32.4: The Disaster Management Cycle
By the end of this section, you should be able to:
- Describe the disaster management cycle.
- Explain the nurse’s role in the disaster management cycle.
- Discuss how community partners work together to prevent, prepare for, respond to, and recover from disasters.
A planned, systematic approach to disaster management allows community members, health care providers, emergency response agencies, and community organizations to respond to disasters and mass causalities effectively and efficiently. Community assessment identifies potential disasters, enabling planning to prevent, prepare, and respond to reduce their impact. Nurses play an integral role in all phases of the disaster management cycle. Multisector, interprofessional collaboration enhances disaster prevention, preparation, and response.
Phases of the Disaster Management Cycle
To establish a plan for disasters, the nurse must understand the phases of the disaster management cycle (Figure 32.6). The public health/community health nurse has an important role in each phase to promote optimal health outcomes for the people involved.
Mitigation , an action to prevent or reduce the cause, impact, or consequences of a disaster, is the first phase of the disaster management cycle. A community assessment identifies community risk factors . This includes determining what natural disasters the community is likely to face. For example, communities geographically situated in coastal regions are at risk for hurricanes or tsunamis. This also includes identification of potential targets of terrorist attacks and potential human-made disasters, such as the release of dangerous chemicals from a factory following an explosion. Community programs focused on preventing the occurrence and mitigating the impact of disasters are developed and implemented (Homeland Security, 2015). For disasters that cannot be prevented, such as earthquakes, hurricanes, and tornadoes, efforts may focus on mitigating the effects of the event on the community. Examples of mitigation include placing sandbags along a river to decrease the amount of floodwater, constructing levees or barriers to prevent or control flooding, boarding windows before a hurricane makes landfall, and improving building infrastructure to reduce the risk of damage during an earthquake.
Preparedness is the second phase of the disaster management cycle. It includes planning, training, and educational activities to address the consequences of disasters that cannot be mitigated. Preparedness is a continuous state of planning for rapid response when a disaster occurs. Task forces made up of community representatives from the local government, health care providers, social services providers, police and fire departments, major industries, local media, schools, and citizens’ groups form to ensure that each area of the community knows its role and has a plan in place for when a disaster occurs. In the preparedness phase, responding personnel from each agency are trained. This training includes drills or simulations of mock disasters. Public education is key in providing community members with information on how they can best prepare their families and whom to contact if they are directly impacted by a disaster (FEMA, 2021). The disaster preparedness plan includes what to do in case of disaster, where to go, who to call for help, identifying home vulnerabilities, and accumulating disaster supplies/equipment.
Personal and family preparedness is a professional nurse disaster competency. In this video, a city councilor from Gresham, Oregon, describes how families can prepare their homes in case of an emergency or disaster.
Watch the video, and then respond to the following questions.
- How does the family plan for disaster within their home?
- How does your personal preparedness home plan compare to the video?
For more information on developing a personal preparedness plan, visit the American Red Cross Disaster Preparedness Plan or Minnesota Department of Public Safety website on Personal and Family Preparedness .
Disaster response is the third phase of the disaster management cycle. Disaster response is the execution of the disaster plan when a disaster event occurs. Disaster response management is a local community responsibility that begins before the announcement of an official disaster declaration (Klein & Irizarry, 2022) and continues for the first 72 hours (deAnda et al., 2022). If local resources are depleted after 72 hours and cannot be managed, the Stafford Disaster Relief and Emergency Assistance Act allows the state governor to request a disaster declaration from the president of the United States. This declaration mobilizes federal resources to assist local agencies in the aftermath of a disaster (deAnda et al., 2022).
Disaster recovery , the fourth phase of the disaster management cycle, begins during the response phase once the threat to human life and infrastructure has passed. Recovery efforts aim to return a community to some degree of normal predisaster functioning (Klein & Irizarry, 2022). The simultaneous work of rebuilding and resuming regular operations and activities may require a prolonged period. Disaster recovery includes rescue work to provide relief, primarily immediate medical care, food and water, clothing, and shelter (Polcarová & Pupíková, 2020). Once people’s immediate needs are met, restoring public services such as communication and transportation begins the process of returning to normalcy. Rebuilding community infrastructure may be a long-term course but may also present improvement opportunities (Polcarová & Pupíková, 2020).
These Healthy People 2030 objectives aim to improve emergency preparedness and response by building community resilience. The following general emergency preparedness objectives are in the developmental stage:
- Increase the proportion of parents and guardians who know the emergency or evaluation plan for their children’s school — PREP-DO1
- Increase the proportion of adults who prepare for a disease outbreak after getting preparedness information — PREP-DO2
- Increase the proportion of adults who know how to evacuate in case of a hurricane, flood, or wildfire — PREP-DO3
- Increase the proportion of adults who have an emergency plan for disasters — PREP-DO4
The Nurse’s Role in the Disaster Management Cycle
The ANA (2016) considers disaster preparedness and response a basic competency of nursing practice. While most nurses may not have extensive experience in disaster response, they are expected to have minimum knowledge about disaster management. They participate in preparedness planning and training drills, and they have skills and abilities to provide client care and community support during a disaster. Community health nurses collaborate with health care professionals and partner with health system leaders, individuals, and families to improve population health outcomes and promote community resiliency when disasters occur (Chegini et al., 2021).
Nurses are an important part of all phases in the disaster management cycle. Understanding the inherent risks a community faces for different types of disasters is critical in the mitigation phase. If a community is at a higher risk of a particular type of disaster, then more time and resources can be focused on activities that will prevent losses. For example, community health nurses who work in areas at risk for tropical weather may need to spend more time mitigating the effects of hurricanes than the risk of forest fires. Public safety education helps people in the community understand the potential hazards and effects of the disasters for which their community is at greatest risk. Community education on strategies to mitigate hazards and how to respond in case of a disaster raises individuals’ awareness and increases the likelihood that they will develop a disaster plan. Families should know what to do, who to call, and where to go in emergencies. When individual community members are generally disaster-aware, they can immediately implement the personal disaster response plan as needed. The nurse plays a key role in educating community members on disaster preparedness.
Community education for disaster preparedness includes:
- Create a family preparedness plan that includes an evacuation plan . Review the plan annually.
- Determine a family communication plan that includes an emergency meeting place. How will you communicate if phone lines and towers are not working?
- Assemble a disaster kit . Inventory the kit at least annually, replacing expired items.
- Assemble an emergency kit for your vehicle.
- Store important documents in a waterproof, portable container.
- Make a plan for your pets and animals .
- Secure property prior to a disaster.
(See U.S. Department of Homeland Security, 2023.)
The nurse must be involved in preparedness activities and participate in disaster management education and training from the beginning of nursing education programs (ANA, 2016). Many organizations provide practicing nurses opportunities for further education to build the skills to prepare for and respond to emergencies. The FEMA Emergency Management Institute has a curriculum of disaster preparedness and response online courses available to the public free of charge at this link .
Collaboration with federal, state, and local relief agencies and organizations is critical in preparedness activities. Organizations like FEMA, the American Red Cross, state officials, and local emergency medical services come together to practice disaster response. Mock drills are simulation exercises that are regularly conducted to ensure community members and responders have practiced how to respond appropriately in disasters. Nurses should help plan and participate in these mock drills that simulate the type of disaster the community is most likely to have. Tabletop exercises are also useful for frequently reviewing disaster policies and procedures.
Tabletop exercises are discussion sessions where community members discuss their roles and responses during a specific emergency or disaster. This video discusses the benefits of conducting tabletop exercises.
Watch the video, and then respond to the following questions.
- Why does this community complete hurricane tabletop exercises annually?
- What are the benefits of testing the emergency plan using a tabletop exercise?
As care providers, nurses are often among the first to respond to health care needs in disaster situations. Lifesaving actions, injury treatment, and minimizing the effects of the disaster are the priority. Immediate needs such as medical treatment, shelter, food, water, and psychological support for survivors are the primary focus. The COVID-19 pandemic brought nursing disaster response to the forefront. Nurses worked tirelessly in their community hospitals, providing client care at the bedside, and the need for nurses to provide client care on the front lines was greater than the number of nurses available in many areas. In January 2020, the CDC activated the Incident Management System in response to the COVID-19 crisis. The CDC and Agency for Toxic Substances and Disease Registry Nurses’ Work Group (CNWG) helped to distribute and meet requests for nurse deployments to support the COVID-19 response (Zauche, 2022). The group used an Emergency Operations Management System application to track the work hours of those who contributed to the response. Among 190 CNWG members, 146 (76.8 percent) were deployed to work between January 21, 2020, and September 18, 2021. Combined, they recorded more than 24,600 days and nearly 198,000 hours on the response. Nurses worked in local facilities and with state public health agencies and departments throughout the pandemic as policy experts, contact tracers , researchers, epidemiologists , nurse consultants, and client advocates (Zauche, 2022).
Nurses also have a role in helping the community return to normalcy in the recovery period. They work collaboratively across disciplines to restore health care to optimal operations along the recovery continuum, adjusting according to the community’s needs. Nursing assessment and intervention during the recovery stage include monitoring and screening for communicable and infectious diseases, educating on water and food safety, educating on cleanup and disposal of debris and deceased livestock/animals, eliminating safety risks, and screening and treatment for mental health issues. Nurses have been essential in the post-pandemic phase of the COVID-19 pandemic. Nurses led the way in vaccine education, distribution, and coordination of administration in vaccine clinics. These efforts were essential to reestablishing health care to its modified baseline, and they have been active participants in creating new policies and practices to help prevent a resurgence.
Read the scenario, and then answer the questions that follow.
Juanita, a public health nurse on the Florida Coast, is a member of the disaster management team. In anticipation of the approaching hurricane season, Juanita reviews disaster prevention and mitigation plans for the community. She assesses the community for the most vulnerable individuals (those with physical and mental disabilities) and maps their locations. She also plans nursing interventions to mitigate the effects of the hurricane.
Juanita meets with the multiagency, interprofessional team during a tabletop exercise on hurricane disasters. She shares the location of the most vulnerable individuals in the community because they may require well-being checks or evaluation during power and phone outages. She provides input from a nursing perspective on potential sheltering locations, shelter resource needs, and the triage and transport process. Juanita suggests including surrounding communities in the hurricane disaster plan to prepare for increased population and health needs in their communities due population displacement before and following hurricane landfall.
1. What is another vulnerable population that may require well-being checks during emergency events and disasters?
- Older adults
- Single parents
- Persons with substance-use disorder
- Young adults
2. Which nursing intervention would be appropriate to mitigate the effects of a hurricane disaster?
- Conduct a simulation to practice triage in mass casualty situations.
- Teach classes on family safety planning prior to hurricane season.
- Draw a shelter blueprint to prepare for mass food, water, and health needs.
- Lead a tabletop exercise on hurricanes with health department staff.
Community Collaboration and Disasters
With the rising incidence of disasters, it is essential that public health professionals engage internal and external partners to promote community resilience and return to normalcy. Disasters present complex challenges for which no single discipline, agency, organization, or jurisdiction can or should bear sole responsibility (Association of Public Health Nurses [APHN], 2014). Community resilience after disasters is dependent on coordinated care and resource distribution. To gain the respect and confidence of the public, local agencies and leaders must manage disaster response and collaborate effectively to achieve optimal outcomes for the community (Charney et al., 2018).
