13.2: Oligopoly in Practice
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Collusion and Competition
Firms in an oligopoly can increase their profits through collusion, but collusive arrangements are inherently unstable.
learning objectives
- Assess the considerations involved in the oligopolist’s decision about whether to compete or cooperate
Oligopoly is a market structure in which there are a few firms producing a product. When there are few firms in the market, they may collude to set a price or output level for the market in order to maximize industry profits. As a result, price will be higher than the market-clearing price, and output is likely to be lower. At the extreme, the colluding firms may act as a monopoly, reducing their individual output so that their collective output would equal that of a monopolist, allowing them to earn higher profits.
OPEC : The oil-producing countries of OPEC have at times cooperated to raise world oil prices in order to secure a steady income for themselves.
If oligopolists individually pursued their own self-interest, then they would produce a total quantity greater than the monopoly quantity, and charge a lower price than the monopoly price, thus earning a smaller profit. The promise of bigger profits gives oligopolists an incentive to cooperate. However, collusive oligopoly is inherently unstable, because the most efficient firms will be tempted to break ranks by cutting prices in order to increase market share.
Several factors deter collusion. First, price-fixing is illegal in the United States, and antitrust laws exist to prevent collusion between firms. Second, coordination among firms is difficult, and becomes more so the greater the number of firms involved. Third, there is a threat of defection. A firm may agree to collude and then break the agreement, undercutting the profits of the firms still holding to the agreement. Finally, a firm may be discouraged from collusion if it does not perceive itself to be able to effectively punish firms that may break the agreement.
In contrast to price-fixing, price leadership is a type of informal collusion which is generally legal. Price leadership, which is also sometimes called parallel pricing, occurs when the dominant competitor publishes its price ahead of other firms in the market, and the other firms then match the announced price. The leader will typically set the price to maximize its profits, which may not be the price that maximized other firms’ profits.
Game Theory Applications to Oligopoly
Game theory provides a framework for understanding how firms behave in an oligopoly.
learning objectives
- Explain how game theory applies to oligopolies
In an oligopoly, firms are interdependent; they are affected not only by their own decisions regarding how much to produce, but by the decisions of other firms in the market as well. Game theory offers a useful framework for thinking about how firms may act in the context of this interdependence. More specifically, game theory can be used to model situations in which each actor, when deciding on a course of action, must also consider how others might respond to that action.
For example, game theory can explain why oligopolies have trouble maintaining collusive arrangements to generate monopoly profits. While firms would be better off collectively if they cooperate, each individual firm has a strong incentive to cheat and undercut their competitors in order to increase market share. Because the incentive to defect is strong, firms may not even enter into a collusive agreement if they don’t perceive there to be a way to effectively punish defectors.
The prisoner’s dilemma is a specific type of game in game theory that illustrates why cooperation may be difficult to maintain for oligopolists even when it is mutually beneficial. In the game, two members of a criminal gang are arrested and imprisoned. The prisoners are separated and left to contemplate their options. If both prisoners confess, each will serve a two-year prison term. If one confesses, but the other denies the crime, the one that confessed will walk free, while the one that denied the crime would get a three-year sentence. If both deny the crime, they will both serve only a one year sentence. Betraying the partner by confessing is the dominant strategy; it is the better strategy for each player regardless of how the other plays. This is known as a Nash equilibrium. The result of the game is that both prisoners pursue individual logic and betray, when they would have collectively gotten a better outcome if they had both cooperated.
Prisoner’s Dilemma : In a prisoner’s dilemma game, the dominant strategy for each player is to betray the other, even though cooperation would have led to a better collective outcome.
The Nash equilibrium is an important concept in game theory. It is the set of strategies such that no player can do better by unilaterally changing his or her strategy. If a player knew the strategies of the other players (and those strategies could not change), and could not benefit by changing his or her strategy, then that set of strategies represents a Nash equilibrium. If any player would benefit by changing his or her strategy, then that set of strategies is not a Nash equilibrium.
While game theory is important to understanding firm behavior in oligopolies, it is generally not needed to understand competitive or monopolized markets. In competitive markets, firms have such a small individual effect on the market, that taking other firms into account is simply not necessary. A monopolized market has only one firm, and thus strategic interactions do not occur.
