12.3: Structure and Function of RNA
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- Ying Liu
- City College of San Francisco
Learning Objectives
- Describe the biochemical structure of ribonucleotides
- Describe the similarities and differences between RNA and DNA
- Describe the functions of the three main types of RNA used in protein synthesis
- Explain how RNA can serve as hereditary information
Structurally speaking, ribonucleic acid (RNA), is quite similar to DNA. However, whereas DNA molecules are typically long and double stranded, RNA molecules are much shorter and are typically single stranded. RNA molecules perform a variety of roles in the cell but are mainly involved in the process of protein synthesis (translation) and its regulation.
RNA Structure
RNA is typically single stranded and is made of ribonucleotides that are linked by phosphodiester bonds. A ribonucleotide in the RNA chain contains ribose (the pentose sugar), one of the four nitrogenous bases (A, U, G, and C), and a phosphate group. The subtle structural difference between the sugars gives DNA added stability, making DNA more suitable for storage of genetic information, whereas the relative instability of RNA makes it more suitable for its more short-term functions.
The RNA-specific pyrimidine uracil forms a complementary base pair with adenine and is used instead of the thymine used in DNA. Even though RNA is single stranded, most types of RNA molecules show extensive intramolecular base pairing between complementary sequences within the RNA strand, creating a predictable three-dimensional structure essential for their function (Figure \(\PageIndex{1}\) and Figure \(\PageIndex{2}\)).
Exercise \(\PageIndex{1}\)
How does the structure of RNA differ from the structure of DNA?
Functions of RNA in Protein Synthesis
Cells access the information stored in DNA by creating RNA to direct the synthesis of proteins through the process of translation. Proteins within a cell have many functions, including building cellular structures and serving as enzyme catalysts for cellular chemical reactions that give cells their specific characteristics. The three main types of RNA directly involved in protein synthesis are messenger RNA (mRNA), ribosomal RNA (rRNA), and transfer RNA (tRNA).
In 1961, French scientists François Jacob and Jacques Monod hypothesized the existence of an intermediary between DNA and its protein products, which they called messenger RNA. 1 Evidence supporting their hypothesis was gathered soon afterwards showing that information from DNA is transmitted to the ribosome for protein synthesis using mRNA. If DNA serves as the complete library of cellular information, mRNA serves as a photocopy of specific information needed at a particular point in time that serves as the instructions to make a protein.
The mRNA carries the message from the DNA, which controls all of the cellular activities in a cell. If a cell requires a certain protein to be synthesized, the gene for this product is “turned on” and the mRNA is synthesized through the process of transcription (see RNA Transcription ). The mRNA then interacts with ribosomes and other cellular machinery (Figure \(\PageIndex{3}\)) to direct the synthesis of the protein it encodes during the process of translation (see Protein Synthesis ). mRNA is relatively unstable and short-lived in the cell, especially in prokaryotic cells, ensuring that proteins are only made when needed.
rRNA and tRNA are stable types of RNA. In prokaryotes and eukaryotes, tRNA and rRNA are encoded in the DNA, then copied into long RNA molecules that are cut to release smaller fragments containing the individual mature RNA species. In eukaryotes, synthesis, cutting, and assembly of rRNA into ribosomes takes place in the nucleolus region of the nucleus, but these activities occur in the cytoplasm of prokaryotes. Neither of these types of RNA carries instructions to direct the synthesis of a polypeptide, but they play other important roles in protein synthesis.
Ribosomes are composed of rRNA and protein. As its name suggests, rRNA is a major constituent of ribosomes, composing up to about 60% of the ribosome by mass and providing the location where the mRNA binds. The rRNA ensures the proper alignment of the mRNA, tRNA, and the ribosomes; the rRNA of the ribosome also has an enzymatic activity (peptidyl transferase) and catalyzes the formation of the peptide bonds between two aligned amino acids during protein synthesis. Although rRNA had long been thought to serve primarily a structural role, its catalytic role within the ribosome was proven in 2000. 2 Scientists in the laboratories of Thomas Steitz (1940–) and Peter Moore(1939–) at Yale University were able to crystallize the ribosome structure from Haloarcula marismortui , a halophilic archaeon isolated from the Dead Sea. Because of the importance of this work, Steitz shared the 2009 Nobel Prize in Chemistry with other scientists who made significant contributions to the understanding of ribosome structure.
Transfer RNA is the third main type of RNA and one of the smallest, usually only 70–90 nucleotides long. It carries the correct amino acid to the site of protein synthesis in the ribosome. It is the base pairing between the tRNA and mRNA that allows for the correct amino acid to be inserted in the polypeptide chain being synthesized (Figure \(\PageIndex{4}\)). Any mutations in the tRNA or rRNA can result in global problems for the cell because both are necessary for proper protein synthesis (Table \(\PageIndex{1}\)).
| mRNA | rRNA | tRNA | |
|---|---|---|---|
| Structure | Short, unstable, single-stranded RNAcorresponding to a gene encoded within DNA | Longer, stable RNA molecules composing 60% of ribosome’s mass | Short (70-90 nucleotides), stable RNA with extensive intramolecular base pairing; contains an amino acid binding site and an mRNA binding site |
| Function | Serves as intermediary between DNA and protein; used by ribosome to direct synthesis of protein it encodes | Ensures the proper alignment of mRNA, tRNA, and ribosome during protein synthesis; catalyzes peptide bond formation between amino acids | Carries the correct amino acid to the site of protein synthesis in the ribosome |
Exercise \(\PageIndex{1}\)
What are the functions of the three major types of RNA molecules involved in protein synthesis?
RNA as Hereditary Information
Although RNA does not serve as the hereditary information in most cells, RNA does hold this function for many viruses that do not contain DNA. Thus, RNA clearly does have the additional capacity to serve as genetic information. Although RNA is typically single stranded within cells, there is significant diversity in viruses. Rhinoviruses, which cause the common cold; influenza viruses; and the Ebola virus are single-stranded RNA viruses. Rotaviruses, which cause severe gastroenteritis in children and other immunocompromised individuals, are examples of double-stranded RNA viruses. Because double-stranded RNA is uncommon in eukaryotic cells, its presence serves as an indicator of viral infection. The implications for a virus having an RNA genome instead of a DNA genome are discussed in more detail in Viruses .
Key Concepts and Summary
- Ribonucleic acid (RNA) is typically single stranded and contains ribose as its pentose sugar and the pyrimidine uracil instead of thymine. An RNA strand can undergo significant intramolecular base pairing to take on a three-dimensional structure.
- There are three main types of RNA, all involved in protein synthesis.
- Messenger RNA ( mRNA ) serves as the intermediary between DNA and the synthesis of protein products during translation.
- Ribosomal RNA ( rRNA ) is a type of stable RNA that is a major constituent of ribosomes. It ensures the proper alignment of the mRNA and the ribosomes during protein synthesis and catalyzes the formation of the peptide bonds between two aligned amino acids during protein synthesis.
- Transfer RNA ( tRNA ) is a small type of stable RNA that carries an amino acid to the corresponding site of protein synthesis in the ribosome. It is the base pairing between the tRNA and mRNA that allows for the correct amino acid to be inserted in the polypeptide chain being synthesized.
- Although RNA is not used for long-term genetic information in cells, many viruses do use RNA as their genetic material.
Footnotes
- 1 A. Rich. “The Era of RNA Awakening: Structural Biology of RNA in the Early Years.” Quarterly Reviews of Biophysics 42 no. 2 (2009):117–137.
- 2 P. Nissen et al. “The Structural Basis of Ribosome Activity in Peptide Bond Synthesis.” Science 289 no. 5481 (2000):920–930.


7.7: What Are the Odds?
- Compute odds.
- Determine odds from probabilities.
- Determine probabilities from odds.
A particular lottery instant-win game has 2 million tickets available. Of those, 500,000 win a prize. If there are 500,000 winners, then it follows that there are 1,500,000 losing tickets. When we evaluate the risk associated with a game like this, it can be useful to compare the number of ways to win the game to the number of ways to lose. In the case of this game, we would compare the 500,000 wins to the 1,500,000 losses. In other words, there are 3 losing tickets for every winning ticket. Comparisons of this type are the focus of this section.
Computing Odds
The ratio of the number of equally likely outcomes in an event to the number of equally likely outcomes not in the event is called the odds for (or odds in favor of) the event. The opposite ratio (the number of outcomes not in the event to the number in the event to the number in the event is called the odds against the event.
Both odds and probabilities are calculated as ratios. To avoid confusion, we will always use fractions, decimals, or percents for probabilities, and we’ll use colons to indicate odds. The rules for simplifying fractions apply to odds, too. Thus, the odds for winning a prize in the game described in the section opener are and the odds against winning a prize are . These would often be described in words as “the odds of winning are one to three in favor” or “the odds of winning are three to one against.” .
Notice that, while probabilities must always be between zero and one inclusive, odds can be any (non-negative) number, as we’ll see in the next example.
- If you roll a fair 6-sided die, what are the odds for rolling a 5 or higher?
- If you roll two fair 6-sided dice, what are the odds against rolling a sum of 7?
- If you draw a card at random from a standard deck, what are the odds for drawing a ?