Multisector, interdisciplinary collaboration is necessary at the community level during each stage of the disaster management cycle. Most often, the public health agency leads prevention and preparedness efforts within a community. Community partners who respond to and are affected by disasters are invited to participate in disaster management activities. This includes emergency response coordinators; emergency department managers; epidemiologists; nurses; emergency medical services; law enforcement; fire services; academic institutions; Medical Reserve Corps; private businesses; faith-based organizations; humanitarian organizations; nonprofit organizations; medical, health, and behavioral providers; and other community-based organizations (RHIhub, 2023). The disaster management team collaborates to assess the community for risk factors, plans efforts to mitigate disaster effects, and prepares for potential disaster through tabletop exercises and disaster simulations. Prevention and preparedness activities identify areas of strength, accessible resources, and areas for improvement.
The disaster management team utilizes the NIMS framework and ICS during disaster response and recovery to establish leadership, organize resources, and communicate effectively. When local resources are depleted, state partners, such as the National Guard, state emergency management agency, state health department, and governor’s office, and federal partners, such as FEMA, National Disaster Medical System, Environment Protection Agency, and U.S. Department of Health and Human Services Incident Response Coordination Team, may join disaster response efforts. Table 32.10 describes the role of various disaster response organizations and partners.
| Local Partners | |
|
|
| Emergency response coordinators |
|
| Medical and mental health providers including nurses |
|
| Emergency medical services |
|
| Law enforcement |
|
| Fire services |
|
| Medical Reserve Corps |
|
| Hospitals and health care services |
|
| Humanitarian organizations |
|
| Faith-based organizations |
|
| State Partners | |
|
|
| State National Guard |
|
| State Public Health Associations |
|
| National and Federal Partners | |
|
|
| U.S. National Response Team (NRT) |
|
| U.S. Environmental Protection Agency (EPA) |
|
| National Guard |
|
| Federal Emergency Management Agency (FEMA) National Incident Management System ( NIMS ) |
|
| Department of Health and Human Services (HHS) |
|
| Homeland Security |
|
| Centers for Disease Control and Prevention (CDC) |
|
| American Red Cross |
|


Meet the Periodic Table
Skills to Develop
- Associate family name to description (especially for groups 1 - 2 and 16 - 18)
The Periodic Table is an essential tool for chemists. I have provided a simple version here . It shows the symbols, atomic numbers, and average atomic masses for each element. If you point at a symbol, it will show the element name. You can search for an element by name or symbol and it will be highlighted so it's easy to find.
How do you use the periodic table? It can help you predict many important properties of elements. To make these predictions, you will need to know a little about the different families or groups , which are the columns of the table. The term for the rows is period . Here is some info about the important groups.
| Group Number | Family Name | Description |
|---|---|---|
| 1 | Alkali Metals | soft, extremely reactive metals, valence 1, almost always M + ions |
| 2 | Alkaline Earth Metals | soft, less reactive metals, valence 2, almost always M 2+ ions |
| 13 | Boron Group | non-metals and metals, valence 3 |
| 14 | Carbon Group | non-metals and metals, valence 4 |
| 15 | Pnictogens | non-metals, valence 3 |
| 16 | Chalcogens | non-metals, valence 2, often X 2– ions |
| 17 | Halogens | non-metals, very reactive, volatile elements, valence 1, usually X – ions |
| 18 | Noble Gases | very unreactive monatomic gases, valence 0 |
| 3-12 | Transition Metals | metals with multiple valences and ionic forms, initially hard to fit into periodic table, many exist as M 2+ |
| *, ** | Rare Earth Metals | (lanthanoides, actinoides) similar to each other, most have M 3+ ionic form |
To learn your way around the table, try going to this much fancier periodic table . Notice how metals are on the left and bottom of the periodic table, while non-metals are on the right and top. Metals are shiny, and they conduct heat and electricity. Non-metals don't conduct, and are often softer or easier to break than metals. Some elements are called metalloids because they are in between metals and non-metals, and you can see that the metalloids are also in between the metals and non-metals in the periodic table. The most reactive elements are on the edges of the table (groups 1 and 17), and the most reactive non-metals are O and F, in the top right corner.


4.5: Nucleophilic acyl substitution reactions
- Understand nucleophilic acyl substitution mechanisms, and factors that affect them.
- Apply nucleophilic acyl substitution to reactions of acid halides, anhydrides, carboxylic acids, esters, and amides.
- Wright the reactions with reagents to convert carboxylic acids into acid halides and phosphate esters in living things.
- Apply nucleophilic acyl substitution reactions to synthesize condensation polymers.
What is a nucleophilic acyl substitution reaction?
A carbon double bonded with oxygen, i.e., \(\ce{-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-}\) is a carbonyl group . The \(\ce{C}\) of the carbonyl group is bonded to two other groups. If one of the group bonded to the carbonyl \(\ce{C}\) is an alkyl (\(\ce{R{-}}\)), or hydrogen (\(\ce{H{-}}\)), it become an acyl group , i.e., \(\ce{R-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-}\). The acyl group has a polar double bond, i.e., \(\ce{R-\!\!{\overset{\overset{\huge\enspace\!{\overset{\Large{\delta{-}}}{O}}}|\!\!|\enspace}{\overset{\delta{+}}{C}}}\!\!-}\) where the carbonyl \(\ce{C}\) is partial positive (\(\delta{+}\)), i.e., it is an electrophile. If the acyl group is attached to a nucleophile that can act as a leaving group (\(\ce{-Lv}\)), a stronger nucleophile (\(\ce{Nu^{-}}\)) can substitute it from the acyl group in reactions called nucleophilic acyl substitution reactions , as shown below in a generalized reaction.
\[\ce{Nu^{-} + R-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-Lv -> R-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-Nu + Lv^{-}}\nonumber \]
Mechanisms of nucleophilic acyl substitution reactions
In nucleophilic acyl substitution reactions, there are two reactants, the nucleophile and the acyl group containing substrate. Nucleophiles can exist in anionic base form \(\ce{Nu^{-}}\) in a basic medium and neutral acid form \(\ce{HNu}\) in a neutral or acidic medium. The anionic form is a better nucleophile than its neutral acid form. Similarly, the substrate can exist as neutral \(\ce{R-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-Lv}\) in neutral or basic medium or as protonated \(\ce{R-\!\!\!\!\!\!{\overset{\overset{\huge\enspace\enspace{\overset{\Large{+}}{O}H}}|\!\!\!\!\!\!\!|\enspace\enspace}{\overset{\delta{+}}{C}}}\!\!\!\!\!\!-\overset{\delta{-}}{Lv}}\) form in an acidic medium. The protonated form has more \(\delta{+}\) charge and is a better electrophile. The anionic nucleophile cannot coexist with protonated substrate, because of their acid-base reaction. The other three combinations, i.e., basic nucleophile + neutral substrate, neutral nucleophile + protonated substrate, and neutral nucleophile + neutral substrate, can coexist as described below.
Base-promoted mechanism
The nucleophile in its more reactive basic form \(\ce{Nu^{-}}\) and neutral substrate \(\ce{R-\!\!{\overset{\overset{\huge\enspace\!{\overset{\Large{\delta{-}}}{O}}}|\!\!|\enspace}{\overset{\delta{+}}{C}}}\!\!-}\) can coexist in a basic medium. The nucleophile \(\ce{Nu^{-}}\) attacks the electrophilic carbonyl \(\ce{\overset{\delta{+}}{C}}\) and, simultaneously, the \(\ce{\overset{\delta{+}}{C}}\) breaks the weakest \(\pi\)-bond, as shown in step#1 of the mechanism below. The electrophilic \(\ce{C}\) changes hybridization from sp 2 to sp 3 , i.e., a tetrahedral intermediate .
The nucleophilic \(\ce{O^{-}}\) created in the first step attacks the \(\ce{\overset{\delta{+}}{C}}\) to re-establish the \(\pi\)-bond, and either \(\ce{\overset{\delta{+}}{C}{-}\overset{\delta{-}}{Nu}}\) or \(\ce{\overset{\delta{+}}{C}{-}\overset{\delta{-}}{Lv}}\)-bond breaks. This step is also called the collapse of the tetrahedral intermediate . Breakage of \(\ce{\overset{\delta{+}}{C}{-}\overset{\delta{-}}{Nu}}\)-bond reverses the first step and breakage of \(\ce{\overset{\delta{+}}{C}{-}\overset{\delta{-}}{Lv}}\)-bond leads to the products of step#2. Later will likely happen when the \(\ce{-Lv}\) is a good leaving group. The second step is usually not reversible because \(\ce{Lv^{-}}\) is usually a poor nucleophile. The leaving group ultimately picks up a proton from any acid molecule (\(\ce{HB^{+}}\)) in the medium, as shown in step#3.
The rate of reaction is increased by converting the neutral or acid form of the nucleophile (\(\ce{HNu}\)) to its more reactive conjugate base form (\(\ce{Nu^{-}}\)) in the basic medium in this mechanism. Therefore, it is called base promoted nucleophilic acyl substitution mechanism .
Acid-catalyzed mechanism
In this mechanism, the nucleophile is in less reactive neutral \(\ce{HNu}\) form, but the substrate is in a more reactive protonated \(\ce{R-\!\!\!\!\!\!{\overset{\overset{\huge\enspace\enspace{\overset{\Large{+}}{O}H}}|\!\!\!\!\!\!\!|\enspace\enspace}{\overset{\delta{+}}{C}}}\!\!\!\!\!\!-\overset{\delta{-}}{Lv}}\) form, in an acidic medium. Acid in the medium protonate the carbonyl \(\ce{O}\) in step#1, as shown in the mechanics below.
Nucleophile \(\ce{HNu}\) attacks the protonated electrophilic carbonyl \(\ce{\overset{\delta{+}}{C}}\) and, simultaneously, the \(\pi\)-bond breaks in step#2, leading to a tetrahedral intermediate-I. The neutral nucleophile becomes +ve charged after donating its lone pair of electrons. It carries an acid proton. Although the medium is acidic in this case, it is usually amphoteric, with some basic groups still present. The incoming nucleophile donates its acidic proton to any basic molecule (\(\ce{:B}\) in the medium in step#3, leading to a tetrahedral intermediate-II. Then the acidic medium protonates either \(\ce{-Nu}\) or \(\ce{-Lv}\)-groups. Protonation of \(\ce{-Nu}\) reverses the reaction but protonation of \(\ce{-Lv}\) makes it a better-leaving group in tetrahedral intermediate-III of step#4. The lone pair on \(\ce{O}\) re-establishes the \(\pi\)-bond with the electrophilic \(\ce{C}\), and at the same time, the leaving group leaves, leading to a protonated acyl product in the step#5. The protonated acyl donates its proton to any base \(\ce{:B}\) in the medium and becomes the product in step#6.
