4: Cell Structure
Your body has many kinds of cells, each specialized for a specific purpose. Just as a home is made from a variety of building materials, the human body is constructed from many cell types. For example, epithelial cells protect the surface of the body and cover the organs and body cavities within. Bone cells help to support and protect the body. Cells of the immune system fight invading bacteria. Additionally, blood and blood cells carry nutrients and oxygen throughout the body while removing carbon dioxide. Each of these cell types plays a vital role during the growth, development, and day-to-day maintenance of the body. In spite of their enormous variety, however, cells from all organisms—even ones as diverse as bacteria, onion, and human—share certain fundamental characteristics.
-
- 4.0: Prelude to Cell Structure
- Your body has many kinds of cells, each specialized for a specific purpose. Just as a home is made from a variety of building materials, the human body is constructed from many cell types. For example, epithelial cells protect the surface of the body and cover the organs and body cavities within. Bone cells help to support and protect the body. Cells of the immune system fight invading bacteria. Additionally, blood and blood cells carry nutrients and oxygen throughout the body while removing CO2
-
- 4.1: Studying Cells
- A cell is the smallest unit of a living thing. A living thing, whether made of one cell (like bacteria) or many cells (like a human), is called an organism.  There are many types of cells, all grouped into one of two broad categories: prokaryotic and eukaryotic. For example, both animal and plant cells are classified as eukaryotic cells, whereas bacterial cells are classified as prokaryotic.
-
- 4.2: Prokaryotic Cells
- Cells fall into one of two broad categories: prokaryotic and eukaryotic. Only the predominantly single-celled organisms of the domains Bacteria and Archaea are classified as prokaryotes (pro- = “before”; -kary- = “nucleus”). Cells of animals, plants, fungi, and protists are all eukaryotes (eu- = “true”) and are made up of eukaryotic cells.
-
- 4.3: Eukaryotic Cells
- Our natural world also utilizes the principle of form following function, especially in cell biology, and this will become clear as we explore eukaryotic cells. Unlike prokaryotic cells, eukaryotic cells have: 1) a membrane-bound nucleus; 2) numerous membrane-bound organelles such as the endoplasmic reticulum, Golgi apparatus, chloroplasts, mitochondria, and others; and 3) several, rod-shaped chromosomes. Because a eukaryotic cell’s nucleus is surrounded by a membrane, it is has “true nucleus.”
-
- 4.4: The Endomembrane System and Proteins
- The endomembrane system is a group of membranes and organelles in eukaryotic cells that works together to modify, package, and transport lipids and proteins. It includes the nuclear envelope, lysosomes, and vesicles, which we’ve already mentioned, and the endoplasmic reticulum and Golgi apparatus. Although not technically within the cell, the plasma membrane is included in the endomembrane system because, as you will see, it interacts with the other endomembranous organelles.
-
- 4.5: The Cytoskeleton
- Within the cytoplasm, there are ions and organic molecules, plus a network of protein fibers that help maintain the shape of the cell, secure some organelles in specific positions, allow cytoplasm and vesicles to move within the cell, and enable cells within multicellular organisms to move. Collectively, this network of protein fibers is known as the cytoskeleton. There are three types of fibers within the cytoskeleton: microfilaments, intermediate filaments, and microtubules.
-
- 4.6: Connections between Cells and Cellular Activities
- You already know that a group of similar cells working together is called a tissue. As you might expect, if cells are to work together, they must communicate with each other, just as you need to communicate with others if you work on a group project. Let’s take a look at how cells communicate with each other.
Thumbnail: The generalized structure of a prokaryotic cell. (CC BY 4.0; OpenStax ).


7.8: Homework- Integration by Parts
-
Solve each of the following using integration by parts:
-
\(\int x \cos(x)dx\)
\(x \sin(x) + \cos(x) + C\)ans
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\(\int (4x - 1) \cos(x)dx\)
\((4 x - 1) \sin(x) + 4 \cos(x) + C\)ans
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\(\int x \sin(x)dx\).
\(-x \cos(x) + \sin(x) + C\)ans
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\(\int x e^xdx\).
\(x e^x - e^x + C\)ans
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\(\int \ln(x)dx\). (Hint: Let \(u = \ln(x)\) and \(v' = 1\))
\(x \ln(x) - x\)ans
-
\(\int x \cos(x)dx\)
- Watch the following Khan Academy video: Integration by parts twice
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Use integration by parts to solve \(\int x^2 \cos(x)dx\).
\(x^2 \sin(x) + 2 x \cos(x) - 2\sin(x) + C\)ans
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Use integration by parts to solve \(\int x^3 e^xdx\).
\(x^3 e^x - 3x^2 e^x + 6x e^x - 6 e^x + C\)ans
- Watch the following Khan Academy video: Integration by parts with e and cos together.
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Use integration by parts to find \(\int e^x \sin(x)dx\).
\(\frac{\sin(x) e^x - \cos(x) e^x}{2} + C\)ans
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Two part question:
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Use \(u\)-substitution to find \(\int \sin(2x)dx\) and \(\int \cos(2x)dx\).
\(-\frac{1}{2} \cos(2x)\) and \(\frac{1}{2} \sin(2x)\)ans
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Use integration by parts to find \(\int x \sin(2x)dx\).
\(-\frac{1}{2} x \cos(2x) - \frac{1}{4} \sin(2x)\)ans
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Use \(u\)-substitution to find \(\int \sin(2x)dx\) and \(\int \cos(2x)dx\).


12.5: Cranial Nerves
By the end of this section, you will be able to:
- Describe the structures found in the PNS
- Distinguish between somatic and autonomic structures, including the special peripheral structures of the enteric nervous system
- Name the twelve cranial nerves and explain the functions associated with each
The PNS is not as contained as the CNS because it is defined as everything that is not the CNS. Some peripheral structures are incorporated into the other organs of the body. In describing the anatomy of the PNS, it is necessary to describe the common structures, the nerves and the ganglia, as they are found in various parts of the body. Many of the neural structures that are incorporated into other organs are features of the digestive system; these structures are known as the enteric nervous system and are a special subset of the PNS.
Nerves
Bundles of axons in the PNS are referred to as nerves. These structures in the periphery are different than the central counterpart, called a tract. Nerves are composed of more than just nervous tissue. They have connective tissues invested in their structure, as well as blood vessels supplying the tissues with nourishment. The outer surface of a nerve is a surrounding layer of fibrous connective tissue called the epineurium . Within the nerve, axons are further bundled into fascicles , which are each surrounded by their own layer of fibrous connective tissue called perineurium . Finally, individual axons are surrounded by loose connective tissue called the endoneurium (Figure \(\PageIndex{1}\) and Figure \(\PageIndex{2}\)). These three layers are similar to the connective tissue sheaths for muscles. Nerves are associated with the region of the CNS to which they are connected, either as cranial nerves connected to the brain or spinal nerves connected to the spinal cord.
Interactive Link
Peripheral Nerve
View a virtual slide of a nerve in longitudinal section at the University of Michigan WebScope to explore the tissue sample in greater detail. With what structures in a skeletal muscle are the endoneurium, perineurium, and epineurium comparable?
- Answer
-
The endoneurium surrounding individual nerve fibers is comparable to the endomysium surrounding myofibrils, the perineurium bundling axons into fascicles is comparable to the perimysium bundling muscle fibers into fascicles, and the epineurium surrounding the whole nerve is comparable to the epimysium surrounding the muscle.
Cranial Nerves
The nerves attached to the brain are the cranial nerves, which are primarily responsible for the sensory and motor functions of the head and neck (with the exception of one that targets organs in the thoracic and abdominal cavities as part of the parasympathetic nervous system). There are twelve cranial nerves, which are designated CNI through CNXII for “Cranial Nerve,” using Roman numerals for 1 through 12, based on the anatomical location on the inferior view of the brain, from anterior to posterior (Figure \(\PageIndex{3}\)). They can be classified as sensory nerves, motor nerves, or a combination of both, meaning that the axons in these nerves originate out of sensory ganglia external to the cranium or motor nuclei within the brainstem. Sensory axons enter the brain to synapse in a nucleus. Motor axons connect to skeletal muscles of the head or neck. Three of the nerves are solely composed of sensory fibers; five are strictly motor; and the remaining four are mixed nerves.
Learning the cranial nerves is a tradition in anatomy courses, and students have always used mnemonic devices to remember the nerve names. A traditional mnemonic is the rhyming couplet, “Oh Oh Oh To Touch And Feel Very Good Velvet Ah Heaven,” in which the initial letter of each word corresponds to the initial letter in the name of each nerve. The names of the nerves have changed over the years to reflect current usage and more accurate naming. An exercise to help learn this sort of information is to generate a mnemonic using words that have personal significance. The names of the cranial nerves are listed in Table \(\PageIndex{1}\) along with a brief description of their function, their source (sensory ganglion or motor nucleus), and their target (sensory nucleus or skeletal muscle). The olfactory nerve and optic nerve are responsible for the sense of smell and vision, respectively. The oculomotor nerve is responsible for eye movements by controlling four of the extraocular muscles. It is also responsible for lifting the upper eyelid when the eyes point up, and for pupillary constriction. The trochlear nerve and the abducens nerve are both responsible for eye movement, but do so by controlling different extraocular muscles. The trigeminal nerve is responsible for cutaneous sensations of the face and controlling the muscles of mastication. The facial nerve is responsible for the muscles involved in facial expressions, as well as part of the sense of taste and the production of saliva. The vestibulocochlear nerve is responsible for the senses of hearing and balance. The glossopharyngeal nerve is responsible for controlling muscles in the oral cavity and upper throat, as well as part of the sense of taste and the production of saliva. The vagus nerve is responsible for contributing to homeostatic control of the organs of the thoracic and upper abdominal cavities. The accessory nerve is responsible for controlling the muscles of the neck, along with cervical spinal nerves. The hypoglossal nerve is responsible for controlling the muscles of the lower throat and tongue.
Four of these cranial nerves make up the cranial component of the autonomic nervous system responsible for pupillary constriction (oculomotor nerve), salivation and lacrimation (facial and glossopharyngeal nerves), and the regulation of the organs of the thoracic and upper abdominal cavities (vagus nerve).