- If you draw 2 cards at random from a standard deck, what are the odds against them both being ?
- Answer
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- The sample space for this experiment is {1, 2, 3, 4, 5, 6}. Two of those outcomes are in the event “roll a five or higher,” while four are not. So, the odds for rolling a five or higher are .
- In Example 7.18, we found the sample space for this experiment using the following table (Figure ):
There are 6 outcomes in the event “roll a sum of 7,” and there are 30 outcomes not in the event. So, the odds against rolling a 7 are .
- There are 13 in a standard deck, and others. So, the odds in favor of drawing a are .
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There are
13 C 2 = 78 13 C 2 ways to draw 2 , and= 78 52 C 2 − 78 = 1, 248 52 C 2 ways to draw 2 cards that are not both . So, the odds against drawing 2 are .− 78 = 1,248
You roll a pair of 4-sided dice with faces labeled 1 through 4.
What are the odds for rolling a sum greater than 3?
What are the odds against both dice giving the same number?
Odds as a Ratio of Probabilities
We can also think of odds as a ratio of probabilities. Consider again the instant-win game from the section opener, with 500,000 winning tickets out of 2,000,000 total tickets. If a player buys one ticket, the probability of winning is , and the probability of losing is . Notice that the ratio of the probability of winning to the probability of losing is , which matches the odds in favor of winning.
For an event ,
We can use these formulas to convert probabilities to odds, and vice versa.
Given the following probabilities of an event, find the corresponding odds for and odds against that event.
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3 5 P ( E ) = 3 5
- Answer
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Using the formula, we have:
(Note that in the last step, we simplified by multiplying both terms in the ratio by 5.)for E = P ( E ) : ( 1 − P ( E ) ) = 3 5 : ( 1 − 3 5 ) = 3 5 : 2 5 = 3 : 2. odds for E = P ( E ) : ( 1 − P ( E ) ) = 3 5 : ( 1 − 3 5 ) = 3 5 : 2 5 = 3 : 2.
Since the odds for are , the odds against must be . -
Again, we’ll use the formula:
(In the last step, we simplified by dividing both terms in the ratio by 0.17.)for E = P ( E ) : ( 1 − P ( E ) ) = 0.17 : ( 1 − 0.17 ) = 0.17 : 0.83 ≈ 1 : 4.88. odds for E = P ( E ) : ( 1 − P ( E ) ) = 0.17 : ( 1 − 0.17 ) = 0.17 : 0.83 ≈ 1 : 4.88.
It follows that the odds against are approximately .
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Using the formula, we have:
If the probability of an event \(E\) is 80%, find the odds for and the odds against \(E\).
Now, let’s convert odds to probabilities. Let’s say the odds for an event are
Let’s put this result in a formula we can use.
If the odds in favor of are , then
.
Find if :
- The odds of are in favor
- The odds of are against
- Answer
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Using the formula we just found, we have
2 3 P .( E ) = 2 2 + 1 = 2 3 -
If the odds against are
, then the odds for are
. Thus, using the formula,
1 7 P .( E ) = 1 1 + 6 = 1 7
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Using the formula we just found, we have
Find \(P(E)\) if \(E\):
The odds of \(E\) are \(15{:}1\) against
The odds of \(E\) are \(2.5{:}1\) in favor
Some places, particularly state lottery websites, will use the words “odds” and “probability” interchangeably. Never assume that the word “odds” is being used correctly! Compute one of the odds/probabilities yourself to make sure you know how the word is being used!

- What are the odds in favor of rolling a green face?
- What are the odds against rolling a blue face?
- What are the odds in favor of rolling an orange face?
- What are the odds in favor of an event with probability \(\frac{3}{8}\)?
- What are the odds against an event with probability \(\frac{2}13\)?
- What is the probability of an event with odds \(9{:}4\) against?
- What is the probability of an event with odds \(5{:}7\) in favor?


30.2: Chapter 5
Unfolding Case Study
1. A. Rationale: The patient appears to be stable currently. The nurse should recognize the need for an interpreter as the priority to improve communication because the patient does not speak English.
2. C. Rationale: Pain is whatever the patient says it is. In this case, it is likely that the patient’s culture processes pain and the meaning of pain differently than what the nurse expects to see. In some cultures, it is common to remain stoic, regardless of how severe the pain is. This highlights the importance of obtaining a thorough history and assessment so that cues are not missed.
3. Health disparities are health outcomes that are worse in certain populations. The social determinants of health are external factors such as the neighborhood people live in or access to health care and have a direct impact on health disparities. In this particular case, the patient reports that income is inconsistent, likely classifying the patient as from a low socioeconomic status, which is a social determinant of health. Having a limited (or inconsistent) amount of income directly impacts the patient’s ability to provide for her family and obtain healthcare services, both of which contribute to the development of health disparities.
4. First, the nurse should attempt to obtain more information about the patient’s situation. This needs to be done with care as this can be sensitive to talk about. In this case, the nurse also needs to request assistance from the interpreter because the patient does not speak English. Once more information about the patient’s finance and home situation obtained, the nurse can initiate appropriate referrals. Often this includes a referral to a social worker or counselor who can assist the patient in finding community and other resources to address economic or social needs.
5. First, the nurse will determine what the patient considers to be the problem and what they call it. In this case, the nurse may ask the patient, “What do you call your illness?” or “What does your illness mean to you?” Second, the nurse should ask the patient what they think caused the illness. In some cultures, it is believed that illness is caused by imbalances in the body or is a punishment from God. The nurse should accept the patient’s answer without judgment, even if it does not align with the nurse’s personal beliefs. Next, the nurse should ask what the patient is doing to cope with the problem. The nurse should take note of any cultural differences in the way the patient handles the illness, such as use of folk remedies or nontraditional healers, which could impact the patient’s care plan. Last, the nurse will ask the patient how concerned they are about the illness. Some cultures view illness as “God’s will” and are not concerned, while others express more concern. In this case, specifically, the nurse may also want to ask how concerned the patient is about their finances and access to health care as it has been determined that the SDOH are likely impacting the patient’s overall health.
6. Many times, nurses provide patients with resources and referrals while in the hospital but then have no idea if they followed through and got the assistance they needed. To combat this, the nurse can do a few different things. First, the nurse should work closely with the social worker to ensure that appropriate resources are available and that the patient’s contact information is provided and all follow-ups are scheduled. This helps ensure that contact with the patient is not lost after discharge. Additionally, the nurse should continuously assess the patient’s social needs while they are in the hospital, initiating appropriate referrals as needed. In many cases, nurses and patients build a rapport during the hospital stay which allows the patient to confide in the nurse and allows the nurse to assist the patient with resources. The nurse can ask the patient questions such as “Did I address all of your concerns?” or “Is there anything you are worried about that we haven’t talked about yet?” to determine whether the patient’s social needs have been completely addressed.


13.3: Putting It Together- Ethics in Business
Let’s return to our opening scenario and apply what you’ve learned in this module. We started by considering the impact of ethical decisions related to the housing market leading up to the Great Recession. Mortgage brokers, ratings agencies, and financial institutions sacrificed good ethics for significant profits.
But how important were these ethical lapses, and how much focus should have been given to these errors in judgment?
In this module, you learned about the factors that determine the ethical intensity of an issue. Because ethical dilemmas can vary, this is a framework that helps us understand how important it is to address a particular topic. Let’s use this framework to evaluate the ethical problems during the housing meltdown.
Magnitude of Consequences
The size of benefits that passed to the companies involved in creating these high-risk loans during the housing crisis was measured in the billions of dollars. Likewise, the magnitude of the impact on the economy when the tipping point was reached was even greater. Millions of people lost their jobs and homes. So the magnitude of this dimension shows an ethically intense decision.
Probability of Effect
Experts leading up to the housing crash were split on the probability dimension. Many believed that housing would not go down in value and that even if customers couldn’t repay their loans, the financial institutions could simply foreclose and resell the assets. On the other hand, there were many investors who were very vocal in their certainty of a coming collapse. For a lender uncertain of the probability, this dimension would add little to the intensity.
Social Consensus
While industry insiders knew that “everyone was doing it” when it came to making bad loans, it wasn’t openly discussed. Ultimately, those involved knew that the social consensus would be opposed to what was done—especially given that the motivation was almost exclusively profit. Therefore, the social consensus dimension would suggest a high-intensity issue.
Temporal Immediacy
Lenders knew that even if the loans they made went bad, it would be some time before the effects were realized. Most customers would make payments for several months in order to keep their home. When they started getting behind, the servicing companies would make concessions to keep the borrower in the home. Likewise, foreclosure tends to be a relatively slow process. All of these factors suggest a low temporal immediacy, lessening the intensity of the ethical issue.
Proximity
Did the decisions the lenders made impact those close to them or strangers? Though the lenders likely assumed that the impact would be only to strangers, the economic storm that followed affected most households to some extent. This misjudgment put less weight on proximity.
Concentration of Effect
The benefits of these lending decisions accrued to only a relatively small number of individuals. However, the negative consequences were felt by millions. Given the asymmetrical weight to the negative consequences, the effects were broadly felt. As such, that would add intensity to the ethical problem.