Acid is a catalyst in this mechanism as it is consumed in step#1 but re-generated in step#6, and accelerates two slow steps in this mechanism, i.e., step#2 and step#5. That is why it is called the acid-catalyzed nucleophilic substitution mechanism . Specifically, the protonation of the substrates \(\ce{R-\!\!{\overset{\overset{\huge\enspace\!{\overset{\Large{\delta{-}}}{O}}}|\!\!|\enspace}{\overset{\delta{+}}{C}}}\!\!-}\) converts it to a more reactive \(\ce{R-\!\!\!\!\!\!{\overset{\overset{\huge\enspace\enspace{\overset{\Large{+}}{O}H}}|\!\!\!\!\!\!\!|\enspace\enspace}{\overset{\delta{+}}{C}}}\!\!\!\!\!\!-\overset{\delta{-}}{Lv}}\) form that accelerates step#2, and protonation of \(\ce{-Lv}\) in tetrahedra intermediate-II converts it to a good leaving \(\ce{-Lv^{+}H}\) group in tetrahedral intermediate-III that accelerates step#5. The other steps in the mechanism are acid-base reactions which are inherently fast.
All the steps in this mechanism are reversible. It means there is an equilibrium between reactants and products. Three situations can arise, which are the following.
- If the incoming nucleophile (\(\ce{HNu}\)) is a poor nucleophile compared to the leaving group (\(\ce{HLv}\)), the reactants dominate,
- If the incoming nucleophile (\(\ce{HNu}\)) is a good nucleophile compared to the leaving group (\(\ce{HLv}\)), the products dominate, and
- If the nucleophilicity of the incoming nucleophile (\(\ce{HNu}\)) is comparable to that of the leaving group (\(\ce{HLv}\)), about an equal-equal mixture of reactants and products exists.
In situation#3, the equilibrium can be manipulated to favor products or reactants based on Le Chatelier’s principle. Employing one of the reactants in excess or removing one of the products drives the reaction forward. Similarly, adding one of the products in excess or removing one of the reactants drives the reaction in the reverse direction.
Neutral nucleophile and neutral substrate reaction mechanism
In this mechanism, both the nucleophile \(\ce{HNu}\) and the substrate \(\ce{R-\!\!{\overset{\overset{\huge\enspace\!{\overset{\Large{\delta{-}}}{O}}}|\!\!|\enspace}{\overset{\delta{+}}{C}}}\!\!-}\) are in their less reactive neutral forms in a neutral medium. Nucleophile \(\ce{HNu}\) attacks the electrophilic carbonyl \(\ce{\overset{\delta{+}}{C}}\) and, simultaneously, the \(\pi\)-bond breaks in step#1, leading to tetrahedral intermediate-I. The acidic proton on the incoming nucleophile in intermediate-I is removed by any base molecule (\(\ce{:B}\) in the medium in step#2, leading to tetrahedral intermediate-II, as shown below.
The nucleophilic \(\ce{O^{-}}\) in tetrahedral intermediate-II re-establishes the \(\pi\)-bond, and, simultaneously, either \(\ce{Nu^{-}}\) or \(\ce{Lv^{-}}\) leaves. Departure of \(\ce{Nu^{-}}\) reverses step#2 and departure of \(\ce{Lv^{-}}\) leads to the product in step#3. \(\ce{Lv^{-}}\) is neutralized by any acid molecule present in the medium in step#4.
Since both the nucleophile and the substrate are in their less reactive neutral forms, it works only in situations where a good leaving group (\(\ce{-Lv}\)) is attached to the acyl substrate. Step#3 is usually irreversible because the \(\ce{-Lv}\) groups chosen for these reactions are usually good leaving groups and poor nucleophiles.
Effect of leaving group on nucleophilic acyl substitution reactions
Good leaving groups (\(\ce{Lv^{-}}\)) are usually weak bases and poor nucleophiles at the same time. Weak bases do not fully share their electrons with acidic protons, and poor nucleophiles do not fully share their electrons with electrophilic \(\ce{C's}\). Poor nucleophiles increase the rates of slow steps in the following ways:
- their bond with \(\ce{C}\) is strongly polar, making the acyl \(\ce{\overset{\delta{+}}{C}}\) a more reactive electrophile, and
- they leave easily increasing the rate of collapse of tetrahedral intermediates.
Leaving groups commonly encountered in nucleophilic acyl substitution reactions are halogens like chlorine (\(\ce{-Cl}\)) or bromine (\(\ce{-Br}\)) , carboxylate (\(\ce{-O-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-R}\)), hydroxyl (\(\ce{-OH}\)), alkoxy (\(\ce{-OR}\)), and amine (\(\ce{-NH2}\)) groups. Their basicity and nucleophilicity increases in this order: \(\overrightarrow{\ce{Cl^{-} < R-COO^{-} < HO^{-} ≈ RO^{-} < ^{-}NH2}}\). Therefore, their ability to leave increases in the opposite order \(\overrightarrow{\ce{^{-}NH2 < HO^{-} ≈ RO^{-} < R-COO^{-} < Cl^{-}}}\), i.e., halogens like chlorine (\(\ce{-Cl}\)) are the best-leaving groups and amines (\(\ce{-NH2}\)) are the worst-leaving group.
These leaving groups are found in the following subclasses of acyl substrates: amides (\(\ce{R-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-NH2}\)), carboxylic acids (\(\ce{R-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-OH}\)), easters (\(\ce{R-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-OR}\)), acid anhydrides (\(\ce{R-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-O-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-R}\)), and acid halides (\(\ce{R-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-Cl}\)). These substrates are carboxylic acids and their derivatives. Their reactivity as acyl substrates in nucleophilic acyl substitution reactions increases in this order: \(\overrightarrow{\ce{R-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-NH2} < \ce{R-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-OH} ≈ \ce{R-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-OR} < \ce{R-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-O-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-R} < \ce{R-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-Cl }}\). It means a group that is higher in the reactivity order can be easily converted to a group that is lower in the order, but the reverse does not happen normally. For example, the following reaction is feasible:
,
but the following reaction, i.e., the reverse of it, does not happen:
.
Examples of nucleophilic acyl substitution reactions
Reactions of acid halides
The acid halides, such as acid chlorides or bromides are the most reactive class of acyl substrates that can react with neutral nucleophiles and neutral substrates in a neutral medium. For example, acid chlorides can react with carboxylic acid, water, alcohol, or amines to form acid anhydrides, carboxylic acids, esters, and amines, respectively, as shown below.
Reactions of acid anhydrides
The acid anhydrides are reactive acyl substrates, second only to acid halides. Anhydride can react with water to produce two equivalents of carboxylic acid; with alcohols to produce one equivalent of ester and one equivalent of carboxylic acid; and with two amines to produce one equivalent of amide and one equivalent of an amine salt of carboxylic acid, as shown below.
Two moles of amine are needed in the last reaction because initially formed carboxylic acid reacts with the amine base producing an amine salt, as shown below.
Reactions of carboxylic acids
Carboxylic acids react with alcohols to produce ester but slowly. These reactions are reversible because the reactivity of the incoming nucleophiles is about the same as that of leaving nucleophoiles. Acid catalysis is applied to accelerate these reactions, as shown in the example below.
Since there is an equilibrium, excess alcohol is used, or water is removed to drive the reaction forward. These relations are called alcoholysis reactions because the reactant alcohol is also solvent. If a reverse reaction is desired, water is employed in excess as a solvent and the reaction is called hydrolysis of esters.
Carboxylic acids do not react with halides and another carboxylic acid because the incoming nucleophiles are poor relative to the leaving group. Base can not be added to accelerate the reactions of carboxylic acids because the base neutralizes the substrate, as shown below.
The carboxylate anion ( \(\ce{R-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-O^{-}}\) is the least reactive acyl substrate that is below amides in the reactivity order. Carboxylic acids do not react with amines for the same reason, i.e., amine base neutralizes the carboxylic acid producing ammonium salts, as shown below.
Protonated amines in the ammonium salts are not nucleophiles as they do not have lone pair of electrons on the \(\ce{N}\).
Reactions of esters
Esters are comparable in reactivity with carboxylic acids but without the acidic protons. Therefore, both acid-catalysis and base-promoted reactions can be carried out with esters. For example, water hydrolyzes esters under acidic conditions, as shown below.
It is a reversible reaction that can be driven forward by employing excess water or by removing the alcohol product. These reactions are called acid-catalyzed hydrolysis of esters.
The alkoxy group of esters can be substituted by an alkoxy group of another alcohol, as shown below.
It is a reversible reaction that can be driven forward by employing excess reactant alcohol or by removing the product alcohol. These reactions are called acid-catalyzed transesterification reactions.
Base-promoted reactions of esters
Transesterification of esters can be performed by employing a conjugated base of the alcohos, as shown in the following example.
Similarly, the example below shows that esters can be hydrolyzed using alkali, i.e., hydroxide ions.
The base-promoted hydrolysis of esters is called saponification which is used to hydrolyze fats, as shown in Figure \(\PageIndex{1}\). Fats are tri-esters of glycerol and fatty acids. Saponification of fasts produces sodium salts of fatty acids and glycerol. Sodium salts of fatty acids are soaps.
Reactions of amides
Amides are the least reactive among the acyl substrates. Amides do not react with halides, water, or alcohols under neutral conditions because the incoming nucleophiles are poor than the leaving group. However, water hydrolyzes amides under acid catalysis and heating, as shown below.
Acid converts poor leaving \(\ce{-NRR'}\) group to a good leaving \(\ce{-\overset{+}{N}HRR'}\) group, and also removes the product amine \(\ce{NRR'}\) by acid-base reaction, as illustrated below.
The ammonium ion (\(\ce{RR'N^{+}HCl^{-}}\) does not have a lone pair on \(\ce{N}\) and is not a nucleophile, that makes the reaction irreversible.
As shown below, alcohols react with amides under acid catalysis and heating.
Activating carboxylic acids in the laboratory and in biochemical systems
Carboxylic acids are common raw materials but must be converted into reactive derivatives, like acid halide, before conversion to other carboxylic acid derivates. Reagents like thionyl chloride (\(\ce{SOCl2}\)), phosphorous trichloride (\(\ce{PCl3}\)), or phosphorous pentachloride (\(\ce{PCl5}\)) are used to convert carboxylic acids into acid chlorides for this purpose, as shown below.
The above reagents are not available in biochemical systems. Adenosine triphosphate, shown below is a common reagent available in biochemical systems.
As illustrated below, adenosine triphosphate is used in biochemical systems to convert acylates into acyl phosphates or acyl adenylates.
The phosphate and adenylate are good leaving groups. For example, coenzyme-A (\(\ce{CoASH}\)) containing nucleophilic thiol (\(\ce{-SH}\)) group is common in biochemical systems.
\(\ce{CoASH}\) displaces adenylate group from acyls through nucleophilic acyl substitution mechanism, as illustrated below.
Thioesters, like \(\ce{-SCoA}\), are also good leaving groups. For example, the neurotransmitter acetylcholine is synthesized by reacting acetyl\(\ce{-SCoA}\) with choline through a nucleophilic acyl substitution mechanism, as shown below.
Condensation polymerization reactions
When two molecules combine to form a single molecule, usually with a loss of a small molecule like water or ammonia, the reaction is called a condensation reaction. For example, nucleophilic acyl substitution reactions between carboxylic acids and alcohols are condensation reactions, as shown below.
Polymers called polyesters are produced when molecules with two carboxylic acids and others with two alcohols condense to form esters. For example, benzene-1,4-dicarboxylic acid, commonly known as terephthalic acid, and ethane-1,2-diol, known as ethylene glycol, condense to produce a polyester called Polyethylene terephthalate (PET), as shown in Figure \(\PageIndex{2}\). Polyethylene terephthalate is used to make drink bottles and polyester fabrics.
