2.3: Final Toulmin Analysis Essay
Focus
Strong thesis, sentence structure, paragraph cohesion & conceptual ﬂow, integration of relevant & authoritative sources, focused intro & conclusion. Use of Toulmin Analysis as the basis for your evaluation.
In class we will discuss and evaluate the method of argument analysis developed by the British philosopher Stephen Toulmin. Afterwards, you will select one of the essays from the assigned section of our Blackboard page and write a Toulmin analysis of the argument(s) presented in the essay.
5-7 Pages, Double-spaced, MLA format.
- Select an essay from the assigned section of our Blackboard page.
- In your introduction, explore the overall topic, the main thesis of the article you are analyzing and your thesis on whether or not you find the argument effective and convincing. What, in Toulmin terms, makes the argument strong or weak?
- Try to find the central claim of the article you are analyzing. How do you know it is the claim? Is it stated or implied? Is it a claim of fact , value or policy ? Does the author provide a qualifier and/or a rebuttal for their claim? If not, does the absence of these elements weaken the claim or not?
- Look at the data/evidence the author provides for their claim. Is the evidence clear? Is it fair? Is it relevant, reliable and representative? Do you get a sense that the author has selectively chosen evidence and ignored other evidence in order to strengthen their claim? Can you demonstrate this? Does the evidence support the particular type of claim the author has made (fact, value or policy)? Build upon the research you did in your Annotated Bibliography to help you to help you determine how relevant, authoritative and reliable the data is which the author of your chosen ar- ticle provides in support of its claim.
- Use examples from relevant sources (in the form of quotation, summary, and paraphrase) and your own discussion to create this thesis-driven essay. Try to move beyond just retelling your experience or opinion. You need to look at the underlying claims ABOUT the topic as well as what the most reliable and authoritative sources you can find have to say about it.
- Can you identify the warrant (the unstated assumption that connects the claim and the data) of the essay you are reading? Does the warrant do a sufficient job of connecting the data to the warrant? Why or why not? Is it a substantive (logos-based), authoritative (ethos-based) or motivational (pathos-based) warrant? Does this type of warrant best suit the type of claim the author is making and the type of evidence they provide to support this claim?
- Does the author provide backing (evidence) for their warrant? If not, would the presence of evidence for their warrant strengthen the argument in any way? Why or why not?
- Is there anything in the essay that needs an extended definition? How might this affect the overall argument with regard to the Toulmin-labeled relationships between the ideas?
- What is your assessment of the overall effectiveness of the argument? Is it persuasive? Why or why not? What, in Toulmin terms, could be changed to make it more persuasive?
Goals
- Be able to analyze a larger cultural issue and respond to it.
- Be able to clearly present your interpretation of the issue, without personal bias or with clear, relevant and authoritative support for the bias using the Toulmin model of argument analysis.
- Be able to use summary, paraphrase, and quotation of relevant sources in your analysis.
- Be able to clearly express your own thoughts about the topic under discussion.
- Be able to create graceful transitions, introduction, and conclusion to tie together the various parts of your essay.
- Present your argument or position in a clear thesis statement in the introduction.
- Support your claim or focus with evidence and examples from relevant, reliable and representative sources. The body of your essay will explore the major points related to your thesis.
- Present one major point (presented in a topic sentence) per paragraph and explain it fully, with detailed support and examples, before you move on to your next point/paragraph.
- Wrap up your essay with a conclusion that revisits your overall topic and theme.
Research
- Take advantage of the WR 122 Library Guide (on our Blackboard page) for help with finding relevant sources, getting your essay into MLA format and all other aspects of the research and writing process.
- Consult the MHCC databases for relevant articles related to your research.
- You may also use popular search engines, interviews and other reliable, authoritative means of support to develop your thesis.
Format & Length
- 5-7 pages, double-spaced, 12.pt font. (Times New Roman, Arial or Helvetica)
- MLA format. (Please consult your textbook and our Library page for help with MLA. Newer versions of Microsoft Word come with an MLA template as well. You have access to Microsoft Office as a student.
- Consult the Class Syllabus and Overview for instructions on how to download it. Remember, MLA primarily means how you structure the first page headings, proper in-text citations and a properly formatted “Works Cited” page).
- A minimum of 4 reliable, relevant external sources cited in the essay and on a properly formatted Works Cited page.
- Give your paper an original title that is NOT simply “Final Essay” or “Social Media Essay.”
- Grab the reader’s attention, provide a BRIEF summary of the major themes and your overall thesis (main point) in the introduction of your essay.
- Present one major point (presented in a topic sentence) per paragraph and explain it fully, with detailed support and examples, before you move on to your next point/paragraph.
- Wrap up your essay with a conclusion that revisits your overall topic, theme and thesis.
- View Toulmin Worksheet #1
- View Toulmin Worksheet #2
Due Date & Other Details
- Your essay is due at our final class meeting. See calendar and course schedule for more details.
- Take a draft of your essay to the Writing Center (link on our Blackboard Page) for feedback & extra credit. Include a signed document in your submission.
- Please type your paper, double-spaced, in a readable font (about 12-point). Be sure to have someone look over your paper, after you have polished the final draft, to catch any mechanical errors. Final drafts turned in should be entirely free of such errors.
- Make sure you are using the elements of argument construction and analysis we have learned this term to strengthen the presentation of your ideas. For instance, make sure you have established proper authority with your sources, have sufficient and reliable data, make appropriate appeals to pathos, and have a proper warrant that connects your evidence to your thesis.
- Make sure to consider and include substantive responses to the major arguments on both (or all) sides of the argument in question. Especially those in opposition to your thesis.


4.6.1: Classification of Finance Problems (Exercises)
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SECTION 6.6 PROBLEM SET: CLASSIFICATION OF FINANCE PROBLEMS
Let the letters A, B, C, D, E and F be represented as follows:
\[\begin{array}{lll}A=F V \text { of a lump-sum } & C=F V \text { of an annuity } & E=\text { Installment payment } \\ B=P V \text { of a lump-sum } & D=\text { sinking fund payment } & F=P V \text { of an annuity }\end{array} \nonumber \]
Classify each by writing the appropriate letter in the box, and write an equation for solution.
1) What monthly deposits made to an account paying 9% will grow to $10,000 in 4 years?
2) An amount of $4000 is invested at 6% compounded daily. What will the final amount be in 5 years?
3) David has won a lottery paying him $10,000 per month for the next 20 years. He'd rather have the whole amount in one lump sum now. If the current interest rate is 7%, how much money can he hope to get?
4) Each month Linda deposits $250 in an account that pays 9%. How much money will she have in 4 years?
5) Find the monthly payment for a $15,000 car if the loan is amortized over 4 years at a rate of 10%.
6) What lump-sum deposited in an account paying 7% compounded daily will grow to $10,000 in 5 years?
7) What amount of quarterly payments will amount to $250,000 in 5 years at a rate of 8%?
8) The Chang family bought their house 25 years ago. They had their loan financed for 30 years at an interest rate of 11% resulting in a payment of $1350 a month. Find the balance of the loan.
A 10-year $1000 bond pays $35 every six months. If the current interest rate is 8%, in order to find the fair market value of the bond, we need to find the following.
9) The present value of $1000.
10) The present value of the $35 per six month payments.
SECTION 6.6 PROBLEM SET: CLASSIFICATION OF FINANCE PROBLEMS
\[\begin{array}{lll}A=F V \text { of a lump-sum } & C=F V \text { of an annuity } & E=\text { Installment payment } \\ B=P V \text { of a lump-sum } & D=\text { sinking fund payment } & F=P V \text { of an annuity }\end{array} \nonumber \]
11) What lump-sum deposit made today is equal to 33 monthly deposits of $500 if the interest rate is 8%?
12) What monthly deposits made to an account paying 10% will accumulated to $10,000 in six years?
13) A department store charges a finance charge of 1.5% per month on the outstanding balance.
If Ned charged $400 three months ago and has not paid his bill, how much does he owe?
14) What will the value of $300 monthly deposits be in 10 years if the account pays 12% compounded monthly?
15) What lump-sum deposited at 6% compounded daily will grow to $2000 in three years?
16) A company buys an apartment complex for $5,000,000 and amortizes the loan over 10 years.
What is the yearly payment if the interest rate is 14%?
17) In 2002, a house in Rock City cost $300,000. Real estate in Rock City has been increasing in value at the annual rate of 5.3%.. Find the price of that house in 2016.
18) You determine that you can afford to pay $400 per month for a car. What is the maximum price you can pay for a car if the interest rate is 11% and you want to repay the loan in 4 years?
19) A business needs $350,000 in 5 years. How much lump-sum should be put aside in an account that pays 9% so that five years from now the company will have $350,000?
20) A person wishes to have $500,000 in a pension fund 20 years from now. How much should he deposit each month in an account paying 9% compounded monthly?


5.1: Introduction
- Describe stress management techniques
- Demonstrate the basic principles of organization and time management
- Describe conflict resolution techniques
- Describe effective problem solving and decision making
Food service and hospitality businesses can be stressful places to work. The pressure of producing high-quality meals in short periods of time can cause stress. Stress management techniques allow you to manage this stress. Part of managing stress is having an enjoyable job. An enjoyable job is one that fits into your personal goals. To feel in control of your life, you must be able to set personal goals and work toward them. For most people, work is an integral part of personal goals. In order to realize these goals, you must be able to conduct a job search, prepare resumés and cover letters, and participate in job interviews.
The success of food service and hospitality businesses depends on how effectively the entire staff of an operation can cooperate to create a well-prepared meal served in pleasant surroundings. The staff must work together as a team. A team does not just happen. It is created through the actions of supervisors who select the right staff, motivate them to do their best, build a sense of pride and common purpose, and resolve problems and conflicts that arise.


1.6: Privacy
Learning Objectives
- Discuss the challenges to privacy in the business computing environment
Information privacy , or data privacy , is the relationship between collection and dissemination of data, technology, the public expectation of privacy, and the legal and political issues surrounding them.
Privacy concerns exist wherever personally identifiable information or other sensitive information is collected and stored—in digital form or otherwise. Improper or non-existent disclosure control can be the root cause for privacy issues. Data privacy issues can arise in response to information from a wide range of sources, such as:
- Healthcare records
- Criminal justice investigations and proceedings
- Financial institutions and transactions
- Biological traits, such as genetic material
- Residence and geographic records
- Ethnicity
- Privacy breach
- Location-based service and geolocation
The challenge in data privacy is to share data while protecting personally identifiable information. The fields of data security and information security design and utility software, hardware and human resources exist to address this issue. As the laws and regulations related to Data Protection are constantly changing, it is important to keep abreast of any changes in the law and continually reassess your compliance with data privacy and security regulations.