Using the framework that you learned, you can understand why the government stepped in with a massive overhaul of the regulatory framework of financial institutions. If the individuals involved had used a similar framework when making their personal decisions, perhaps the crisis would never have occurred. Those in positions of leadership and authority should be held to a high standard of integrity, which is why the study of ethics is foundational to the field of management.
In this module, we have covered creating a company ethics program both to avoid fines when someone breaks the law but also to encourage better behavior toward stakeholders. Research strongly suggests that stakeholders will reward good citizens. [1] Like Whole Foods’ flooding experience, stakeholders will look out for firms that look out for them.
- Webley, S., & More, E. (n.d.). Does Business Ethics Pay? Retrieved September 19, 2017, from http://www.ibe.org.uk/userfiles/doesbusethicpaysumm.pdf ↵
Contributors and Attributions
- Putting It Together: Ethics in Business. Authored by : Jeff Heflin and Lumen Learning. License : CC BY: Attribution
- Image: Business Ethics. Authored by : Nick Youngson. Located at : http://www.thebluediamondgallery.com/highlighted/b/business-ethics.html . License : CC BY-SA: Attribution-ShareAlike


2.4: Ancient Observations
Many of the ancient civilizations observed positions of specific stars and planets, like the rising and setting times of these objects. Ancient observatories were built to do this fairly accurately. For example, the Mayan observatory at Chichén Itzá had windows placed for observations of Venus. The Polynesians were some of the best observers for their island-to-island sailing navigation. The Chinese made incredibly detailed observations of the skies. This was done for their Emperor, the “son of the heavens. ” They needed to accurately predict events to show the Emperor’s “divineness. ” The Chinese were the first to record observations of comets, meteor showers, meteorites, eclipse predictions, and supernovae. It was the Chinese as well who built instruments to conduct these studies.
Other contributions of these early civilizations included the Mesopotamians , who were the first to develop a comprehensive catalog of the night sky, circa 750 BC, and the Babylonians , who combined the practice of astrology and astronomy.
The Egyptians also had an infatuation with the heavens. They developed the first recorded sundials in the form of Obelisks, and around 3500 BC recognized the seasons, and had implemented a day clock. The Pyramids also had astronomical implications; the Great Pyramid at Giza completed in 2680 BC, aligned with stars of Osiris and in a specific compass direction.


1.5: Activities
Activity \(\PageIndex{1}\)
Consider the following statement.
If the game is on and the popcorn is ready, then Joe is happy.
- Assign statement variables and rewrite the statement in symbolic language.
- Write out the truth table for your symbolic statement.
- You visit Joe's residence room and found that Joe is unhappy even though the game is on. Assuming that the above conditional statement is a true statement about Joe, what can you conclude about the popcorn? Which rows in your truth table justify this conclusion?
Activity \(\PageIndex{2}\)
Consider the logical statement
- Make up an English language statement that has the same logical structure as this symbolic statement. (Do not just make a word-for-word translation of the logical connectives — make sure you have a reasonable-sounding English sentence when read out loud.)
- Argue convincingly that this symbolic statement is a tautology, not by writing out its truth table, but by arguing that it is not possible for the statement to be false.
- Don't Skip Ahead
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In Chapter 2 , we will learn that the two substatements involved in this conditional are logically equivalent . If you have already read ahead into that chapter, do not just use this equivalence of these two statements to carry out this task.
- Hint
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Start with the assumption that this conditional statement is false, and then work backwards from the statement to the possible truth values of \(p\) and \(q\) based on that assumption to conclude that the statement being false is not actually possible.
Activity \(\PageIndex{3}\)
Consider the logical statement
- Make the statement simpler by assigning new variables to represent compound statements and rewriting the statement in terms of the new variables.
- Argue that your new statement is a tautology. What does this mean about the original statement?
Activity \(\PageIndex{4}\)
First, re-familiarize yourself with what it means when one statement logically implies another.
Suppose that \(A\) logically implies \(B\) and \(B\) logically implies \(C\text{.}\) Must \(A\) logically imply \(C\text{?}\) Argue convincingly in support of your answer by arguing that the technical definition of logically implies is satisfied.
Activity \(\PageIndex{5}\)
If there is still time: work through Exercise 1.6.3 from this chapter.


2.1: Introducing the Market System
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Defining a Market System
A market system is a way to match buyers and sellers.
Learning objectives
Identify the characteristics of a market system
In an economy, a market system is any systematic process that enables many market players to bid and ask. In other words, a market system is a place (virtual or physical) that facilitates the matching of buyers and sellers. Many markets exist, and each can be defined based on a number of characteristics, such as what is being exchanged in the market, the regulations, who is allowed to participate, and how transactions occur.
One defining component of markets is the medium of exchange, or the price. In most American markets, the medium of exchange is dollars. Both buyers and sellers look at the price to determine whether or not they want to trade. A seller has a certain minimum price at which s/he is willing to sell, though s/he would happily accept more. Likewise, a buyer has a certain maximum price at which s/he is willing to buy, though s/he would happily pay less. If the minimum the seller would accept is less than the maximum a buyer would pay, a transaction can occur. Markets help such buyers and sellers meet to trade.
In market systems, prices are discoverable; both buyers and sellers are capable of finding out the current price at which a transaction could occur. Publishing current prices is a key component with a market system. The chosen prices impact the immediate group of buyers and sellers, but also may impact long term supply and demand decisions within the market.
There are many examples of market systems. Perhaps the most famous is the stock market in which buyers and sellers trade stocks. The prices at which those sales occur is recorded, and is the basis for the stock price you may have seen in the newspaper or on TV. There are markets for many types of products other than stocks: the global oil market, your local farmers’ market, and eBay are all forms of markets with their own defining characteristics.
NASDAQ Stock Market Display : The NASDAQ is a stock market where buyers and sellers of stocks can meet and trade.
Another important component of market systems is that there is competition, which serves as the main regulatory mechanism. Based on the level of competition in a market system, economists have identified a number of different types of structures, such as monopoly, oligopoly, and perfect competition. We will go into more detail on different market structures later in the book.
Gains from Markets
Gains in a market are referred to as total welfare or economic surplus.
Learning objectives
Explain how to calculate total welfare
Gains within a market are referred to as total welfare or economic surplus. Within total welfare, economists look at consumer surplus and producer surplus. A surplus is defined as the excess of a good or service when the quantity supplied exceeds the quantity demanded; this occurs when the price is above the equilibrium price.
Economic Surpluses : The total welfare (or economic surplus) is the sum of the consumer surplus and the producer surplus.
Consumer surplus is the monetary gain that consumers receive when they purchase a good for less than the highest price they are willing to pay. For example, a customer is willing to pay $50 for a new pair of running shoes. They are able to purchase the pair for $35 and consumer surplus is $15.
Producer surplus is the amount that producers benefit by selling a good at a market price that is higher than the least that they would be willing to sell it for. An example would be a manufacturer that makes jeans. The lowest price the producer is willing to sell a pair of jeans for is $40, but the jeans actually sell for $50. The producer surplus is $10.
In order to calculate the total welfare, the supply and demand of the good must be used to determine the economic gain. On a demand and supply curve graph, the consumer surplus is located under the demand curve and above a horizontal line that shows the actual price of a good (equilibrium price).
When the supply of a good increases, the price falls which increases consumer surplus. When the demand for a good increases, the price increases and the supply decreases resulting in producer surplus. When a good is in high demand, consumers are willing to pay more in order to obtain the good.
Production Possibility Frontier
A production-possibility frontier (PPF) graphs the combinations for the production of two commodities with which the same amounts are used.
Learning objectives
Explain the benefits of trade and exchange using the production possibilities frontier (PPF)
Within a market system, economists use the production possibility frontier (PPF) to graph the combinations of the amounts of two commodities that can be produced using the same amount of each factor of production. A PPF graph chooses specific input quantities. As a result, it shows the maximum production level for one commodity for any production level of the other commodity. PPF is used to define production efficiency.
A common PPF : A common PPF where there is an increase in opportunity cost.
Within a PPF graph, the use of a curve or line acts as a benchmark for measuring efficiency. If a point on the graph is above the curve it indicates efficiency, while a point below the curve signifies inefficiency. For further analysis, additional information is always supplied with a PPF including the period of time taken for the observation, production technologies, and the amounts of inputs that were available.
Economists can use a PPF to illustrate a number of economic concepts including scarcity, opportunity cost, productive efficiency, allocative efficiency, and economies of scale.
When an economy is operating on the PPF curve it is efficient. It is not possible to produce more of one good without decreasing the amount produced for the other good. Likewise, if the economy is operating below the PPF curve, it is inefficient. In this case, the economy can reallocate resources and produce more of both the goods.
The PPF graph shows how resources must be shared among goods during the production process. The points of the graph show the trade -off that takes place between two goods. For example, if more of Good A needs to be produced, the amount of resources in use by Good B must be reduced and transferred to Good A. The sacrifice in production of Good B is called opportunity cost. When graphing PPF there are three types: the common, the straight line, and the inverted PPF. All three of the PPF graphs are directly influenced by the opportunity cost.
An inverted PPF : An inverted PPF where the opportunity cost is decreasing.
A straight line PPF : A straight line PPF where the opportunity cost is constant.