Nucleophilic acyl substitution reactions between carboxylic acids and amines are another example of condensation reactions that can produce polyamides. For example, condensation of hexanedioic acid and hexane-1,6-diamine, known as hexamethylene diamine, produces a polyamide called nylon 66, as illustrated in Figure \(\PageIndex{3}\).
The first digit of 66 in the name nylon 66, tells number of carbons in one monomer (hexanedioic acid) and the second digit for the other monomer (hexane-1,6-diamine). Nylone 66 is commonly used in clothing, fishing lines, and guitar strings. Kevlar used to make bulletproof jackets is another example of polyamide, shown in Figure \(\PageIndex{4}\)
Condensation reaction between diisocyanates (\(\ce{O=C=N-R-N=C=O}\)) and alcohols (\(\ce{HO-R-OH}\)) produces polyurethane where ureththane groups ( \(\ce{-{\overset{\Large{H}}{N}}-\!\!{\overset{\overset{\huge\enspace\!{O}}|\!\!|\enspace}{C}}\!\!-O\bond{-}}\) ) link the monomer units in the polymer molecule, as illustrated in Figure \(\PageIndex{5}\).
Polyurethane makes foam, e.g., mattresses foam.
Nucleic acids, i.e., DNA and RNA are biopolymers in which phosphate diester linkages hold the monomer units together, as illustrated in the figure on the right (copyright: Public domain). Nucleic acids are described in detail in a separate chapter. Proteins are polyamide bipolymers. For example, cinnamycin (lanthiopeptin), shown in the figure on the left (copyright: Public domain), is an antibacterial peptide produced by Streptomyces cinnamoneus containing 19 amino acids. Proteins are also described in detail in a separate chapter.


4.2: The Selection Process - A Practical Guide
The Selection Process - A Practical Guide
What is the selection process? In this video, we explain what the selection process is, and how to optimize it in your organization 🌟 A good selection process is key in finding talent, and it forms the backbone for effective performance management.
There are several important metrics that should be tracked during the selection process in order to keep improving and optimizing your selection funnel. In discussing these, we touch upon the following:
💡job application
💡screening & pre-selection
💡job interview
💡assessment
💡references and background check
💡hiring decision
💡job offer and contract
Watch this video to learn more.


6.6: Oxidation-Reduction (Redox) Reactions
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Skills to Develop
- Define oxidation and reduction in terms of a gain or loss of oxygen or hydrogen atoms.
- Identify the substances involved in oxidation and reduction in a reaction.
- Identify the oxidizing and reducing agents in a redox reaction.
- To identify a chemical reaction as an oxidation-reduction reaction.
Oxygen's Role in Reactions
Oxygen is an element that has been known for centuries. In its pure elemental form, oxygen is highly reactive, and it readily makes compounds with most other elements. It is also the most abundant element by mass in the Earth's crust. The class of reactions called oxidation and reduction were originally defined with respect to the element oxygen.
Early scientists viewed oxidation as a process in which a substance combines with oxygen to produce oxides. Examples of oxidation are shown below: magnesium, methane, iron and copper are being oxidized.
The heating of magnesium in air allows it to combine with oxygen from the air to form magnesium oxide.
\[2 \ce{Mg} \left( s \right) + \ce{O_2} \left( g \right) \rightarrow 2 \ce{MgO} \left( s \right)\]
The burning (combustion) of methane allows the fuel (methane) to react with oxygen (from the air), producing carbon dioxide and water (and heat).
\[\ce{CH_4} \left( g \right) + 2 \ce{O_2} \left( g \right) \rightarrow \ce{CO_2} \left( g \right) + 2 \ce{H_2O} \left( g \right)\]
The oxidation (commonly known as rusting or corrosion) of iron is shown below. Iron that is exposed to air (and water) is slowly oxidized to produce rust (an oxide of iron).
\[4 \ce{Fe} \left( s \right) + 3\ce{O_2} \left( g \right) \rightarrow 2 \ce{Fe_2O_3} \left( s \right)\]
The Statue of Liberty is a landmark every American recognizes. The Statue of Liberty is easily identified by its height, stance, and unique blue-green color. When this statue was first delivered from France, its appearance was not green. It was brown, the color of its copper “skin.” So how did the Statue of Liberty change colors? The change in appearance was a direct result of corrosion. The copper that is the primary component of the statue slowly underwent oxidation from the air. The oxidation-reduction reactions of copper metal in the environment occur in several steps. Copper metal is oxidized to copper(I) oxide Cu 2 O), which is red, and then to copper(II) oxide (CuO), which is black.
\[4 \ce{Cu} \left( s \right) + \ce{O_2} \left( g \right) \rightarrow 2 \ce{Cu_2O} \left( s \right)\]
\[2 \ce{Cu_2O} \left( s \right) + \ce{O_2} \left( g \right) \rightarrow 4 \ce{CuO} \left( s \right)\]
Coal, which was often high in sulfur, was burned extensively in the early part of the last century. As a result, sulfur trioxide, carbon dioxide, and water all reacted with the CuO. These three compounds are responsible for the characteristic blue-green patina seen today. Fortunately, formation of the patina created a protective layer on the surface, preventing further corrosion of the copper skin.
Bleaches contain various compounds such as sodium hypochlorite (NaClO) that can oxidize stains by the transfer of oxygen atoms, making the molecules in the stains more water-soluble and therefore easier to rinse off. Hydrogen peroxide (H 2 O 2 ) releases oxygen as it spontaneously decomposes. It also acts as a bleach and is used as an antiseptic that kills bacteria by oxidizing them.
We worry a lot about our smiles. Over the years teeth do discolor some, so the procedure of teeth bleaching has become more and more popular. Best done in a dentist's office, various chemical preparations containing peroxides are used to whiten teeth. Less effective, but easier to use are "teeth-whitening" toothpastes (also containing peroxides) that promise to give you a brighter smile and improve your life in every way.
Oxidation and Reduction
The opposite of oxidation is called reduction . Since oxidation was originally defined as the addition of oxygen , reduction was therefore the removal of oxygen from a substance. Many naturally occurring metal ores are present as oxides. The pure metals can be extracted by reduction. For example, pure iron is obtained from iron (III) oxide by reacting it with carbon at high temperatures.
\[2 \ce{Fe_2O_3} \left( s \right) + 3 \ce{C} \left( s \right) \rightarrow 4 \ce{Fe} \left( s \right) + 3 \ce{CO_2} \left( g \right)\]
The removal of oxygen from the Fe 2 O 3 molecule means that it is being reduced to Fe. Note that an oxidation process is also occurring simultaneously in this reaction; the carbon reactant is being oxidized to CO 2 . This is an important concept. In looking at organic compounds, we can describe oxidation and reduction in terms of the loss or gain of oxygen or the loss or gain of hydrogen. The changes in number of oxygen atoms, and number of hydrogen atoms is summarized below for oxidation and reduction reactions.
| Process | Change in oxygen (some reactions) | Change in hydrogen (some reactions) |
|---|---|---|
| Oxidation | gain | lose |
| Reduction | lose | gain |
If we look at the oxidation of ethanol (CH 3 CH 2 OH, an alcohol) to form ethanal (CH 3 CHO, an aldehyde), we see that the number of bonds to oxygen has increased and the number of hydrogen atoms has decreased from six to four. Either or both of these indicate that an oxidation has occurred.
For a reduction , we look for the removal of oxygen or bonds to oxygen or the addition or hydrogen atoms . In the following reaction, we see ethanoic acid (CH 3 COOH, a carboxylic acid) being reduced to ethanal (CH 3 CHO, an aldehyde). Note that the number of oxygen atoms is reduced from two to one. There is no change in the number of hydrogen atoms, but both changes do not need to happen for evidence of oxidation or reduction.
If we see evidence of an oxidation or reduction, we know the other must have happened as well. Oxidation and reduction must happen together. Neither can happen alone in a reaction.
Oxidation and Reduction Must Occur at the same time
Note that an oxidation process is simultaneously occurring with a reduction process. Oxidation cannot happen alone, without reduction. If reactant A is gaining oxygen (getting oxidized), reactant B must be losing the O (getting reduced). If reactant A is undergoing the removal of hydrogen (getting oxidized), reactant B must be undergoing the addition of hydrogen (getting reduced).
Redox reactions are also commonly described in terms of oxidizing and reducing agents. In the reaction below, C is being oxidized by gaining oxygen. The source of oxygen is Fe 2 O 3 . In other words, Fe 2 O 3 is causing the C to be oxidized. Consequently, Fe 2 O 3 is referred to as the oxidizing agent. Conversely, the C causes the Fe 2 O 3 to lose oxygen and become reduced, so C is the reducing agent. An oxidizing agent ( OA ) is a substance that causes oxidation by releasing oxygen , and a reducing agent ( RA ) is a substance that causes reduction by gaining oxygen . Said another way, the oxidizing agent (OA) is the substance that is reduced, while the reducing agent (RA) is the substance that is oxidized. Focusing on the reaction below, Fe 2 O 3 is the substance being reduced, while C is the substance that is being oxidized. Hence, Fe 2 O 3 is the oxidizing agent (OA) and C is the reducing agent (RA).
\[2 \ce{Fe_2O_3} \left( s \right) + 3 \ce{C} \left( s \right) \rightarrow 4 \ce{Fe} \left( s \right) + 3 \ce{CO_2} \left( g \right)\]
Fe 2 O 3 = substance reduced C = substance oxidized
Fe 2 O 3 = oxidizing agent (OA) C = reducing agent (RA)
Note
The substance oxidized and substance reduced are both reactants in the reaction.
Several classes of organic compounds are related to one another by oxidation and reduction reactions. Alkanes, alkenes, and alkynes represent different levels of oxidation of a hydrocarbon. When an alkane is heated in the presence of an appropriate catalyst, it can be oxidized to the corresponding alkene in a reaction called a dehydrogenation reaction. Two hydrogen atoms are removed in the process. The alkene can be further oxidized to an alkyne by the removal of two more hydrogen atoms.
oxidation: CH 3 CH 3 →CH 2 =CH 2 →CH≡CH
The reactions are reversible, so an alkyne can be reduced first to an alkene and then to an alkane. These are addition reactions.
reduction: CH≡CH→CH 2 =CH 2 →CH 3 CH 3
The alkane is the most reduced form of a hydrocarbon, while the alkyne is the most oxidized form.
Oxidation reactions in organic chemistry often involve the addition of oxygen to a compound (or an increase in the number of bonds to oxygen), which changes the functional group that is present. The following sequence shows how methane can be oxidized first to methanol (alcohol), then to methanal (aldehyde), then to methanoic acid (carboxylic acid), and finally to carbon dioxide. Each step in the process is either a gain of oxygen or a loss of hydrogen. Each step also represents energy, which explains why the complete combustion of alkanes to carbon dioxide is an extremely exothermic reaction. Hence, alkanes are used as fuels.
The opposite process is the reduction of a carboxylic acid to an alkane which involves the loss of oxygen and the gain of hydrogen.
When a primary alcohol is oxidized to an aldehyde in the presence of water, the reaction can be difficult to stop because the aldehyde can be further oxidized to the corresponding carboxylic acid. For example, the oxidation of ethanal produces ethanoic (acetic) acid. Ethanol-containing beverages such as wine are susceptible to such oxidation if kept for long periods of time after having been opened and exposed to the air. Wine that has become oxidized will have an unpleasant vinegary taste due to the production of acetic acid.