Three of the cranial nerves also contain autonomic fibers, and a fourth is almost purely a component of the autonomic system. The oculomotor, facial, and glossopharyngeal nerves contain fibers that contact autonomic ganglia. The oculomotor fibers initiate pupillary constriction, whereas the facial and glossopharyngeal fibers both initiate salivation. The vagus nerve primarily targets autonomic ganglia in the thoracic and upper abdominal cavities.
Another important aspect of the cranial nerves that lends itself to a mnemonic is the functional role each nerve plays. The nerves fall into one of three basic groups. They are sensory, motor, or both (see Table \(\PageIndex{1}\)). The sentence, “Some Say Marry Money But My Brother Says Brains Beauty Matter More,” corresponds to the basic function of each nerve. The first, second, and eighth nerves are purely sensory: the olfactory (CNI), optic (CNII), and vestibulocochlear (CNVIII) nerves. The three eye-movement nerves are all motor: the oculomotor (CNIII), trochlear (CNIV), and abducens (CNVI). The accessory (CNXI) and hypoglossal (CNXII) nerves are also strictly motor. The remainder of the nerves contain both sensory and motor fibers. They are the trigeminal (CNV), facial (CNVII), glossopharyngeal (CNIX), and vagus (CNX) nerves. The nerves that convey both are often related to each other. The trigeminal and facial nerves both concern the face; one concerns the sensations and the other concerns the muscle movements. The facial and glossopharyngeal nerves are both responsible for conveying gustatory, or taste, sensations as well as controlling salivary glands. The vagus nerve is involved in visceral responses to taste, namely the gag reflex. This is not an exhaustive list of what these combination nerves do, but there is a thread of relation between them.
| Mnemonic | # | Name | Function (S/M/B) | Central connection (nuclei) | Peripheral connection (ganglion or muscle) |
|---|---|---|---|---|---|
| Oh | I | Olfactory | Smell (S) | Olfactory bulb | Olfactory epithelium |
| Oh | II | Optic | Vision (S) | Hypothalamus/thalamus/midbrain | Retina (retinal ganglion cells) |
| Oh | III | Oculomotor | Eye movements (M) | Oculomotor nucleus | Extraocular muscles (other 4), levator palpebrae superioris, ciliary ganglion (autonomic) |
| To | IV | Trochlear | Eye movements (M) | Trochlear nucleus | Superior oblique muscle |
| Touch | V | Trigeminal | Sensory/motor – face (B) | Trigeminal nuclei in the midbrain, pons, and medulla | Trigeminal |
| And | VI | Abducens | Eye movements (M) | Abducens nucleus | Lateral rectus muscle |
| Feel | VII | Facial | Motor – face, Taste (B) | Facial nucleus, solitary nucleus, superior salivatory nucleus | Facial muscles, Geniculate ganglion, Pterygopalatine ganglion (autonomic) |
| Very | VIII | Vestibulocochlear | Hearing/balance (S) | Cochlear nucleus, Vestibular nucleus/cerebellum | Spiral ganglion (hearing), Vestibular ganglion (balance) |
| Good | IX | Glossopharyngeal | Motor – throat Taste (B) | Solitary nucleus, inferior salivatory nucleus, nucleus ambiguus | Pharyngeal muscles, Geniculate ganglion, Otic ganglion (autonomic) |
| Velvet | X | Vagus | Motor/sensory – viscera (autonomic) (B) | Medulla | Terminal ganglia serving thoracic and upper abdominal organs (heart and small intestines) |
| Ah | XI | Accessory | Motor – head and neck (M) | Spinal accessory nucleus | Neck muscles |
| Heaven | XII | Hypoglossal | Motor – lower throat (M) | Hypoglossal nucleus | Muscles of the larynx and lower pharynx |
Cranial Nerve Ganglia
A ganglion (ganglia for plural) is a group of neuron cell bodies in the peripheral nervous system. Ganglia can be categorized, for the most part, as either sensory ganglia or autonomic ganglia, referring to their primary functions. A type of sensory ganglion is a cranial nerve ganglion . The roots of cranial nerves are within the cranium, whereas the ganglia are outside the skull. For example, the trigeminal ganglion is superficial to the temporal bone whereas its associated nerve is attached to the mid-pons region of the brainstem. The neurons of cranial nerve ganglia are also unipolar in shape with associated satellite cells.
Interactive Link
Vision Loss
Read this article about a man who wakes with a headache and a loss of vision . His regular doctor sent him to an ophthalmologist to address the vision loss. The ophthalmologist recognizes a greater problem and immediately sends him to the emergency room. Once there, the patient undergoes a large battery of tests, but a definite cause cannot be found. A specialist recognizes the problem as meningitis, but the question is what caused it originally. How can that be cured? The loss of vision comes from swelling around the optic nerve, which probably presented as a bulge on the inside of the eye. Why is swelling related to meningitis going to push on the optic nerve?
- Answer
-
The optic nerve enters the CNS in its projection from the eyes in the periphery, which means that it crosses through the meninges. Meningitis will include swelling of those protective layers of the CNS, resulting in pressure on the optic nerve, which can compromise vision.
AGING AND THE...
Nervous System: Anosmia
Anosmia is the loss of the sense of smell. It is often the result of the olfactory nerve being severed, usually because of blunt force trauma to the head. The sensory neurons of the olfactory epithelium have a limited lifespan of approximately one to four months, and new ones are made on a regular basis. The new neurons extend their axons into the CNS by growing along the existing fibers of the olfactory nerve. The ability of these neurons to be replaced is lost with age. Age-related anosmia is not the result of impact trauma to the head, but rather a slow loss of the sensory neurons with no new neurons born to replace them.
Smell is an important sense, especially for the enjoyment of food. There are only five tastes sensed by the tongue, and two of them are generally thought of as unpleasant tastes (sour and bitter). The rich sensory experience of food is the result of odor molecules associated with the food, both as food is moved into the mouth, and therefore passes under the nose, and when it is chewed and molecules are released to move up the pharynx into the posterior nasal cavity. Anosmia results in a loss of the enjoyment of food.
As the replacement of olfactory neurons declines with age, anosmia can set in. Without the sense of smell, many sufferers complain of food tasting bland. Often, the only way to enjoy food is to add seasoning that can be sensed on the tongue, which usually means adding table salt. The problem with this solution, however, is that this increases sodium intake, which can lead to cardiovascular problems through water retention and the associated increase in blood pressure.
Concept Review
The PNS is composed of the groups of neurons (ganglia) and bundles of axons (nerves) that are outside of the brain and spinal cord. Nerves are organized into structures by layers of connective tissue that cover them. The epineurium covers the nerve, the perineurium covers the fascicles and the endoneurium covers the individual axon.
Nerves are classified as cranial nerves or spinal nerves on the basis of their connection to the brain or spinal cord, respectively. The twelve cranial nerves can be strictly sensory in function, strictly motor in function, or a combination of the two functions. The olfactory nerve (CN I) and optic nerve (CN II) are responsible for the sense of smell and vision, respectively. The oculomotor nerve (CN III) is responsible for eye movements, lifting the upper eyelid and size of the pupil. The trochlear nerve (CN IV) and the abducens nerve (CN VI) are both responsible for eye movement, but do so by controlling different extraocular muscles. The trigeminal nerve (CN V) is responsible for cutaneous sensations of the face and controlling the muscles of mastication. The facial nerve (VII) is responsible for the muscles involved in facial expressions, as well as part of the sense of taste and the production of saliva. The vestibulocochlear nerve (VIII) is responsible for the senses of hearing and balance. The glossopharyngeal nerve (IX) is responsible for controlling muscles in the oral cavity and upper throat, as well as part of the sense of taste and the production of saliva. The vagus nerve (CN X) is responsible for contributing to homeostatic control of the organs of the thoracic and upper abdominal cavities. The accessory nerve (CN XI) is responsible for controlling the muscles of the neck, along with cervical spinal nerves. The hypoglossal nerve (CN XII) is responsible for controlling the muscles of the lower throat and tongue.
Ganglia are of two types, sensory or autonomic. Sensory ganglia contain unipolar sensory neurons and are associated with many of the cranial nerves.
Review Questions
Q. What type of ganglion contains neurons that control homeostatic mechanisms of the body?
A. sensory ganglion
B. dorsal root ganglion
C. autonomic ganglion
D. ventral root ganglion
- Answer
-
C
Q. Which ganglion is responsible for cutaneous sensations of the face?
A. otic ganglion
B. vestibular ganglion
C. geniculate ganglion
D. trigeminal ganglion
- Answer
-
D
Q. What is the name for a bundle of axons within a nerve?
A. fascicle
B. tract
C. nerve root
D. epineurium
- Answer
-
A
Q. Which cranial nerve does not control organs in the head and neck?
A. olfactory
B. trochlear
C. glossopharyngeal
D. vagus
- Answer
-
D
Critical Thinking Questions
Q. Why are ganglia and nerves not surrounded by protective structures like the meninges of the CNS?
A. The peripheral nervous tissues are out in the body, sometimes part of other organ systems. There is not a privileged blood supply like there is to the brain and spinal cord, so peripheral nervous tissues do not need the same sort of protections.
Q. Testing for neurological function involves a series of tests of functions associated with the cranial nerves. What functions, and therefore which nerves, are being tested by asking a patient to follow the tip of a pen with their eyes?
A. The contraction of extraocular muscles is being tested, which is the function of the oculomotor, trochlear, and abducens nerves.
