way to answer your research question without it. Therefore, if your research design includes any form of active deception, you should consider whether it is truly necessary.
In general, it is considered acceptable to wait until debriefing before you reveal your research question as long as you describe the procedure, risk and benefits during the informed consent process.
Once the risks of the research have been identified and minimized, you need to weigh them against the benefits. This requires identifying all the benefits. Remember to consider benefits to the research participants, to science, and to society. If you are a student researcher, remember that one of the benefits is the knowledge you will gain about how to conduct scientific research, knowledge you can then use to complete your studies and succeed in graduate school or in your career.
If the research poses minimal risk, no more than in people’s daily lives or in routine physical or psychological examinations, then even a small benefit to participants, science, or society is generally considered enough to justify it. If it poses more than minimal risk, then there should be more benefits. If the research has the potential to upset some participants, for example, then it becomes more important that the study be well designed and answer a scientifically interesting research question or have clear practical implications. It would be unethical to subject people to pain, fear, or embarrassment for minimal scientific gain. Research that has the potential to cause harm that is more than minor, or lasts for more than a short time, is rarely considered justified by its benefits.
Once you have settled on a research design, you need to create your informed consent and debriefing procedures. Start by deciding whether informed consent is necessary according to APA Standard 8.05 . If informed consent is necessary, there are several things you should do. First, when you recruit participants whether it is through word of mouth, posted advertisements, or a participant pool, provide them with as much information about the study as you can. This will allow those who might find the study objectionable to avoid it. Second, prepare a script or talking points to help you explain the study to your participants in simple everyday language. This should include a description of the procedure, the risks and benefits, and their right to withdraw at any time. Third, create an informed consent form that covers all the points in APA Standard 8.02a that participants can read and sign after you have described the study to them. Your university, department, or course instructor may have a sample consent form that you can adapt for your own study. If not, an Internet Search will turn up several samples. Remember that if appropriate, both the oral and written parts of the informed consent process should include the fact that you are keeping some information about the design or purpose of the study from them but that you will reveal it during debriefing.
Debriefing is similar to informed consent in that you cannot necessarily expect participants to read and understand written debriefing forms. So again, it is best to write a script or set of talking points, with the goal of being able to explain the study in simple everyday language. During debriefing, you should reveal the research question and full design of the study. For example, if participants are tested under only one condition, then you should explain what happened in the other conditions. If you deceived your participants, you should reveal this as soon as possible, apologize for the deception, explain why it was necessary, and correct any misconceptions that participants might have as a result. Debriefing is also a good time to provide additional benefits to research participants, by giving them relevant practical information or referrals to other sources of help. For example, in a study of attitudes toward domestic abuse, you could provide pamphlets about domestic abuse and referral information to the university counseling center for those who might want it.
Remember to schedule plenty of time for the informed consent and debriefing processes. They cannot be effective if you have to rush through them. The next step is to get institutional approval for your research, based on the specific policies and procedures at your institution or for your course. This will generally require writing a protocol that describes the purpose of the study, the research design and procedure, the risks and benefits, the steps taken to minimize risks, and the informed consent and debriefing procedures. Do not think of the institutional approval process as merely an obstacle to overcome but as an opportunity to think through the ethics of your research and to consult with others who are likely to have more experience or different perspectives than you. If the IRB has questions or concerns about your research, address them promptly and in good faith. This might even mean making further modifications to your research design and procedure before resubmitting your protocol.
Your concern with ethics should not end when your study receives institutional approval. It now becomes important to stick to the protocol you submitted, or to seek additional approval for anything other than a minor change. During the research, you should monitor your participants for unanticipated reactions, and seek feedback from them during debriefing. Be alert also for potential violations of confidentiality. Keep the consent forms and the data safe and separate from each other and make sure that no one, intentionally or unintentionally, has access to participants personal information.
Finally, you must maintain your integrity through the publication process and beyond. Address publication credit; who will be authors on the research and the order of authors with your collaborators early and avoid plagiarism in your writing. Remember that our scientific goal is to learn about the way the world actually is and that your scientific duty is to report on your results honestly and accurately. So do not be tempted to fabricate data or alter your results in any way. Besides, unexpected results are often as interesting or more so than expected ones.