The Internet and Security
The ability to control the information one reveals about oneself over the Internet, and who can access that information, has become a growing concern. These concerns include whether email can be stored or read by third parties without consent, or whether third parties can continue to track the web sites someone has visited. Another concern is web sites which are visited collect, store, and possibly share personally identifiable information about users.
The advent of various search engines and the use of data mining created a capability for data about individuals to be collected and combined from a wide variety of sources very easily. The FTC has provided a set of guidelines that represent widely accepted concepts concerning fair information practices in an electronic marketplace called the Fair Information Practice Principles.
Everything is accessible over the Internet nowadays. A major issue with privacy relates back to social networking. For example, there are millions of users on Facebook, Twitter, and other social media platforms, and regulations have changed. People may be tagged in photos or have valuable information exposed about themselves either by choice or most of the time unexpectedly by others.
It is important to be cautious of what is being said over the Internet and what information is being displayed as well as photos because this all can searched across the web and used to access private databases making it easy for anyone to quickly go online and profile a person.
Practice Question
Contributors and Attributions
- Practice Question. Authored by : Sherri Pendleton. Provided by : Lumen Learning. License : CC BY: Attribution
- Modification of Information Privacy. Provided by : Northern Virginia Community College Extended Learning Institute. Located at : https://courses.lumenlearning.com/zeliite115/chapter/reading-information-privacy/ . License : CC BY-SA: Attribution-ShareAlike


11.2: Conflict Management Styles
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Would you describe yourself as someone who prefers to avoid conflict? Do you like to get your way? Are you good at working with someone to reach a solution that is mutually beneficial? Odds are that you have been in situations where you could answer yes to each of these questions. You may also find that one of the above approaches is preferable for most situations you face, making it your primary or “go-to” conflict style. Conflict management styles are the communication strategies we use that attempt to avoid, address, or resolve a conflict. In this section, we will describe five approaches for managing conflict: competing, avoiding, accommodating, compromising, and collaborating.
Each of these conflict styles emphasize a dynamic between concern for self and others. In order to better understand the elements of the five styles of conflict management, we will apply each to the following scenario. Amal and Vaughn have been married for seventeen years. Amal is growing frustrated because Vaughn continues to give money to their teenager, Sasha, even though they both decided to keep Sasha on a fixed allowance. While conflicts regarding money and child rearing are very common, we will see the numerous ways that Amal and Vaughn could approach this problem as we address each of the five styles. We rarely are conscious of our approach. Rather, unless we’ve been trained, we tend to handle conflict habitually, in the default ways we’ve been conditioned through observing others (e.g., family, culture).
Competing
The competing style indicates a high concern for self and a low concern for other. When we compete, we are striving to “win” the conflict, potentially at the expense or “loss” of the other person. One way we may gauge our win is by being granted or taking concessions from the other person. For example, if Vaughn gives Sasha extra money behind Amal’s back, then Vaughn is taking an indirect competitive route. The competing style also involves the use of power, which can be noncoercive or coercive (Sillars, 1980). Noncoercive strategies include requesting and persuading. Amal could try to persuade Vaughn to stop giving Sasha extra allowance money by bringing up their fixed budget or by reminding Vaughn that they are saving for a summer vacation. Coercive strategies may include aggressive communication directed at rousing your partner’s emotions through insults, profanity, and yelling, or through threats of punishment if you do not get your way. If Amal is the primary income earner in the family, they could use that power to threaten to take Vaughn’s ATM card away if Vaughn continues to give Sasha money. In all these scenarios, the “win” that could result is only short term and can lead to conflict escalation, even in the seemingly innocuous noncoercive situation, because the core of the conflict was not resolved. Each parent’s goals for Sasha may still be incompatible.
Interpersonal conflict is rarely isolated, meaning ripple effects can occur that connect the current conflict to previous and future conflicts. Vaughn’s behind-the-scenes money giving or Amal’s confiscation of the ATM card could lead to built-up negative emotions that could further test the relationship.
Avoiding
The avoiding style of conflict management often indicates a low concern for self and a low concern for other, and no overt or direct communication about the conflict takes place. However, in some cultures that emphasize group harmony over individual interests, and even in some situations in the United States, avoiding a conflict can indicate a high level of concern for the other. In general, avoiding doesn’t mean that there is no communication about the conflict. Remember, you cannot not communicate. Even when we try to avoid conflict, we may intentionally or unintentionally give away our feelings through our verbal and nonverbal communication. Amal’s sarcastic tone when telling Vaughn “You are soooo good with money!” and subsequent eye roll both bring the conflict to the surface without specifically addressing it. The avoiding style is either passive or indirect, which may make this style less effective than others.
We may decide to avoid conflict for many different reasons, some of which are better than others. If you view the conflict as having little importance to you, it may be better to ignore it. If the person with whom you’re in conflict with will only be working in your office for a week, you may perceive a conflict to be temporary and choose to avoid it, hoping it will solve itself. If you are not emotionally invested in the conflict, you may be able to reframe your perspective and see the situation in a different way, therefore resolving the issue. In all these cases, avoiding doesn’t really require an investment of time, emotion, or communication skill, so there is not much at stake to lose. However, while it may be easy to tolerate a problem when you’re not personally invested in it or view it as temporary, when faced with a situation like Amal and Vaughn’s, avoidance may just make the problem worse. For example, avoidance could first manifest as changing the subject, and then progress from avoiding the issue to avoiding the person altogether, to even ending the relationship.
Indirect strategies of hinting and joking also fall under the avoiding style. When we hint, we drop clues that we hope our partner will find and piece together to see the problem and hopefully change, thereby solving the problem without any direct communication. However, often the person dropping the hints overestimates their partner’s detective abilities. For example, when Amal leaves the bank statement on the kitchen table in hopes that Vaughn will realize how much extra money is being given to Sasha, Vaughn may simply ignore it. We also overestimate our partner’s ability to decode the jokes we make about a conflict situation. It is more likely the receiver of the jokes will think you’re genuinely trying to be funny or feel provoked or insulted than realize the conflict situation that you are referencing. More frustration may develop when the hints and jokes are not accurately decoded, which often leads to a more extreme form of hinting/joking: passive-aggressive behavior.
Passive-aggressive behavior is a way of dealing with conflict in which one person indirectly communicates negative thoughts or feelings through nonverbal behaviors, such as not completing a task. For example, Amal may wait a few days to deposit money into the bank so Vaughn can’t withdraw it to give to Sasha. Although passive-aggressive behavior can feel rewarding in the moment, it is one of the most unproductive ways to deal with conflict. However, as noted above, avoidance can be appropriate in some situations—for example, when the conflict is temporary, when the stakes are low or there is little personal investment, or when there is the potential for violence or retaliation.
Accommodating
The accommodating conflict management style indicates a low concern for self and a high concern for other, and is often viewed as passive or submissive, in that someone complies with or obliges another. Basically, accommodating entails doing what the other wants, whereas avoiding is doing nothing in a situation. It should be noted that sometimes avoiding often leads to accommodating indirectly as not addressing a problem or voicing our opinion can lead others to perceive that we are okay with doing things their way.
The context for and motivation behind accommodating play an important role in whether or not it is an appropriate approach. Generally, we accommodate because we are being generous, we are obeying, we are yielding (Bobot, 2010). If we are being generous, we accommodate because we genuinely want to. If we are obeying, we don’t have a choice but to accommodate (perhaps due to the potential for negative consequences or punishment). If we yield, we may have our own views or goals but give up on them due to fatigue, time constraints, or because a better solution has been offered.
Accommodating can be appropriate when there is little chance that our own goals can be achieved, when we don’t have much to lose by accommodating, when we feel we are wrong, or when advocating for our own needs could negatively affect the relationship (Isenhart & Spangle, 2000). The occasional accommodation can be useful in maintaining a relationship. For example, Amal may say, “It’s OK that you gave Sasha some extra money. Sasha did have to spend more on gas this week since the prices went up.” However, being a team player can slip into being a pushover, which people generally do not appreciate. If Amal keeps telling Vaughn, “It’s OK this time,” the family may find themselves short on spending money at the end of the month.
Compromising
The compromising style shows a moderate concern for self and other. Even though we often hear that the best way to handle a conflict is to compromise, the compromising style isn’t a win/win solution; it is a partial win/lose. In essence, when we compromise, we give up some or most of what we want. It’s true that the conflict gets resolved temporarily, but lingering thoughts of what you gave up could lead to a future conflict. Compromising may be a good strategy when there are time limitations or when prolonging a conflict may lead to relationship deterioration. Compromise may also be good when both parties have equal power or when other resolution strategies have not worked (Macintosh & Stevens, 2008). Compromising may help conflicting parties come to a resolution, but neither may be completely satisfied if they each had to give something up.
Compromising versus Accommodating
People often get accommodating and compromising confused. Accommodating means sacrificing your needs/wants/desires for what the other wants, without them giving anything in return. When you compromise, both parties give something and gain something.
A negative of compromising is that it may be used as an easy way out of a conflict. The compromising style is most effective when both parties find the solution reasonably agreeable. If Amal values using allowance to teach responsibility and Vaughn wants to give Sasha an extra twenty dollars a week to make Sasha’s life easier, they may decide to compromise by giving Sasha ten more dollars a week.
Collaborating
The collaborating style involves a high degree of concern for self and other, and usually indicates investment in the conflict situation and the relationship. Although the collaborating style takes the most work in terms of communication competence, it can ultimately lead to a win/win situation in which neither party has to make concessions because a mutually beneficial solution is discovered or created. The obvious advantage is that both parties are satisfied, which could strengthen the overall relationship and may lead to positive problem-solving in the future. For example, Amal and Vaughn may agree that Sasha’s allowance needs to be increased and may decide to give Sasha twenty more dollars a week in exchange for babysitting their five-year old sibling. In this case, they didn’t make the conflict personal but focused on the situation and came up with a solution that may end up saving them money. The disadvantage is that this style is often time consuming, and only one person may be willing to use this approach while the other person may be eager to either compete or accommodate.
Problem-Solving
When trying to collaborate on solving a conflict, it is useful to use the following five-step problem-solving sequence:
- Identify the problem(s).
- Analyze the problem(s), the causes, and symptoms. In other words, how did the problem come about and why are you having this conflict?
- Identify the goals/needs of each person in the conflict. In other words, what does each person want?
- Identify solutions that might solve the problem and meet the goals/needs of the conflict participants. Be creative and think outside the box if necessary.
- Evaluate the solutions that were identified. When evaluating the solutions, you should consider the following: Will it solve the problem? Will it satisfy the goals/needs of the conflict participants? What are some potential issues that might arise when the choice is implemented?
Here are some tips for collaborating and achieving a win/win outcome (Hargie, 2011):
- Avoid viewing the conflict as a contest you are trying to win.
- Remain flexible and realize there are solutions yet to be discovered.