Summary
- Oxidation is the gain of oxygen (O) while reduction is the loss of oxygen (O).
- Oxidation is the loss of hydrogen (H) while reduction is the gain of hydrogen (H).
- Oxidation and reduction reactions must occur together.
- The reactant oxidized is the reducing agent.
- The reactant reduced is the oxidizing agent.
T o Your Health: Redox Reactions and Pacemaker Batteries
All batteries use redox reactions to supply electricity because electricity is basically a stream of electrons being transferred from one substance to another. Pacemakers—surgically implanted devices for regulating a person’s heartbeat—are powered by tiny batteries, so the proper operation of a pacemaker depends on a redox reaction.
Pacemakers used to be powered by NiCad batteries, in which nickel and cadmium (hence the name of the battery) react with water according to this redox reaction:
Cd(s) + 2NiOOH(s) + 2H 2 O(ℓ) → Cd(OH) 2 (s) + 2Ni(OH) 2 (s)
The cadmium is oxidized, while the nickel atoms in NiOOH are reduced. Except for the water, all the substances in this reaction are solids, allowing NiCad batteries to be recharged hundreds of times before they stop operating. Unfortunately, NiCad batteries are fairly heavy batteries to be carrying around in a pacemaker. Today, the lighter lithium/iodine battery is used instead. The iodine is dissolved in a solid polymer support, and the overall redox reaction is as follows:
2Li(s) + I 2 (s) → 2LiI (s)
Lithium is oxidized, and iodine is reduced. Although the lithium/iodine battery cannot be recharged, one of its advantages is that it lasts up to 10 years. Thus, a person with a pacemaker does not have to worry about periodic recharging; about once per decade a person requires minor surgery to replace the pacemaker/battery unit. Lithium/iodine batteries are also used to power calculators and watches.
Example \(\PageIndex{1}\)
In each conversion, indicate whether oxidation or reduction is occurring.
- N 2 → NH 3
- CH 3 CH 2 OHCH 3 → CH 3 COCH 3
- HCHO → HCOOH
SOLUTION
- Hydrogen is being added to the original reactant molecule, so reduction is occurring.
- Hydrogen is being removed from the original reactant molecule, so oxidation is occurring.
- Oxygen is being added to the original reactant molecule, so oxidation is occurring.
Exercise \(\PageIndex{1}\)
In each conversion, indicate whether oxidation or reduction is occurring.
- CH 4 → CO 2 + H 2 O
- NO 2 → N 2
- CH 2 =CH 2 → CH 3 CH 3
- Answer
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- oxidation
- reduction
- reduction
Summary
Chemical reactions in which oxygen and/or hydrogen are transferred are called oxidation-reduction, or redox, reactions. Oxidation is the gain of O or loss of H. Reduction is the loss of O or gain of H. Oxidation and reduction always occur together, even though they can be written as separate chemical equations.
Concept Review Exercises
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Give two different definitions for oxidation and reduction.
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Give an example of each definition of oxidation and reduction.
Answers
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Oxidation is the addition of oxygen or the loss of hydrogen; reduction is the loss of oxygen or the addition of hydrogen.
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C → CO 2 (oxidation); C 2 H 4 + H 2 → C 2 H 6 (reduction) (answers will vary)
Exercises
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For the following redox reactions, identify the substance oxidized and the substance reduced.
- 2C 2 H 2 + 5O 2 →4CO 2 + 2H 2 O
- 4Al + 3O 2 → 2Al 2 O 3
- H 2 O 2 + H 2 → 2H 2 O
- 2CH 3 OH + 3O 2 → 2CO 2 + 4H 2 O
2. Identify the oxidizing and reducing agent in each redox reaction below.
- 2C 2 H 2 + 5O 2 →4CO 2 + 2H 2 O
- 4Al + 3O 2 → 2Al 2 O 3
- H 2 O 2 + H 2 → 2H 2 O
- 2CH 3 OH + 3O 2 → 2CO 2 + 4H 2 O
Answers
1. Look at the reactants, and examine which is the substance oxidized and the substance reduced.
- substance oxidized:C 2 H 2 since it lost H (and gained O) substance reduced: O 2 since it gained H
- substance oxidized:Al since Al gained O substance reduced: O 2
- substance oxidized:H 2 since it gained O substance reduced: H 2 O 2 since it lost O
- substance oxidized:CH 3 OH since it lost H (and gained O) substance reduced: O 2 since it gained H
2. Identify the oxidizing and reducing agent in each reaction below.
The substance oxidized is the RA while the substance reduced is the OA.
- RA: C 2 H 2 OA: O 2
- RA: Al OA: O 2
- RA: H 2 OA: H 2 O 2
- RA: CH 3 OH OA: O 2


3.6: Augmented Matrices
In this section, we will see how to use matrices to solve systems of equations. In both the graphical method and the expected value method, you have had to solve a system of equations. In the graphical method, you had systems consisting of two lines such as Example \(3.6.1\).
An example of a system of two lines:
\(\begin{array} sy &= \dfrac{3}{5}x-\dfrac{1}{5}\\ y &= -x+3. \end{array}\)
In the expected value method we had systems of three equations such as Example \(3.6.2\).
An example of a system of three equations where the variables are \(E_1(A), E_1(B), P_2(C), P_2(D)\text{:}\)
\( \begin{array}sE_1(A) &= P_2(C)-P_2(D)\\ E_1(B) &= 2P_2(D)\\ 1 &= P_2(C)+P_2(D). \end{array}\)
In Example \(3.6.2\), even after setting \(E_1(A)=E_1(B)\) so that there were only \(2\) variables, the algebra began to get cumbersome. What if we wanted to solve a much larger game, such as a \(5 \times 5\) game?
We've used matrices to represent our games, but now we want to use them as a mathematical tool to help us solve systems of equations. In order to use matrices to solve our systems of equations, we want to write all our equations in the same form: we will have all the variable terms on the left of the equals sign and all constants on the right.
Thus, in Example \(3.6.1\) we can write the system as
In fact, we can simplify the first equation by multiplying both sides by \(5\):
We can use the coefficients and constants to create a matrix:
In this matrix, you have a column for the coefficients of each variable. So the coefficients of \(x\) are in the first column, the coefficients of \(y\) are in the second. The constant terms are always in the last column. Each row corresponds to one equation. This matrix is called an augmented matrix . It is really just a matrix, but we call it augmented if we include information from both sides of the equation (the coefficients and the constants).
The algebraic method for solving the system of equations (finding the \(x\) and \(y\) values that satisfy both equations) is called row reduction . It is based on the elimination method that you may have seen in a precalculus or college algebra course. We won't go through the algebra here, as we really don't need it. Since our goal is to be able to easily solve larger systems of equations, we will rely on technology to do the algebra.
Computer algebra systems such as Sage , Mathematica , and Maple , as well as graphing calculators, can easily do the row reduction for us.
On a graphing calculator, use the matrix function to enter the matrix. Then use the “rref” function (this stands for “reduced row echelon form”). The result will be the following matrix:
We can solve this even more directly using Sage .
Recall that the first column is for \(x\) and the second is for \(y\text{.}\) Each row represents an equation. So translating each row back to equations, we have the following system of equations:
or
By plugging these values back into the original equations, you can verify that this is in fact the solution.
Since the technology does all the algebra for us, our job is to translate the equations into an appropriate matrix and then translate the final matrix back into the solution to the system of equations. Remember, when using a matrix to solve a game, the matrix is only a tool, it is not the solution to the game.
Now, let's try the equations for the expected value method using Example \(3.6.2\). As presented, how many variables does the system have?
It has \(4\): \(E_1(A), E_1(B), P_2(C)\) and \(P_2(D)\text{.}\) But when we solved these equations, we set the expected values equal. This gives us the two equations
Now if we put these into the form with all variables on the left and constants on the right, we get
Putting these equations into an augmented matrix, gives us
\[\begin{bmatrix} 1 & -3 & 0\\ 1 & 1 & 1 \end{bmatrix}. \nonumber \]
where the first column corresponds to \(P_2(C)\) and the second column corresponds to \(P_2(D)\text{.}\)
Using row reduction, we get
\[\begin{bmatrix} 1 & 0 & \dfrac{3}{4}\\ 0 & 1 & \dfrac{1}{4} \end{bmatrix}. \nonumber \]
Thus, recalling Column 1 is for \(P_2(C)\) and Column 2 is for \(P_2(D)\text{,}\) our solution is \(P_2(C)= \dfrac{3}{4} \text{,}\) and \(P_2(D)= \dfrac{1}{4} \text{.}\)
Here are some more systems of equations to practice solving using augmented matrices. If you want to use the above Sage cells just edit the values for each row in the cell.
Solve the system of equations.
\(\begin{array} s2x-2y &=6 \\ x+3y &=7 \end{array}.\)
Solve the system of equations.
\(\begin{array} s4p_1-2p_2&=0 \\ p_1+p_2&=1 \end{array}.\)
For larger matrices, you can edit the Sage cell by adding additional terms in each row, and adding more rows. For example, you can replace \([3,-5,1],[1,1,3]\) with \([4,8,-4,4],[3,8,5,-11],[-2,1,12,-17]\text{.}\)
Consider the system of equations
\(\begin{array} s4x+8y-4z&=4\\3x+8y+5z&=-11\\-2x+y+12z&=-17. \end{array}\)
- Set up the augmented matrix for this system.
- Use row reduction to find the solution.
Consider the system of equations
\(\begin{array}s2x+y-4z&=10\\3x+5z&=-5\\y+2z&=7. \end{array}\)
- Set up the augmented matrix for this system.
- Use row reduction to find the solution.
Consider the system of equations
\(\begin{array}sa+b-5c&=0\\-4a-b+6c&=0\\a+b+c&=1. \end{array}\)
- Set up the augmented matrix for this system.
- Use row reduction to find the solution.
Consider the system of equations
\(\begin{array}s3x+2y-w-v&=0\\2x-y+3z+w+5v&=0\\x+2y+6z-w&=0\\ -y+z-3w+v&=0\\x+y+z+w+v&=1. \end{array}\)
- Set up the augmented matrix for this system.
- Use row reduction to find the solution.
Now we are ready to apply everything we have learned about solving repeated zero-sum games to a much more challenging game in the next section!


8.0: Prelude to Linear Momentum and Collisions
We use the term momentum in various ways in everyday language, and most of these ways are consistent with its precise scientific definition. We speak of sports teams or politicians gaining and maintaining the momentum to win. We also recognize that momentum has something to do with collisions. For example, looking at the rugby players in the photograph colliding and falling to the ground, we expect their momenta to have great effects in the resulting collisions. Generally, momentum implies a tendency to continue on course—to move in the same direction—and is associated with great mass and speed.
Momentum, like energy, is important because it is conserved. Only a few physical quantities are conserved in nature, and studying them yields fundamental insight into how nature works, as we shall see in our study of momentum.


1: What is Game Theory?
“Game Theory is not about 'playing games.' It is about conflict resolution among rational but distrustful beings.” Poundstone, Prisoner's Dilemma , p. \(39\).