The Prisoner’s Dilemma and Oligopoly
The prisoner’s dilemma shows why two individuals might not cooperate, even if it is collectively in their best interest to do so.
learning objectives
- Analyze the prisoner’s dilemma using the concepts of strategic dominance, Pareto optimality, and Nash equilibria
Sometimes firms fail to cooperate with each other, even when cooperation would bring about a better collective outcome. The prisoner’s dilemma is a canonical example of a game analyzed in game theory that shows why two individuals might not cooperate, even if it appears that it is in their best interest to do so.
In the game, two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. The police offer each prisoner a bargain:
Prisoner’s Dilemma : Betrayal in the dominant strategy for both players, as it provides for a better individual outcome regardless of what the other player does. However, the resulting outcome is not Pareto-optimal. Both players would clearly have been better off if they had cooperated.
- If Prisoner A and Prisoner B both confess to the crime, each of them will serve two years in prison.
- If A confesses but B denies the crime, A will be set free, while B will serve three years in prison (and vice versa).
- If both A and B deny the crime, both of them will only serve one year in prison.
For both players, the choice to betray the partner by confessing has strategic dominance in this situation; it is the better strategy for each player regardless of what the other player does. This set of strategies is thus a Nash equilibrium in the game–no player would be better off by changing his or her strategy. As a result, all purely self-interested prisoners would betray each other, resulting in a two year prison sentence for both. This outcome is not Pareto optimal; it is clearly possible to improve the outcomes for both players through cooperation. If both players had denied the crime, they would each be serving only one year in prison.
Similarly to the prisoner’s dilemma scenario, cooperation is difficult to maintain in an oligopoly because cooperation is not in the best interest of the individual players. However, the collective outcome would be improved if firms cooperated, and were thus able to maintain low production, high prices, and monopoly profits.
One traditional example of game theory and the prisoner’s dilemma in practice involves soft drinks. Coca-Cola and Pepsi compete in an oligopoly, and thus are highly competitive against one another (as they have limited other competitive threats). Considering the similarity of their products in the soft drink industry (i.e. varying types of soda), any price deviation on part of one competitor is seen as an act of non-conformity or betrayal of an established status quo.
In such a scenario, there are a number of plausible reactions and outcomes. If Coca-Cola reduces their prices, Pepsi may follow to ensure they do not lose market share. In this situation, defection results in a lose-lose. Which is to say that, due to the initial price reduction by Coca-Cola (betrayal of status quo), both companies likely see reduced profit margins. On the other hand, Pepsi could uphold the price point despite Coca-Cola’s deviation, sacrificing market share to Coca-Cola but maintaining the established price point. Prisoner dilemma scenarios are difficult strategic choices, as any deviation from established competitive practice may result in less profits and/or market share.
Duopoly Example
The Cournot model, in which firms compete on output, and the Bertrand model, in which firms compete on price, describe duopoly dynamics.
learning objectives
- Discuss the characteristics of a duopoly
A true duopoly is a specific type of oligopoly where only two producers exist in a market. There are two principle duopoly models: Cournot duopoly and Bertrand duopoly.
Cournot Duopoly
Cournot duopoly is an economic model that describes an industry structure in which firms compete on output levels. The model makes the following assumptions:
- There are two firms, which produce a homogeneous product;
- The number of firms is fixed;
- Firms do not cooperate (there is no collusion);
- Firms have market power, and each firm’s output decision affects the good’s price;
- Firms are economically rational and act strategically, seeking to maximize profit given their competitor’s decisions; and
- Firms compete on quantity, and choose quantity simultaneously.
The Cournot model focuses on the production output decision of a single firm. The firm determines its rival’s output level, evaluates the residual market demand, and then changes its own output level to maximize profits. It is assumed that the firm’s output decision will not affect the output decision of its competitor.
For example, suppose that there are two firms in the market for toasters with a given demand function. Firm A will determine the output of Firm B, hold it constant, and then determine the remainder of the market demand for toasters. Firm A will then determine its profit-maximizing output for that residual demand as if it were the entire market, and produce accordingly. Firm B will be conducting similar calculations with respect to Firm A at the same time.
Bertrand Duopoly
The Bertrand model describes interactions among firms that compete on price. Firms set profit-maximizing prices in response to what they expect a competitor to charge. The model rests on the following assumptions:
- There are two firms producing homogeneous products;
- Firms do not cooperate;
- Firms compete by setting prices simultaneously; and
- Consumers buy everything from a firm with a lower price. If all firms charge the same price, consumers randomly select among them.