- Separate between the person and the problem (don’t make it personal).
- Determine the underlying needs driving the other person’s demands.
- Identify areas of common ground or shared interests that you can work from to develop solutions.
- Ask questions to allow them to clarify and to help you understand their perspective.


Redox Reactions
Skills to Develop
- Describe what occurs in a redox reaction
- Identify redox patterns in the periodic table
Redox reactions are reactions in which electrons shift allegiance. Allegiance means loyalty or commitment to a group, like your allegiance to your family or your country. If you decide to leave your home and become a citizen of a new country, you have shifted allegiance. Of course, you might be able to share your allegiance between your old country and your new country by being a citizen of both countries at the same time. This is similar to what electrons do. They have allegiance to a nucleus, and sometimes during a reaction they will shift allegiance to a different nucleus. They might shift allegiance completely, if the new nucleus is much better for them than the old one, or they might shift partially, and share their loyalty between the two. (Also, electrons, like people, can spend some time near nuclei they don't belong to, just like you can visit countries where you don't live, and then go back near their main nucleus, although electrons' trips are much shorter than ours!)
Redox Patterns in the Periodic Table
All neutral atoms have a population of electrons equal to their number of protons. Atoms, just like countries, always have elite citizens that they treat really well (power, money...) and these elite citizens are way too comfortable to shift allegiance unless they get kidnapped or something. Unlike countries, atoms always have only a small number of non-elite citizens, between 0 and 8, roughly, but usually no more than 3 or 4. These non-elite electrons will easily shift allegiance if they get a better offer.
What are the good parts of the periodic table? Fluorine is like heaven for electrons, they will basically never leave. If fluorine is accepting immigrants, electrons will leave anywhere else to move to F. Oxygen is second best. (Fluorine seems nasty to us because we don't want our citizen electrons to leave us for fluorine. We have jobs for them to do!) The noble gases are aloof nations, their populations of electrons hardly ever change. Their citizens don't want to leave, and they don't want any troublesome immigrants either. In general, the upper right corner of the periodic table are good comfortable nations to live in. The bottom left is such a bad place to live that electrons will destroy property in their rush to leave. (Watch this video for an example) This shows "standard of living" across the periodic table. Red elements will tend to keep their own electrons and attract electrons from other nuclei; yellow elements will tend to lose their non-elite electrons.
Ionic substances are like alliances between nations with really different standards of living. The non-elite electrons of the metal will try to shift allegiance to the non-metal. In molecular substances, also called covalent, the electrons will have dual citizenship, although they might have slightly greater loyalty to one nuclei than another, because these are alliances between similarly comfortable non-metal nuclei.
Oxidation States: Redox from the Nuclei's Perspective
With all these dual citizen electrons, with partial loyalty to several nuclei, it can get hard to count the population of electrons at each nucleus. Oxidation number is one procedure for counting. In this procedure, electrons are counted as belonging completely to the nuclei they feel the greatest loyalty to. The number of electrons that feel primary loyalty to each nucleus is counted, and subtracted the number of protons in that nucleus. But there are shortcuts to make this faster using valence rules. For example, in CO 2 , oxygen has a normal valence of 2. This means that it can accept 2 extra citizen electrons per nucleus, so it has an oxidation number of -2. Therefore, because there are 2 O nuclei, carbon loses 4 citizen electrons, who transfer their primary loyalty to O. Thus, carbon has an oxidation number of +4, because it has 4 fewer truly loyal electrons than protons.
How to count oxidation states:
- In elemental form, all nuclei have oxidations number 0
- In single-nucleus ions, the oxidation number is the charge
- Alkali nuclei always have an oxidation number of +1, unless in metallic form
- Alkaline earth nuclei always have an oxidation number of +2, unless in metallic form
- Oxygen nearly always has an oxidation number of -2, unless it is in an O-O unit like peroxide
- Non-metals have negative oxidation numbers when bound to metals and positive oxidation numbers when bound to O or F.
- Halogens have oxidation number of -1 unless bound to O or F. F is always -1 unless in elemental form.
- The sum of all the oxidation numbers of a molecule or ion is its overall charge (0 if neutral, +2 if a dipositive cation, etc)
Types of Redox Reactions
Oxidation refers to a process in which something loses electrons, and has its oxidation number increase. This usually happens to compounds that react with oxygen gas, which is why it is called oxidation. Reduction refers to a process in which something gains electrons, and its oxidation number is reduced. Actually this is not quite where the word came from. When a metal oxide is reduced to the elemental metal and an elemental non-metal or non-metal compound, the mass of solid decreases because the non-metal usually leaves as a gas. This reduction in quantity is where the word comes from. When you think about it, for one thing to be oxidized, another thing must be reduced (because electrons can't appear out of nowhere), which is why we often use the combination word redox.
Many of the reaction types we've already seen involve redox. For instance, combination of elements is a redox reaction. Decomposition reactions are often redox reactions. Combustion reactions are redox reactions. (Dissolution/precipitation and acid-base are not redox reactions.) The examples of redox you've already seen involve electrons shifting loyalty at the same time that new bonds are made between nuclei. It's also possible for electrons to abandon one nucleus for another separate nucleus that isn't bonded to the first one. For example, if an elemental metal is placed in a solution of salt or acid, the metal can lose some electrons to the cation in the solution. The metal forms cations, and the former cation usually leaves the solution, forming hydrogen gas or elemental metal. Here are some examples:
\[Zn(s) + 2HBr(aq) \rightarrow ZnBr_{2}(aq) + H_{2}(g)\]
\[Fe(s) + Pb(NO_{3})_{2}(aq) \rightarrow Fe(NO_{3})_{2}(aq) + Pb(s)\]
All that happened in these reactions is that the disloyal electrons left a less-comfortable home country for a more comfortable one. These are called displacement reactions because one cation is displaced (replaced) by another. To predict what reactions will occur on this pattern, you can use the activity series , which is like a ranking of "worst places to live". It's pretty good for making predictions, because in general electrons will move to places lower on the list, but it doesn't take into account all the specific circumstances, like current population pressures. You'll learn more reliable ways to make predictions later. The activity series is:
- Worst place to live/easiest to oxidize: the alkali and alkaline earth metals
- Bad: the active metals, aluminum, zinc, iron, and others from the middle of the transition metals
- Better: tin, lead, hydrogen from acid, copper
- Best place to live, hardest to oxidize: the noble metals, silver, mercury, platinum, gold
Copper metal doesn't react with acid, because copper comes after hydrogen in the list. However, some acids have other oxidants in them that are stronger than hydrogen ions. For instance, nitrate ions are oxidizing, and Cu will react with nitric acid. Even gold reacts with a concentrated acid mixture, aqua regia, made of nitric and hydrochloric acids.


1.5: Multiply and Divide Integers
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Learning Objectives
By the end of this section, you will be able to:
- Multiply integers
- Divide integers
- Simplify expressions with integers
- Evaluate variable expressions with integers
- Translate English phrases to algebraic expressions
- Use integers in applications
A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers .
Multiply Integers
Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers . Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.
We remember that \(a\cdot b\) means add \(a,\, b\) times. Here, we are using the model just to help us discover the pattern.
The next two examples are more interesting.
What does it mean to multiply \(5\) by \(−3\)? It means subtract \(5, 3\) times. Looking at subtraction as “taking away,” it means to take away \(5, 3\) times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away \(5\) three times.
In summary:
\[\begin{array} {ll} {5 \cdot 3 = 15} &{-5(3) = -15} \\ {5(-3) = -15} &{(-5)(-3) = 15} \end{array} \nonumber\]
Notice that for multiplication of two signed numbers, when the:
- signs are the same , the product is positive .
- signs are different , the product is negative .
We’ll put this all together in the chart below.
MULTIPLICATION OF SIGNED NUMBERS
For multiplication of two signed numbers:
| Same signs | Product | Example |
|---|---|---|
| Two positives | Positive | \(7\cdot 4 = 28\) |
| Two negatives | Positive | \(-8(-6) = 48\) |
| Different signs | Product | Example |
|---|---|---|
| Positive \(\cdot\) negative | Negative | \(7(-9) = -63\) |
| Negative \(\cdot\) positive | Negative | \(-5\cdot 10= -50\) |
Example \(\PageIndex{1}\)
Multiply:
- \(-9\cdot 3\)
- \(-2(-5)\)
- \(4(-8)\)
- \(7\cdot 6\)
Solution
- \(-9\cdot 3 = -27 \quad \text{Multiply, noting that the signs are different, so the product is negative.}\)
- \( -2(-5) = 10 \quad \text{Multiply, noting that the signs are same, so the product is positive.}\)
- \( \quad 4(-8) = -32 \quad \text{Multiply, with different signs.} \)
- \( \quad 7\cdot 6 = 42 \quad \text{Multiply, with the same signs.} \)
Try It \(\PageIndex{2}\)
Multiply:
- \(-6\cdot 8\)
- \(-4(-7)\)
- \(9(-7)\)
- \(5\cdot 12\)
- Answer
-
- \(-48\)
- \(28\)
- \(-63\)
- \(60\)
Try It \(\PageIndex{3}\)
Multiply:
- \(-8\cdot 7\)
- \(-6(-9)\)
- \(7(-4)\)
- \(3\cdot 13\)
- Answer
-
- \(-56\)
- \(54\)
- \(-28\)
- \(39\)
When we multiply a number by \(1\), the result is the same number. What happens when we multiply a number by \(−1\)? Let’s multiply a positive number and then a negative number by \(−1\) to see what we get.
\[\begin{array} {ll} {\text{Multiply.}} & {\text{Multiply.}} \\ {-1\cdot 4 = -4} & {-1(-3)= 3} \\ {} & {} \\ {\text{Note that } -4\text{ is the opposite of 4.}} & {\text{Note that }3\text{ is the opposite of } -3} \end{array} \nonumber\]
Each time we multiply a number by \(−1\), we get its opposite!
MULTIPLICATION BY −1
\[−1a=−a \nonumber\]
Multiplying a number by \(−1\) gives its opposite.