Glossary
- abducens nerve
- sixth cranial nerve; responsible for contraction of one of the extraocular muscles
- accessory nerve
- eleventh cranial nerve; responsible for contraction of neck muscles
- cranial nerve
- one of twelve nerves connected to the brain that are responsible for sensory or motor functions of the head and neck
- cranial nerve ganglion
- sensory ganglion of cranial nerves
- endoneurium
- innermost layer of connective tissue that surrounds individual axons within a nerve
- epineurium
- outermost layer of connective tissue that surrounds an entire nerve
- facial nerve
- seventh cranial nerve; responsible for contraction of the facial muscles and for part of the sense of taste, as well as causing saliva production
- fascicle
- small bundles of nerve or muscle fibers enclosed by connective tissue
- glossopharyngeal nerve
- ninth cranial nerve; responsible for contraction of muscles in the tongue and throat and for part of the sense of taste, as well as causing saliva production
- hypoglossal nerve
- twelfth cranial nerve; responsible for contraction of muscles of the tongue
- oculomotor nerve
- third cranial nerve; responsible for contraction of four of the extraocular muscles, the muscle in the upper eyelid, and pupillary constriction
- olfactory nerve
- first cranial nerve; responsible for the sense of smell
- optic nerve
- second cranial nerve; responsible for visual sensation
- perineurium
- layer of connective tissue surrounding fascicles within a nerve
- trigeminal ganglion
- sensory ganglion that contributes sensory fibers to the trigeminal nerve
- trigeminal nerve
- fifth cranial nerve; responsible for cutaneous sensation of the face and contraction of the muscles of mastication
- trochlear nerve
- fourth cranial nerve; responsible for contraction of one of the extraocular muscles
- vagus nerve
- tenth cranial nerve; responsible for the autonomic control of organs in the thoracic and upper abdominal cavities
- vestibulocochlear nerve
- eighth cranial nerve; responsible for the sensations of hearing and balance
Contributors and Attributions
OpenStax Anatomy & Physiology (CC BY 4.0). Access for free at https://openstax.org/books/anatomy-and-physiology


9.9: References
Agency for Healthcare Research and Quality. (2013). Which fall prevention practices do you want to use? https://www.ahrq.gov/patient-safety/...practices.html
Albanese, B., Piredda, M., Bravi, M., Bressi, F., Gualandi, R., Marchetti, A., Facchinetti, G., Ianni, A., Cordella, F., Zollo, L., & De Marinis, M. G. (2022). Interventions to prevent and reduce work-related musculoskeletal injuries and pain among healthcare professionals: A comprehensive systematic review of the literature. Journal of Safety Research, 82 , 124–143. https://doi.org/10.1016/j.jsr.2022.05.004
American Nurses Association. (2012). Position statement: Reduction of patient restraint and seclusion in health care settings . https://www.nursingworld.org/practic...care-settings/
Anderson, T. B., & Duong, H. (2023). Weight bearing. In StatPearls [Internet]. StatPearls Publishing. www.ncbi.nlm.nih.gov/books/NBK551573/
Bergman, R., & De Jesus, O. (2022). Patient care transfer technique. In StatPearls [Internet]. StatPearls Publishing. www.ncbi.nlm.nih.gov/books/NBK564305/
Brennan, M. M. (2023). Movement is muscle in hospitalized adults. Geriatric Nursing, In Press . www.sciencedirect.com/scienc...97457223002999
Campbell, B. J. (2021). Healthy bones at every age. OrthoInfo. https://orthoinfo.aaos.org/en/stayin...-at-every-age/
Centers for Disease Control and Prevention. (2023). Disability impacts all of us . https://www.cdc.gov/ncbddd/disabilit...pacts-all.html
Cleveland Clinic. (2022). Sarcopenia. my.clevelandclinic.org/healt...167-sarcopenia
Fuller-Thompson, E., Ferreirinha, J., & Ahlin, K. M. (2023). Temporal trends (from 2008 to 2017) in functional limitations and limitations in activities of daily living: Findings from a nationally representative sample of 5.4 million older Americans. International Journal of Environmental Research and Public Health , 20 (3), 2665. https://doi.org/10.3390/ijerph20032665
Gyasi, R. M., Asante, F., Hambali, M. G., Odei, J., Jacob, L., Obeng, B., Peprah, P., Asamoah, E., Agyemang-Duah, W., Abass, K., Asiki, G., & Adam, A. M. (2023). Mobility limitations and emotional dysfunction in old age: The moderating effects of physical activity and social ties. International Journal of Geriatric Psychiatry, 38 (7), e5969. doi.org/10.1002/gps.5969
Joint Commission Resources. (2018). Environmental infection prevention: Guidance for continuously maintaining a safe patient care and survey-ready environment. https://store.jcrinc.com/assets/1/7/..._%28002%29.pdf
Kiyoshi-Teo, H., Northrup-Snyder, K., Cohen, D. J., Dieckmann, N., Stoyles, S., Winters-Stone, K., & Eckstrom, E. (2019). Older hospital inpatients’ fall risk factors, perceptions, and daily activities to prevent falling. Geriatric Nursing , 40 (3), 290–295. https://doi.org/10.1016/j.gerinurse.2018.11.005
National Council of State Boards of Nursing. (n.d). Clinical judgement measurement model . Next Generation NCLEX. https://www.nclex.com/clinical-judgm...ent-model.page
National Institute on Aging. (2020). Research highlights: Maintaining mobility and preventing disability are key to living independently as we age . www.nia.nih.gov/news/maintai...ndently-we-age
Orellano-Colon, E. M., Mann, W. C., Rivero, M., Torres, M., Jutai, J., Santiago, A., & Varas, N. (2016). Hispanic older adult’s perceptions of personal, contextual, and technology-related barriers for using assistive technology devices. Journal of Racial and Ethnic Health Disparities, 3 (4), 678–686. www.ncbi.nlm.nih.gov/pmc/articles/PMC4911321/
Parkes, D., & Tadi, P. (2022). Patient restraint and seclusion. In StatPearls [Internet]. StatPearls Publishing. www.ncbi.nlm.nih.gov/books/NBK565873/
Physiopedia. (n.d.). Assessing range of motion . www.physio-pedia.com/Assessi...ange_of_Motion
Piercy, K. L., & Troiano, R. P. (2018). Physical activity guidelines for Americans from the US Department of Health and Human Services. Circulation , 11 (11), e005263. doi.org/10.1161/CIRCOUTCOMES.118.005263
Sheehan, M. C., & Tucker-Drob, E. M. (2019). Gendered expectations distort male-female differences in instrumental activities of daily living in later adulthood. Journals of Gerontology. Series B, Psychological Sciences and Social Sciences, 74 (4). https://doi.org/10.1093/geronb/gbw209
Stenhold, S., Shardell, M., Bandinelli, S., Guralnik, J. M., & Ferrucci, L. (2015). Physiological factors contributing to mobility loss over 9 years of follow-up – results from the InCHIANTI study. Journal of Gerentology, 70 (5), 591–597. https://doi.org/doi:10.1093/gerona/glv004
The Joint Commission. (2022). Quick safety 25: Preventing pressure injuries . https://www.jointcommission.org/reso...sure-injuries/
The Joint Commission. (2023a). Standard PC.02.09. Comprehensive Accreditation Manual for Hospitals .
The Joint Commission. (2023b). Standard PC.03.03.13. Comprehensive Accreditation Manual for Hospitals .
U.S. Department of Health and Human Services & Office of Disease Prevention and Health Promotion. (n.d.). Social determinants of health. Healthy People 2030 . https://health.gov/healthypeople/pri...minants-health
U.S. Department of Health and Human Services. (n.d.) Think culture health: Culturally and linguistically appropriate services . https://thinkculturalhealth.hhs.gov/clas/what-is-clas
Vickery, J., Compton, L., Allard, J., Beeson, T., Howard, J., & Pittman, J. (2020). Pressure injury prevention and wound management for the patient who is actively dying: Evidence-based recommendations to guide care. Journal of Wound, Ostomy, & Continence Nursing, 47 (6), 569–575.


18.6: Summary
18.1 Nursing Care During the First Stage of Labor
Starting at the time the laboring person arrives in the obstetric triage unit, the nurse establishes a relationship with the person during this first stage in the process of becoming a parent. The nurse’s role throughout the first stage of labor can be a simple task, such as obtaining ice chips for dry mucous membranes, or more complex, such as facilitating epidural anesthesia for pain control measures. Whether the care is as simple as getting ice chips or as complicated as fetal monitor interpretation, the nurse is the care provider at the bedside. Knowledge of the process of labor and birth, pain management techniques, fetal and contraction monitoring, and monitoring for complications is applied by the nurse caring for a person throughout the first stage of labor.
18.2 Nursing Care During the Second Stage of Labor
The second stage of labor is an exciting time for families. The birthing person is now fully dilated, and the birth of the newborn is near. When it is time for pushing, the nurse will be available to direct the birthing person with pushing efforts when necessary. During the second stage, the nurse will also assist the birthing person with pain management, positioning, and the birth itself while monitoring the fetus and birthing person continuously.
18.3 Nursing Care During the Third Stage of Labor
The third stage of labor starts immediately after the delivery of the newborn and ends with the delivery of the placenta. Nursing care during the third stage of labor includes assessment of the birthing person and newborn. The nurse will need to assess the birthing person’s vital signs, pain or discomfort, bleeding, uterus, and bladder. For the newborn, the nurse will assess vital signs, assign an Apgar score, and monitor the newborn’s temperature throughout the third stage of labor. The nurse may also initiate skin-to-skin contact between the newborn and the birthing person, while also coordinating clamping of the umbilical cord. The nurse must be aware of potential complications during the third stage of labor related to placental separation and expulsion, such as postpartum hemorrhage.
18.4 Nursing Care During the Fourth Stage of Labor
The fourth stage of labor is a vital time for the nurse to prevent complications while promoting family bonding. Skin-to-skin contact between the newborn and parent can promote bonding and breast-feeding. The nurse will continue to monitor for postpartum bleeding and other postdelivery complications. Patient education is also important during this time, as the birthing person and their partner are learning about their newborn and may be new to parenting. The nurse must provide clinical and emotional support for the birthing person and their family, keeping in mind cultural and personal preferences.


14.18: Marketing Communication Program (SMART)
- Outline the steps involved in the creation of a marketing communication program (SMART)
As you consider your advertising and promotion plan, it’s important to keep three (3) things in-mind:
- How can we maximize coverage, given budgets?
- What channels will be the most effective?
- How will we assess campaign performance?
The first should be self-evident. While there may be temptation to match the proliferation of channels with an effort to be present in each, there is a finite limit to what organizations can and should invest in their marketing activity, given their strategy, resources and the competitive environment. Considering the costs of developing content and securing access in each channel, it would be incredibly expensive to run campaigns in all channels.
Further, being present in every channel may lead to diminishing returns. It’s rarely possible and seldom advisable to spread messages across all channels, knowing that the increased reach will likely mean sacrificing frequency and the need for multiple impressions. Consider the question we posed earlier, “Was each ad beneficial, or did some get ‘tuned out’ because the consumer had already decided that the product was right for them?”