Although we will play many games throughout this book, our goal is to understand how rational, distrustful players would play the game. These games are meant to serve as models for situations of conflict. We will explore how to “solve” games under certain assumptions about our players. As with any mathematical model, we will need to make assumptions about how our players will behave, what information they have, and the constraints of the game. For example, we will assume that our players will use all information available to them, and that players will follow the rules of the game.
Games can provide hours of recreational enjoyment, and are worth studying for this alone. However, even simple games can be used to model political, social, and economic interactions. Understanding some foundations of game theory can help us interpret, predict, and respond in competitive situations.
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- 1.1: Players and Strategies
- In this book, most of the games will be played by two players. Each player must decide how he or she will play the game. In order to study games mathematically, we need to make some assumptions about how the players should play the game. This allows us to be able to better predict what our players should do. The example given in this section illustrates the characteristics we will assume about our players.


12.1: Representing Ratios with Tables
Lesson
Let's use tables to represent equivalent ratios.
Exercise \(\PageIndex{1}\): How Is It Growing?
Look for a pattern in the figures.
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How many total tiles will be in:
- the 4th figure?
- the 5th figure?
- the 10th figure?
- How do you see it growing?
Exercise \(\PageIndex{2}\): A Huge Amount of Sparkling Orange Juice
Noah’s recipe for one batch of sparkling orange juice uses 4 liters of orange juice and 5 liters of soda water.
- Use the double number line to show how many liters of each ingredient to use for different-sized batches of sparkling orange juice.
- If someone mixes 36 liters of orange juice and 45 liters of soda water, how many batches would they make?
- If someone uses 400 liters of orange juice, how much soda water would they need?
- If someone uses 455 liters of soda water, how much orange juice would they need?
- Explain the trouble with using a double number line diagram to answer the last two questions.
Exercise \(\PageIndex{3}\): Bacthes of Trail Mix
A recipe for trail mix says: “Mix 7 ounces of almonds with 5 ounces of raisins.” Here is a table that has been started to show how many ounces of almonds and raisins would be in different-sized batches of this trail mix.
| almonds (oz) | raisins (oz) |
|---|---|
| \(7\) | \(5\) |
| \(28\) | |
| \(10\) | |
| \(3.5\) | |
| \(250\) | |
| \(56\) |
- Complete the table so that ratios represented by each row are equivalent.
- What methods did you use to fill in the table?
- How do you know that each row shows a ratio that is equivalent to \(7:5\)? Explain your reasoning.
Are you ready for more?
You have created a best-selling recipe for chocolate chip cookies. The ratio of sugar to flour is \(2:5\).
Create a table in which each entry represents amounts of sugar and flour that might be used at the same time in your recipe.
- One entry should have amounts where you have fewer than 25 cups of flour.
- One entry should have amounts where you have between 20–30 cups of sugar.
- One entry can have any amounts using more than 500 units of flour.
Summary
A table is a way to organize information. Each horizontal set of entries is called a row , and each vertical set of entries is called a column . (The table shown has 2 columns and 5 rows.) A table can be used to represent a collection of equivalent ratios.
Here is a double number line diagram and a table that both represent the situation: “The price is $2 for every 3 mangos.”
Glossary Entries
Definition: Table
A table organizes information into horizontal rows and vertical columns . The first row or column usually tells what the numbers represent.
For example, here is a table showing the tail lengths of three different pets. This table has four rows and two columns.
| pet | tail length (inches) |
|---|---|
| dog | \(22\) |
| cat | \(12\) |
| mouse | \(2\) |
Practice
Exercise \(\PageIndex{4}\)
Complete the table to show the amounts of yellow and red paint needed for different-sized batches of the same shade of orange paint.
| yellow paint (quarts) | red paint (quarts) |
|---|---|
| \(5\) | \(6\) |
Explain how you know that these amounts of yellow paint and red paint will make the same shade of orange as the mixture in the first row of the table.
Exercise \(\PageIndex{5}\)
A car travels at a constant speed, as shown on the double number line.
How far does the car travel in 14 hours? Explain or show your reasoning.
Exercise \(\PageIndex{6}\)
The olive trees in an orchard produce 3,000 pounds of olives a year. It takes 20 pounds of olives to make 3 liters of olive oil. How many liters of olive oil can this orchard produce in a year? If you get stuck, consider using the table.
| olives (pounds) | olive oil (liters) |
|---|---|
| \(20\) | \(3\) |
| \(100\) | |
| \(3,000\) |
Exercise \(\PageIndex{7}\)
At a school recess, there needs to be a ratio of 2 adults for every 24 children on the playground. The double number line represents the number of adults and children on the playground at recess.
- Label each remaining tick mark with its value.
- How many adults are needed if there are 72 children? Circle your answer on the double number line.
(From Unit 2.3.1)
Exercise \(\PageIndex{8}\)
While playing basketball, Jada’s heart rate goes up to 160 beats per minute. While jogging, her heart beats 25 times in 10 seconds. Assuming her heart beats at a constant rate while jogging, which of these activities resulted in a higher heart rate? Explain your reasoning.
(From Unit 2.3.5)
Exercise \(\PageIndex{9}\)
A shopper bought the following items at the farmer’s market:
- 6 ears of corn for $1.80. What was the cost per ear?
- 12 apples for $2.88. What was the cost per apple?
- 5 tomatoes for $3.10. What was the cost per tomato?
(From Unit 2.3.3)


3.3: Affine Notions and Parallel Transport
The Affine Parameter in Curved Spacetime: A Rough Sketch
We want to be able to measure things in curved spacetime. There turn out to be two complementary systems of measurement we can apply: affine measure and metric measure . Affine measure in a flat geometry was introduced in Section 2.1. Surprisingly, it turns out to be quite easy to generalize this to the curved case. Our construction of the affine parameter with a scaffolding of parallelograms depended on the existence and uniqueness of parallels expressed by axiom A1, so we might imagine that there was no point in trying to generalize the construction to curved spacetime. But the equivalence principle tells us that spacetime is locally affine to some approximation. Concretely, clock-time is one example of an affine parameter, and the curvature of spacetime clearly can’t prevent us from building a clock and releasing it on a free-fall trajectory.
More generally, we can use the fact that every segment of a geodesic is geometrically similar to every other segment. For example, consider an arc of the earth’s equator spanning one degree of longitude. That arc could be slid along the equator to a different location, then expanded to cover 3 degrees of longitude. The two arcs are similar.
Example 2: Geodesics are special
The following three non-examples show that this is a special property of geodesics.
The property is not enjoyed by a non-geodesic curve. A segment of a pentagon that encompasses one of the vertices is not similar to some other segment that is straight.
Another non-example involving non-geodesics is the curve that we get in 1+1-dimensional spacetime by joining together the positive x axis and the positive t axis. We can never take a one-year segment of the t axis and, through any combination of boosts and rotations, make it coincide with a one-light-year piece of the x axis. The original segment is timelike, and any boost or rotation will preserve its timelike character.
Furthermore, it is not true in general, when curvature exists, that we can take any geometrical figure, transport it wherever we like, and also scale it as we like. For example, Euclidean geometry is a good approximation on small portions of the Earth’s spherical surface, so a roadmap can be made in the shape of a rectangle with four right-angle corners. However, it is not possible to scale up such a rectangle; to map a large portion of the world, we have to introduce distortions of the type used in map projections.
Because geodesics have this special property, we can slide any portion of a geodesic to anywhere else on the geodesic and employ it as a standard of measure. This gives us a complete system of measurement along that geodesic, and it works regardless of whether the geodesic is timelike, lightlike, or spacelike. But as in flat geometry, affine measurement does not allow us to compare lengths along one geodesic to lengths along another.
The Affine Parameter in More Detail
When we originally defined affine measure in Section 2.1, for a flat space, we did it through the explicit construction of a scaffolding. An important example of the differential, i.e., local, nature of our geometry is the generalization of the scaffolding construction from to a context broader than affine geometry.
To generalize the recipe for the construction (Figure 3.2.1), the first obstacle is the ambiguity of the instruction to construct parallelogram \((0,1,q_0,q_1)\), which requires us to draw 1q 1 parallel to 0q 0 . Suppose we construe this as an instruction to make the two segments initially parallel, i.e., parallel as they depart the line at 0 and 1. By the time they get to q 0 and q 1 , they may be converging or diverging.
Because parallelism is only approximate here, there will be a certain amount of error in the construction of the affine parameter. One way of detecting such an error is that lattices constructed with different initial distances will get out of step with one another. For example, we can define \(\frac{1}{2}\) as before by requiring that the lattice constructed with initial segment 0 \(\frac{1}{2}\) line up with the original lattice at 1. We will find, however, that they do not quite line up at other points, such as 2. Let’s use this discrepancy \(\epsilon\) = 2 − 2' as a numerical measure of the error. It will depend on both \(\delta_{1}\), the distance 01, and on \(\delta_{2}\), the distance between 0 and q 0 . Since \(\epsilon\) vanishes for either \(\delta_{1}\) = 0 or \(\delta_{2}\) = 0, and since the equivalence principle guarantees smooth behavior on small scales, the leading term in the error will in general be proportional to the product \(\delta_{1} \delta_{2}\). In the language of infinitesimals, we can replace \(\delta_{1}\) and \(\delta_{2}\) with infinitesimally short distances, which for simplicity we assume to be equal, and which we call d\(\lambda\). Then the affine parameter \(\lambda\) is defined as \(\lambda = \int d \lambda\), where the error of order d\(\lambda^{2}\) is, as usual, interpreted as the negligible discrepancy between the integral and its approximation as a Riemann sum.
Parallel Transport
If you were alert, you may have realized that I cheated you at a crucial point in this construction. We were to make 1q 1 and 0q 0 “initially parallel” as they left 01. How should we even define this idea of “initially parallel?” We could try to do it by making angles q 0 01 and q 1 12 equal, but this doesn’t quite work, because it doesn’t specify whether the angle is to the left or the right on the two-dimensional plane of the page. In three or more dimensions, the issue becomes even more serious. The construction workers building the lattice need to keep it all in one plane, but how do they do that in curved spacetime?
A mathematician’s answer would be that our geometry lacks some additional structure called a connection , which is a rule that specifies how one locally flat neighborhood is to be joined seamlessly onto another locally flat neighborhood nearby. If you’ve ever bought two maps and tried to tape them together to make a big map, you’ve formed a connection. If the maps were on a large enough scale, you also probably noticed that this was impossible to do perfectly, because of the curvature of the earth.
Physically, the idea is that in flat spacetime, it is possible to construct inertial guidance systems like the ones discussed earlier. Since they are possible in flat spacetime, they are also possible in locally flat neighborhoods of spacetime, and they can then be carried from one neighborhood to another. In three space dimensions, a gyroscope’s angular momentum vector maintains its direction, and we can orient other vectors, such as 1q 1 , relative to it. Suppose for concreteness that the construction of the affine parameter above is being carried out in three space dimensions. We place a gyroscope at 0, orient its axis along 0q 0 , slide it along the line to 1, and then construct 1q 1 along that axis.
In 3+1 dimensions, a gyroscope only does part of the job. We now have to maintain the direction of a four-dimensional vector. Four-vectors will not be discussed in detail until section 4.2, but similar devices can be used to maintain their orientations in spacetime. These physical devices are ways of defining a mathematical notion known as parallel transport , which allows us to take a vector from one point to another in space. In general, specifying a notion of parallel transport is equivalent to specifying a connection.