Example \(\PageIndex{4}\)
Multiply:
- \(-1 \cdot 7\)
- \(-1(-11)\)
Solution
- \[\begin{array} {ll} {} &{-1\cdot 7} \\ {\text{Multiply, noting that the signs are different}} &{-7} \\ {\text{so the product is negative.}} &{-7\text{ is the opposite of 7.}} \end{array} \nonumber\]
- \[\begin{array} {ll} {} &{-1(-11)} \\ {\text{Multiply, noting that the signs are the same}} &{11} \\ {\text{so the product is positive.}} &{11\text{ is the opposite of -11.}} \end{array} \nonumber\]
Try It \(\PageIndex{5}\)
Multiply:
- \(-1\cdot 9\)
- \(-1\cdot(-17)\)
- Answer
-
- \(-9\)
- \(17\)
Try It \(\PageIndex{6}\)
Multiply:
- \(-1\cdot 8\)
- \(-1\cdot(-16)\)
- Answer
-
- \(-8\)
- \(16\)
Divide Integers
What about division ? Division is the inverse operation of multiplication. So, \(15\div 3=5\) because \(5 \cdot 3 = 15\). In words, this expression says that \(15\) can be divided into three groups of five each because adding five three times gives \(15\). Look at some examples of multiplying integers, to figure out the rules for dividing integers.
\[ \begin{align*} 5 \cdot 3 &= 15 & \text{so} & & 15 \div 3 &= 5 \\
-5 \cdot 3 &= -15 & \text{so} & & -15 \div 3 &= -5 \\
(5)(-3) &= -15 & \text{so} & & -15 \div (-3) &= 5 \\
(-5)(-3) &= 15 & \text{so} & & 15 \div (-3) &= -5 \end{align*} \]
Division follows the same rules as multiplication!
For division of two signed numbers, when the:
- signs are the same , the quotient is positive .
- signs are different , the quotient is negative .
And remember that we can always check the answer of a division problem by multiplying.
MULTIPLICATION AND DIVISION OF SIGNED NUMBERS
For multiplication and division of two signed numbers:
- If the signs are the same, the result is positive.
- If the signs are different, the result is negative.
| Same signs | Result |
|---|---|
| Two positives | Positive |
| Two negatives | Positive |
| If the signs are the same, the result is positive. |
| Different signs | Result |
|---|---|
| Positive and negative | Negative |
| Negative and positive | Negative |
| If the signs are different, the result is negative. |
Example \(\PageIndex{7}\)
- \(-27\div 3\)
- \(-100\div (-4)\)
Solution
- \[\begin{array} {ll} {} &{-27 \div 3} \\ {\text{Divide, with different signs, the quotient is}} &{-9} \\ {\text{negative.}} &{} \end{array} \nonumber\]
- \[\begin{array} {ll} {} &{-100 \div (-4)} \\ {\text{Divide, with signs that are the same the}} &{25} \\ {\text{ quotient is negative.}} &{} \end{array} \nonumber\]
Try It \(\PageIndex{8}\)
Divide:
- \(-42\div 6\)
- \(-117\div (-3)\)
- Answer
-
- \(-7\)
- \(39\)
Try It \(\PageIndex{9}\)
Divide:
- \(-63\div 7\)
- \(-115\div (-5)\)
- Answer
-
- \(-9\)
- \(23\)
Simplify Expressions with Integers
What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?
Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.
Example \(\PageIndex{10}\)
Simplify:
\(7(-2)+4(-7)-6\)
Solution
\[\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\ {\text{Multiply first.}} &{-14+(-28)-6} \\ {\text{Add.}} &{-42-6} \\{\text{Subtract}} &{-48} \end{array} \nonumber\]
Try It \(\PageIndex{11}\)
Simplify:
\(8(-3)+5(-7)-4\)
- Answer
-
\(-63\)
Try It \(\PageIndex{12}\)
Simplify:
\(9(-3)+7(-8)-1\)
- Answer
-
\(-84\)
Example \(\PageIndex{13}\)
Simplify:
- \((-2)^{4}\)
- \(-2^{4}\)
Solution
- \[\begin{array} {ll} {} &{(-2)^{4}} \\ {\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\ {\text{Multiply}} &{4(-2)(-2)} \\{\text{Multiply}} &{-8(-2)} \\{\text{Multiply}} &{16} \end{array} \nonumber\]
- \[\begin{array} {ll} {} &{-2^{4}} \\ {\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\cdot 2\cdot 2 \cdot 2)} \\ {\text{Multiply}} &{-(4\cdot 2\cdot 2)} \\{\text{Multiply}} &{-(8\cdot 2)} \\{\text{Multiply}} &{-16} \end{array} \nonumber\]
Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the \((−2)\) to the \(4^{th}\) power. In part (2), the exponent means to raise just the \(2\) to the \(4^{th}\) power and then take the opposite.
Try It \(\PageIndex{14}\)
Simplify:
- \((-3)^{4}\)
- \(-3^{4}\)
- Answer
-
- \(81\)
- \(-81\)
Try It \(\PageIndex{15}\)
Simplify:
- \((-7)^{2}\)
- \(-7^{2}\)
- Answer
-
- \(49\)
- \(-49\)
The next example reminds us to simplify inside parentheses first.
Example \(\PageIndex{16}\)
Simplify:
\(12-3(9 - 12)\)
Solution
\[\begin{array} {llll} {12-3(9 - 12)} & {}\\ {= 12-3(-3)} & {\text{Subtract in parentheses first.}}\\ {= 12-(-9)} & {\text{Multiply.}}\\ {= 12 + 9} & {\text{Add the opposite.}} \\ {=21} & {}\end{array} \nonumber\]
Try It \(\PageIndex{17}\)
Simplify:
\(17 - 4(8 - 11)\)
- Answer
-
\(29\)
Try It \(\PageIndex{18}\)
Simplify:
\(16 - 6(7 - 13)\)
- Answer
-
\(52\)
Example \(\PageIndex{19}\)
Simplify:
\(8(-9)\div (-2)^{3}\)
Solution
\[\begin{array} {ll} {} &{8(-9)\div(-2)^{3}} \\ {\text{Exponents first}} &{8(-9)\div(-8)} \\ {\text{Multiply.}} &{-72\div (-8)} \\{\text{Divide}} &{9} \end{array} \nonumber\]
Try It \(\PageIndex{20}\)
Simplify:
\(12(-9)\div (-3)^{3}\)
- Answer
-
\(4\)
Try It \(\PageIndex{21}\)
Simplify:
\(18(-4)\div (-2)^{3}\)
- Answer
-
\(9\)
Example \(\PageIndex{22}\)
Simplify:
\(-30\div 2 + (-3)(-7)\)
Solution
\[\begin{array} {ll} {} &{-30\div 2 + (-3)(-7)} \\ {\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\ {\text{Multiply.}} &{-15+ 21} \\{\text{Add}} &{6} \end{array} \nonumber\]
Try It \(\PageIndex{23}\)
Simplify:
\(-27\div 3 + (-5)(-6)\)
- Answer
-
\(21\)
Try It \(\PageIndex{24}\)
Simplify:
\(-32\div 4 + (-2)(-7)\)
- Answer
-
\(6\)
Evaluate Variable Expressions with Integers
Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.
Example \(\PageIndex{25}\)
When \(n=−5\), evaluate:
- \(n+1\)
- \(−n+1\).
Solution
- \[\begin{array} {ll} {} &{n+ 1} \\ {\text{Substitute }{ \color{red}{-5}}\text{ for } n} &{\color{red}{-5}}+1 \\ {\text{Simplify.}} &{-4} \end{array} \nonumber\]
- \[\begin{array} {ll} {} &{n+ 1} \\ {\text{Substitute }{ \color{red}{-5}}\text{ for } n} &{- {\color{red}{(-5)}} +1} \\ {\text{Simplify.}} &{5+1} \\{\text{Add.}} &{6} \end{array} \nonumber\]
Try It \(\PageIndex{26}\)
When \(n=−8\), evaluate:
- \(n+2\)
- \(−n+2\).
- Answer
-
- \(-6\)
- \(10\)
Try It \(\PageIndex{27}\)
When \(y=−9\), evaluate:
- \(y+8\)
- \(−y+8\).
- Answer
-
- \(-1\)
- \(17\)
Example \(\PageIndex{28}\)
Evaluate \((x+y)^{2}\) when \(x = -18\) and \(y = 24\).
Solution
\[\begin{array} {ll} {} &{(x+y)^{2}} \\ {\text{Substitute }-18\text{ for }x \text{ and } 24 \text{ for } y} &{(-18 + 24)^{2}} \\ {\text{Add inside parentheses}} &{(6)^{2}} \\{\text{Simplify.}} &{36} \end{array} \nonumber\]
Try It \(\PageIndex{29}\)
Evaluate \((x+y)^{2}\) when \(x = -15\) and \(y = 29\).
- Answer
-
\(196\)
Try It \(\PageIndex{30}\)
Evaluate \((x+y)^{3}\) when \(x = -8\) and \(y = 10\).
- Answer
-
\(8\)
Example \(\PageIndex{31}\)
Evaluate \(20 -z \) when
- \(z = 12\)
- \(z = -12\)
Solution
- \[\begin{array} {ll} {} &{20 - z} \\ {\text{Substitute }12\text{ for }z.} &{20 - 12} \\ {\text{Subtract}} &{8} \end{array} \nonumber\]
- \[\begin{array} {ll} {} &{20 - z} \\ {\text{Substitute }-12\text{ for }z.} &{20 - (-12)} \\ {\text{Subtract}} &{32} \end{array} \nonumber\]
Try It \(\PageIndex{32}\)
Evaluate \(17 - k\) when
- \(k = 19\)
- \(k = -19\)
- Answer
-
- \(-2\)
- \(36\)
Try It \(\PageIndex{33}\)
Evaluate \(-5 - b\) when
- \(b = 14\)
- \(b = -14\)
- Answer
-
- \(-19\)
- \(9\)
Example \(\PageIndex{34}\)
Evaluate:
\(2x^{2} + 3x + 8\) when \(x = 4\).
Solution
Substitute \(4\) for \(x\). Use parentheses to show multiplication.
\[\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\ {\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\ {\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\ {\text{Multiply.}} &{32 + 12 + 8} \\{\text{Add.}} &{52} \end{array} \nonumber\]
Try It \(\PageIndex{35}\)
Evaluate:
\(3x^{2} - 2x + 6\) when \(x =-3\).
- Answer
-
\(39\)
Try It \(\PageIndex{36}\)
Evaluate:
\(4x^{2} - x - 5\) when \(x = -2\).
- Answer
-
\(13\)
Translate Phrases to Expressions with Integers
Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.
Example \(\PageIndex{37}\)
Translate and simplify: the sum of \(8\) and \(−12\), increased by \(3\).
Solution
\[\begin{array} {ll} {} &{\text{the } \textbf{sum} \text{of 8 and -12, increased by 3}} \\ {\text{Translate.}} &{[8 + (-12)] + 3} \\ {\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\{\text{brackets with an absolute value sign.}} \\{\text{Add.}} &{-1} \end{array} \nonumber\]
Try It \(\PageIndex{38}\)
Translate and simplify: the sum of \(9\) and \(−16\), increased by \(4\).
- Answer
-
\((9 + (-16)) + 4 - 3\)
Try It \(\PageIndex{39}\)
Translate and simplify: the sum of \(-8\) and \(−12\), increased by \(7\).