While these standards of ethics put forth by the APA are seen as a definitive guide in the pursuit of ethical research, even the organization itself, the APA has failed to abide by them in large scale research that has been widely published, cited and used as justification for policy and funding decisions nationwide. Many historically relevant psychological studies are clear violations of the current standards set by the American Psychological Association (APA) and were founded on the basis of systemic racism. Experiments like the aforementioned Tuskegee Syphilis Study, Jane Elliot’s Blue/Brown Eyes Exercise examined in the film, A Class Divided (full film) | FRONTLINE , and The Doll Test Landmark Cases: Brown v Board Doll Test (C-SPAN) hold a variety of unethical practices in the way they were produced, but are still considered to be groundbreaking contributions to the current psychological landscape.
Additionally, many early thinkers including Ronald Fisher, a key player in the development of statistical science, also supported racism and eugenics. There is a laundry list of connections between systemic racism and psychology, which is why the APA’s decision to apologize for its role in incorporating racism within the discipline of psychology is critical to the development of psychological practices to come, but not the only solution.
The APA cannot completely correct the wrongs of the racist practices put forth in the past, however, the Association has a responsibility to put forth more ethical practices that do not further marginalize and misdiagnose people of color. To this day, there is an overrepresentation of Black Americans who are admitted into psychiatric hospitals , suicide rates for Black Youth continue to rise , and Black Americans are more likely to be diagnosed with disorders like schizophrenia as compared to White Americans .
As American culture strives to become more centered around the emotional wellness of all people and seeks to condemn dehumanization, institutions must develop better practices for building spaces of healing.


By the end of this section, you should be able to:
- Define trauma-informed care (TIC).
- List the six key elements to TIC.
- Demonstrate how the nurse can utilize TIC during a crisis.
Trauma-informed care (TIC) is a therapeutic approach that acknowledges trauma’s long-lasting emotional, neurological, psychological, social, and biological effects on a person’s present and future health. For many people, adverse childhood experience s ( ACEs ) ultimately hinder emotional, social, and even physical health, even if they seem unrelated to present problems (Centers for Disease Control and Prevention [CDC], 2023a). Trauma-informed care is unique as it reimagines the client-therapist relationship as a partnership. The nurse or therapist provides support throughout a client’s journey of healing and growth, using six guiding principles that ensure every interaction is mindful and aware. However, first and foremost, at the core of trauma-informed care is the tenet of primum non nocere, which translates from Latin as “first, do no harm” (Sommers-Flanagan & Sommers-Flanagan, 2021).
Six Key Elements to Trauma-Informed Care
The CDC and the Substance Abuse and Mental Health Services Administration (SAMHSA, 2020) collaborated to develop six principles for TIC that negate the potential for re-traumatizing the client. These core principles are safety, trustworthiness and transparency, peer support, collaboration and mutuality, empowerment and choice, and cultural, historical, and gender issues (Table 31.1).
| Principle | Definition |
|---|---|
| Safety |
|
| Trustworthiness and transparency |
|
| Peer support |
|
| Collaboration and mutuality |
|
| Empowerment and choice |
|
| Cultural, historical, and gender issues |
|
A printable infographic of “6 Guiding Principles to a Trauma-Informed Approach” is available from the CDC and SAMHSA here.
The CDC and SAMHSA note that a trauma-informed approach is not accomplished through a single technique or checklist. A trauma-informed approach requires constant attention, caring awareness, sensitivity, and possibly a cultural change at an organizational level. An internal organizational assessment and quality improvement, as well as engagement with community partners, will help to embed this approach, which can be augmented with organizational development and practice improvement. However, with thoughtful intention by the nurse, it is easy to generalize and apply the six core principles across the various clinical settings where clients may receive treatment.
The Role of Nurses and Trauma-Informed Care in Clinical Practice
A nurse can apply the principles of TIC in daily interactions with all clients. TIC requires a nurse to be mindful, sensitive, and responsive; to be sensitive to the impact of trauma on others and oneself; to understand and use tools to support self and others during times of stress; and to identify and support the system change needed to reduce re-traumatization. In alignment with the principles of safety, respect, and trust, nurses can ask themselves three simple questions as a first step to applying a trauma-informed lens to their practice (Fleishman et al., 2019):
Safety: Does this cultivate a sense of safety?
Respect: Am I, and are others, showing respect?
Trust: Does this build trust?
The nurse can mindfully employ strategies in their daily practice to engender trustworthiness, transparency, empowerment, and the other facets of TIC; for example (Fleishman et al., 2019):
- Introduce yourself and your role in every client interaction: When a client understands who you are and your role in their care, they can feel empowered to be more actively engaged in their own care.
- Use open and nonthreatening body positioning: Using nonthreatening body positioning helps prevent the threat-detection areas of the brain from taking over, which helps clients stay regulated.