The slope of the PPF shows the rate at which the production of one good can be transferred to another. The slope is called the marginal rate of transformation (MRT).
Within an economy, if the capacity to produce both goods increases, the result is economic growth. Factors that influence economic capacity include technology, an increase in the supply of factors of production, and production interactions such as trade and exchange. When any of these factors are used it allows for an increase in capacity so that the production of neither good has to be sacrificed.
PPF graphs help economists study the current state of production as well as possible production scenarios. The output of the economy is impacted by many factors. When production can be graphed and monitored it allows adjustments to be made to work towards attaining economic growth and stability.
The slope of the PPF shows the rate at which the production of one good can be transferred to another. The slope is called the marginal rate of transformation (MRT).
Within an economy, if the capacity to produce both goods increases, the result is economic growth. Factors that influence economic capacity include technology, an increase in the supply of factors of production, and production interactions such as trade and exchange. When any of these factors are used it allows for an increase in capacity so that the production of neither good has to be sacrificed.
PPF graphs help economists study the current state of production as well as possible production scenarios. The output of the economy is impacted by many factors. When production can be graphed and monitored it allows adjustments to be made to work towards attaining economic growth and stability.
The Circular Flow Model
In economics, a circular flow model is a diagram that is used to represent the monetary transactions in an economy.
Learning objectives
State the function of the circular flow diagram and the production possibilities frontier
In economics, a circular flow model is a diagram that is used to represent the monetary transactions in an economy. There are two flows present within the model including flows of physical things (goods or labor) and flows of money (what pays for physical things). A circular flow model depicts the inner workings of a market system and specific portions of the economy.
The basic circular flow model consists of two sectors that determine income, expenditure, and output. A state of equilibrium is reached when there is no tendency for the levels of income (YY), expenditure (EE), and output (OO) to change (Y=E=OY=E=O). This equation means that the expenditure of buyers (households) becomes income for sellers ( firms ). The firms spend the income on factors of production, which “transfers” the income to the factor owners. The factor owners spend the income on goods which leads to the circular flow of payments.
Circular flow of goods income : The circular flow model shows the flow of payments between households and firms.
The circular flow of payments is important within an economy because it 1) measures the national income, 2) provides knowledge of interdependence, 3) illustrates the unending nature of economic activities, and 4) shows injections and leakages.
The circular flow of income follows a specific pattern: Production → Income → Expenditure → Production. This circular flow is ongoing between households and firms.
The circular flow of income can also be analyzed using the production possibility frontier (PPF). The PPF is a graph that shows the various combinations of amounts of two commodities that could be produced using the same fixed total amount of each of the factors of production. The graph shows the maximum possible production level of one commodity for any production level of the other, based on the state of technology. The PPF defines production efficiency. A point of the frontier line indicates the efficient use of available inputs, while a point beneath the curve shows inefficiency. A PPF graphs shows opportunity cost, actual output, potential output, and economic growth.
Production Possibilities Frontier Curve : The graph illustrates a typical production possibilities frontier curve. When a market is operating on the PPF it is said to be efficient.
Key Points
- Publishing current prices is a key component with a market system.
- Competition is the primary regulatory mechanism in a market system.
- Economists recognize a number of different structures of market systems based on characteristics such as the level of competition.
- Within total welfare, economists look at consumer surplus and producer surplus.
- Consumer surplus is the monetary gain that consumers receive when they purchase a good for less than the highest price they are willing to pay.
- Producer surplus is the amount that producers benefit by selling a good at a market price that is higher than the least that they would be willing to sell it for.
- In order to calculate the total welfare, the supply and demand of the good must be used to determine the economic gain.
- When the supply of a good increases, the price falls which increases consumer surplus. When the demand for a good increases, the price increases and the supply decreases resulting in producer surplus.
- A PPF graph shows the maximum production level for one commodity for any production level of the other commodity.
- If a point on the graph is above the curve it indicates efficiency, while a point below the curve signifies inefficiency.
- The PPF graph shows how resources must be shared among goods during the production process.
- Within an economy, if the capacity to produce both goods increases which results in economic growth.
- There are two flows present within the model including flows of physical things (goods or labor) and flows of money (what pays for physical things).
- The circular flow of income follows a specific pattern: Production → Income → Expenditure → Production.
- The production possibility frontier can be used to illustrate the circular flow model.
- Economists use data, statistics, and natural experiments in order to make economic “laws” that explain general patterns.
Key Terms
- price : The quantity of payment or compensation given by one party to another in return for goods or services.
- welfare : Health, safety, happiness and prosperity; well-being in any respect.
- commodity : Raw materials, agricultural and other primary products as objects of large-scale trading in specialized exchanges.
- marginal : Of, relating to, or located at or near a margin or edge; also figurative usages of location and margin (edge).
- expenditure : Act of expending or paying out.
- Factors of production : In economics, factors of production are inputs. They may also refer specifically to the primary factors, which are stocks including land, labor, and capital goods applied to production.
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5.1: Audiences and Audience Research
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- Media Hack Team
- University of Otago and others via Creative Commons
This section discusses the different ways we think about receivers, audiences, and users, and how communication and media scholars might approach thinking about and studying audiences. It is worth breaking these concepts down into separate components to try to understand them. However, these concepts are artificial constructs, and the line between receivers, users, and audiences is becoming increasingly soft and blurred and they may have faded in their usefulness.
When dealing with one-to-one or one-to-few communication, we usually use the label of receiver to describe those whom the message targets. As articulated in the transmission models of communication, a receiver can also be a sender, but for now, it’s easiest to start with the simplest articulation of a receiver, where a receiver can be defined as anyone targeted for or taking in information or communication. Leaving the reality that receivers often become senders aside, we can still regard receivers as not merely passive sponges of information – they can interpret material, bring to bear their own experiences, and ‘read’ messages in a number of different ways.
In terms of information received, we can generally say that information is decoded in three ways. There is the dominant reading, where the message intended is the message received. There is a negotiated reading, where the receiver accepts some of the intended message, and rejects other parts of it. Finally, there are oppositional readings, where the reader completely rejects the message intended. And of course, there is also the possibility that the message is not received – through miscommunication or a failure to communicate. It is also worth remembering here that communication is never ideologically neutral.
All of this forms the core of Stuart Hall’s encoding/decoding model. Hall argued that any piece of information can be encoded in multiple ways, and every message has more than one meaning. We refer to this as the polysemy of messages – literally, the ‘many meanings.’ Therefore, communication is always subject to decisions made within the context and systems employed – meaning-making is not natural but cultural. There isn’t such a thing as a ‘right’ way to read a message, only the preferred way within a particular context.
A YouTube element has been excluded from this version of the text. You can view it online here: https://opentextbc.ca/mediastudies101/?p=93
16 Rude and Interesting Gestures around the World – Standard Youtube License (Video uploaded by SkyscannerLtd)
The cultural aspect and interpretation of a message shows in the different non-verbal signs described in the video above. The depend on national, thus cultural backgrounds and might have different meanings in other cultures or might simply not be understood. Also, each gesture would only make sense in a certain context.
When thinking about mass communication, and broadcasting to a large number of people (many-to-many communication), we need to ask whether or not an audience is just an aggregate of receivers. Is there anything special about an audience?
It’s a surprisingly difficult question. A very basic view might see the audience merely as a group of people who are receiving the same message at the same time, whether it be a musical performance or a TV show. But the breadth of communicative options these days quickly exposes the problems with such a simple definition. If one person records a show on MySky™, another downloads the show off a torrent, another watches it when it is first broadcast on ‘free-to-air’ TV, and another buys the DVD, are they collectively still an audience? They’re all receiving the same text, but in different modes and at different times.
We might instead try to think about audiences both as cultural constructs and as responses to particular media or texts. We can define audiences in terms of location (such as the spectators at a rugby game), by population demographics (for example, children), by medium or channel (such as television or Youtube), or by message content (political speeches or soap operas), as well as by time of transmission or reception.
How we perceive audiences has also changed over time, from passive to active to fragmented. We started off in pre-modern times thinking about the audience in terms of the public. Publics, such as those at agora and fora, were generally constrained in time, space and culture. That is, they tended to be in the same place, at the same time, with participants sharing similar values and culture. It wasn’t until the rise of the media and mass communication that audiences started to become more dispersed and hetereogeneous. It is this mass audience that is perhaps more relevant to media scholars.
The mass audience is an interesting phenomena, given how dispersed they are. For much of the 20th century, the mass audience was characterized by a lack of self-generated identity – only with a few rare texts, like certain bands, or shows or sports teams – did people share a common identity in belonging to that audience. For the most part, members of the mass audience, particularly the mediated mass audience, were dispersed across time, space, and culture. Furthermore, what identities they did have were classified by externally generated markers such as demographics, like age, gender (and perhaps most importantly in capitalist media markets,) economic profile.
This era of audience research was very much focused around marketing demographics. Leaving aside public service broadcasting for the moment, a lot of information captured about audiences, and indeed the language we used to describe audiences, came from marketers and program buyers, who determined which mass media texts made it into general circulation based on what kind of audience it would attract (children, housewives, retirees) and therefore what kind of advertising they could sell on it.