- Answer
-
\((-8 + (-12)) + 7 - 13\)
When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.
| \(a−b\) |
|---|
|
\(a\) minus \(b\)
the difference of \(a\) and \(b\) \(b\) subtracted from \(a\) \(b\) less than \(a\) |
Be careful to get a and b in the right order!
Example \(\PageIndex{40}\)
Translate and then simplify
- the difference of \(13\) and \(−21\)
- subtract \(24\) from \(−19\).
Solution
- \[\begin{array} {ll} {} &{\text{the } \textbf{difference } \text{of 13 and -21}} \\ {\text{Translate.}} &{13 - (-21)} \\ {\text{Simplify.}} &{34} \end{array} \nonumber\]
- \[\begin{array} {ll} {} &\textbf{subtract }24 \textbf{ from }-19 \\ {\text{Translate.}} &{-19 - 24} \\ {\text{Remember, subtract b from a means }a - b} &{} \\{\text{Simplify.}} &{-43} \end{array} \nonumber\]
Try It \(\PageIndex{41}\)
Translate and simplify
- the difference of \(14\) and \(−23\)
- subtract \(21\) from \(−17\).
- Answer
-
- \(14 - (-23); 37\)
- \(-17 - 21; -38\)
Try It \(\PageIndex{42}\)
Translate and simplify
- the difference of \(11\) and \(−19\)
- subtract \(18\) from \(−11\).
- Answer
-
- \(11 - (-19); 30\)
- \(-11 - 18; -29\)
Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “ product ” and for division is “ quotient .”
Example \(\PageIndex{43}\)
Translate to an algebraic expression and simplify if possible: the product of \(−2\) and \(14\).
Solution
\[\begin{array} {ll} {} &{\text{the product of }-2 \text{ and } 14} \\ {\text{Translate.}} &{(-2)(14)} \\{\text{Simplify.}} &{-28} \end{array} \nonumber\]
Try It \(\PageIndex{44}\)
Translate to an algebraic expression and simplify if possible: the product of \(−5\) and \(12\).
- Answer
-
\(-5(12); -60\)
Try It \(\PageIndex{45}\)
Translate to an algebraic expression and simplify if possible: the product of \(8\) and \(-13\).
- Answer
-
\(-8(13); -104\)
Example \(\PageIndex{46}\)
Translate to an algebraic expression and simplify if possible: the quotient of \(−56\) and \(−7\).
Solution
\[\begin{array} {ll} {} &{\text{the quotient of }-56 \text{ and } -7} \\ {\text{Translate.}} &{-56\div(-7)} \\{\text{Simplify.}} &{8} \end{array}\]
Try It \(\PageIndex{47}\)
Translate to an algebraic expression and simplify if possible: the quotient of \(−63\) and \(−9\).
- Answer
-
\(-63\div (-9); 7\)
Try It \(\PageIndex{48}\)
Translate to an algebraic expression and simplify if possible: the quotient of \(−72\) and \(−9\).
- Answer
-
\(-72\div (-9); 8\)
Use Integers in Applications
We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.
How to Apply a Strategy to Solve Applications with Integers
Example \(\PageIndex{49}\)
The temperature in Urbana, Illinois one morning was \(11\) degrees. By mid-afternoon, the temperature had dropped to \(−9\) degrees. What was the difference of the morning and afternoon temperatures?
Solution
| Step 1 . Read the problem. Make sure all the words and ideas are understood. | |
| Step 2 . Identify what we are asked to find. | the difference of the morning and afternoon temperatures |
| Step 3 . Write a phrase that gives the information to find it. | the difference of \(11\) and \(-9\) |
| Step 4. Translate the phrase to an expression. | \(11 - (-9)\) |
| Step 5 . Simplify the expression. | \(20\) |
| Step 6 . Write a complete sentence that answers the question. | The difference in temperatures was 20 degrees. |
Try It \(\PageIndex{50}\)
The temperature in Anchorage, Alaska one morning was \(15\) degrees. By mid-afternoon the temperature had dropped to \(30\) degrees below zero. What was the difference in the morning and afternoon temperatures?
- Answer
-
The difference in temperatures was \(45\) degrees.
Try It \(\PageIndex{51}\)
The temperature in Denver was \(−6\) degrees at lunchtime. By sunset the temperature had dropped to \(−15\) degrees. What was the difference in the lunchtime and sunset temperatures?
- Answer
-
The difference in temperatures was \(9\) degrees.
APPLY A STRATEGY TO SOLVE APPLICATIONS WITH INTEGERS.
- Read the problem. Make sure all the words and ideas are understood
- Identify what we are asked to find.
- Write a phrase that gives the information to find it.
- Translate the phrase to an expression.
- Simplify the expression.
- Answer the question with a complete sentence.
Example \(\PageIndex{52}\)
The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?
Solution
| Step 1 . Read the problem. Make sure all the words and ideas are understood. | |
| Step 2 . Identify what we are asked to find. | the number of yards lost |
| Step 3 . Write a phrase that gives the information to find it. | three times a \(15\)-yard penalty |
| Step 4. Translate the phrase to an expression. | \(3(-15)\) |
| Step 5 . Simplify the expression. | \(-45\) |
| Step 6 . Write a complete sentence that answers the question. | The team lost \(45\) yards. |
Try It \(\PageIndex{53}\)
The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of \(15\) yards. What is the number of yards lost due to penalties?
- Answer
-
The Bears lost \(105\) yards.
Try It \(\PageIndex{54}\)
Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?
- Answer
-
A $16 fee was deducted from his checking account.
Key Concepts
-
Multiplication and Division of Two Signed Numbers
- Same signs—Product is positive
- Different signs—Product is negative
-
Strategy for Applications
- Identify what you are asked to find.
- Write a phrase that gives the information to find it.
- Translate the phrase to an expression.
- Simplify the expression.
- Answer the question with a complete sentence.


20: Electric Circuits
- Understand how a battery works.
- Understand Kirchhoff rules and how to apply them.
- Understand how to model a circuit with resistors and/or capacitors.
- Understand how an ammeter and voltmeter function, and how to model them.
In this chapter, we develop the tools to model electric circuits. This will allow us to determine the current and voltages across different components, such as resistors and capacitors, within a circuit. We will also discuss how a battery can provide a current at a fixed potential difference, and how one can construct devices to measure current and voltages.
If two outlets in your house are connected to the same circuit, are the outlets connected in series or in parallel?
- series
- parallel
-
- 20.1: Batteries and Simple Circuits
- A battery is an electric component that provides a constant electric potential difference (a fixed voltage) across its terminals. Luigi Galvani was the first to realize that certain combination of metals placed into contact with each other can lead to an electric potential difference (or rather, they can make the legs of a dead frog twitch, which we now understand to be from the potential difference due to the metals). Effectively, Galvani created the first “electrochemical cell”.
-
- 20.2: Kirchhoff’s rules
- Kirchhoff’s rules correspond to concepts that we have already covered, but allow us to easily model more complex circuits, for instance, those where there is more than one path for the current to take. Kirchhoff’s rules refer to “junctions” and “loops”. Junctions and loops depend only on the shape of the circuit, and not on the components in the circuit.
-
- 20.3: Applying Kirchhoff’s rule to model circuits
- In this section, we show how to model a circuit using Kirchhoff’s rules. In general, one can consider a circuit to be fully modeled if one can determine the current in each segment of the circuit. We will show how one can apply the same procedure to model any circuit that contains batteries and resistors.
-
- 20.4: Measuring current and voltage
- In this section, we describe how one can build devices to measure current and voltage. A device that measures current is called an “ammeter” and a device that measured voltage is called a “voltmeter”. Nowadays, these are usually found within the same physical device (a “multimeter”), which can also measure resistance (by measuring voltage and current, resistance can easily determined). We will limit our description to the design of simple analogue ammeters and voltmeters.


5.17: Multi-Step Calculations
Sometimes multi-step calculations are required, especially for medications used in critical care. There are many different ways to solve multi-step calculations, so it is important to select a method that works for you that is consistently accurate. Let’s practice a multi-step calculation for a medication supplied in mg/mL but is prescribed based on micrograms (mcg) per kilogram (kg) per minute, and the patient’s weight is provided in pounds.
Practice Problem: Multi-Step Calculations
Patient Information:
Name: Ideen Hanson, DOB: 09/29/19xx, Allergies: NKDA, Weight: 180 lbs
Diagnosis: Hypertension
Prescription: Begin initial infusion of Nipride at 0.5 mcg/kg/min
Medication Supplied: See Figure \(\PageIndex{1}\) [1] for the drug label of the medication supplied.
Problem: What rate (in mL/hr) should the nurse set the pump to begin the infusion?
1. Set up the goal units to solve for, which is mL/hr.
\[ \frac{mL}{hr}~= \]
2. Review the problem. The prescription is based on mcg/kg/min. Having three elements can create confusion when setting up the equation using dimensional analysis, so it can be easier to eliminate one element by doing some preliminary steps. First convert the patient weight to kilograms (kg) by dividing 180 pounds/2.2= 81.8181 kg. Now multiplying the micrograms (mcg) ordered by weight in kg to determine the amount of medication to administer per minute: 81.8181 x 0.5 mcg = 40.9090 mcg/minute. Use this information as you set up your problem.
3. Start by identifying the unit you are solving for, which is mL/hour. Then set up the first fraction. Match the numerator to mL. Look to the problem for information related to mL. On the drug label, we see that 50 mg of Nipride is supplied in 250 mL of D5W. Plug in 250 in the numerator, and then 50 mg in the denominator:
\[ mL~=~\frac{250~mL}{50~mg} \]
4. Set up the second fraction with the intent to cross off mg by placing 1 mg in the numerator. Based on the known equivalency that 1000 mcg ar e equal to 1 mg, place 1000 mcg in the denominator:
\[ mL~=~\frac{250~mL}{50~mg}~\times~\frac{1~mg}{1000~mcg} \]
5. Cross off mg diagonally. Set up the third equation with the intent to cross off mcg by placing it in the numerator. Plug in the information previously calculated, which was 40.9090 mcg/minute:
\[ mL~=~\frac{250~mL}{50~\cancel{mg}}~\times~\frac{1~\cancel{mg}}{1000~mcg}~\times~\frac{40.9090~mcg}{1~min} \]
6. Cross off mcg diagonally. Set up the fourth fraction with the intent to cross off minutes. Based on the known equivalency of 60 minutes in 1 hour, plug in 60 in the numerator and 1 hour in the denominator:
\[ mL~=~\frac{250~mL}{50~\cancel{mg}}~\times~\frac{1~\cancel{mg}}{1000~\cancel{mcg}}~\times~\frac{40.9090~\cancel{mcg}}{1~min}~\times~\frac{60~min}{1~hour}~=~12.2727 \]
7. Cross off min diagonally. Review the equation to ensure the goal unit has been met. It has been met, so multiply across the numerators and the denominators, and then divide the final fraction:
\[ \frac{mL}{Hr}~\times~\frac{250~mL}{50~\cancel{mg}}~\times~\frac{1~\cancel{mg}}{1000~\cancel{mcg}}~\times~\frac{40.9090~\cancel{mcg}}{1~\cancel{min}}~\times~\frac{60~\cancel{min}}{1~hour}~=~12.2727 \]
8. Depending on the agency policy and available pump settings, round to 12.27 mL/hour.
Heparin Drip [2]
Dopamine Drip [3]
- "Nipride Label" by Jody Myhre-Oechsle, Chippewa Valley Technical College , Open RN is licensed under CC BY 4.0 ↵
- RegisteredNurseRN. (2018, July 26). Heparin drip calculation practice problems for nurses | Dosage calculations nursing . [Video]. YouTube. All rights reserved. Video used with permission. https://youtu.be/10y4gRdnCy8 ↵
- RegisteredNurseRN. (2019, February 8). Dopamine IV drip calculation & nursing considerations pharmacology implications. [Video]. YouTube. All rights reserved. Video used with permission. https://youtu.be/R2XMro13dD0 ↵


6: Linear Correlation and Regression
Linear Correlation and Regression
This calculator creates a scatter plot, the regression equation, \(r\) and \(r^{2}\), and performs the hypothesis test for a nonzero correlation. Please report the error to Dr. Jessica Kuang at jkuangATvcccd.edu.