- Protect client privacy: Protect client privacy and safety by ensuring that the client desires those present to hear about their care. This ensures compliance with the HIPAA Privacy Rule (U.S. Department of Health and Human Services [HHS], 2022).
- Use plain language and teach-back: Using clear language and teach-back empowers clients with knowledge and understanding about their care.
For a health care system to thoroughly implement and embody TIC, policies, procedures, and culture need to be trauma-informed. Nurses who utilize a trauma-informed approach in clinical practice can enhance job satisfaction, reduce risk for burnout, and improve client experiences and outcomes (Wolotira, 2023).
In this video, nurse Cheryl Martin describes trauma-informed care in her community as a responsive approach to providing care for individuals who have experienced trauma.
Watch the video, and then respond to the following questions:
- How would you distinguish positive stress, tolerable stress, and toxic stress?
- What are some examples of how people respond to toxic stress? Why might each individual respond differently to a similar situation?
- How would you describe a “safe environment”? What is the nurse’s role in creating a safe environment for clients?
- Cheryl Martin makes the point that only the client is the “expert” about their own disease. Why is this viewpoint essential for providing TIC?
Read the scenario, and then respond to the questions that follow.
Nellie, 21, is a survivor of interpersonal violence. After successfully transitioning from an abusive household to a shelter, Nellie lives independently in rent-assisted housing. She is employed as a waitress at a local restaurant. However, she struggles with chronic headaches and neck pain from past abuse by her former boyfriend, who repeatedly beat her and tried to strangle her while he was intoxicated. Since becoming a “survivor,” Nellie frequently visits the local clinic for follow-up appointments related to these past injuries. After enduring this physical and emotional abuse, she has misused substances for both anxiety and pain. Nellie wants her physical and emotional pain to go away and wants the nurse to convince the doctor to write her prescriptions because she has been “cut off” and finds the “talk therapy” that they want her to go to to be too “scary and emotional to deal with.”
- Why is it important for Nellie to feel that she is in a safe space? What can the nurse do to facilitate this?
- How might the nurse use a TIC approach to open a dialogue with Nellie to find out what she really needs?
- What alternate pain-management techniques might the nurse suggest for Nellie?


2.6: The Vector Projection of One Vector onto Another
Projection
Let’s project vector \(\overrightarrow{u}\boldsymbol{=}\left\langle u_x,\left.u_y\right\rangle \right.\) onto the vector \(\overrightarrow{v}\boldsymbol{=}\left\langle v_x,\left.v_y\right\rangle \right.\) .
To do so, imagine a light bulb above \(\overrightarrow{u}\) shining perpendicular onto \(\overrightarrow{v}\) .
The light from the bulb will cast a shadow of \(\overrightarrow{u}\) onto \(\overrightarrow{v}\) , and it is this shadow that we are looking for. The shadow is the projection of \(\overrightarrow{u}\) onto \(\overrightarrow{v}\) .
The red vector is the projection of \(\overrightarrow{u}\) onto \(\overrightarrow{v}\) . The notation commonly used to represent the projection of \(\overrightarrow{u}\) onto \(\overrightarrow{v}\) is \({\mathrm{proj}}_{\overrightarrow{\mathrm{v}}}\overrightarrow{u}\) .
Vector parallel to \(\overrightarrow{v}\) with magnitude \(\frac{\overrightarrow{u}\bullet \overrightarrow{v}}{\left\|\overrightarrow{v}\right\|}\) in the direction of \(\overrightarrow{v}\) is called projection of \(\overrightarrow{u}\) onto \(\overrightarrow{v}\) .
The formula for \({\mathrm{proj}}_{\overrightarrow{\mathrm{v}}}\overrightarrow{u}\) is
\[\operatorname{proj}_{\overrightarrow{\mathrm{v}}} \vec{u}=\frac{\vec{u} \cdot \vec{v}}{\|\vec{v}\|^2} \vec{v} \nonumber \]
To find the projection of \(\vec{u}=\langle 4,3\rangle\) onto \(\vec{v}=\langle 2,8\rangle\), we need to compute both the dot product of \(\overrightarrow{u}\) and \(\overrightarrow{v}\) , and the magnitude of \(\overrightarrow{v}\) , then apply the formula.