In the Bertrand model, Firm A’s optimum price depends on where it believes Firm B will set its price. Pricing just below the other firm will obtain full market demand, though this choice is not optimal if the other firm is pricing below marginal cost, as this would result in negative profits. If Firm B is setting the price below marginal cost, Firm A will set the price at marginal cost. If Firm B is setting the price above marginal cost but below monopoly price, then Firm A will set the price just below that of Firm B. If Firm B sets the price above monopoly price, Firm A will set the price at monopoly level.
Bertrand Duopoly : The diagram shows the reaction function of a firm competing on price. When P2 (the price set by Firm 2) is less than marginal cost, Firm 1 prices at marginal cost (P1=MC). When Firm 2 prices above MC but below monopoly prices, Firm 1 prices just below Firm 2. When Firm 2 prices above monopoly price (PM), Firm 1 prices at monopoly level (P1=PM).
Imagine if both firms set equal prices above marginal cost. Each firm would get half the market at a higher than marginal cost price. However, by lowering prices just slightly, a firm could gain the whole market. As a result, both firms are tempted to lower prices as much as they can. However, it would be irrational to price below marginal cost, because the firm would make a loss. Therefore, both firms will lower prices until they reach the marginal cost limit. According to this model, a duopoly will result in an outcome exactly equivalent to what prevails under perfect competition. The result of the firms’ strategies is a Nash equilibrium –a pair or strategies where neither firm can increase profits by unilaterally changing the price.
Colluding to charge the monopoly price and supplying one half of the market each is the best that the firms could do in this scenario. However, not colluding and charging the marginal cost, which is the non-cooperative outcome, is the only Nash equilibrium of this model.
The accuracy of the Cournot or Bertrand model will vary from industry to industry. If capacity and output can be easily changed, Bertrand is generally a better model of duopoly competition. If output and capacity are difficult to adjust, then Cournot is generally a better model.
Cartel Example
A cartel is a formal collusive arrangement among firms with the goal of increasing profits.
learning objectives
- Assess the role of competition and collusion in the formation of cartels
A cartel is an agreement among competing firms to collude in order to attain higher profits. Cartels usually occur in an oligopolistic industry, where the number of sellers is small and the products being traded are homogeneous. Cartel members may agree on such matters are price fixing, total industry output, market share, allocation of customers, allocation of territories, bid rigging, establishment of common sales agencies, and the division of profits.
Game theory suggests that cartels are inherently unstable, because the behavior of cartel members represents a prisoner’s dilemma. Each member of a cartel would be able to make a higher profit, at least in the short-run, by breaking the agreement (producing a greater quantity or selling at a lower price) than it would make by abiding by it. However, if the cartel collapses because of defections, the firms would revert to competing, profits would drop, and all would be worse off.
Whether members of a cartel choose to cheat on the agreement depends on whether the short-term returns to cheating outweigh the long-term losses from the possible breakdown of the cartel. It also partly depends on how difficult it is for firms to monitor whether the agreement is being adhered to by other firms. If monitoring is difficult, a member is likely to get away with cheating for longer; members would then be more likely to cheat, and the cartel will be more unstable.
Perhaps the most globally recognizable and effective cartel is OPEC, the Organization of Petroleum Exporting Countries. In 1973 members of OPEC reduced their production of oil. Because crude oil from the Middle East was known to have few substitutes, OPEC member’s profits skyrocketed. From 1973 to 1979, the price of oil increased by $70 per barrel, an unprecedented number at the time. In the mid 1980s, however, OPEC started to weaken. Discovery of new oil fields in Alaska and Canada introduced new alternatives to Middle Eastern oil, causing OPEC’s prices and profits to fall. Around the same time OPEC members also started cheating to try to increase individual profits.
OPEC : In the 1970s, OPEC members successfully colluded to reduce the global production of oil, leading to higher profits for member countries.
Key Points
- Firms in an oligopoly may collude to set a price or output level for a market in order to maximize industry profits. At an extreme, the colluding firms can act as a monopoly.
- Oligopolists pursuing their individual self-interest would produce a greater quantity than a monopolist, and charge a lower price.
- Collusive arrangements are generally illegal. Moreover, it is difficult for firms to coordinate actions, and there is a threat that firms may defect and undermine the others in the arrangement.
- Price leadership, which occurs when a dominant competitor sets the industry price and others follow suit, is an informal type of collusion which is generally legal.