To learn how to use this calculator, please watch a short video here.
Input
Put the independent variable's data separated by commas in the first row and the dependent variable's data separated by commas in the second row, then click Plot Points and Calculate.
Explanatory/Independent Variable (x):
Response/Dependent Variable (y):
Output
|
Regression Equation: |
\(r\): | \(r^{2}\): | |||
|
Test Statistic (\(t\)):
|
\(p\)-value:
|
(Powered by DESMOS)


13.2: Introduction- Naturalism
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- Berke, Bleil, & Cofer
- Middle Georgia State University, College of Coastal Georgia, & Abraham Baldwin Agricultural College via University of North Georgia Press
The generation of writers that followed William Dean Howells broke with their past, as did the Realists when they rejected Romanticism as a literary style. Frank Norris, Stephen Crane, Jack London, Theodore Dreiser, Harold Frederic, Hamlin Garland, Ellen Glasgow and Kate Chopin, to name a few, rejected the limitations of Realism in terms of subject matter. While they all, to some extent, embraced the Realist style of writing with its attention to detail and authenticity, they rejected Realism’s tendency not to offend the sensibilities of readers in the genteel classes. The new writers were not afraid of provocative subject matters and wrote about the human condition in starker, grimmer contexts. They all, to some extent, were influenced by not only scientific ideas of the day, including Charles Darwin’s views on evolution, but also European writers experimenting with this new style: Naturalism . Émile Zola , a prominent French novelist, had articulated a theory of Naturalism in Le Roman Expérimental (1880). Zola had argued for a kind of intense Realism, one that did not look away from any aspects of life, including the base, dirty, or ugly. Also influenced by Darwin, Zola saw the human in animal terms, and he argued that a novel written about the human animal could be set up as a kind of scientific experiment, where, once the ingredients were added, the story would unfold with scientific accuracy. He was particularly interested in how hereditary traits under the influence of a particular social environment might determine how a human behaves. The American writers Norris, Crane, and London, similarly characterize humans as part of the evolutionary landscape, as beings influenced and even determined by forces of heredity and environment beyond their understanding or control.
With Darwin’s and Zola’s influence apparent, the naturalists sought to push Realism even further, or as Frank Norris argued in his essay “A Plea for Romantic Fiction,” to go beyond the “meticulous presentation of teacups, rag carpets, wall paper, and hair-cloth sofas” or beyond Realism as mere photographic accuracy and to embrace a kind of writing that explores the “unplumbed depths of the human heart, and the mystery of sex, and the black, unsearched penetralia of the souls of men.” Norris is calling for a grittier approach in examining the human being as essentially an upright animal, a kind of walking complex combination of inherited traits, attributes, and habits deeply affected by social and economic forces.
Naturalistic works went where Realistic works did not go, dealing with taboo subjects for the time, subjects such as prostitution, alcoholism, domestic violence, violent deaths, crime, madness, and degeneration. Sometimes defined as pessimistic materialistic determinism, Naturalism sought to look at human nature in a scientific light, and the author often took on the role of scientist, coolly observing the human animal in a variety of plights, at the mercy of forces beyond his control or understanding, compelled by instinct and determined by cause and effect to behave in certain, often self-destructive, ways as a result of heredity and environment. In such works, the plot plays out on the material evolutionary plain, where a benevolent deity or any supernatural form is absent and idealistic concepts, such as justice, liberty, innate goodness, and morality, are shown as illusions, as simple fabrications of the human animal trying to elevate himself above the other animals.
In the Naturalistic works, nature is depicted as indifferent, sometimes even hostile, to humans, and humans are often depicted as small, insignificant, nameless losers in battles against an allpowerful nature. Characters may dream of heroic actions in the midst of a battle to survive extreme conditions, but they are most often trapped by circumstances, unable to summon the will to change their determined outcome. Characters rarely exhibit free will at all; they often stumble through events, victims of their own vices, weaknesses, hereditary traits, and grim social or natural environments. A male character in a Naturalistic novel is often characterized as part “brute,” and he typically exhibits strong impulses, compulsions, or instinctive drives, as he attempts to satiate his greed, his sexual urges, his decadent lusts, or his desire for power or dominance. Female characters also typically exhibit subconscious drives, acting without knowing why, unable to change course.
Naturalistic works are not defined by a region; the characters’ action may take place in the frozen Alaska wilderness, on the raging sea, or within the slums of a city. Stylistically, Naturalistic novels are written from an almost journalistic perspective, with narrative distance from action and the characters. Often characters are not given names as a way to reinforce their cosmic insignificance. The plot of the story often follows the steady decline of a character into degeneration or death (known as the “ plot of decline ”).


2.5: The Basic Units of Measurement
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Learning Objectives
- State the different measurement systems used in chemistry.
- Describe how prefixes are used in the metric system and identify how the prefixes milli-, centi-, and kilo- compare to the base unit.
How long is a yard? It depends on whom you ask and when you asked the question. Today we have a standard definition of the yard, which you can see marked on every football field. If you move the ball ten yards, you get a first down and it does not matter whether you are playing in Los Angeles, Dallas, or Green Bay. But at one time that yard was arbitrarily defined as the distance from the tip of the king's nose to the end of his outstretched hand. Of course, the problem there is simple: new king, new distance (and then you have to re-mark all of those football fields).
SI Base Units
All measurements depend on the use of units that are well known and understood. The English system of measurement units (inches, feet, ounces, etc.) are not used in science because of the difficulty in converting from one unit to another. The metric system is used because all metric units are based on multiples of 10, making conversions very simple. The metric system was originally established in France in 1795. The International System of Units is a system of measurement based on the metric system. The acronym SI is commonly used to refer to this system and stands for the French term, Le Système International d'Unités . The SI was adopted by international agreement in 1960 and is composed of seven base units in Table \(\PageIndex{1}\).
| Quantity | SI Base Unit | Symbol |
|---|---|---|
| Length | meter | \(\text{m}\) |
| Mass | kilogram | \(\text{kg}\) |
| Temperature | kelvin | \(\text{K}\) |
| Time | second | \(\text{s}\) |
| Amount of a Substance | mole | \(\text{mol}\) |
| Electric Current | ampere | \(\text{A}\) |
| Luminous Intensity | candela | \(\text{cd}\) |
The first units are frequently encountered in chemistry. All other measurement quantities, such as volume, force, and energy, can be derived from these seven base units.
Unfortunately, the Metric System is Not Ubiquitous
The map below shows the adoption of the SI units in countries around the world. The United States has legally adopted the metric system for measurements, but does not use it in everyday practice. Great Britain and much of Canada use a combination of metric and imperial units.
Prefix Multipliers
Conversions between metric system units are straightforward because the system is based on powers of ten. For example, meters, centimeters, and millimeters are all metric units of length. There are 10 millimeters in 1 centimeter and 100 centimeters in 1 meter. Metric prefixes are used to distinguish between units of different size. These prefixes all derive from either Latin or Greek terms. For example, mega comes from the Greek word \(\mu \varepsilon \gamma \alpha \varsigma\), meaning "great". Table \(\PageIndex{2}\) lists the most common metric prefixes and their relationship to the central unit that has no prefix. Length is used as an example to demonstrate the relative size of each prefixed unit.
| Prefix | Unit Abbreviation | Meaning | Example |
|---|---|---|---|
| giga | \(\text{G}\) | 1,000,000,000 | 1 gigameter \(\left( \text{Gm} \right)=10^9 \: \text{m}\) |
| mega | \(\text{M}\) | 1,000,000 | 1 megameter \(\left( \text{Mm} \right)=10^6 \: \text{m}\) |
| kilo | \(\text{k}\) | 1,000 | 1 kilometer \(\left( \text{km} \right)=1,000 \: \text{m}\) |
| hecto | \(\text{h}\) | 100 | 1 hectometer \(\left( \text{hm} \right)=100 \: \text{m}\) |
| deka | \(\text{da}\) | 10 | 1 dekameter \(\left( \text{dam} \right)=10 \: \text{m}\) |
| 1 | 1 meter \(\left( \text{m} \right)\) | ||
| deci | \(\text{d}\) | 1/10 | 1 decimeter \(\left( \text{dm} \right)=0.1 \: \text{m}\) |
| centi | \(\text{c}\) | 1/100 | 1 centimeter \(\left( \text{cm} \right)=0.01 \: \text{m}\) |
| milli | \(\text{m}\) | 1/1,000 | 1 millimeter \(\left( \text{mm} \right)=0.001 \: \text{m}\) |
| micro | \(\mu\) | 1/1,000,000 | 1 micrometer \(\left( \mu \text{m} \right)=10^{-6} \: \text{m}\) |
| nano | \(\text{n}\) | 1/1,000,000,000 | 1 nanometer \(\left( \text{nm} \right)=10^{-9} \: \text{m}\) |
| pico | \(\text{p}\) | 1/1,000,000,000,000 | 1 picometer \(\left( \text{pm} \right)=10^{-12} \: \text{m}\) |
There are a couple of odd little practices with the use of metric abbreviations. Most abbreviations are lowercase. We use "\(\text{m}\)" for meter and not "\(\text{M}\)". However, when it comes to volume, the base unit "liter" is abbreviated as "\(\text{L}\)" and not "\(\text{l}\)". So we would write 3.5 milliliters as \(3.5 \: \text{mL}\).