Solution
\(\begin{gathered}
\operatorname{proj}_{\overrightarrow{\mathrm{v}}} \vec{u}=\frac{\vec{u} \cdot \vec{v}}{\|\vec{v}\|^2} \vec{v} \\
\operatorname{proj}_{\vec{v}} \vec{u}=\frac{\langle 4,3\rangle \cdot\langle 2,8\rangle}{\|\langle 2,8\rangle\|^2}\langle 2,8\rangle \\
\operatorname{proj}_{\overrightarrow{\mathrm{v}}} \vec{u}=\frac{4 \cdot 2+3 \cdot 8}{\left(\sqrt{2^2+8^2}\right)^2}\langle 2,8\rangle \\
\operatorname{proj}_{\vec{v}} \vec{u}=\frac{32}{(\sqrt{4+64})^2}\langle 2,8\rangle \\
\operatorname{proj}_{\vec{v}} \vec{u}=\frac{32}{68}\langle 2,8\rangle \\
\operatorname{proj}_{\vec{v}} \vec{u}=\frac{8}{17}\langle 2,8\rangle \\
\operatorname{proj}_{\vec{v}} \vec{u}=\left\langle\frac{16}{17}, \frac{64}{17}\right\rangle
\end{gathered}\)
Using Technology
We can use technology to determine the projection of one vector onto another.
Go to www.wolframalpha.com.
To find the projection of \(\overrightarrow{u}=\left\langle 4,\left.3\right\rangle \right.\) onto \(\vec{v}=\langle 2,8\rangle\), use the “projection” command. In the entry field enter projection of \(\mathrm{<}\) 4, 3 \(\mathrm{>}\) onto \(\mathrm{<}\) 2, 8 \(\mathrm{>}\) .
\\
Wolfram alpha tells you what it thinks you entered, then tells you its answer. In this case, \(\left\langle \frac{16}{17},\left.\frac{64}{17}\right\rangle \right.\) .
As an applied example, suppose a video game has a ball moving near a wall.
Solution
We take the origin at the bottom-left-most corner of the screen. The wall is at a 30 \(\mathrm{{}^\circ}\) angle to the horizontal, and at a point in time, the ball is at position \(\overrightarrow{v}\boldsymbol{=\ }\) \(\mathrm{\langle }\) 4, \(\ \left.7\right\rangle\) . To find the perpendicular distance from the ball to the wall, we use the projection formula to project the vector \(\overrightarrow{v}\boldsymbol{=\ }\) \(\mathrm{\langle }\) 4, \(\ \left.7\right\rangle\) onto the wall.
We begin by decomposing \(\overrightarrow{v}\) into two vectors \({\overrightarrow{v}}_1\) and \({\overrightarrow{v}}_2\) so that \(\overrightarrow{v}=\ {\overrightarrow{v}}_1+{\overrightarrow{v}}_2\) and \({\overrightarrow{v}}_1\) lies along the wall .
The length (magnitude) of the vector \(\overrightarrow{v}\) is then the distance from the ball to the wall.
The vector \({\overrightarrow{v}}_1\) is the projection of \(\overrightarrow{v}\) onto the wall. We can get \({\overrightarrow{v}}_1\) by scaling (multiplying) a unit vector \(\overrightarrow{w}\) that lies along the wall and, thus, along with \({\overrightarrow{v}}_1\) .
Since \(\overrightarrow{w}\) lies at a 30 \(\mathrm{{}^\circ}\) angle to the horizontal, \(\overrightarrow{w}\boldsymbol{=}\left.\boldsymbol{\langle }\mathrm{cos}30{}^\circ ,\ \ \mathrm{sin}30{}^\circ \right\rangle \boldsymbol{=}\left.\boldsymbol{\langle }\mathrm{0.866},\ \mathrm{0.5}\right\rangle\) , using the projection formula, we get the projection of \(\overrightarrow{v}\) that lies along the wall.