- In an oligopoly, firms are affected not only by their own production decisions, but by the production decisions of other firms in the market as well. Game theory models situations in which each actor, when deciding on a course of action, must also consider how others might respond to that action.
- The prisoner’s dilemma is a type of game that illustrates why cooperation is difficult to maintain for oligopolists even when it is mutually beneficial. In this game, the dominant strategy of each actor is to defect. However, acting in self-interest leads to a sub-optimal collective outcome.
- The Nash equilibrium is an important concept in game theory. It is the set of strategies such that no player can do better by unilaterally changing his or her strategy.
- Game theory is generally not needed to understand competitive or monopolized markets.
- In the game, two criminals are arrested and imprisoned. Each criminal must decide whether he will cooperate with or betray his partner. The criminals cannot communicate to coordinate their actions.
- Betrayal is the dominant strategy for both players in the game. Betrayal leads to best individual outcome regardless of what the other person does.
- Both players choosing betrayal is the Nash equilibrium of the game. However, this outcome is not Pareto-optimal. Both players would have clearly been better off if they had cooperated.
- Cooperation by firms in oligopolies is difficult to achieve because defection is in the best interest of each individual firm.
- The Cournot model focuses on the production output decision of a single firm. A firm determines its competitor’s output level and the residual market demand. It then determines its profit -maximizing output for that residual demand as if it were the entire market, and produces accordingly.
- In the Bertrand model, firms set profit-maximizing prices in response to what they expect the competitor to charge. The model predicts that both firms will lower prices until they reach the marginal cost limit, arriving at an outcome equivalent to what prevails under perfect competition.
- The accuracy of the Cournot or Bertrand model will vary from industry to industry, depending on how easy it is to adjust output levels in the industry.
- Cartel members cooperate to set industry price and output.
- Game theory indicates that cartels are inherently unstable. Each individual member has an incentive to cheat in order to make higher profits in the short run.
- Cheating may lead to the collapse of a cartel. With the collapse, firms would revert to competing, which would lead to decreased profits.
- OPEC, the Organization of Petroleum Exporting Countries, provides an example of a historically effective cartel.
Key Terms
- Price leadership : Occurs when one company, usually the dominant competitor among several, leads the way in determining prices, the others soon following.
- collusion : A secret agreement for an illegal purpose; conspiracy.
- price fixing : An agreement between sellers to sell a product only at a fixed price, or maintain the market conditions such that the price is maintained at a given level by controlling supply.
- Prisoner’s dilemma : A game that shows why two individuals might not cooperate, even if it appears that it is in their best interests to do so.
- game theory : A branch of applied mathematics that studies strategic situations in which individuals or organisations choose various actions in an attempt to maximize their returns.
- Nash equilibrium : The set of players’ strategies for which no player can benefit by changing his or her strategy, assuming that the other players keep theirs unchanged.
- Pareto optimal : Describing a situation in which the profit of one party cannot be increased without reducing the profit of another.
- Strategic dominance : Occurs when one strategy is better than another strategy for one player, no matter how that player’s opponents may play.
- Cournot duopoly : An economic model describing an industry in which companies compete on the amount of output they will produce, which they decide on independently of each other and at the same time.
- Bertrand duopoly : A model that describes interactions among firms competing on price.
- Cartel : A group of businesses or nations that collude to limit competition within an industry or market.
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5.1: Why It Matters- Microsoft Word Advanced Skills
Why learn to complete advanced skills in Microsoft Word?
We have previously covered the basics of creating a document, but there is more to business communication than just text.
Using the tools covered in this module will add variety to your document and help convey more information.
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1.4: Requirements for Human Life
Learning Objectives
- Discuss the role of oxygen and nutrients in maintaining human survival
- Explain why extreme heat and extreme cold threaten human survival
- Explain how the pressure exerted by gases and fluids influences human survival
Humans have been adapting to life on Earth for at least the past 200,000 years. Earth and its atmosphere have provided us with air to breathe, water to drink, and food to eat, but these are not the only requirements for survival. Although you may rarely think about it, you also cannot live outside of a certain range of temperature and pressure that the surface of our planet and its atmosphere provides. The next sections explore these four requirements of life.
Oxygen
Atmospheric air is only about 20 percent oxygen, but that oxygen is a key component of the chemical reactions that keep the body alive, including the reactions that produce ATP. Brain cells are especially sensitive to lack of oxygen because of their requirement for a high-and-steady production of ATP. Brain damage is likely within five minutes without oxygen, and death is likely within ten minutes.