Parallel transport is path-dependent, as shown in Figure 3.2.2.
Affine Parameters Defined Only along Geodesics
In the context of flat spacetime, the affine parameter was defined only along lines, not arbitrary curves, and could not be compared between lines running in different directions. In curved spacetime, the same limitation is present, but with “along lines” replaced by “along geodesics.” Figure 3.2.3 shows what goes wrong if we try to apply the construction to a world-line that isn’t a geodesic. One definition of a geodesic is that it’s the course we’ll end up following if we navigate by keeping a fixed bearing relative to an inertial guidance device such as gyroscope; that is, the tangent to a geodesic, when parallel-transported farther along the geodesic, is still tangent. A non-geodesic curve lacks this property, and the effect on the construction of the affine parameter is that the segments nq n drift more and more out of alignment with the curve.


6.1: Evaluate and Graph Exponential Functions
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Graph Exponential Functions
The functions we have studied so far do not give us a model for many naturally occurring phenomena. From the growth of populations and the spread of viruses to radioactive decay and compounding interest, the models are very different from what we have studied so far. These models involve exponential functions.
An exponential function is a function of the form \(f(x)=a^{x}\) where \(a>0\) and \(a≠1\).
Definition \(\PageIndex{1}\)
An exponential function, where \(a>0\) and \(a≠1\), is a function of the form
\(f(x)=a^{x}\)
Notice that in this function, the variable is the exponent. In our functions so far, the variables were the base.
Our definition says \(a≠1\). If we let \(a=1\), then \(f(x)=a^{x}\) becomes \(f(x)=1^{x}\). Since \(1^{x}=1\) for all real numbers, \(f(x)=1\). This is the constant function.
Our definition also says \(a>0\). If we let a base be negative, say \(−4\), then \(f(x)=(−4)^{x}\) is not a real number when \(x=\frac{1}{2}\).
\(\begin{aligned} f(x) &=(-4)^{x} \\ f\left(\frac{1}{2}\right) &=(-4)^{\frac{1}{2}} \\ f\left(\frac{1}{2}\right) &=\sqrt{-4} \text { not a real number } \end{aligned}\)
In fact, \(f(x)=(−4)^{x}\) would not be a real number any time \(x\) is a fraction with an even denominator. So our definition requires \(a>0\).
By graphing a few exponential functions, we will be able to see their unique properties.
Example \(\PageIndex{1}\)
On the same coordinate system graph \(f(x)=2^{x}\) and \(g(x)=3^{x}\).
Solution :
We will use point plotting to graph the functions.
Exercise \(\PageIndex{1}\)
Graph: \(f(x)=4^{x}\).
- Answer
Exercise \(\PageIndex{2}\)
Graph: \(g(x)=5^{x}\)
- Answer
If we look at the graphs from the previous Example 10.2.1 and Exercises 10.2.1 and 10.2.2, we can identify some of the properties of exponential functions.
The graphs of \(f(x)=2^{x}\) and \(g(x)=3^{x}\), as well as the graphs of \(f(x)=4^{x}\) and \(g(x)=5^{x}\), all have the same basic shape. This is the shape we expect from an exponential function where \(a>1\).
We notice, that for each function, the graph contains the point \((0,1)\). This make sense because \(a^{0}=1\) for any \(a\).
The graph of each function, \(f(x)=a^{x}\) also contains the point \((1,a)\). The graph of \(f(x)=2^{x}\) contained \((1,2)\) and the graph of \(g(x)=3^{x}\) contained \((1,3)\). This makes sense as \(a^{1}=a\).
Notice too, the graph of each function \(f(x)=a^{x}\) also contains the point \((−1,\frac{1}{a})\). The graph of \(f(x)=2^{x}\) contained \((−1,\frac{1}{2})\) and the graph of \(g(x)=3^{x}\) contained \((−1,\frac{1}{3})\).This makes sense as \(a^{−1}=\frac{1}{a}\).
What is the domain for each function? From the graphs we can see that the domain is the set of all real numbers. There is no restriction on the domain. We write the domain in interval notation as \((−∞,∞)\).
Look at each graph. What is the range of the function? The graph never hits the \(x\)-axis. The range is all positive numbers. We write the range in interval notation as \((0,∞)\).
Whenever a graph of a function approaches a line but never touches it, we call that line an asymptote . For the exponential functions we are looking at, the graph approaches the \(x\)-axis very closely but will never cross it, we call the line \(y=0\), the \(x\)-axis, a horizontal asymptote.
Properties of the Graph of \(f(x)=a^{x}\) when \(a>1\)
| Domain | \((-\infty, \infty)\) |
| Range | \((0, \infty)\) |
| \(x\)-intercept | None |
| \(y\)-intercept | \((0,1)\) |
| Contains | \((1, a),\left(-1, \frac{1}{a}\right)\) |
| Asymptote | \(x\)-axis, the line \(y=0\) |
Our definition of an exponential function \(f(x)=a^{x}\) says \(a>0\), but the examples and discussion so far has been about functions where \(a>1\). What happens when \(0<a<1\) The next example will explore this possibility.
Example \(\PageIndex{2}\)
On the same coordinate system, graph \(f(x)=\left(\frac{1}{2}\right)^{x}\) and \(g(x)=\left(\frac{1}{3}\right)^{x}\).
Solution :
We will use point plotting to graph the functions.
Exercise \(\PageIndex{3}\)
Graph: \(f(x)=\left(\frac{1}{4}\right)^{x}\).
- Answer
Exercise \(\PageIndex{4}\)
Graph: \(g(x)=\left(\frac{1}{5}\right)^{x}\).
- Answer
Now let’s look at the graphs from the previous Example 10.2.2 and Exercises 10.2.3 and 10.2.4 so we can now identify some of the properties of exponential functions where \(0<a<1\).
The graphs of \(f(x)=\left(\frac{1}{2}\right)^{x}\) and \(g(x)=\left(\frac{1}{3}\right)^{x}\) as well as the graphs of \(f(x)=\left(\frac{1}{4}\right)^{x}\) and \(g(x)=\left(\frac{1}{5}\right)^{x}\) all have the same basic shape. While this is the shape we expect from an exponential function where \(0<a<1\), the graphs go down from left to right while the previous graphs, when \(a>1\), went from up from left to right.
We notice that for each function, the graph still contains the point \((0, 1)\). This make sense because \(a^{0}=1\) for any \(a\).
As before, the graph of each function, \(f(x)=a^{x}\), also contains the point \((1,a)\). The graph of \(f(x)=\left(\frac{1}{2}\right)^{x}\) contained \(\left(1, \frac{1}{2}\right)\) and the graph of \(g(x)=\left(\frac{1}{3}\right)^{x}\) contained \(\left(1, \frac{1}{3}\right)\). This makes sense as \(a^{1}=a\).
Notice too that the graph of each function, \(f(x)=a^{x}\), also contains the point \(\left(-1, \frac{1}{a}\right)\). The graph of \(f(x)=\left(\frac{1}{2}\right)^{x}\) contained \((−1,2)\) and the graph of \(g(x)=\left(\frac{1}{3}\right)^{x}\) contained \((−1,3)\). This makes sense as \(a^{-1}=\frac{1}{a}\).
What is the domain and range for each function? From the graphs we can see that the domain is the set of all real numbers and we write the domain in interval notation as \((−∞,∞)\). Again, the graph never hits the \(x\)-axis. The range is all positive numbers. We write the range in interval notation as \((0,∞)\).
We will summarize these properties in the chart below. Which also include when \(a>1\).
Properties of the Graph of \(f(x)=a^{x}\)
| When \(a>1\) | When \(0<a<1\) | ||
|---|---|---|---|
| 1\)">Domain | \((-\infty, \infty)\) | Domain | \((-\infty, \infty)\) |
| 1\)">Range | \((0, \infty)\) | Range | \((0, \infty)\) |
| 1\)">\(x\)-intercept | none | \(x\)-intercept | none |
| 1\)">\(y\)-intercept | \((0,1)\) | \(y\)-intercept | \((0,1)\) |
| 1\)">Contains | \((1, a),\left(-1, \frac{1}{a}\right)\) | Contains | \((1, a),\left(-1, \frac{1}{a}\right)\) |
| 1\)">Asymptote |
\(x\)-axis, the line \(y=0\) |
Asymptote | \(x\)-axis, the line \(y=0\) |
| 1\)">Basic Shape | increasing | Basic Shape | decreasing |
It is important for us to notice that both of these graphs are one-to-one, as they both pass the horizontal line test. This means the exponential function will have an inverse. We will look at this later.
When we graphed quadratic functions, we were able to graph using translation rather than just plotting points. Will that work in graphing exponential functions?
Example \(\PageIndex{3}\)
On the same coordinate system graph \(f(x)=2^{x}\) and \(g(x)=2^{x+1}\).
Solution :
We will use point plotting to graph the functions.
Exercise \(\PageIndex{5}\)
On the same coordinate system, graph: \(f(x)=2^{x}\) and \(g(x)=2^{x-1}\).
- Answer
Exercise \(\PageIndex{6}\)
On the same coordinate system, graph \(f(x)=3^{x}\) and \(g(x)=3^{x+1}\).
- Answer
Looking at the graphs of the functions \(f(x)=2^{x}\) and \(g(x)=2^{x+1}\) in the last example, we see that adding one in the exponent caused a horizontal shift of one unit to the left. Recognizing this pattern allows us to graph other functions with the same pattern by translation.
Let’s now consider another situation that might be graphed more easily by translation, once we recognize the pattern.
Example \(\PageIndex{4}\)
On the same coordinate system graph \(f(x)=3^{x}\) and \(g(x)=3^{x}-2\).
Solution :
We will use point plotting to graph the functions.
Exercise \(\PageIndex{7}\)
On the same coordinate system, graph \(f(x)=3^{x}\) and \(g(x)=3^{x}+2\).
- Answer
Exercise \(\PageIndex{8}\)
On the same coordinate system, graph \(f(x)=4^{x}\) and \(g(x)=4^{x}-2\).
- Answer
Looking at the graphs of the functions \(f(x)=3^{x}\) and \(g(x)=3^{x}−2\) in the last example, we see that subtracting \(2\) caused a vertical shift of down two units. Notice that the horizontal asymptote also shifted down \(2\) units. Recognizing this pattern allows us to graph other functions with the same pattern by translation.
All of our exponential functions have had either an integer or a rational number as the base. We will now look at an exponential function with an irrational number as the base.
Before we can look at this exponential function, we need to define the irrational number, \(e\). This number is used as a base in many applications in the sciences and business that are modeled by exponential functions. The number is defined as the value of \(\left(1+\frac{1}{n}\right)^{n}\) as \(n\) gets larger and larger. We say, as \(n\) approaches infinity, or increases without bound. The table shows the value of \(\left(1+\frac{1}{n}\right)^{n}\) for several values of \(n\).
| \(n\) | \(\left(1+\frac{1}{n}\right)^{n}\) |
|---|---|
| \(1\) | \(2\) |
| \(2\) | \(2.25\) |
| \(5\) | \(2.48832\) |
| \(10\) | \(2.59374246\) |
| \(100\) | \(2.704813829 \ldots\) |
| \(1,000\) | \(2.716923932 \ldots\) |
| \(10,000\) | \(2.718145927 \ldots\) |
| \(100,000\) | \(2.718268237 \ldots\) |
| \(1,000,000\) | \(2.718280469 \ldots\) |
| \(1,000,000,000\) | \(2.718281827 \ldots\) |
\(e \approx 2.718281827\)
The number \(e\) is like the number \(π\) in that we use a symbol to represent it because its decimal representation never stops or repeats. The irrational number \(e\) is called the natural base .