\(\begin{gathered}
\vec{v}_1=\operatorname{proj}_{\vec{w}} \vec{v}=\frac{\vec{v} \cdot \vec{w}}{\|\vec{w}\|^2} \vec{w} \\
\vec{v}_1 \frac{\langle 4,7\rangle \cdot\langle 0.866,0.5\rangle}{\left\|\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\right\|^2}\langle 0.866,0.5\rangle =\frac{4 \cdot(0.866)+7 \cdot(0.5)}{\left(\sqrt{(0.866)^2+(.5)^2}\right)^2}\langle 0.866,0.5\rangle \\
\vec{v}_1=\frac{6.964}{(\sqrt{1})^2}\langle 0.866,0.5\rangle\\
\vec{v}_1=(6.964)\langle 0.866,0.5\rangle \\
\vec{v}_1=\langle 6.031,3.482\rangle
\end{gathered}\)
Since that \(\vec{v}=\vec{v}_{\mathbf{1}}+\vec{v}_{\mathbf{2}}\), subtraction get us
\(\begin{gathered}
\vec{v}_2=\vec{v}-\vec{v}_1 \\
\vec{v}_2=\langle 4,7\rangle-\langle 6.031,3.482\rangle \\
\vec{v}_2=\langle 4-6.031,7-3.482\rangle \\
\vec{v}_2=\langle-2.031,3.518\rangle
\end{gathered}\)
To get the magnitude of \(\vec{v}_2\), we use
\(\begin{gathered}
\left\|\vec{v}_2\right\|=\sqrt{v_x^2+v_y^2} \\
\left\|\overrightarrow{\boldsymbol{v}}_2\right\|=\sqrt{(-2.031)^2+3.518^2} \\
\left\|\vec{v}_2\right\|=\sqrt{4.125+12.376} \\
\left\|\overrightarrow{\vec{v}_2}\right\|=4.062
\end{gathered}\)
Try These
Find the projection of the vector \(\vec{v}=\langle 3,5\rangle\) onto the vector \(\vec{u}=\langle 6,2\rangle\).
- Answer
-
\(\left\langle \frac{21}{5}\right.,\ \left.\frac{7}{5}\right\rangle\)
Find \(\operatorname{proj}_{\vec{v}} \vec{u}\), with \(\vec{u}=\langle-2,5\rangle\) and \(\vec{v}=\langle 6,-5\rangle\).
- Answer
-
\(\left\langle \frac{-222}{61}\right.,\ \left.\frac{185}{61}\right\rangle\)


16: Extinction
| Era | Period | Epoch | Approximate Duration of Era, Period or Epoch (millions of years before present) | Major Extinction Evens |
|---|---|---|---|---|
| CENOZOIC | Quaternary | Holocene | present - 0.01 |
\(6^{th}\) major extinction ? \(5^{th}\) major extinction (end of Cretaceous; K-T boundary) \(4^{th}\) major extinction (end of Triassic) \(3^{th}\) major extinction (end of Permian) \(2^{nd}\) major extinction (Late Devonian) \(1^{st}\) major extinction (end of Ordovician) |
| Pleistocene | 0.01-1.6 | |||
| Tertiary | Pliocene | 1.6-5.3 | ||
| Miocene | 5.3-23 | |||
| Oligocene | 24-37 | |||
| Eocene | 37-58 | |||
| Paleocene | 58-65 | |||
| MESOZOIC | Cretaceous | 65-144 | ||
| Jurassic | 144-208 | |||
| Triassic | 208-245 | |||
| PALEOZOIC |
Permian |
245-286 | ||
|
(Carboniferous) Pennsylvanian |
286-325 | |||
| (Carboniferous) Mississippian | 325-360 | |||
| Devonian | 360-408 | |||
| Silurian | 408-440 | |||
| Ordovician | 440-505 | |||
| Cambrian | 505-570 | |||
| PRECAMBRIAN | 570-4500 |
Each of the first five mass extinctions shown in Table \(\PageIndex{1}\) represents a significant loss of biodiversity - but recovery has been good on a geologic time scale. Mass extinctions are apparently followed by a sudden burst of evolutionary diversification on the part of the remaining species, presumably because the surviving species started using habitats and resources that were previously "occupied" by more competitively successful species that went extinct. However, this does not mean that the recoveries from mass extinction have been rapid; they have usually required some tens of millions of years (Jablonski, 1995).
It is hypothesized that we are currently on the brink of a "sixth mass extinction," but one that differs from previous events. The five other mass extinctions predated humans and were probably the ultimate products of some physical process (e.g. climate change through meteor impacts), rather than the direct consequence of the action of some other species. In contrast, the sixth mass extinction is the product of human activity over the last several hundred, or even several thousand years. These mass extinctions, and their historic and modern consequences are discussed in more detail in the modules on Historical perspectives on extinction and the current biodiversity crisis, and Ecological consequences of extinctions..
Glossary
- Extinct
- a species is assumed to be extinct when there is no reasonable doubt that the last individual has died (IUCN, 2002)
- Extinction
- the complete disappearance of a species from Earth
- Mass extinction
- a period when there is a sudden increase in the rate of extinction, such that the rate at least doubles, and the extinctions include representatives from many different taxonomic groups of plants and animals