Is the idea of the audience still relevant for the 21st Century? A simple test might be to think about what you did last night? With an increasing plethora of channels, modes, and sources of information and entertainment, our existing notion of the mass audience is fractured more than ever.
While the audience still has some use as a concept, particularly as we define the demographics of our audience more tightly, a new concept has come to the fore in communication research. There are a few different labels given to similar concepts, so let’s start with the idea of the user. A user has a number of definitions given the context, but to start with, we can define a user as an active agent who uses available tools to interact with information.
A user is of interest to us because they move through a number of different communicative positions simultaneously. They can be both senders and receivers, audiences and producers, engaged in interpersonal communication that is also public performance. Users form networks, and act as nodes which both pass on and reinterpret information taken in from multiple sources, often other users. Users interact with other users to form networks along which information is not only sent and received, but also modified and interpreted.
These messy networks quickly go beyond any simple, linear model that we might use for understanding audiences. The rise of the user in both the interpersonal and mass communication landscape has been enabled to a large extent by information communication technologies such as the internet and mobile phones. But a lot of the interplay between users that we’ve been seeing reflects other, small scale social networks, such as the pre-modern idea of the village. What has changed is the scale and speed of the network connections, and the idea of memory – an interpersonal network can subjectively remember things. In an electronic network, the exact words, content or message can often be recalled long after the interaction has ceased – a good demonstration of this is searching for yourself online (egosurfing) to find old posts and messages you had long forgotten.
As an aside, it is interesting to use the ritual view of communication as a lens through which to think about the social network of users as a group of communicators. If users can be both senders and receivers, produce and consume (and reproduce) texts, and always have, at least in theory, equal capacity of channel to play either sender or receiver, then the flow of cultural capital is not only sped up, it is also partially removed from the hands of capitalist content producers (i.e. those producing content to sell advertising around or for monetary gain). We’re currently seeing the early shockwaves of this change playing out in the copyright battles of the TPPA, SOPA, PIPA, ACTA and s21a.)
Overall, audiences are a fascinatingly complex concept, and one we’ll return to in the section on fandom. But now, it might be worth thinking how we approach audience research.
Discussion
- Why has the internet changed the concept of audiences?
- Find examples that state your theory.


6.4: Molecular Structure and Polarity
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- Scott Van Bramer
- Widener University
Learning Objectives
- Predict the structures of small molecules using valence shell electron pair repulsion (VSEPR) theory
- Explain the concepts of polar covalent bonds and molecular polarity
- Assess the polarity of a molecule based on its bonding and structure
Thus far, we have used two-dimensional Lewis structures to represent molecules. However, molecular structure is actually three-dimensional, and it is important to be able to describe molecular bonds in terms of their distances, angles, and relative arrangements in space (Figure \(\PageIndex{1}\)). A bond angle is the angle between two adjacent bonds. A bond length is the distance between the nuclei of two bonded atoms. Bond distances are measured in Ångstroms (1 Å = 10 –10 m) or picometers (1 pm = 10 –12 m, 100 pm = 1 Å).
VSEPR Theory
Valence shell electron-pair repulsion theory (VSEPR theory) is used to predict the structure around a central atom of a molecule based on the number of bonds and lone electron pairs in the Lewis structure. The VSEPR model assumes that electron pairs in the valence shell of a central atom arrange to minimize repulsions by maximizing the distance between these electron pairs. The electrons in the valence shell of a central atom form either bonding pairs of electrons, located primarily between bonded atoms, or lone pairs. The electrostatic repulsion of these electrons is reduced when the various regions of high electron density assume positions as far from each other as possible.
VSEPR theory predicts the arrangement of electron pairs around each central atom and the arrangement of atoms in a molecule. We should understand, however, that the theory is a first approximation and that fully understanding the structure of a molecule is much more complex.
A gaseous BeF 2 molecule is a simple example to introduce VSEPR. The Lewis structure of BeF 2 (Figure \(\PageIndex{2}\)) shows two electron pairs around the central beryllium atom. There are two bonds and no lone pairs of electrons on this beryllium atom so according to VSEPR the bonds should be as far apart as possible. The electron density from the bonds repel and this repulsion is minimized when the bonds are on opposite sides of the central atom so the bond angle is 180° (Figure \(\PageIndex{2}\)).
Figure \(\PageIndex{3}\) illustrates electron-pair geometries for different structures that minimize the repulsions among regions of high electron density (bonds and/or lone pairs). Two regions of electron density around a central atom in a molecule form a linear geometry; three regions form a trigonal planar geometry; four regions form a tetrahedral geometry; five regions form a trigonal bipyramidal geometry; and six regions form an octahedral geometry. The spatial relationship, diagrams for the structure, names for the structures, and bond angles are all shown in this figure.
Electron-pair Geometry versus Molecular Structure
It is important to note that electron-pair geometry around a central atom is not the same thing as its molecular structure. The electron-pair geometries shown in Figure \(\PageIndex{3}\) describe all regions where electrons are located, bonds as well as lone pairs. Molecular structure describes the location of the atoms , not the electrons.
We differentiate between these two situations by naming the geometry that includes all electron pairs the electron-pair geometry . The structure that includes only the placement of the atoms in the molecule is called the molecular structure . The electron-pair geometries are the same as the molecular structures when there are no lone electron pairs around the central atom. When a central atom has lone pair electrons the molecular structure is different from the electron-pair geometry.
For example, the methane molecule, CH 4 , which is the major component of natural gas, has four bonding pairs of electrons around the central carbon atom; the electron-pair geometry is tetrahedral, as is the molecular structure (Figure \(\PageIndex{4}\)). On the other hand, the ammonia molecule, NH 3 , also has four electron pairs associated with the nitrogen atom, and thus has a tetrahedral electron-pair geometry. One of these regions, however, is a lone pair, which is not included in the molecular structure, and this lone pair influences the shape of the molecule (Figure \(\PageIndex{5}\)). Since the electron-pair geometry and the molecular structure are different for ammonia we need two different names for these geometries.
In the ammonia molecule, the three hydrogen atoms attached to the central nitrogen are not arranged in a flat, trigonal planar molecular structure, but rather in a three-dimensional trigonal pyramid (Figure \(\PageIndex{5}\)) with the nitrogen atom at the apex and the three hydrogen atoms forming the base. The ideal bond angles in a trigonal pyramid are based on the tetrahedral electron pair geometry. There are slight deviations from the ideal because lone pairs occupy larger regions of space than bonding electrons so the H–N–H bond angle in NH 3 is slightly smaller than the 109.5° angle tetrahedron. (Figure \(\PageIndex{6}\)) summarizes the possible elecvtron-pair geometries and the corresponding molecular structures for up to six electron pairs.
Please note that because the lone pair-bonding pair repulsion is greater than the bonding pair-bonding pair repulsion the bond angles for real molecules will differ slightly from those shown in Figure \(\PageIndex{6}\). The ideal molecular structures are predicted based on the electron-pair geometries.
Small distortions from these ideal angles are shown for ammonia in Figure \(\PageIndex{5}\). The distortions result from differences in repulsion between various regions of electron density. VSEPR theory predicts the magnitude of these distortions by establishing an order of repulsions and an order of the space occupied by different types of electron pairs. The order of electron-pair repulsions from greatest to least repulsion is:
lone pair-lone pair > lone pair-bonding pair > bonding pair-bonding pair
Different types of electron pairs also need different amounts of space. A lone pair of electrons occupies a larger region of space than the electrons in a triple bond; in turn, electrons in a triple bond occupy more space than those in a double bond, and so on. The order of sizes from largest to smallest is:
lone pair > triple bond > double bond > single bond
Taken together the amount of repulsion and the amount of space can help determine the location of the lone pair electrons in the molecular structure and the amount that expected bond angles are distorted in actual molecules. Consider formaldehyde, H 2 CO shown in Figure \(\PageIndex{1}\). This molecule has three regions of high electron density, two single bonds and one double bond. The basic geometry is trigonal planar with predicted 120° bond angles. However, the double bond takes up more space and pushes the single bonds further away so the angle between the single bonds is slightly smaller (118°) than expected.
According to VSEPR theory, the terminal atom locations (Xs in Figure \(\PageIndex{7}\)) are equivalent within the linear, trigonal planar, and tetrahedral electron-pair geometries (the first three rows of the table). It does not matter which X is replaced with a lone pair because the molecules can be rotated and all the positions are equivilent. For trigonal bipyramidal electron-pair geometries, however, there are two distinct X positions (Figure \(\PageIndex{7}\)a): an axial position (if we hold a model of a trigonal bipyramid by the two axial positions, we have an axis around which we can rotate the model) and an equatorial position (three positions form an equator around the middle of the molecule). The axial position is surrounded by bond angles of 90°, whereas the equatorial position has more space available because of the 120° bond angles. In a trigonal bipyramidal electron-pair geometry, lone pairs always occupy equatorial positions because these more spacious positions can more easily accommodate the larger lone pairs.
The molecule ClF 3 is an example of a compound with trigonal bipyramid electron-pair geometry and two sets of lone pair electrons. Figure \(\PageIndex{7}\) shows the three possible arrangements for the three C-F bonds in the ClF 3 molecule. The most stable structure is with the lone pairs in equatorial locations so they have the most space. This gives ClF 3 a T-shaped molecular structure as shown in Figure \(\PageIndex{7}\)b.