As a practical matter, whenever possible you should express the units in a small and manageable number. If you are measuring the weight of a material that weighs \(6.5 \: \text{kg}\), this is easier than saying it weighs \(6500 \: \text{g}\) or \(0.65 \: \text{dag}\). All three are correct, but the \(\text{kg}\) units in this case make for a small and easily managed number. However, if a specific problem needs grams instead of kilograms, go with the grams for consistency.
Example \(\PageIndex{1}\): Unit Abbreviations
Give the abbreviation for each unit and define the abbreviation in terms of the base unit.
- kiloliter
- microsecond
- decimeter
- nanogram
Solutions
| Explanation | Answer | |
|---|---|---|
| a | The prefix kilo means “1,000 ×,” so 1 kL equals 1,000 L. | kL |
| b | The prefix micro implies 1/1,000,000th of a unit, so 1 µs equals 0.000001 s. | µs |
| c | The prefix deci means 1/10th, so 1 dm equals 0.1 m. | dm |
| d | The prefix nano means 1/1000000000, so a nanogram is equal to 0.000000001 g. | ng |
Exercise \(\PageIndex{1}\)
Give the abbreviation for each unit and define the abbreviation in terms of the base unit.
- kilometer
- milligram
- nanosecond
- centiliter
- Answer a:
- km
- Answer b:
- mg
- Answer c:
- ns
- Answer d:
- cL
Summary
- Metric prefixes derive from Latin or Greek terms. The prefixes are used to make the units manageable.
- The SI system is based on multiples of ten. There are seven basic units in the SI system. Five of these units are commonly used in chemistry.


3.E: Basic Concepts of Chemical Bonding (Exercises)
CHEMICAL BONDS, LEWIS SYMBOLS AND THE OCTET RULE (1~4)
Conceptual Problems
- The Lewis electron system is a simplified approach for understanding bonding in covalent and ionic compounds. Why do chemists still find it useful?
- Is a Lewis dot symbol an exact representation of the valence electrons in an atom or ion? Explain your answer.
- How can the Lewis electron dot system help to predict the stoichiometry of a compound and its chemical and physical properties?
- How is a Lewis dot symbol consistent with the quantum mechanical model of the atom? How is it different?
Conceptual Answer
- Lewis dot symbols allow us to predict the number of bonds atoms will form, and therefore the stoichiometry of a compound. The Lewis structure of a compound also indicates the presence or absence of lone pairs of electrons, which provides information on the compound’s chemical reactivity and physical properties.
IONIC BONDING (5~16)
Conceptual Problems
-
Describe the differences in behavior between NaOH and CH 3 OH in aqueous solution. Which solution would be a better conductor of electricity? Explain your reasoning.
-
What is the relationship between the strength of the electrostatic attraction between oppositely charged ions and the distance between the ions? How does the strength of the electrostatic interactions change as the size of the ions increases?
-
Which will result in the release of more energy: the interaction of a gaseous sodium ion with a gaseous oxide ion or the interaction of a gaseous sodium ion with a gaseous bromide ion? Why?
-
Which will result in the release of more energy: the interaction of a gaseous chloride ion with a gaseous sodium ion or a gaseous potassium ion? Explain your answer.
-
What are the predominant interactions when oppositely charged ions are
- far apart?
- at internuclear distances close to r 0 ?
- very close together (at a distance that is less than the sum of the ionic radii)?
-
Several factors contribute to the stability of ionic compounds. Describe one type of interaction that destabilizes ionic compounds. Describe the interactions that stabilize ionic compounds.
-
What is the relationship between the electrostatic attractive energy between charged particles and the distance between the particles?
Conceptual Answer
-
-
-
The interaction of a sodium ion and an oxide ion. The electrostatic attraction energy between ions of opposite charge is directly proportional to the charge on each ion ( Q 1 and Q 2 in Equation 9.1). Thus, more energy is released as the charge on the ions increases (assuming the internuclear distance does not increase substantially). A sodium ion has a +1 charge; an oxide ion, a −2 charge; and a bromide ion, a −1 charge. For the interaction of a sodium ion with an oxide ion, Q 1 = +1 and Q 2 = −2, whereas for the interaction of a sodium ion with a bromide ion, Q 1 = +1 and Q 2 = −1. The larger value of Q 1 × Q 2 for the sodium ion–oxide ion interaction means it will release more energy.
Numerical Problems
-
How does the energy of the electrostatic interaction between ions with charges +1 and −1 compare to the interaction between ions with charges +3 and −1 if the distance between the ions is the same in both cases? How does this compare with the magnitude of the interaction between ions with +3 and −3 charges?
-
How many grams of gaseous MgCl 2 are needed to give the same electrostatic attractive energy as 0.5 mol of gaseous LiCl? The ionic radii are Li + = 76 pm, Mg +2 = 72 pm, and Cl − = 181 pm.
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Sketch a diagram showing the relationship between potential energy and internuclear distance (from r = ∞ to r = 0) for the interaction of a bromide ion and a potassium ion to form gaseous KBr. Explain why the energy of the system increases as the distance between the ions decreases from r = r 0 to r = 0.
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Calculate the magnitude of the electrostatic attractive energy ( E , in kilojoules) for 85.0 g of gaseous SrS ion pairs. The observed internuclear distance in the gas phase is 244.05 pm.
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What is the electrostatic attractive energy ( E , in kilojoules) for 130 g of gaseous HgI 2 ? The internuclear distance is 255.3 pm.
Numerical Answers
-
According to Equation 9.1, in the first case Q 1 Q 2 = (+1)(−1) = −1; in the second case, Q 1 Q 2 = (+3)(−1) = −3. Thus, E will be three times larger for the +3/−1 ions. For +3/−3 ions, Q 1 Q 2 = (+3)(−3) = −9, so E will be nine times larger than for the +1/−1 ions.
-
-
At r < r 0 , the energy of the system increases due to electron–electron repulsions between the overlapping electron distributions on adjacent ions. At very short internuclear distances, electrostatic repulsions between adjacent nuclei also become important.
COVALENT BONDING (17~30)
Conceptual Problems
- Which would you expect to be stronger—an S–S bond or an Se–Se bond? Why?
- Which element—nitrogen, phosphorus, or arsenic—will form the strongest multiple bond with oxygen? Why?
- Why do multiple bonds between oxygen and period 3 elements tend to be unusually strong?
- What can bond energies tell you about reactivity?
- Bond energies are typically reported as average values for a range of bonds in a molecule rather than as specific values for a single bond? Why?
- If the bonds in the products are weaker than those in the reactants, is a reaction exothermic or endothermic? Explain your answer.
- A student presumed that because heat was required to initiate a particular reaction, the reaction product would be stable. Instead, the product exploded. What information might have allowed the student to predict this outcome?
Numerical Problems
-
What is the bond order about the central atom(s) of hydrazine (N 2 H 4 ), nitrogen, and diimide (N 2 H 2 )? Draw Lewis electron structures for each compound and then arrange these compounds in order of increasing N–N bond distance. Which of these compounds would you expect to have the largest N–N bond energy? Explain your answer.
-
What is the carbon–carbon bond order in ethylene (C 2 H 4 ), BrH 2 CCH 2 Br, and FCCH? Arrange the compounds in order of increasing C–C bond distance. Which would you expect to have the largest C–C bond energy? Why?
-
From each pair of elements, select the one with the greater bond strength? Explain your choice in each case.
- P–P, Sb–Sb
- Cl–Cl, I–I
- O–O, Se–Se
- S–S, Cl–Cl
- Al–Cl, B–Cl
-
From each pair of elements, select the one with the greater bond strength? Explain your choice in each case.
- Te–Te, S–S
- C–H, Ge–H
- Si–Si, P–P
- Cl–Cl, F–F
- Ga–H, Al–H
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Approximately how much energy per mole is required to completely dissociate acetone [(CH 3 ) 2 CO] and urea [(NH 2 ) 2 CO] into their constituent atoms?
-
Approximately how much energy per mole is required to completely dissociate ethanol, formaldehyde, and hydrazine into their constituent atoms?
-
Is the reaction of diimine (N 2 H 2 ) with oxygen to produce nitrogen and water exothermic or endothermic? Quantify your answer.

BOND POLARITY AND ELECRONEGATIVITY (31~40)
Conceptual Problems
-
Why do ionic compounds such as KI exhibit substantially less than 100% ionic character in the gas phase?
-
Of the compounds LiI and LiF, which would you expect to behave more like a classical ionic compound? Which would have the greater dipole moment in the gas phase? Explain your answers.

- RbCl
- S 8
- TiCl 2
- SbCl 3
- LiI
- Br 2
-
Based on relative electronegativities, classify the bonding in each compound as ionic, covalent, or polar covalent. Indicate the direction of the bond dipole for each polar covalent bond.
- NO
- HF
- MgO
- AlCl 3
- SiO 2
- the C=O bond in acetone
- O 3
-
Based on relative electronegativities, classify the bonding in each compound as ionic, covalent, or polar covalent. Indicate the direction of the bond dipole for each polar covalent bond.
- NaBr
- OF 2
- BCl 3
- the S–S bond in CH 3 CH 2 SSCH 2 CH 3
- the C–Cl bond in CH 2 Cl 2
- the O–H bond in CH 3 OH
- PtCl 4 2 −
-
Classify each species as having 0%–40% ionic character, 40%–60% ionic character, or 60%–100% ionic character based on the type of bonding you would expect. Justify your reasoning.
- CaO
- S 8
- AlBr 3
- ICl
- Na 2 S
- SiO 2
- LiBr
-
If the bond distance in HCl (dipole moment = 1.109 D) were double the actual value of 127.46 pm, what would be the effect on the charge localized on each atom? What would be the percent negative charge on Cl? At the actual bond distance, how would doubling the charge on each atom affect the dipole moment? Would this represent more ionic or covalent character?
-
Calculate the percent ionic character of HF (dipole moment = 1.826 D) if the H–F bond distance is 92 pm.
-
Calculate the percent ionic character of CO (dipole moment = 0.110 D) if the C–O distance is 113 pm.
-
Calculate the percent ionic character of PbS and PbO in the gas phase, given the following information: for PbS, r = 228.69 pm and µ = 3.59 D; for PbO, r = 192.18 pm and µ = 4.64 D. Would you classify these compounds as having covalent or polar covalent bonds in the solid state?
DRAWING LEWIS STRUCTURES (41~71)
Conceptual Problems
-
Compare and contrast covalent and ionic compounds with regard to
- volatility.
- melting point.
- electrical conductivity.
- physical appearance.
-
What are the similarities between plots of the overall energy versus internuclear distance for an ionic compound and a covalent compound? Why are the plots so similar?
-
Which atom do you expect to be the central atom in each of the following species?