Think about it this way. If a marketer messages on its website, Facebook page, Twitter feed, then pays for banner ads, billboards, print ads in magazines and television commercials, the message has incredible reach and assumed frequency, likely generating high impressions. What would the further benefit be of incorporating radio into the campaign? What if we messaged on Instagram or added paid search? Would this be duplicative to the television activity? Or, would it be complementary?
One way to answer the question is to think of the second question. That is, examine the Customer Journey to understand how consumers shop the product or service and how the channels function to deliver information. In what channels are the target consumers present and engaged? What are their explicit needs at given contact points and in each channel. How does the channel meet a shopper’s needs for information or support? Instead of spreading marketing investments across all channels, like spreading peanut butter across a slice of bread, most marketers will “over-invest” in specific channels, selecting the ones that provide the greatest benefit. This could mean the widest audience, the highest level of interaction, the best fit with the specific target market or another criterion.
For example, if you’re responsible for a well-known brand that generates organic, i.e. natural, web traffic to your own site, it may mean paid search is less important for you. But, this also means that SEO is critically important, so that your site ranks well and shoppers can find what they’re looking for.
Or, if you find that a high percentage of your followers on social media are prospects, it might be worthwhile to invest in creative and messaging there. (By some estimates, existing customers outnumber prospects on social media by 4:1. However, that also means that 20% of followers are prospective customers—that shouldn’t be overlooked.)
If your product/ service is complex or highly visual, you might want to consider how best to share details with your target. If complex, would a magazine ad work well? What about blogs or video blogs where influencers describe the product and benefits?
If it’s a visual item, would you ever consider radio? Or, would print work best?
And, of course, we need to determine how best to measure the effectiveness of the campaign, i.e. the messaging, channels and impact on consumers. Like allocating a budget, the determination of a campaign’s performance is specific to the firm’s strategy, resources and the competitive environment. For example, if an organization is launching a new product or service, their priority might be upon building initial consumer awareness and trial. We see this commonly in the grocery industry, when new and seasonal items are featured in advertising and in-store merchandising, e.g. Oui™ French style yogurt by Yoplait® and Hershey’s® Gold caramel crème candy bar.
Now, compare this to our evaluation of The Coca-Cola Company. In them we saw a dominant, global brand advertising products with widespread distribution and consumption. Clearly, they aren’t trying to develop awareness and trial. More likely, in this case, they’re advertising to reinforce the emotional benefits of drinking the beverage to support brand affinity and to keep it “top of mind.” These goals are much different than what’s in play for a firm supporting new items or services.
Alternatively, advertising can be a useful tool, if the brand equity has been impaired– if a firm has suffered a public relations issue. In this case, the intent of the marketing activity might be to improve customer perception. After their well-publicized account fraud scandal brought about by the creation of millions of fraudulent savings and checking accounts without customer consent, Wells Fargo has initiated an advertising campaign to “Earn back your trust.” Clearly, these messages are aimed at restoring confidence in the brand and organization, following the scandal and resulting $185 million in fines & penalties.
Regardless of the motivation for or objectives of the advertising campaign, marketers should be guided by the mnemonic acronym SMART. Independent of the specific goals, i.e. reach, impressions, brand awareness or other measures, SMART gives criteria to guide in the setting of objectives. The letters S and M usually mean specific and measurable. Possibly the most common version has the remaining letters referring to achievable, relevant and time-bound.
SMART was introduced in a paper by T. Doran called There’s a S.M.A.R.T. way to write management’s goals and objectives . Subsequent authors have adjusted their meaning slightly to:
- Specific, i.e. targeting a specific area for improvement.
- Measurable, i.e. quantifiable or at least suggest an indicator of progress.
- Achievable, i.e. realistically be achievable, given available resources.
- Relevant, i.e. applicable to the current objective and supportive of the broader strategy
- Time-bound, i.e. specific to a given period when the result(s) can/ should be achieved
It’s important to apply these criteria to assessment of the campaign. Without them and clear understanding of the scope, it becomes too difficult to understand the impact of the marketing activity. For example, without a specific objective, is customer awareness more important or sales (as an indicator of trial)? If there aren’t measures or the objectives aren’t actionable, can we accurately assess correlation or causation? By the same token, if the objectives aren’t realistic, do we get an incorrect impression of the marketing activity’s impact for good or for bad? And, if not time-bound, will we know how quickly to make adjustments to our plan to improve return?
Thus, it’s important to keep three things in-mind, when building an advertising and promotion plan:
- How can we maximize coverage, given budgets?
- What channels will be the most effective?
- How will we assess campaign performance?
Avoid the temptation to match the proliferation of channels with an effort to be present in each, given the costs of development and the risk of diminishing returns. Examine the Customer Journey to understand how consumers shop the product or service and how the channels function to deliver information. Apply the SMART criteria to assess the campaign objectives to manage scope and to understand impact.
Contributors and Attributions
- Marketing Communication Program (SMART). Authored by : Patrick Williams. Provided by : Lumen Learning. License : CC BY: Attribution
- SMART criteria. Provided by : Wikipedia. Located at : https://en.wikipedia.org/wiki/SMART_criteria . License : CC BY-SA: Attribution-ShareAlike


5.11: The F Distribution
In this section we will study a distribution that has special importance in statistics. In particular, this distribution arises from ratios of sums of squares when sampling from a normal distribution, and so is important in estimation and in the two-sample normal model and in hypothesis testing in the two-sample normal model.
Basic Theory
Definition
Suppose that \(U\) has the chi-square distribution with \(n \in (0, \infty)\) degrees of freedom, \(V\) has the chi-square distribution with \(d \in (0, \infty)\) degrees of freedom, and that \(U\) and \(V\) are independent. The distribution of \[ X = \frac{U / n}{V / d} \] is the \(F\) distribution with \(n\) degrees of freedom in the numerator and \(d\) degrees of freedom in the denominator.
The \(F\) distribution was first derived by George Snedecor, and is named in honor of Sir Ronald Fisher. In practice, the parameters \( n \) and \( d \) are usually positive integers, but this is not a mathematical requirement.
Distribution Functions
Suppose that \(X\) has the \( F \) distribution with \( n \in (0, \infty) \) degrees of freedom in the numerator and \( d \in (0, \infty) \) degrees of freedom in the denominator. Then \( X \) has a continuous distribution on \( (0, \infty) \) with probability density function \( f \) given by \[ f(x) = \frac{\Gamma(n/2 + d/2)}{\Gamma(n / 2) \Gamma(d / 2)} \frac{n}{d} \frac{[(n/d) x]^{n/2 - 1}}{\left[1 + (n / d) x\right]^{n/2 + d/2}}, \quad x \in (0, \infty) \] where \( \Gamma \) is the gamma function.
Proof
The trick, once again, is conditioning. The conditional distribution of \( X \) given \( V = v \in (0, \infty) \) is gamma with shape parameter \( n/2 \) and scale parameter \( 2 d / n v \). Hence the conditional PDF is \[ x \mapsto \frac{1}{\Gamma(n/2) \left(2 d / n v\right)^{n/2}} x^{n/2 - 1} e^{-x(nv /2d)} \] By definition, \( V \) has the chi-square distribution with \( d \) degrees of freedom, and so has PDF \[ v \mapsto \frac{1}{\Gamma(d/2) 2^{d/2}} v^{d/2 - 1} e^{-v/2} \] The joint PDF of \( (X, V) \) is the product of these functions: \[g(x, v) = \frac{1}{\Gamma(n/2) \Gamma(d/2) 2^{(n+d)/2}} \left(\frac{n}{d}\right)^{n/2} x^{n/2 - 1} v^{(n+d)/2 - 1} e^{-v( n x / d + 1)/2}; \quad x, \, v \in (0, \infty)\] The PDF of \( X \) is therefore \[ f(x) = \int_0^\infty g(x, v) \, dv = \frac{1}{\Gamma(n/2) \Gamma(d/2) 2^{(n+d)/2}} \left(\frac{n}{d}\right)^{n/2} x^{n/2 - 1} \int_0^\infty v^{(n+d)/2 - 1} e^{-v( n x / d + 1)/2} \, dv \] Except for the normalizing constant, the integrand in the last integral is the gamma PDF with shape parameter \( (n + d)/2 \) and scale parameter \( 2 d \big/ (n x + d) \). Hence the integral evaluates to \[ \Gamma\left(\frac{n + d}{2}\right) \left(\frac{2 d}{n x + d}\right)^{(n + d)/2} \] Simplifying gives the result.
Recall that the beta function \( B \) can be written in terms of the gamma function by \[ B(a, b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a + b)},\ \quad a, \, b \in (0, \infty) \] Hence the probability density function of the \( F \) distribution above can also be written as \[ f(x) = \frac{1}{B(n/2, d/2)} \frac{n}{d} \frac{[(n/d) x]^{n/2 - 1}}{\left[1 + (n / d) x\right]^{n/2 + d/2}}, \quad x \in (0, \infty) \] When \( n \ge 2 \), the probability density function is defined at \( x = 0 \), so the support interval is \( [0, \infty) \) is this case.
In the special distribution simulator, select the \(F\) distribution. Vary the parameters with the scroll bars and note the shape of the probability density function. For selected values of the parameters, run the simulation 1000 times and compare the empirical density function to the probability density function.
Both parameters influence the shape of the \( F \) probability density function, but some of the basic qualitative features depend only on the numerator degrees of freedom. For the remainder of this discussion, let \( f \) denote the \( F \) probability density function with \( n \in (0, \infty) \) degrees of freedom in the numerator and \( d \in (0, \infty) \) degrees of freedom in the denominator.
Probability density function \( f \) satisfies the following properties:
- If \( 0 \lt n \lt 2 \), \( f \) is decreasing with \( f(x) \to \infty \) as \( x \downarrow 0 \).
- If \( n = 2 \), \( f \) is decreasing with mode at \( x = 0 \).
- If \( n \gt 2 \), \(f\) increases and then decreases, with mode at \(x = \frac{(n - 2) d}{n (d + 2)}\).