Similarly for compounds with octahedral electron-pair geometry, when a central atom has lone electron pairs there are several different possible molecular structures. With two lone pairs they orient on opposite sides of the octahedron (180° apart), giving a square planar molecular structure that minimizes lone pair-lone pair repulsions.  All these options are summarized in Figure \(\PageIndex{6}\).
Predicting Electron Pair Geometry and Molecular Structure
The following procedure uses VSEPR theory to determine the electron pair geometries and the molecular structures:
- Write the Lewis structure of the molecule or polyatomic ion.
- Count the number of regions of electron density (lone pairs and bonds) around the central atom. A single, double, or triple bond counts as one region of electron density.
- Identify the electron-pair geometry based on the number of regions of electron density: linear, trigonal planar, tetrahedral, trigonal bipyramidal, or octahedral (Figure \(\PageIndex{7}\), first column).
- Use the number of lone pairs to determine the molecular structure (Figure \(\PageIndex{7}\) ). If more than one arrangement of lone pairs and chemical bonds is possible, choose the one that will minimize repulsions, remembering that lone pairs occupy more space than multiple bonds, which occupy more space than single bonds. In trigonal bipyramidal arrangements, repulsion is minimized when every lone pair is in an equatorial position. In an octahedral arrangement with two lone pairs, repulsion is minimized when the lone pairs are on opposite sides of the central atom.
The following examples illustrate the use of VSEPR theory to predict the molecular structure of molecules or ions that have no lone pairs of electrons. In this case, the molecular structure is identical to the electron pair geometry.
Example \(\PageIndex{1}\): Predicting Electron-pair Geometry and Molecular Structure
Predict the electron-pair geometry and molecular structure for each of the following:
- carbon dioxide, CO 2 , a molecule produced by the combustion of fossil fuels
- boron trichloride, BCl 3 , an important industrial chemical
Solution
(a) We write the Lewis structure of CO 2 as:
This shows us two regions of high electron density around the carbon atom—each double bond counts as one region, and there are no lone pairs on the carbon atom. Using VSEPR theory, we predict that the two regions of electron density arrange themselves on opposite sides of the central atom with a bond angle of 180°. The electron-pair geometry and molecular structure are identical, and CO 2 molecules are linear.
(b) We write the Lewis structure of BCl 3 as:
Thus we see that BCl 3 contains three bonds, and there are no lone pairs of electrons on boron. The arrangement of three regions of high electron density gives a trigonal planar electron-pair geometry. The B–Cl bonds lie in a plane with 120° angles between them. BCl 3 also has a trigonal planar molecular structure.
The electron-pair geometry and molecular structure of BCl 3 are both trigonal planar. Note that the VSEPR geometry indicates the correct bond angles (120°), unlike the Lewis structure shown above.
Exercise \(\PageIndex{1}\)
Carbonate, \(\ce{CO3^2-}\), is a common polyatomic ion found in various materials from eggshells to antacids. What are the electron-pair geometry and molecular structure of this polyatomic ion?
- Answer
-
The electron-pair geometry is trigonal planar and the molecular structure is trigonal planar. Due to resonance, all three C–O bonds are identical. Whether they are single, double, or an average of the two, each bond counts as one region of electron density.
Example \(\PageIndex{2}\): Predicting Electron-pair Geometry and Molecular Structure
Two of the top 50 chemicals produced in the United States, ammonium nitrate and ammonium sulfate, both used as fertilizers, contain the ammonium ion. Predict the electron-pair geometry and molecular structure of the \(\ce{NH4+}\) cation.
Solution
We write the Lewis structure of \(\ce{NH4+}\) as:
Exercise \(\PageIndex{2}\)
Identify a molecule with trigonal bipyramidal molecular structure.
- Answer
-
Any molecule with five electron pairs around the central atoms including no lone pairs will be trigonal bipyramidal. \(\ce{PF5}\) is a common example
The next several examples illustrate the effect of lone pairs of electrons on molecular structure.
Example \(\PageIndex{3}\): Lone Pairs on the Central Atom
Predict the electron-pair geometry and molecular structure of a water molecule.
Solution
The Lewis structure of H 2 O indicates that there are four regions of high electron density around the oxygen atom: two lone pairs and two chemical bonds:
Exercise \(\PageIndex{3}\)
The hydronium ion, H 3 O + , forms when acids are dissolved in water. Predict the electron-pair geometry and molecular structure of this cation.
- Answer
-
electron pair geometry: tetrahedral; molecular structure: trigonal pyramidal
Example \(\PageIndex{4}\): SF 4 Sulfur tetrafluoride,
Predicting Electron-pair Geometry and Molecular Structure: SF 4 , is extremely valuable for the preparation of fluorine-containing compounds used as herbicides (i.e., SF 4 is used as a fluorinating agent). Predict the electron-pair geometry and molecular structure of a SF 4 molecule.
Solution
The Lewis structure of SF 4 indicates five regions of electron density around the sulfur atom: one lone pair and four bonding pairs:
Exercise \(\PageIndex{4}\)
Predict the electron pair geometry and molecular structure for molecules of XeF 2 .
- Answer
-
The electron-pair geometry is trigonal bipyramidal. The molecular structure is linear.
Example \(\PageIndex{4}\): XeF 4
Of all the noble gases, xenon is the most reactive, frequently reacting with elements such as oxygen and fluorine. Predict the electron-pair geometry and molecular structure of the XeF 4 molecule.
Solution
The Lewis structure of XeF 4 indicates six regions of high electron density around the xenon atom: two lone pairs and four bonds:
Exercise \(\PageIndex{4}\)
In a certain molecule, the central atom has three lone pairs and two bonds. What will the electron pair geometry and molecular structure be?
- Answer
-
electron pair geometry: trigonal bipyramidal; molecular structure: linear
Molecular Structure for Multicenter Molecules
When a molecule or polyatomic ion has only one central atom, the molecular structure completely describes the shape of the molecule. Larger molecules do not have a single central atom, but are connected by a chain of interior atoms that each possess a “local” geometry. The way these local structures are oriented with respect to each other also influences the molecular shape, but such considerations are largely beyond the scope of this introductory discussion. For our purposes, we will only focus on determining the local structures.
Example \(\PageIndex{5}\): Predicting Structure in Multicenter Molecules
The Lewis structure for the simplest amino acid, glycine, H 2 NCH 2 CO 2 H, is shown here. Predict the local geometry for the nitrogen atom, the two carbon atoms, and the oxygen atom with a hydrogen atom attached:
Solution
Consider each central atom independently. The electron-pair geometries:
- nitrogen––four regions of electron density; tetrahedral
- carbon ( C H 2 )––four regions of electron density; tetrahedral
- carbon ( C O 2 )—three regions of electron density; trigonal planar
- oxygen ( O H)—four regions of electron density; tetrahedral
The local structures:
- nitrogen––three bonds, one lone pair; trigonal pyramidal
- carbon ( C H 2 )—four bonds, no lone pairs; tetrahedral
- carbon ( C O 2 )—three bonds (double bond counts as one bond), no lone pairs; trigonal planar
- oxygen ( O H)—two bonds, two lone pairs; bent (109°)
Exercise \(\PageIndex{5}\)
Another amino acid is alanine, which has the Lewis structure shown here. Predict the electron-pair geometry and local structure of the nitrogen atom, the three carbon atoms, and the oxygen atom with hydrogen attached:
- Answer
-
electron-pair geometries: nitrogen––tetrahedral; carbon ( C H)—tetrahedral; carbon ( C H 3 )—tetrahedral; carbon ( C O 2 )—trigonal planar; oxygen ( O H)—tetrahedral; local structures: nitrogen—trigonal pyramidal; carbon ( C H)—tetrahedral; carbon ( C H 3 )—tetrahedral; carbon ( C O 2 )—trigonal planar; oxygen ( O H)—bent (109°)
Example \(\PageIndex{6}\): Molecular Simulation
Using this molecular shape simulator allows us to control whether bond angles and/or lone pairs are displayed by checking or unchecking the boxes under “Options” on the right. We can also use the “Name” checkboxes at bottom-left to display or hide the electron pair geometry (called “electron geometry” in the simulator) and/or molecular structure (called “molecular shape” in the simulator).
Build the molecule HCN in the simulator based on the following Lewis structure:
Click on each bond type or lone pair at right to add that group to the central atom. Once you have the complete molecule, rotate it to examine the predicted molecular structure. What molecular structure is this?
Solution
The molecular structure is linear.
Exercise \(\PageIndex{6}\)
Build a more complex molecule in the simulator. Identify the electron-group geometry, molecular structure, and bond angles. Then try to find a chemical formula that would match the structure you have drawn.
- Answer
-
Answers will vary. For example, an atom with four single bonds, a double bond, and a lone pair has an octahedral electron-group geometry and a square pyramidal molecular structure. XeOF 4 is a molecule that adopts this structure.