- SO 4 2 −
- NH 4 +
- BCl 3
- SO 2 Cl 2
-
Which atom is the central atom in each of the following species?
- PCl 3
- CHCl 3
- SO 2
- IF 3
-
What is the relationship between the number of bonds typically formed by the period 2 elements in groups 14, 15, and 16 and their Lewis electron structures?
-
Although formal charges do not represent actual charges on atoms in molecules or ions, they are still useful. Why?
Numerical Problems
-
Give the electron configuration and the Lewis dot symbol for the following. How many more electrons can each atom accommodate?
- Se
- Kr
- Li
- Sr
- H
-
Give the electron configuration and the Lewis dot symbol for the following. How many more electrons can each atom accommodate?
- Na
- Br
- Ne
- C
- Ga
-
Based on Lewis dot symbols, predict the preferred oxidation state of Be, F, B, and Cs.
-
Based on Lewis dot symbols, predict the preferred oxidation state of Br, Rb, O, Si, and Sr.
-
Based on Lewis dot symbols, predict how many bonds gallium, silicon, and selenium will form in their neutral compounds.
-
Determine the total number of valence electrons in the following.
- Cr
- Cu +
- NO +
- XeF 2
- Br 2
- CH 2 Cl 2
- NO 3 −
- H 3 O +
-
Determine the total number of valence electrons in the following.
- Ag
- Pt 2 +
- H 2 S
- OH −
- I 2
- CH 4
- SO 4 2 −
- NH 4 + .
-
Draw Lewis electron structures for the following.
- F 2
- SO 2
- AlCl 4 −
- SO 3 2 −
- BrCl
- XeF 4
- NO +
- PCl 3
-
Draw Lewis electron structures for the following.
- Br 2
- CH 3 Br
- SO 4 2 −
- O 2
- S 2 2 −
- BF 3
-
Draw Lewis electron structures for CO 2 , NO 2 − , SO 2 , and NO 2 + . From your diagram, predict which pair(s) of compounds have similar electronic structures.
-
Write Lewis dot symbols for each pair of elements. For a reaction between each pair of elements, predict which element is the oxidant, which element is the reductant, and the final stoichiometry of the compound formed.
- K, S
- Sr, Br
- Al, O
- Mg, Cl
-
Write Lewis dot symbols for each pair of elements. For a reaction between each pair of elements, predict which element is the oxidant, which element is the reductant, and the final stoichiometry of the compound formed.
- Li, F
- Cs, Br
- Ca, Cl
- B, F
-
Use Lewis dot symbols to predict whether ICl and NO 4 − are chemically reasonable formulas.
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Draw a plausible Lewis electron structure for a compound with the molecular formula Cl 3 PO.
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Draw a plausible Lewis electron structure for a compound with the molecular formula CH 4 O.
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While reviewing her notes, a student noticed that she had drawn the following structure in her notebook for acetic acid:
Why is this structure not feasible? Draw an acceptable Lewis structure for acetic acid. Show the formal charges of all nonhydrogen atoms in both the correct and incorrect structures.
-
A student proposed the following Lewis structure shown for acetaldehyde.
Why is this structure not feasible? Draw an acceptable Lewis structure for acetaldehyde. Show the formal charges of all nonhydrogen atoms in both the correct and incorrect structures.
-
Draw the most likely structure for HCN based on formal charges, showing the formal charge on each atom in your structure. Does this compound have any plausible resonance structures? If so, draw one.
-
Draw the most plausible Lewis structure for NO 3 − . Does this ion have any other resonance structures? Draw at least one other Lewis structure for the nitrate ion that is not plausible based on formal charges.
-
At least two Lewis structures can be drawn for BCl 3 . Using arguments based on formal charges, explain why the most likely structure is the one with three B–Cl single bonds.
-
Using arguments based on formal charges, explain why the most feasible Lewis structure for SO 4 2 − has two sulfur–oxygen double bonds.
-
At least two distinct Lewis structures can be drawn for N 3 − . Use arguments based on formal charges to explain why the most likely structure contains a nitrogen–nitrogen double bond.
-
Is H–O–N=O a reasonable structure for the compound HNO 2 ? Justify your answer using Lewis electron dot structures.
-
Is H–O=C–H a reasonable structure for a compound with the formula CH 2 O? Use Lewis electron dot structures to justify your answer.
-
Explain why the following Lewis structure for SO 3 2 − is or is not reasonable.
Numerical Answers
-
-
[Ar]4 s 2 3 d 10 4 p 4
Selenium can accommodate two more electrons, giving the Se 2 − ion.
-
[Ar]4 s 2 3 d 10 4 p 6
Krypton has a closed shell electron configuration, so it cannot accommodate any additional electrons.
-
1 s 2 2 s 1
Lithium can accommodate one additional electron in its 2 s orbital, giving the Li − ion.
-
[Kr]5 s 2
Strontium has a filled 5 s subshell, and additional electrons would have to be placed in an orbital with a higher energy. Thus strontium has no tendency to accept an additional electron.
-
1 s 1
Hydrogen can accommodate one additional electron in its 1 s orbital, giving the H − ion.
 The final stoichiometry is K 2 S.
-
Sr is the reductant; Br is the oxidant. The final stoichiometry is SrBr 2 .
-
Al is the reductant; O is the oxidant. The final stoichiometry is Al 2 O 3 .
-
Mg is the reductant; Cl is the oxidant. The final stoichiometry is MgCl 2 .
-
-
-
-
-
The only structure that gives both oxygen and carbon an octet of electrons is the following:
-
-
The student’s proposed structure has two flaws: the hydrogen atom with the double bond has four valence electrons (H can only accommodate two electrons), and the carbon bound to oxygen only has six valence electrons (it should have an octet). An acceptable Lewis structure is
The formal charges on the correct and incorrect structures are as follows:
-
-
The most plausible Lewis structure for NO 3 − is:
There are three equivalent resonance structures for nitrate (only one is shown), in which nitrogen is doubly bonded to one of the three oxygens. In each resonance structure, the formal charge of N is +1; for each singly bonded O, it is −1; and for the doubly bonded oxygen, it is 0.
The following is an example of a Lewis structure that is not plausible:
This structure nitrogen has six bonds (nitrogen can form only four bonds) and a formal charge of –1.
-
-
With four S–O single bonds, each oxygen in SO 4 2 − has a formal charge of −1, and the central sulfur has a formal charge of +2. With two S=O double bonds, only two oxygens have a formal charge of –1, and sulfur has a formal charge of zero. Lewis structures that minimize formal charges tend to be lowest in energy, making the Lewis structure with two S=O double bonds the most probable.
-
-
Yes. This is a reasonable Lewis structure, because the formal charge on all atoms is zero, and each atom (except H) has an octet of electrons.
RESONANCE STRUCTURES (72~81)
Conceptual Problems
- Why are resonance structures important?
- In what types of compounds are resonance structures particularly common?
- True or False, The picture below is a resonance structure?
Numerical Problems
-
Draw all the resonance structures for each ion.
- HSO 4 −
- HSO 3 −
- Draw the Lewis Dot Structure for SO 4 2 - and all possible resonance structures. Which of the following resonance structure is not favored among the Lewis Structures? Explain why. Assign Formal Charges.
- Draw the Lewis Dot Structure for CH 3 COO - and all possible resonance structures. Assign Formal Charges. Choose the most favorable Lewis Structure.
- Draw the Lewis Dot Structure for H PO 3 2 - and all possible resonance structures. Assign Formal Charges.
- Draw the Lewis Dot Structure for CHO 2 1 - and all possible resonance structures. Assign Formal Charges.
- Draw the Resonance Hybrid Structure for P O 4 3 - .
- Draw the Resonance Hybrid Structure for N O 3 - .
Numerical Answers
- False, because the electrons were not moved around, only the atoms (this violates the Resonance Structure Rules).
-
Below are the all Lewis dot structure with formal charges (in red) for Sulfate (
SO
4
2
-
). There isn't a most favorable resonance of the Sulfate ion because they are all identical in charge and there is no change in Electronegativity between the Oxygen atoms.
-
Below is the resonance for
CH
3
COO
-
, formal charges are displayed in red. The Lewis Structure with the most formal charges is not desirable, because we want the Lewis Structure with the least formal charge.
-
The resonance for
HPO
3
2
-
, and the formal charges (in red).
-
The resonance for
CHO
2
1
-
, and the formal charges (in red).
-
The resonance hybrid for
PO
4
3
-
, hybrid bonds are in red.
-
The resonance hybrid for
NO
3
-
, hybrid bonds are in red.
EXCEPTIONS TO THE OCTET RULE (82~98)
Conceptual Problems
- What regions of the periodic table contain elements that frequently form molecules with an odd number of electrons? Explain your answer.
- How can atoms expand their valence shell? What is the relationship between an expanded valence shell and the stability of an ion or a molecule?
- What elements are known to form compounds with less than an octet of electrons? Why do electron-deficient compounds form?
- List three elements that form compounds that do not obey the octet rule. Describe the factors that are responsible for the stability of these compounds.
Numerical Problems
-
What is the major weakness of the Lewis system in predicting the electron structures of PCl 6 − and other species containing atoms from period 3 and beyond?
-
The compound aluminum trichloride consists of Al 2 Cl 6 molecules with the following structure (lone pairs of electrons removed for clarity):
Does this structure satisfy the octet rule? What is the formal charge on each atom? Given the chemical similarity between aluminum and boron, what is a plausible explanation for the fact that aluminum trichloride forms a dimeric structure rather than the monomeric trigonal planar structure of BCl 3 ?
-
Draw Lewis electron structures for ClO 4 − , IF 5 , SeCl 4 , and SbF 5 .
-
Draw Lewis electron structures for ICl 3 , Cl 3 PO, Cl 2 SO, and AsF 6 − .
-
Draw plausible Lewis structures for the phosphate ion, including resonance structures. What is the formal charge on each atom in your structures?
-
Draw an acceptable Lewis structure for PCl 5 , a compound used in manufacturing a form of cellulose. What is the formal charge of the central atom? What is the oxidation number of the central atom?
-
Using Lewis structures, draw all of the resonance structures for the BrO 3 − ion.
-
Draw an acceptable Lewis structure for xenon trioxide (XeO 3 ), including all resonance structures.
Numerical Answers
-
-
-
ClO 4 − (one of four equivalent resonance structures)
-
-
The formal charge on phosphorus is 0, while three oxygen atoms have a formal charge of −1 and one has a formal charge of zero.
-
-
Practice Problems
- Draw the Lewis structure for the molecule I 3 - .
- Draw the molecule ClF 3 .
- The central atom for an expanded octet must have an atomic number larger than what?
- Draw the Lewis structure for the molecule NO 2 .
- Which Lewis structure is more likely?
or
Practice Answers
- 10 (Sodium and higher)