Proof
These properties follow from standard calculus. The first derivative of \( f \) is \[ f^\prime(x) = \frac{1}{B(n/2, d/2)} \left(\frac{n}{d}\right)^2 \frac{[(n/d)x]^{n/2-2}}{[1 + (n/2)x]^{n/2 + d/2 + 1}} [(n/2 - 1) - (n/d)(d/2 + 1)x], \quad x \in (0, \infty) \]
Qualitatively, the second order properties of \( f \) also depend only on \( n \), with transitions at \( n = 2 \) and \( n = 4 \).
For \( n \gt 2 \), define \begin{align} x_1 & = \frac{d}{n} \frac{(n - 2)(d + 4) - \sqrt{2 (n - 2)(d + 4)(n + d)}}{(d + 2)(d + 4)} \\ x_2 & = \frac{d}{n} \frac{(n - 2)(d + 4) + \sqrt{2 (n - 2)(d + 4)(n + d)}}{(d + 1)(d + 4)} \end{align} The probability density function \( f \) satisfies the following properties:
- If \( 0 \lt n \le 2 \), \( f \) is concave upward.
- If \( 2 \lt n \le 4 \), \( f \) is concave downward and then upward, with inflection point at \( x_2 \).
- If \( n \gt 4 \), \( f \) is concave upward, then downward, then upward again, with inflection points at \( x_1 \) and \( x_2 \).
Proof
These results follow from standard calculus. The second derivative of \( f \) is \[ f^{\prime\prime}(x) = \frac{1}{B(n/2, d/2)} \left(\frac{n}{d}\right)^3 \frac{[(n/d)x]^{n/2-3}}{[1 + (n/d)x]^{n/2 + d/2 + 2}}\left[(n/2 - 1)(n/2 - 2) - 2 (n/2 - 1)(d/2 + 2) (n/d) x + (d/2 + 1)(d/2 + 2)(n/d)^2 x^2\right], \quad x \in (0, \infty) \]
The distribution function and the quantile function do not have simple, closed-form representations. Approximate values of these functions can be obtained from the special distribution calculator and from most mathematical and statistical software packages.
In the special distribution calculator, select the \(F\) distribution. Vary the parameters and note the shape of the probability density function and the distribution function. In each of the following cases, find the median, the first and third quartiles, and the interquartile range.
- \(n = 5\), \(d = 5\)
- \(n = 5\), \(d = 10\)
- \(n = 10\), \(d = 5\)
- \(n = 10\), \(d = 10\)
The general probability density function of the \( F \) distribution is a bit complicated, but it simplifies in a couple of special cases.
Special cases.
- If \( n = 2 \), \[ f(x) = \frac{1}{(1 + 2 x / d)^{1 + d / 2}}, \quad x \in (0, \infty) \]
- If \( n = d \in (0, \infty)\), \[ f(x) = \frac{\Gamma(n)}{\Gamma^2(n/2)} \frac{x^{n/2-1}}{(1 + x)^n}, \quad x \in (0, \infty)\]
- If \( n = d = 2 \), \[ f(x) = \frac{1}{(1 + x)^2}, \quad x \in (0, \infty) \]
- If \( n = d = 1 \), \[ f(x) = \frac{1}{\pi \sqrt{x}(1 + x)}, \quad x \in (0, \infty) \]
Moments
The random variable representation in the definition , along with the moments of the chi-square distribution can be used to find the mean, variance, and other moments of the \( F \) distribution. For the remainder of this discussion, suppose that \(X\) has the \(F\) distribution with \(n \in (0, \infty)\) degrees of freedom in the numerator and \(d \in (0, \infty)\) degrees of freedom in the denominator.
Mean
- \(\E(X) = \infty\) if \(0 \lt d \le 2\)
- \(\E(X) = \frac{d}{d - 2}\) if \(d \gt 2\)
Proof
By independence, \( \E(X) = \frac{d}{n} \E(U) \E\left(V^{-1}\right) \). Recall that \( \E(U) = n \). Similarly if \( d \le 2 \), \( \E\left(V^{-1}\right) = \infty \) while if \( d \gt 2 \), \[ \E\left(V^{-1}\right) = \frac{\Gamma(d/2 - 1)}{2 \Gamma(d/2)} = \frac{1}{d - 2} \]
Thus, the mean depends only on the degrees of freedom in the denominator.
Variance
- \(\var(X)\) is undefined if \(0 \lt d \le 2\)
- \(\var(X) = \infty\) if \(2 \lt d \le 4\)
- If \(d \gt 4\) then \[ \var(X) = 2 \left(\frac{d}{d - 2} \right)^2 \frac{n + d - 2}{n (d - 4)} \]
Proof
By independence, \( \E\left(X^2\right) = \frac{d^2}{n^2} \E\left(U^2\right) \E\left(V^{-2}\right) \). Recall that \[ E(\left(U^2\right) = 4 \frac{\Gamma(n/2 + 2)}{\Gamma(n/2)} = (n + 2) n \] Similarly if \( d \le 4 \), \( \E\left(V^{-2}\right) = \infty \) while if \( d \gt 4 \), \[ \E\left(V^{-2}\right) = \frac{\Gamma(d/2 - 2)}{4 \Gamma(d/2)} = \frac{1}{(d - 2)(d - 4)} \] Hence \( \E\left(X^2\right) = \infty \) if \( d \le 4 \) while if \( d \gt 4 \), \[ \E\left(X^2\right) = \frac{(n + 2) d^2}{n (d - 2)(d - 4)} \] The results now follow from the previous result on the mean and the computational formula \( \var(X) = \E\left(X^2\right) - \left[\E(X)\right]^2 \).
In the simulation of the special distribution simulator, select the \(F\) distribution. Vary the parameters with the scroll bar and note the size and location of the mean \( \pm \) standard deviation bar. For selected values of the parameters, run the simulation 1000 times and compare the empirical mean and standard deviation to the distribution mean and standard deviation..
General moments . For \( k \gt 0 \),
- \(\E\left(X^k\right) = \infty\) if \(0 \lt d \le 2 k\)
- If \(d \gt 2 k\) then \[ \E\left(X^k\right) = \left( \frac{d}{n} \right)^k \frac{\Gamma(n/2 + k) \, \Gamma(d/2 - k)}{\Gamma(n/2) \Gamma(d/2)} \]
Proof
By independence, \( \E\left(X^k\right) = \left(\frac{d}{n}\right)^k \E\left(U^k\right) \E\left(V^{-k}\right) \). Recall that \[ \E\left(U^k\right) = \frac{2^k \Gamma(n/2 + k)}{\Gamma(n/2)} \] On the other hand, \( \E\left(V^{-k}\right) = \infty \) if \( d/2 \le k \) while if \( d/2 \gt k \), \[ \E\left(V^{-k}\right) = \frac{2^{-k} \Gamma(d/2 - k)}{\Gamma(d/2)} \]
If \( k \in \N \), then using the fundamental identity of the gamma distribution and some algebra, \[ \E\left(X^{k}\right) = \left(\frac{d}{n}\right)^k \frac{n (n + 2) \cdots [n + 2(k - 1)]}{(d - 2)(d - 4) \cdots (d - 2k)} \] From the general moment formula, we can compute the skewness and kurtosis of the \( F \) distribution.
Skewness and kurtosis
- If \( d \gt 6 \), \[ \skw(X) = \frac{(2 n + d - 2) \sqrt{8 (d - 4)}}{(d - 6) \sqrt{n (n + d - 2)}} \]
- If \( d \gt 8 \), \[ \kur(X) = 3 + 12 \frac{n (5 d - 22)(n + d - 2) + (d - 4)(d-2)^2}{n(d - 6)(d - 8)(n + d - 2)} \]
Proof
These results follow from the formulas for \( \E\left(X^k\right) \) for \( k \in \{1, 2, 3, 4\} \) and the standard computational formulas for skewness and kurtosis.
Not surprisingly, the \( F \) distribution is positively skewed. Recall that the excess kurtosis is \[ \kur(X) - 3 = 12 \frac{n (5 d - 22)(n + d - 2) + (d - 4)(d-2)^2}{n(d - 6)(d - 8)(n + d - 2)}\]
In the simulation of the special distribution simulator, select the \(F\) distribution. Vary the parameters with the scroll bar and note the shape of the probability density function in light of the previous results on skewness and kurtosis. For selected values of the parameters, run the simulation 1000 times and compare the empirical density function to the probability density function.
Relations
The most important relationship is the one in the definition , between the \( F \) distribution and the chi-square distribution. In addition, the \( F \) distribution is related to several other special distributions.
Suppose that \(X\) has the \(F\) distribution with \(n \in (0, \infty)\) degrees of freedom in the numerator and \(d \in (0, \infty)\) degrees of freedom in the denominator. Then \(1 / X\) has the \(F\) distribution with \(d\) degrees of freedom in the numerator and \(n\) degrees of freedom in the denominator.
Proof
This follows easily from the random variable interpretation in the definition . We can write \[ X = \frac{U/n}{V/d} \] where \( U \) and \( V \) are independent and have chi-square distributions with \( n \) and \( d \) degrees of freedom, respectively. Hence \[ \frac{1}{X} = \frac{V/d}{U/n} \]
Suppose that \(T\) has the \(t\) distribution with \(n \in (0, \infty)\) degrees of freedom. Then \(X = T^2\) has the \(F\) distribution with 1 degree of freedom in the numerator and \(n\) degrees of freedom in the denominator.
Proof
This follows easily from the random variable representations of the \( t \) and \( F \) distributions. We can write \[ T = \frac{Z}{\sqrt{V/n}} \] where \( Z \) has the standard normal distribution, \( V \) has the chi-square distribution with \( n \) degrees of freedom, and \( Z \) and \( V \) are independent. Hence \[ T^2 = \frac{Z^2}{V/n} \] Recall that \( Z^2 \) has the chi-square distribution with 1 degree of freedom.
Our next relationship is between the \( F \) distribution and the exponential distribution.
Suppose that \( X \) and \( Y \) are independent random variables, each with the exponential distribution with rate parameter \( r \in (0, \infty) \). Then \(Z = X / Y\). has the \( F \) distribution with \( 2 \) degrees of freedom in both the numerator and denominator.