Definition \(\PageIndex{2}\)
Natural Base \(e\)
The number \(e\) is defined as the value of \(\left(1+\frac{1}{n}\right)^{n}\), as \(n\) increases without bound. We say, as \(n\) approaches infinity,
\(e \approx 2.718281827\)
The exponential function whose base is \(e\), \(f(x)=e^{x}\) is called the natural exponential function .
Definition \(\PageIndex{3}\)
Natural Exponential Function
The natural exponential function is an exponential function whose base is \(e\)
\(f(x)=e^{x}\)
The domain is \((−∞,∞)\) and the range is \((0,∞)\).
Let’s graph the function \(f(x)=e^{x}\) on the same coordinate system as \(g(x)=2^{x}\) and \(h(x)=3^{x}\).
Notice that the graph of \(f(x)=e^{x}\) is “between” the graphs of \(g(x)=2^{x}\) and \(h(x)=3^{x}\).Does this make sense as \(2<e<3\)?
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Use Exponential Models in Applications
Exponential functions model many situations. If you own a bank account, you have experienced the use of an exponential function. There are two formulas that are used to determine the balance in the account when interest is earned. If a principal, \(P\), is invested at an interest rate, \(r\), for \(t\) years, the new balance, \(A\), will depend on how often the interest is compounded. If the interest is compounded \(n\) times a year we use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\). If the interest is compounded continuously, we use the formula \(A=Pe^{rt}\). These are the formulas for compound interest .
Definition \(\PageIndex{5}\)
Compound Interest
For a principal, \(P\), invested at an interest rate, \(r\), for \(t\) years, the new balance, \(A\), is:
\(\begin{array}{ll}{A=P\left(1+\frac{r}{n}\right)^{n t}} & {\text { when compounded } n \text { times a year. }} \\ {A=P e^{r t}} & {\text { when compounded continuously. }}\end{array}\)
As you work with the Interest formulas, it is often helpful to identify the values of the variables first and then substitute them into the formula.
Example \(\PageIndex{7}\)
A total of $\(10,000\) was invested in a college fund for a new grandchild. If the interest rate is \(5\)%, how much will be in the account in \(18\) years by each method of compounding?
- compound quarterly
- compound monthly
- compound continuously
Solution :
Identify the values of each variable in the formulas. Remember to express the percent as a decimal.
\(\begin{aligned} A &=? \\ P &=\$ 10,000 \\ r &=0.05 \\ t &=18 \text { years } \end{aligned}\)
a. For quarterly compounding, \(n=4\). There are \(4\) quarters in a year.
\(A=P\left(1+\frac{r}{n}\right)^{n t}\)
Substitute the values in the formula.
\(A=10,000\left(1+\frac{0.05}{4}\right)^{4 \cdot 18}\)
Compute the amount. Be careful to consider the order of operations as you enter the expression into your calculator.
\(A=\$ 24,459.20\)
b. For monthly compounding, \(n=12\).There are \(12\) months in a year.
\(A=P\left(1+\frac{r}{n}\right)^{n t}\)
Substitute the values in the formula.
\(A=10,000\left(1+\frac{0.05}{12}\right)^{12 \cdot 18}\)
Compute the amount.
\(A=\$ 24,550.08\)
c. For compounding continuously,
\(A=P e^{r t}\)
Substitute the values in the formula.
\(A=10,000 e^{0.05 \cdot 18}\)
Compute the amount.
\(A=\$ 24,596.03\)
Exercise \(\PageIndex{13}\)
Angela invested $\(15,000\) in a savings account. If the interest rate is \(4\)%, how much will be in the account in \(10\) years by each method of compounding?
- compound quarterly
- compound monthly
- compound continuously
- Answer
-
- $\(22,332.96\)
- $\(22,362.49\)
- $\(22,377.37\)
Exercise \(\PageIndex{14}\)
Allan invested $\(10,000\) in a mutual fund. If the interest rate is \(5\)%, how much will be in the account in \(15\) years by each method of compounding?
- compound quarterly
- compound monthly
- compound continuously
- Answer
-
- $\(21,071.81\)
- $\(21,137.04\)
- $\(21,170.00\)
Other topics that are modeled by exponential functions involve growth and decay. Both also use the formula \(A=Pe^{rt}\) we used for the growth of money. For growth and decay, generally we use \(A_{0}\), as the original amount instead of calling it \(P\), the principal. We see that exponential growth has a positive rate of growth and exponential decay has a negative rate of growth.
Definition \(\PageIndex{6}\)
Exponential Growth and Decay
For an original amount, \(A_{0}\), that grows or decays at a rate, \(r\), for a certain time, \(t\), the final amount, \(A\), is:
\(A=A_{0} e^{r t}\)
Exponential growth is typically seen in the growth of populations of humans or animals or bacteria. Our next example looks at the growth of a virus.
Example \(\PageIndex{8}\)
Chris is a researcher at the Center for Disease Control and Prevention and he is trying to understand the behavior of a new and dangerous virus. He starts his experiment with \(100\) of the virus that grows at a rate of \(25\)% per hour. He will check on the virus in \(24\) hours. How many viruses will he find?
Solution :
Identify the values of each variable in the formulas. Be sure to put the percent in decimal form. Be sure the units match--the rate is per hour and the time is in hours.
\(\begin{aligned} A &=? \\ A_{0} &=100 \\ r &=0.25 / \text { hour } \\ t &=24 \text { hours } \end{aligned}\)
Substitute the values in the formula: \(A=A_{0} e^{r t}\).
\(A=100 e^{0.25 \cdot 24}\)
Compute the amount.
\(A=40,342.88\)
Round to the nearest whole virus.
\(A=40,343\)
The researcher will find \(40,343\) viruses.
Exercise \(\PageIndex{15}\)
Another researcher at the Center for Disease Control and Prevention, Lisa, is studying the growth of a bacteria. She starts his experiment with \(50\) of the bacteria that grows at a rate of \(15\)% per hour. He will check on the bacteria every \(8\) hours. How many bacteria will he find in \(8\) hours?
- Answer
-
She will find \(166\) bacteria.
Exercise \(\PageIndex{16}\)
Maria, a biologist is observing the growth pattern of a virus. She starts with \(100\) of the virus that grows at a rate of \(10\)% per hour. She will check on the virus in \(24\) hours. How many viruses will she find?
- Answer
-
She will find \(1,102\) viruses.
Access these online resources for additional instruction and practice with evaluating and graphing exponential functions.
Key Concepts
- Properties of the Graph of \(f(x)=a^{x}\):
| When \(a>1\) | When \(0<a<1\) | ||
|---|---|---|---|
| 1\)">Domain | \((-\infty, \infty)\) | Domain | \((-\infty, \infty)\) |
| 1\)">Range | \((0, \infty)\) | Range | \((0, \infty)\) |
| 1\)">\(x\)-intercept | none | \(x\)-intercept | none |
| 1\)">\(y\)-intercept | \((0,1)\) | \(y\)-intercept | \((0,1)\) |
| 1\)">Contains | \((1, a),\left(-1, \frac{1}{a}\right)\) | Contains | \((1, a),\left(-1, \frac{1}{a}\right)\) |
| 1\)">Asymptote |
\(x\)-axis, the line \(y=0\) |
Asymptote | \(x\)-axis, the line \(y=0\) |
| 1\)">Basic Shape | increasing | Basic Shape | decreasing |
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One-to-One Property of Exponential Equations:
For \(a>0\) and \(a≠1\),\(A=A_{0} e^{r t}\)
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How to Solve an Exponential Equation
- Write both sides of the equation with the same base, if possible.
- Write a new equation by setting the exponents equal.
- Solve the equation.
- Check the solution.
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Compound Interest:
For a principal, \(P\), invested at an interest rate, \(r\), for \(t\) years, the new balance, \(A\), is
\(\begin{array}{ll}{A=P\left(1+\frac{r}{n}\right)^{n t}} & {\text { when compounded } n \text { times a year. }} \\ {A=P e^{r t}} & {\text { when compounded continuously. }}\end{array}\) - Exponential Growth and Decay: For an original amount, \(A_{0}\) that grows or decays at a rate, \(r\), for a certain time \(t\), the final amount, \(A\), is \(A=A_{0}e^{rt}\).
Glossary
- asymptote
- A line which a graph of a function approaches closely but never touches.
- exponential function
- An exponential function, where \(a>0\) and \(a≠1\), is a function of the form \(f(x)=a^{x}\).
- natural base
- The number \(e\) is defined as the value of \((1+\frac{1}{n})^{n}\), as \(n\) gets larger and larger. We say, as \(n\) increases without bound, \(e≈2.718281827...\)
- natural exponential function
- The natural exponential function is an exponential function whose base is \(e\): \(f(x)=e^{x}\). The domain is \((−∞,∞)\) and the range is \((0,∞)\).


6.10C: Microbial Growth at Low or High pH
Microorganisms live and thrive within specific pH levels.
LEARNING OBJECTIVES
Differentiate microbial growth at high or low pHs
Key Takeaways
In chemistry, pH is a measure of the activity of the (solvated) hydrogen ion. In other words, it is a measure of hydrogen ion concentration. Pure water has a pH very close to 7 at 25°C. Solutions with a pH less than 7 are said to be acidic, and solutions with a pH greater than 7 are said to be basic or alkaline. The pH scale is traceable to a set of standard solutions whose pH is established by international agreement. The pH of different cellular compartments, body fluids, and organs is usually tightly regulated in a process called acid-base homeostasis. Microorganisms live and thrive within specific pH levels.
Neutrophiles are organisms that thrive in neutral (pH 7) environments; extromophiles are organisms that thrive in extreme pH environments.
Alkaliphiles are microbes that thrive in alkaline environments with a pH of 9 to 11, such as playa lakes and carbonate-rich soils. To survive, alkaliphiles maintain a relatively low alkaline level of about 8 pH inside their cells by constantly pumping hydrogen ions in the form of hydronium ions (H3O+) across their cell membranes and into their cytoplasm.
Acidophilic organisms are those that thrive under highly acidic conditions (usually at pH 2.0 or below). Most acidophile organisms have evolved extremely efficient mechanisms to pump protons out of the intracellular space in order to keep the cytoplasm at or near neutral pH. Therefore, intracellular proteins do not need to develop acid stability through evolution. However, other acidophiles, such as Acetobacter aceti, have an acidified cytoplasm which forces nearly all proteins in the genome to evolve acid stability.
Key Points
- Neutrophiles are organisms that thrive in neutral (pH 7) environments.
- Alkaliphiles are microbes that thrive in alkaline (pH 9-11) environments.
- Acidophilic organisms are those that thrive under highly acidic conditions (usually at pH 2.0 or below).
Key Terms
- neutrophile : any organism that thrives in a relatively neutral pH
- alkaliphile : any organism that lives and thrives in an alkaline environment, such as a soda lake; a form of extremophile
- acidophile : an organism that lives and thrives under acidic conditions; a form of extremophile