Molecular Polarity and Dipole Moment
As discussed previously, polar covalent bonds connect two atoms with differing electronegativities, leaving one atom with a partial positive charge (δ+) and the other atom with a partial negative charge (δ–), as the electrons are pulled toward the more electronegative atom. This separation of charge gives rise to a bond dipole moment. This bond moment can be represented as a vector, a quantity having both direction and magnitude (Figure \(\PageIndex{12}\)). Dipole vectors are shown as arrows pointing along the bond from the less electronegative atom toward the more electronegative atom. A small plus sign is drawn on the less electronegative end to indicate the partially positive end of the bond. The length of the arrow is proportional to the magnitude of the electronegativity difference between the two atoms.
A whole molecule may also have a separation of charge, depending on its molecular structure and the polarity of each of its bonds. If such a charge separation exists, the molecule is said to be a polar molecule (or dipole); otherwise the molecule is said to be nonpolar. The dipole moment measures the extent of net charge separation in the molecule as a whole. We determine the dipole moment by adding the bond moments in three-dimensional space, taking into account the molecular structure.
For diatomic molecules, there is only one bond, so its bond dipole moment determines the molecular polarity. Homonuclear diatomic molecules such as Br 2 and N 2 have no difference in electronegativity, so their dipole moment is zero. For heteronuclear molecules such as CO, there is a small dipole moment. For HF , there is a larger dipole moment because there is a larger difference in electronegativity.
When a molecule contains more than one bond, the geometry must be taken into account. If the bonds in a molecule are arranged such that their bond moments cancel (vector sum equals zero), then the molecule is nonpolar. This is the situation in CO 2 (Figure \(\PageIndex{13A}\)). Each of the bonds is polar, but the molecule as a whole is nonpolar. From the Lewis structure, and using VSEPR theory, we determine that the CO 2 molecule is linear with polar C=O bonds on opposite sides of the carbon atom. The bond moments cancel because they are pointed in opposite directions. In the case of the water molecule (Figure \(\PageIndex{13B}\)), the Lewis structure again shows that there are two bonds to a central atom, and the electronegativity difference again shows that each of these bonds has a nonzero bond moment. In this case, however, the molecular structure is bent because of the lone pairs on O, and the two bond moments do not cancel. Therefore, water does have a net dipole moment and is a polar molecule (dipole).
The OCS molecule has a structure similar to CO 2 , but a sulfur atom has replaced one of the oxygen atoms. To determine if this molecule is polar, we draw the molecular structure. VSEPR theory predicts a linear molecule:
The C–O bond is considerably polar. Although C and S have very similar electronegativity values, S is slightly more electronegative than C, and so the C-S bond is just slightly polar. Because oxygen is more electronegative than sulfur, the oxygen end of the molecule is the negative end.
Chloromethane, CH 3 Cl, is another example of a polar molecule. Although the polar C–Cl and C–H bonds are arranged in a tetrahedral geometry, the C–Cl bonds have a larger bond moment than the C–H bond, and the bond moments do not completely cancel each other. All of the dipoles have a upward component in the orientation shown, since carbon is more electronegative than hydrogen and less electronegative than chlorine:
When we examine the highly symmetrical molecules BF 3 (trigonal planar), CH 4 (tetrahedral), PF 5 (trigonal bipyramidal), and SF 6 (octahedral), in which all the polar bonds are identical, the molecules are nonpolar. The bonds in these molecules are arranged such that their dipoles cancel. However, just because a molecule contains identical bonds does not mean that the dipoles will always cancel. Many molecules that have identical bonds and lone pairs on the central atoms have bond dipoles that do not cancel. Examples include H 2 S and NH 3 . A hydrogen atom is at the positive end and a nitrogen or sulfur atom is at the negative end of the polar bonds in these molecules:
To summarize, to be polar, a molecule must:
- Contain at least one polar covalent bond.
- Have a molecular structure such that the sum of the vectors of each bond dipole moment does not cancel.
Properties of Polar Molecules
Polar molecules tend to align when placed in an electric field with the positive end of the molecule oriented toward the negative plate and the negative end toward the positive plate (Figure \(\PageIndex{14}\)). We can use an electrically charged object to attract polar molecules, but nonpolar molecules are not attracted. Also, polar solvents are better at dissolving polar substances, and nonpolar solvents are better at dissolving nonpolar substances.
Example \(\PageIndex{7}\): Polarity Simulations
Open the molecule polarity simulation and select the “Three Atoms” tab at the top. This should display a molecule ABC with three electronegativity adjustors. You can display or hide the bond moments, molecular dipoles, and partial charges at the right. Turning on the Electric Field will show whether the molecule moves when exposed to a field, similar to Figure \(\PageIndex{14}\).
Use the electronegativity controls to determine how the molecular dipole will look for the starting bent molecule if:
- A and C are very electronegative and B is in the middle of the range.
- A is very electronegative, and B and C are not.
Solution
- Molecular dipole moment points immediately between A and C.
- Molecular dipole moment points along the A–B bond, toward A.
Exercise \(\PageIndex{7}\)
Determine the partial charges that will give the largest possible bond dipoles.
- Answer
-
The largest bond moments will occur with the largest partial charges. The two solutions above represent how unevenly the electrons are shared in the bond. The bond moments will be maximized when the electronegativity difference is greatest. The controls for A and C should be set to one extreme, and B should be set to the opposite extreme. Although the magnitude of the bond moment will not change based on whether B is the most electronegative or the least, the direction of the bond moment will.
Summary
VSEPR theory predicts the three-dimensional arrangement of atoms in a molecule. It states that valence electrons will assume an electron-pair geometry that minimizes repulsions between areas of high electron density (bonds and/or lone pairs). Molecular structure, which refers only to the placement of atoms in a molecule and not the electrons, is equivalent to electron-pair geometry only when there are no lone electron pairs around the central atom. A dipole moment measures a separation of charge. For one bond, the bond dipole moment is determined by the difference in electronegativity between the two atoms. For a molecule, the overall dipole moment is determined by both the individual bond moments and how these dipoles are arranged in the molecular structure. Polar molecules (those with an appreciable dipole moment) interact with electric fields, whereas nonpolar molecules do not.
Glossary
- axial position
- location in a trigonal bipyramidal geometry in which there is another atom at a 180° angle and the equatorial positions are at a 90° angle
- bond angle
- angle between any two covalent bonds that share a common atom
- bond distance
- (also, bond length) distance between the nuclei of two bonded atoms
- bond dipole moment
- separation of charge in a bond that depends on the difference in electronegativity and the bond distance represented by partial charges or a vector
- dipole moment
- property of a molecule that describes the separation of charge determined by the sum of the individual bond moments based on the molecular structure
- electron-pair geometry
- arrangement around a central atom of all regions of electron density (bonds, lone pairs, or unpaired electrons)
- equatorial position
- one of the three positions in a trigonal bipyramidal geometry with 120° angles between them; the axial positions are located at a 90° angle
- linear
- shape in which two outside groups are placed on opposite sides of a central atom
- molecular structure
- structure that includes only the placement of the atoms in the molecule
- octahedral
- shape in which six outside groups are placed around a central atom such that a three-dimensional shape is generated with four groups forming a square and the other two forming the apex of two pyramids, one above and one below the square plane
- polar molecule
- (also, dipole) molecule with an overall dipole moment
- tetrahedral
- shape in which four outside groups are placed around a central atom such that a three-dimensional shape is generated with four corners and 109.5° angles between each pair and the central atom
- trigonal bipyramidal
- shape in which five outside groups are placed around a central atom such that three form a flat triangle with 120° angles between each pair and the central atom, and the other two form the apex of two pyramids, one above and one below the triangular plane
- trigonal planar
- shape in which three outside groups are placed in a flat triangle around a central atom with 120° angles between each pair and the central atom
- valence shell electron-pair repulsion theory (VSEPR)
- theory used to predict the bond angles in a molecule based on positioning regions of high electron density as far apart as possible to minimize electrostatic repulsion
- vector
- quantity having magnitude and direction


3.2: Vector Addition and Subtraction- Graphical Methods
Learning Objectives
By the end of this section, you will be able to:
- Understand the rules of vector addition, subtraction, and multiplication.
- Apply graphical methods of vector addition and subtraction to determine the displacement of moving objects.
Vector Addition and Subtraction: Graphical Methods
Vectors in Two Dimensions
A vector is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, are all vectors. In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign. In two dimensions (2-d), however, we specify the direction of a vector relative to some reference frame (i.e., coordinate system), using an arrow having length proportional to the vector’s magnitude and pointing in the direction of the vector.
Figure shows such a graphical representation of a vector , using as an example the total displacement for the person walking in a city considered in Kinematics in Two Dimensions: An Introduction. We shall use the notation that a boldface symbol, such as \(D\), stands for a vector. Its magnitude is represented by the symbol in italics, \(D\), and its direction by \(θ\).
VECTORS IN THIS TEXT
In this text, we will represent a vector with a boldface variable. For example, we will represent the quantity force with the vector \(F\), which has both magnitude and direction. The magnitude of the vector will be represented by a variable in italics, such as \(F\), and the direction of the variable will be given by an angle \(θ\).
Figure \(\PageIndex{2}\): A person walks 9 blocks east and 5 blocks north. The displacement is 10.3 blocks at an angle .1 º north of east.