Proof
We first find the distribution function \( F \) of \( Z \) by conditioning on \( X \): \[ F(z) = \P(Z \le z) = \P(Y \ge X / z) = \E\left[\P(Y \ge X / z \mid X)\right] \] But \( \P(Y \ge y) = e^{-r y} \) for \( y \ge 0 \) so \( F(z) = \E\left(e^{-r X / z}\right) \). Also, \( X \) has PDF \( g(x) = r e^{-r x} \) for \( x \ge 0 \) so \[ F(z) = \int_0^\infty e^{- r x / z} r e^{-r x} \, dx = \int_0^\infty r e^{-r x (1 + 1/z)} \, dx = \frac{1}{1 + 1/z} = \frac{z}{1 + z}, \quad z \in (0, \infty) \] Differentiating gives the PDF of \( Z \) \[ f(z) = \frac{1}{(1 + z)^2}, \quad z \in (0, \infty) \] which we recognize as the PDF of the \( F \) distribution with 2 degrees of freedom in the numerator and the denominator.
A simple transformation can change a variable with the \( F \) distribution into a variable with the beta distribution, and conversely.
Connections between the \( F \) distribution and the beta distribution.
- If \( X \) has the \( F \) distribution with \( n \in (0, \infty) \) degrees of freedom in the numerator and \( d \in (0, \infty) \) degrees of freedom in the denominator, then \[ Y = \frac{(n/d) X}{1 + (n/d) X} \] has the beta distribution with left parameter \( n/2 \) and right parameter \( d/2 \).
- If \( Y \) has the beta distribution with left parameter \( a \in (0, \infty) \) and right parameter \( b \in (0, \infty) \) then \[ X = \frac{b Y}{a(1 - Y)} \] has the \( F \) distribution with \( 2 a \) degrees of freedom in the numerator and \( 2 b \) degrees of freedom in the denominator.
Proof
The two statements are equivalent and follow from the standard change of variables formula. The function \[ y = \frac{(n/d) x}{1 + (n/d) x} \] maps \( (0, \infty) \) one-to-one onto (0, 1), with inverse \[ x = \frac{d}{n}\frac{y}{1 - y} \] Let \( f \) denote the PDF of the \( F \) distribution with \( n \) degrees of freedom in the numerator and \( d \) degrees of freedom in the denominator, and let \( g \) denote the PDF of the beta distribution with left parameter \( n/2 \) and right parameter \( d/2 \). Then \( f \) and \( g \) are related by
- \( g(y) = f(x) \frac{dx}{dy} \)
- \( f(x) = g(y) \frac{dy}{dx} \)
The \( F \) distribution is closely related to the beta prime distribution by a simple scale transformation.
Connections with the beta prime distributions.
- If \( X \) has the \( F \) distribution with \( n \in (0, \infty) \) degrees of freedom in the numerator and \( d \in (0, \infty) \) degrees of freedom in the denominator, then \( Y = \frac{n}{d} X \) has the beta prime distribution with parameters \( n/2 \) and \( d/2 \).
- If \( Y \) has the beta prime distribution with parameters \( a \in (0, \infty) \) and \( b \in (0, \infty) \) then \( X = \frac{b}{a} X \) has the \( F \) distribution with \( 2 a \) degrees of the freedom in the numerator and \( 2 b \) degrees of freedom in the denominator.
Proof
Let \( f \) denote the PDF of \( X \) and \( g \) the PDF of \( Y \).
- By the change of variables formula, \[ g(y) = \frac{d}{n} f\left(\frac{d}{n} y\right), \quad y \in (0, \infty) \] Substituting into the beta \( F \) PDF shows that \( Y \) has the appropriate beta prime distribution.
- Again using the change of variables formula, \[ f(x) = \frac{a}{b} g\left(\frac{a}{b} x\right), \quad x \in (0, \infty) \] Substituting into the beta prime PDF shows that \( X \) has the appropriate \( F \) PDF.
The Non-Central \( F \) Distribution
The \( F \) distribution can be generalized in a natural way by replacing the ordinary chi-square variable in the numerator in the definition above with a variable having a non-central chi-square distribution. This generalization is important in analysis of variance.
Suppose that \(U\) has the non-central chi-square distribution with \(n \in (0, \infty) \) degrees of freedom and non-centrality parameter \(\lambda \in [0, \infty)\), \(V\) has the chi-square distribution with \(d \in (0, \infty)\) degrees of freedom, and that \(U\) and \(V\) are independent. The distribution of \[ X = \frac{U / n}{V / d} \] is the non-central \(F\) distribution with \(n\) degrees of freedom in the numerator, \(d\) degrees of freedom in the denominator, and non-centrality parameter \( \lambda \).
One of the most interesting and important results for the non-central chi-square distribution is that it is a Poisson mixture of ordinary chi-square distributions. This leads to a similar result for the non-central \( F \) distribution.
Suppose that \( N \) has the Poisson distribution with parameter \( \lambda / 2 \), and that the conditional distribution of \( X \) given \( N \) is the \( F \) distribution with \( N + 2 n \) degrees of freedom in the numerator and \( d \) degrees of freedom in the denominator, where \( \lambda \in [0, \infty) \) and \( n, \, d \in (0, \infty) \). Then \( X \) has the non-central \( F \) distribution with \( n \) degrees of freedom in the numerator, \( d \) degrees of freedom in the denominator, and non-centrality parameter \( \lambda \).
Proof
As in the theorem, let \( N \) have the Poisson distribution with parameter \( \lambda / 2 \), and suppose also that the conditional distribution of \( U \) given \( N \) is chi-square with \( n + 2 N \) degrees of freedom, and that \( V \) has the chi-square distribution with \( d \) degrees of freedom and is independent of \( (N, U) \). Let \( X = (U / n) \big/ (V / d) \). Since \( V \) is independent of \( (N, U) \), the variable \( X \) satisfies the condition in the theorem; that is, the conditional distribution of \( X \) given \( N \) is the \( F \) distribution with \( n + 2 N \) degrees of freedom in the numerator and \( d \) degrees of freedom in the denominator. But then also, (unconditionally) \( U \) has the non-central chi-square distribution with \( n \) degrees of freedom in the numerator and non-centrality parameter \( \lambda \), \( V \) has the chi-square distribution with \( d \) degrees of freedom, and \( U \) and \( V \) are independent. So by definition \( X \) has the \( F \) distribution with \( n \) degrees of freedom in the numerator, \( d \) degrees of freedom in the denominator, and non-centrality parameter \( \lambda \).
From the last result, we can express the probability density function and distribution function of the non-central \( F \) distribution as a series in terms of ordinary \( F \) density and distribution functions. To set up the notation, for \( j, k \in (0, \infty) \) let \( f_{j k} \) be the probability density function and \( F_{j k} \) the distribution function of the \( F \) distribution with \( j \) degrees of freedom in the numerator and \( k \) degrees of freedom in the denominator. For the rest of this discussion, \( \lambda \in [0, \infty) \) and \( n, \, d \in (0, \infty) \) as usual.
The probability density function \( g \) of the non-central \( F \) distribution with \( n \) degrees of freedom in the numerator, \( d \) degrees of freedom in the denominator, and non-centrality parameter \( \lambda \) is given by \[ g(x) = \sum_{k = 0}^\infty e^{-\lambda / 2} \frac{(\lambda / 2)^k}{k!} f_{n + 2 k, d}(x), \quad x \in (0, \infty) \]
The distribution function \( G \) of the non-central \( F \) distribution with \( n \) degrees of freedom in the numerator, \( d \) degrees of freedom in the denominator, and non-centrality parameter \( \lambda \) is given by \[ G(x) = \sum_{k = 0}^\infty e^{-\lambda / 2} \frac{(\lambda / 2)^k}{k!} F_{n + 2 k, d}(x), \quad x \in (0, \infty) \]


5: Molecular Orbitals
Molecular Orbital Theory
Molecular Orbital (MO) Theory is a sophisticated bonding model. It is generally considered to be more powerful than Lewis and Valence Bond Theories for predicting molecular properties; however, this power comes at the price of complexity. In its full development, MO Theory requires complex mathematics, though the ideas behind it are simple. Atomic orbitals (AOs) that are localized on individual atoms combine to make molecular orbitals (MOs) that are distributed over the molecule. The simplest example is the molecule dihydrogen (H 2 ), in which two independent hydrogen 1s orbitals combine to form the \(\sigma\) bonding MO and the \(\sigma\) antibonding MO of the dihydrogen molecule (see figure). The MO’s are also called Linear Combinations of Atomic Orbitals (LCAO).
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- 5.1: Formation of Molecular Orbitals from Atomic Orbitals
- Molecular orbital theory extends from quantum theory and the atomic orbital wavefunctions ( ψ ) described by the Schrödinger equation. While the Schrödinger equation defines a Ψ for electrons in individual atoms, we can approximate a the molecular wavefunction ( Ψ would look like if we combined the ψ of individual atoms. The addition or subtraction of wavefunctions is termed linear combination of atomic orbitals (LCAO). Molecular orbital theory applied LCAO to describe bonding.
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- 5.3: Larger (Polyatomic) Molecules
- We can extend the method we used for diatomic molecules to draw the molecular orbitals of more complicated, polyatomic molecules (molecules with more than two atoms). To combine several different atoms in a molecular orbital diagram, we will group orbitals from different atoms into sets that match the symmetry of a central atom. These group orbitals are also referred to as symmetry adapted linear combinations (SALCs).
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- 5.4: BeH₂ is Linear and H₂O is Bent
- Walsh correlation diagram is a plot of molecular orbital energy as a function of some systematic change in molecular geometry. For example, the correlation between orbital energies and bond angle for an \(AH_2\) molecule. The geometry of a molecule is determined by which possible structure is lowest in energy. We can use the Walsh diagram to determine the energy trends based on which orbitals are occupied.
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- 5.5: Photoelectron Spectroscopy
- A photoelecton spectrum can show the relative energies of occupied molecular orbitals by ionization. (i.e. ejection of an electron). A photoelectron spectrum can also be used to determine energy spacing between vibrational levels of a given electronic state. Each orbital energy band has a structure showing ionization to different vibrational levels.


7.4: How to Establish a Virtual Onboarding Program
How to Establish a Virtual Onboarding Program
The COVID-19 pandemic has caused employers to switch up their hiring and onboarding practices. Whether you are hiring employees who will work remotely or welcoming new staff while you work from home yourself, the circumstances are quite different from what many of us are used to. Follow these steps to ensure a smooth process.
Step 1. Use Technology Effectively
Technology is the cornerstone of virtual onboarding, and ensuring that adequate technology is available to all employees is essential. Start with identifying needed software and providing hardware to new hires that includes the necessary network access and configuration for the individual to perform his or her job.
- Determine how the hardware will be delivered. Will it be mailed to the employee or will pickup be arranged?
- Identify and procure software that will enable employees to conduct and/or participate in virtual meetings and other onboarding tasks. Apps such as Zoom, WebEx and Microsoft Teams, which allow for video meetings with screen sharing and other collaborative tools, are popular options.
- Provide easy-to-follow printed instructions for setting up the hardware and software.
- Schedule a one-on-one virtual meeting for each new hire with a member of the IT staff to conduct the initial setup of the hardware and software and to troubleshoot any issues.
In addition to hardware and software needs, create digital versions of documents such as benefit enrollment forms, employee handbooks, etc. that were previously provided to new employees in hard-copy format. Ensure that employees have the technology to access this information and electronically complete required forms and/or acknowledgments.
Step 2. Design an Orientation Schedule
Onboarding employees is more than a one-day or even one-week event. In all circumstances, onboarding should be a strategic, long-term process coordinated with all levels of the organization. In our "new normal," however, it is necessary to create a virtual orientation program that will replace the in-person orientation that typically occurs during a new hire's first few days.
Start with determining how much time each day individuals should be asked to participate in online meetings versus working at their own pace reviewing policies, procedures, etc., and then create a schedule for the first week or days with detailed agendas. All members of the leadership team should be involved in the orientation schedule, and video introductions should be utilized as much as possible.
If the resources are available, create a series of short videos on training topics, which can be sent to the new hire each day to complement virtual sessions. Create training videos or PowerPoint presentations on common topics such as completing timesheets, using job-specific software, etc.
Step 3. Communicate Often
New hires often feel lost during the first few days at a new job, and physical distance can heighten this isolation. To lessen the impact of this for remote employees, be sure to:
- Touch base regularly and assign a co-worker as a buddy to be the point of contact for routine questions and frequent check-ins.
- Provide new hires with a "who-to-turn-to list" that includes names, contact information, and descriptions of each employee's role and responsibilities in the company.
- Ensure managers have the training and tools needed to communicate regularly with new employees to help replace in-person contact, and require managers to establish frequent one-on-one meetings.
- Communicate clear performance expectations during the orientation period and beyond.
Step 4. Focus on Engagement
Establish ways to keep new employees engaged with management and their team members, including:
- Starting each virtual meeting with a discussion question that each participant answers about how they are handling remote work, what they like most and least about telework, or how they are spending their free time.
- Encouraging coworkers to join virtual meetings with video to allow new employees to put faces to names.
- Organizing virtual happy hours where employees can interact with new hires in an informal and relaxed environment.
- Making engagement fun by quizzing other employees on information about the new hire and award small gifts or other rewards to encourage individuals to interact with new team members.
- Training each new hire's buddy on ways to engage with their new co-worker, and encourage socialization—from a distance, of course.
Step 5. Follow Up with New Hires
After the first day, reach out to the new hire to determine how the process is going. Because virtual onboarding is likely not a one-size-fits-all situation, find out what each person needs and/or wants individually to make his or her onboarding successful. Be willing to adjust practices when feasible to make each new hire's experience a positive one.
After the first few weeks on the job, send new hires a short survey about their orientation to the company.
Source: "How to Establish a Virtual Onboarding Program", Society for Human Resource Management (SHRM), 2024.


5.1.1: Interpreting Negative Numbers
Lesson
Let's review what we know about signed numbers.
Exercise \(\PageIndex{1}\): Using the Thermometer
Here is a weather thermometer. Three of the numbers have been left off.
- What numbers go in the boxes?
- What temperature does the thermometer show?
Exercise \(\PageIndex{2}\): Fractions of a Degree
- What temperature is shown on each thermometer?
- Which thermometer shows the highest temperature?
- Which thermometer shows the lowest temperature?
- Suppose the temperature outside is \(-4^{\circ}\text{C}\). Is that colder or warmer than the coldest temperature shown? How do you know?
Exercise \(\PageIndex{3}\): Seagulls Soar, SHarks Swim
Here is a picture of some sea animals. The number line on the left shows the vertical position of each animal above or below sea level, in meters.
- How far above or below sea level is each animal? Measure to their eye level.
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A mobula ray is 3 meters above the surface of the ocean. How does its vertical position compare to the height or depth of:
The jumping dolphin?
The flying seagull?
The octopus? -
An albatross is 5 meters above the surface of the ocean. How does its vertical position compare to the height or depth of:
The jumping dolphin?
The flying seagull?
The octopus? -
A clownfish is 2 meters below the surface of the ocean. How does its vertical position compare to the height or depth of:
The jumping dolphin?
The flying seagull?
The octopus? - The vertical distance of a new dolphin from the dolphin in the picture is 3 meters. What is its distance from the surface of the ocean?
Are you ready for more?
The north pole is in the middle of the ocean. A person at sea level at the north pole would be 3,949 miles from the center of Earth. The sea floor below the north pole is at an elevation of approximately -2.7 miles. The elevation of the south pole is about 1.7 miles. How far is a person standing on the south pole from a submarine at the sea floor below the north pole?
Exercise \(\PageIndex{4}\): Card Sort: Rational Numbers
- Your teacher will give your group a set of cards. Order the cards from least to greatest.
- Pause here so your teacher can review your work. Then, your teacher will give you a second set of cards.
- Add the new set of cards to the first set so that all of the cards are ordered from least to greatest.
Summary
We can use positive numbers and negative numbers to represent temperature and elevation.
When numbers represent temperatures, positive numbers indicate temperatures that are warmer than zero and negative numbers indicate temperatures that are colder than zero. This thermometer shows a temperature of -1 degree Celsius, which we write \(-1^{\circ}\text{C}\).
When numbers represent elevations, positive numbers indicate positions above sea level and negative numbers indicate positions below sea level.
We can see the order of signed numbers on a number line.
A number is always less than numbers to its right. So \(-7<-3\).
We use absolute value to describe how far a number is from 0. The numbers 15 and -15 are both 15 units from 0, so \(|15|=15\) and \(|-15|=15\). We call 15 and -15 opposites. They are on opposite sides of 0 on the number line, but the same distance from 0.
Glossary Entries
Definition: Absolute Value
The absolute value of a number is its distance from 0 on the number line.
The absolute value of -7 is 7, because it is 7 units away from 0. The absolute value of 5 is 5, because it is 5 units away from 0.
Definition: Negative Number
A negative number is a number that is less than zero. On a horizontal number line, negative numbers are usually shown to the left of 0.
Definition: Positive Number
A positive number is a number that is greater than zero. On a horizontal number line, positive numbers are usually shown to the right of 0.
Practice
Exercise \(\PageIndex{5}\)
It was \(-5^{\circ}\text{C}\) in Copenhagen and \(-12^{\circ}\text{C}\) in Oslo. Which city was colder?
Exercise \(\PageIndex{6}\)
- A fish is 12 meters below the surface of the ocean. What is its elevation?
- A sea bird is 28 meters above the surface of the ocean. What is its elevation?
- If the bird is directly above the fish, how far apart are they?
Exercise \(\PageIndex{7}\)
Compare using >, =, or <.
- \(3\) _____ \(-3\)
- \(12\) _____ \(24\)
- \(-12\) _____ \(-24\)
- \(5\) _____ \(-(-5)\)
- \(7.2\) _____ \(7\)
- \(-7.2\) _____ \(-7\)
- \(-1.5\) _____ \(\frac{-3}{2}\)
- \(\frac{-4}{5}\) _____ \(\frac{-5}{4}\)
- \(\frac{-3}{5}\) _____ \(\frac{-6}{10}\)
- \(\frac{-2}{3}\) _____ \(\frac{1}{3}\)
Exercise \(\PageIndex{8}\)
Han wants to buy a $30 ticket to a game, but the pre-order tickets are sold out. He knows there will be more tickets sold the day of the game, with a markup of 200%. How much should Han expect to pay for the ticket if he buys it the day of the game?
(From Unit 4.2.2)
Exercise \(\PageIndex{9}\)
A type of green paint is made by mixing 2 cups of yellow with 3.5 cups of blue.
- Find a mixture that will make the same shade of green but a smaller amount.
- Find a mixture that will make the same shade of green but a larger amount.
- Find a mixture that will make a different shade of green that is bluer.
- Find a mixture that will make a different shade of green that is more yellow.
(From Unit 2.1.1)


1.7.2: Health Effects of Alcohol Abuse
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Drinking too much, on a single occasion or over time, can take a serious toll on your health. Here’s how alcohol can affect your body:
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Brain
- Alcohol interferes with the brain’s communication pathways, and can affect the way the brain looks and works. These disruptions can change mood and behavior, and make it harder to think clearly and move with coordination.
- Learning and memory problems, including dementia and poor school performance.
- Mental health problems, including depression and anxiety.
- Social problems, including lost productivity, family problems, and unemployment.
- Violence, including homicide, suicide, sexual assault, and intimate partner violence.
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Heart
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Drinking a lot over a long time or too much on a single occasion can damage the heart, causing problems including:
- Cardiomyopathy – Impaired ability of the heart to deliver blood to the body, can lead to heart failure
- Arrhythmias – Irregular heart beat
- Stroke
- High blood pressure
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Drinking a lot over a long time or too much on a single occasion can damage the heart, causing problems including:
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Liver
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Heavy drinking takes a toll on the liver, and can lead to a variety of problems and liver inflammations including:
- Steatosis, or fatty liver: A condition where normal liver tissue is replaced by more than 5-6 percent fat. The accumulation of fat can cause inflammation, cell death, and scarring, which can lead to liver fibrosis.
- Alcoholic hepatitis: Inflammation of the liver caused by drinking alcohol that can result in scar tissue.
- Liver Fibrosis: Occurs when the liver is damaged and the organ develops scar tissue in response to the inflammation. Advanced liver fibrosis results in cirrhosis, liver failure, and portal hypertension and often requires liver transplantation.