Vector Addition: Head-to-Tail Method
The head-to-tail method is a graphical way to add vectors, described in Figure below and in the steps following. The tail of the vector is the starting point of the vector, and the head (or tip) of a vector is the final, pointed end of the arrow.
Step 1. Draw an arrow to represent the first vector (9 blocks to the east) using a ruler and protractor .
Step 2. Now draw an arrow to represent the second vector (5 blocks to the north). Place the tail of the second vector at the head of the first vector.
Step 3. If there are more than two vectors, continue this process for each vector to be added. Note that in our example, we have only two vectors, so we have finished placing arrows tip to tail .
Step 4. Draw an arrow from the tail of the first vector to the head of the last vector . This is the resultant , or the sum, of the other vectors.
Step 5. To get the magnitude of the resultant, measure its length with a ruler. (Note that in most calculations, we will use the Pythagorean theorem to determine this length.)
Step 6. To get the direction of the resultant, measure the angle it makes with the reference frame using a protractor. (Note that in most calculations, we will use trigonometric relationships to determine this angle.)
The graphical addition of vectors is limited in accuracy only by the precision with which the drawings can be made and the precision of the measuring tools. It is valid for any number of vectors.
Example \(\PageIndex{1}\):Adding Vectors Graphically Using the Head-to-Tail Method: A Woman Takes a Walk
Use the graphical technique for adding vectors to find the total displacement of a person who walks the following three paths (displacements) on a flat field. First, she walks 25.0 m in a direction north of east. Then, she walks 23.0 m heading north of east. Finally, she turns and walks 32.0 m in a direction 68.0° south of east.
Strategy
Represent each displacement vector graphically with an arrow, labeling the first , the second , and the third , making the lengths proportional to the distance and the directions as specified relative to an east-west line. The head-to-tail method outlined above will give a way to determine the magnitude and direction of the resultant displacement, denoted .
Solution
(1) Draw the three displacement vectors.
(2) Place the vectors head to tail retaining both their initial magnitude and direction.
(3) Draw the resultant vector, .
(4) Use a ruler to measure the magnitude of , and a protractor to measure the direction of . While the direction of the vector can be specified in many ways, the easiest way is to measure the angle between the vector and the nearest horizontal or vertical axis. Since the resultant vector is south of the eastward pointing axis, we flip the protractor upside down and measure the angle between the eastward axis and the vector.
In this case, the total displacement is seen to have a magnitude of 50.0 m and to lie in a direction south of east. By using its magnitude and direction, this vector can be expressed as = 50.0 m and = 7 . 0 º south of east.
Discussion
The head-to-tail graphical method of vector addition works for any number of vectors. It is also important to note that the resultant is independent of the order in which the vectors are added. Therefore, we could add the vectors in any order as illustrated in Figure and we will still get the same solution.
Here, we see that when the same vectors are added in a different order, the result is the same. This characteristic is true in every case and is an important characteristic of vectors. Vector addition is commutative . Vectors can be added in any order.
\(A+B=B+A.\)
(This is true for the addition of ordinary numbers as well—you get the same result whether you add + 3 or + 2 , for example).
Vector Subtraction
Vector subtraction is a straightforward extension of vector addition. To define subtraction (say we want to subtract from , written – B , we must first define what we mean by subtraction. The negative of a vector is defined to be ; that is, graphically the negative of any vector has the same magnitude but the opposite direction , as shown in Figure. In other words, has the same length as , but points in the opposite direction. Essentially, we just flip the vector so it points in the opposite direction.
The subtraction of vector from vector is then simply defined to be the addition of to . Note that vector subtraction is the addition of a negative vector. The order of subtraction does not affect the results.
This is analogous to the subtraction of scalars (where, for example, ( –2 ) ) . Again, the result is independent of the order in which the subtraction is made. When vectors are subtracted graphically, the techniques outlined above are used, as the following example illustrates.
Example \(\PageIndex{1}\):Subtracting Vectors Graphically: A Woman Sailing a Boat
A woman sailing a boat at night is following directions to a dock. The instructions read to first sail 27.5 m in a direction north of east from her current location, and then travel 30.0 m in a direction north of east (or west of north). If the woman makes a mistake and travels in the opposite direction for the second leg of the trip, where will she end up? Compare this location with the location of the dock.
Strategy
We can represent the first leg of the trip with a vector , and the second leg of the trip with a vector . The dock is located at a location + B . If the woman mistakenly travels in the opposite direction for the second leg of the journey, she will travel a distance (30.0 m) in the direction – 112 º = 68 º south of east. We represent this as , as shown below. The vector has the same magnitude as but is in the opposite direction. Thus, she will end up at a location + ( – B ) , or – B .
We will perform vector addition to compare the location of the dock, + B , with the location at which the woman mistakenly arrives, + ( – B ) .
Solution
(1) To determine the location at which the woman arrives by accident, draw vectors and .
(2) Place the vectors head to tail.
(3) Draw the resultant vector .
(4) Use a ruler and protractor to measure the magnitude and direction of .
In this case, = 23 . 0 m and = 7 . 5 º south of east.
(5) To determine the location of the dock, we repeat this method to add vectors and . We obtain the resultant vector ' :
In this case = 52.9 m and = 90.1 º north of east.
We can see that the woman will end up a significant distance from the dock if she travels in the opposite direction for the second leg of the trip.
Discussion
Because subtraction of a vector is the same as addition of a vector with the opposite direction, the graphical method of subtracting vectors works the same as for addition.
Multiplication of Vectors and Scalars
If we decided to walk three times as far on the first leg of the trip considered in the preceding example, then we would walk × 27 . 5 m , or 82.5 m, in a direction . 0 º north of east. This is an example of multiplying a vector by a positive scalar . Notice that the magnitude changes, but the direction stays the same.
If the scalar is negative, then multiplying a vector by it changes the vector’s magnitude and gives the new vector the opposite direction. For example, if you multiply by –2, the magnitude doubles but the direction changes. We can summarize these rules in the following way: When vector is multiplied by a scalar ,
- the magnitude of the vector becomes the absolute value of ,
- if is positive, the direction of the vector does not change,
- if is negative, the direction is reversed.
In our case, = 3 and = 27.5 m . Vectors are multiplied by scalars in many situations. Note that division is the inverse of multiplication. For example, dividing by 2 is the same as multiplying by the value (1/2). The rules for multiplication of vectors by scalars are the same for division; simply treat the divisor as a scalar between 0 and 1.
Resolving a Vector into Components
In the examples above, we have been adding vectors to determine the resultant vector. In many cases, however, we will need to do the opposite. We will need to take a single vector and find what other vectors added together produce it. In most cases, this involves determining the perpendicular components of a single vector, for example the x - and y -components, or the north-south and east-west components.
For example, we may know that the total displacement of a person walking in a city is 10.3 blocks in a direction .0 º north of east and want to find out how many blocks east and north had to be walked. This method is called finding the components (or parts) of the displacement in the east and north directions, and it is the inverse of the process followed to find the total displacement. It is one example of finding the components of a vector. There are many applications in physics where this is a useful thing to do. We will see this soon in Projectile Motion, and much more when we cover forces in Dynamics: Newton’s Laws of Motion. Most of these involve finding components along perpendicular axes (such as north and east), so that right triangles are involved. The analytical techniques presented in Vector Addition and Subtraction: Analytical Methods are ideal for finding vector components.
PHET EXPLORATIONS: MAZE GAME
Learn about position, velocity, and acceleration in the "Arena of Pain". Use the green arrow to move the ball. Add more walls to the arena to make the game more difficult. Try to make a goal as fast as you can.
- The graphical method of adding vectors and involves drawing vectors on a graph and adding them using the head-to-tail method. The resultant vector is defined such that + B = R . The magnitude and direction of are then determined with a ruler and protractor, respectively.
- The graphical method of subtracting vector from involves adding the opposite of vector , which is defined as B . In this case, – B = A + ( – B ) = R . Then, the head-to-tail method of addition is followed in the usual way to obtain the resultant vector .
- Addition of vectors is commutative such that + B = B + A .
- The head-to-tail method of adding vectors involves drawing the first vector on a graph and then placing the tail of each subsequent vector at the head of the previous vector. The resultant vector is then drawn from the tail of the first vector to the head of the final vector.
- If a vector is multiplied by a scalar quantity , the magnitude of the product is given by . If is positive, the direction of the product points in the same direction as ; if is negative, the direction of the product points in the opposite direction as .
Glossary
- component (of a 2-d vector)
- a piece of a vector that points in either the vertical or the horizontal direction; every 2-d vector can be expressed as a sum of two vertical and horizontal vector components
- commutative
- refers to the interchangeability of order in a function; vector addition is commutative because the order in which vectors are added together does not affect the final sum
- direction (of a vector)
- the orientation of a vector in space
- head (of a vector)
- the end point of a vector; the location of the tip of the vector’s arrowhead; also referred to as the “tip”
- head-to-tail method
- a method of adding vectors in which the tail of each vector is placed at the head of the previous vector
- magnitude (of a vector)
- the length or size of a vector; magnitude is a scalar quantity
- resultant
- the sum of two or more vectors
- resultant vector
- the vector sum of two or more vectors
- scalar
- a quantity with magnitude but no direction
- tail
- the start point of a vector; opposite to the head or tip of the arrow