- Cirrhosis of the liver: A condition in which your liver is scarred and permanently damaged.
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Heavy drinking takes a toll on the liver, and can lead to a variety of problems and liver inflammations including:
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Pancreas
- Alcohol causes the pancreas to produce toxic substances that can eventually lead to pancreatitis, a dangerous inflammation and swelling of the blood vessels in the pancreas that prevents proper digestion.
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Cancer
- Drinking too much alcohol can increase your risk of developing certain cancers, including cancers of the mouth, esophagus, throat, liver, and breast
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Immune System
- Drinking too much can weaken your immune system, making your body a much easier target for disease. Chronic drinkers are more liable to contract diseases like pneumonia and tuberculosis than people who do not drink too much. Drinking a lot on a single occasion slows your body’s ability to ward off infections, even up to 24 hours after getting drunk.
 These behaviors can result in unintended pregnancy or sexually transmitted diseases, including HIV.


12.4: Copying – and Not Copying – Arrays
Now, a surprise for the unwary. Suppose I write this code:
Code \(\PageIndex{1}\) (Python):
stooges = np.array(['Larry','Beavis','Moe'])
funny_people = stooges
stooges[1] = 'Curly'
print("The stooges are: {}.".format(stooges))
print("The funny people are: {}.".format(funny_people))
Take a moment and predict what you think the output will be. Then, read it and (possibly) weep:
| The stooges are: ['Larry' 'Curly' 'Moe'].
| The funny people are: ['Larry' 'Curly' 'Moe'].
Note carefully: no Beavis .
Now the question is why. To understand this (and virtually any other tricky programming problem) you have to return once again to the memory picture. Figure 9.4.1 shows the situation immediately before, and after, the line “ stooges[1] = 'Curly' ” executes. Crucially, there is only one array in memory. Both variables – stooges and funny_people – are pointing at it.
Figure \(\PageIndex{1}\): The code on p. 79 immediately before (left side) and after (right side) the line “ stooges[1] = 'Curly' ” is reached.
You see, if y contains aggregate (instead of atomic) data, the line “ x = y ” does not perform a copy operation. Instead, it just points the x variable name to the same place y is pointing to.
Once you grasp this, it’s easy to see why " Beavis " completely disappeared. There’s only one array at all, so changing stooges has the side effect of implicitly changing funny_people as well.
Actually Copying
The “point the variable to the same thing, but don’t do a copy” behavior is the default, because such copy operations are expensive (in terms of memory usage and time to execute). They’re normally not what you want anyway. Sometimes, however, you do want to produce an entire separate copy of an array, so you can modify the copy yet preserve the original. To do so, you use the .copy() method:
Code \(\PageIndex{2}\) (Python):
orig_beatles = np.array(['John', 'Paul', 'George', 'Pete'])
beatles = orig_beatles.copy()
beatles[3] = 'Ringo'
print("The Beatles were originally {}.".format(orig_beatles))
print("But the ones we all know were {}.".format(beatles))
Look carefully at that second line: it makes all the difference. Instead of making the new variable beatles point to the same array in memory that orig_beatles did, we explicitly copied the array and made beatles point to that new copy. The final memory picture is thus as per Figure 9.4.2, and the output is of course:
| The Beatles were originally ['John' 'Paul' 'George' 'Pete'].
| But the ones we all know were ['John' 'Paul' 'George' 'Ringo'].
Figure \(\PageIndex{2}\): The memory picture after calling the .copy() method, instead of simply assigning to a new variable.


13.3: Interactions of the sulfur and oxygen cycles
The sulfur cycle of marine sediments is primarily driven by the dissimilatory sulfate reduction (DSR) to sulfide by anaerobic microorganisms (e.g., Jørgensen and Kasten, 2006). This process links the complex food web of organic matter degradation to the terminal organic carbon oxidation to CO 2 . Most of the sulfide is ultimately reoxidized back to sulfate, via diverse sulfur intermediates, by geochemical or microbial reactions that involve oxygen, nitrate, manganese [Mn(IV)], iron [Fe(III)], and other potential oxidants (e.g., Rickard, 2012). A fraction of the sulfide precipitates with iron and other metals or reacts with organic matter and is buried deeply into the seabed. The microbial sulfur transformations affect the isotopic composition of sulfate and sulfides and the resulting isotope fractionation is thereby diagnostic for both process rates and pathways of the sulfur cycle (e.g., Canfield, 2001).
Figure \(\PageIndex{1}\) presents the sulfur cycle of marine sediments. The processes include chemical reactions, microbially catalyzed pathways, and a combination of both. Sulfate (SO 4 2 - ) reduction to sulfide (H 2 S + HS - + S 2- ) is driven by the oxidation of buried organic carbon (C org ), supplemented by the anaerobic oxidation of methane (CH 4 ) at the subsurface sulfate-methane transition (SMT). Manganese and iron reduction are focused toward the surface sediment, but Fe(III) is also buried and acts as an oxidant for sulfide in the deeper sediment layers where it partly binds the produced sulfide as iron sulfide (FeS) and pyrite (FeS 2 ). Pyrite is the end product of iron-sulfide mineral formation and provides a deep sink for sulfur. Two pathways of pyrite formation are the “polysulfide pathway” (1) and the “H 2 S pathway” (2) (Figure \(\PageIndex{1}\)). The sulfidization of buried organic matter provides an additional deep sink for sulfur. Intermediate sulfur species, such as elemental sulfur (S 0 ), thiosulfate (S 2 O 3 2 - ), tetrathionate (S 4 O 6 2 - ), and sulfite (SO 3 2 - ), are formed during the oxidation of sulfide by, for example, buried Fe(III). These intermediates may be reduced back to sulfide, oxidized further to sulfate, or disproportionated to form both sulfide and sulfate. In very sulfidic sediments, a part of the sulfide diffuses up to the surface sediment where it may be oxidized by cable bacteria, by large sulfur bacteria such as Beggiatoa spp., or by other, less conspicuous sulfide oxidizers. The different pathways of sulfide oxidation ultimately depend on oxygen (and less on nitrate) as the ultimate oxidant, and thereby consume a considerable part of the total oxygen uptake of the seabed (Jørgensen, 1982b). The oxygen flux into the sediment is enhanced by bioirrigation (ventillation of burrows) by the benthic macrofauna (e.g., Kristensen et al., 2013).
Sulfate Reduction
Organic Matter Degradation
Organic matter deposited on the seafloor provides food for the benthic communities, either at the sediment surface or upon burial into the sediment layers below. Oxygen is available for respiration and chemical reactions near the surface and through faunal burrows. Beneath this mixed surface zone, marine sediments constitute an anoxic world inhabited by anaerobic microorganisms. These subsurface organisms become increasingly sparse with depth, yet they account for half of all microbial cells in the ocean (Kallmeyer et al., 2012). Their energy source in most of the seabed is the buried organic matter, which they oxidize to CO 2 and inorganic nutrients. Due to the high concentration of sulfate in seawater (28 mM at an ocean salinity of 35), sulfate generally penetrates meters down into the seabed and supplies the sulfate reducing microorganisms (SRM) with an electron acceptor for their respiration. As the sediment ages with increasing burial depth beneath the seafloor, the remaining organic matter becomes steadily more refractory to microbial degradation. The time-course of organic matter degradation in the sediment, and thus of sulfate reduction rates (SRR), can be described by the sum of several exponential decay functions relating to different organic matter components, each of which is being degraded by first-order kinetics (Westrich and Berner, 1984).
The anaerobic degradation of organic matter involves complex microbial food chains, starting with the hydrolysis of macromolecular structures by extracellular enzymes and the formation of organic molecules small enough (generally < ca. 600 dalton, but for polysaccharides possibly larger) to be taken up by bacteria or archaea (Arnosti, 2011; Reintjes et al., 2017). It is this initial hydrolysis of the complex organic material that is rate-limiting for the overall degradation rate of organic matter (Kristensen and Holmer, 2001; Arnosti, 2004; Beulig et al., 2018). Microbial cells, which take up the small organic molecules such as sugars, amino acids, lipids, organic acids etc., conserve energy and grow by multistep fermentation processes that produce a range of volatile fatty acids (VFAs), such as formate, acetate, propionate and butyrate, plus H 2 and CO 2 .
Sulfide Oxidation
Mass balance estimates and diffusion gradients of sulfide indicate that a significant fraction of the sulfide produced by sulfate reduction in marine sediments is reoxidized (Jørgensen, 1982b; Canfield et al., 1992; Pellerin et al., 2015b). This reoxidation occurs through diverse biological and geochemical pathways, forming a variety of reactive intermediates (Figure \(\PageIndex{2}\)). The extent of sulfide reoxidation depends upon the quantity and type of available oxidant as well as the presence of microorganisms (e.g., Luther et al., 2011).
Formation of Pyrite
The formation of pyrite (FeS 2 ) represents the main burial of sulfur, and thereby of reducing potential, in marine sediments, as pyrite is stable over geological timescales under anoxic conditions (Bottrell and Newton, 2006; Fike et al., 2015). Very generally, pyrite forms from the reaction of sulfide with buried ferric iron minerals, initially forming a mixture of elemental sulfur, polysulfides and ferrous iron minerals. Different overall reactions leading to pyrite formation in marine sediments have been proposed over the years, depending upon the initial reacting iron, and sulfur species. However, it has been argued that despite this potential variety only two reaction mechanisms are important: the reaction between FeS and H 2 S (“H 2 S pathway”; Equation 3) (Rickard and Luther, 1997; Thiel et al., 2019) and the reaction between FeS and polysulfide (“polysulfide pathway”; Equation 4) (Rickard and Luther, 2007).
\[\mathrm{FeS}+\mathrm{H}_2 \mathrm{~S} \rightarrow \mathrm{FeS}_2+\mathrm{H}_2\]
\[\mathrm{FeS}+\mathrm{S}_{\mathrm{x}}^{2-} \rightarrow \mathrm{FeS}_2+\mathrm{S}_{\mathrm{x}-1}{ }^{2-} \]
References
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Bottrell, S. H., and Newton, R. J. (2006). Reconstruction of changes in global sulfur cycling from marine sulfate isotopes. Earth Sci. Rev. 75, 59–83. doi: 10.1016/j.earscirev.2005.10.004
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Excerpted from:
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