Nutrients
A nutrient is a substance in foods and beverages that is essential to human survival. The three basic classes of nutrients are water, the energy-yielding and body-building nutrients, and the micronutrients (vitamins and minerals).
The most critical nutrient is water. Depending on the environmental temperature and our state of health, we may be able to survive for only a few days without water. The body’s functional chemicals are dissolved and transported in water, and the chemical reactions of life take place in water. Moreover, water is the largest component of cells, blood, and the fluid between cells, and water makes up about 70 percent of an adult’s body mass. Water also helps regulate our internal temperature and cushions, protects, and lubricates joints and many other body structures.
The energy-yielding nutrients are primarily carbohydrates and lipids, while proteins mainly supply the amino acids that are the building blocks of the body itself. You ingest these in plant and animal foods and beverages, and the digestive system breaks them down into molecules small enough to be absorbed. The breakdown products of carbohydrates and lipids can then be used in the metabolic processes that convert them to ATP. Although you might feel as if you are starving after missing a single meal, you can survive without consuming the energy-yielding nutrients for at least several weeks.
Water and the energy-yielding nutrients are also referred to as macronutrients because the body needs them in large amounts. In contrast, micronutrients are vitamins and minerals. These elements and compounds participate in many essential chemical reactions and processes, such nerve impulses, and some, such as calcium, also contribute to the body’s structure. Your body can store some of the micronutrients in its tissues, and draw on those reserves if you fail to consume them in your diet for a few days or weeks. Some others micronutrients, such as vitamin C and most of the B vitamins, are water-soluble and cannot be stored, so you need to consume them every day or two.
Narrow Range of Temperature
You have probably seen news stories about athletes who died of heat stroke, or hikers who died of exposure to cold. Such deaths occur because the chemical reactions upon which the body depends can only take place within a narrow range of body temperature, from just below to just above 37°C (98.6°F). When body temperature rises well above or drops well below normal, certain proteins (enzymes) that facilitate chemical reactions lose their normal structure and their ability to function and the chemical reactions of metabolism cannot proceed.
That said, the body can respond effectively to short-term exposure to heat (Figure \(\PageIndex{1}\)) or cold. One of the body’s responses to heat is, of course, sweating. As sweat evaporates from skin, it removes some thermal energy from the body, cooling it. Adequate water (from the extracellular fluid in the body) is necessary to produce sweat, so adequate fluid intake is essential to balance that loss during the sweat response. Not surprisingly, the sweat response is much less effective in a humid environment because the air is already saturated with water. Thus, the sweat on the skin’s surface is not able to evaporate, and internal body temperature can get dangerously high.
The body can also respond effectively to short-term exposure to cold. One response to cold is shivering, which is random muscle movement that generates heat. Another response is increased breakdown of stored energy to generate heat. When that energy reserve is depleted, however, and the core temperature begins to drop significantly, red blood cells will lose their ability to give up oxygen, denying the brain of this critical component of ATP production. This lack of oxygen can cause confusion, lethargy, and eventually loss of consciousness and death. The body responds to cold by reducing blood circulation to the extremities, the hands and feet, in order to prevent blood from cooling there and so that the body’s core can stay warm. Even when core body temperature remains stable, however, tissues exposed to severe cold, especially the fingers and toes, can develop frostbite when blood flow to the extremities has been much reduced. This form of tissue damage can be permanent and lead to gangrene, requiring amputation of the affected region.
EVERYDAY CONNECTION
Controlled Hypothermia
As you have learned, the body continuously engages in coordinated physiological processes to maintain a stable temperature. In some cases, however, overriding this system can be useful, or even life-saving. Hypothermia is the clinical term for an abnormally low body temperature (hypo- = “below” or “under”). Controlled hypothermia is clinically induced hypothermia performed in order to reduce the metabolic rate of an organ or of a person’s entire body.
Controlled hypothermia often is used, for example, during open-heart surgery because it decreases the metabolic needs of the brain, heart, and other organs, reducing the risk of damage to them. When controlled hypothermia is used clinically, the patient is given medication to prevent shivering. The body is then cooled to 25–32°C (79–89°F). The heart is stopped and an external heart-lung pump maintains circulation to the patient’s body. The heart is cooled further and is maintained at a temperature below 15°C (60°F) for the duration of the surgery. This very cold temperature helps the heart muscle to tolerate its lack of blood supply during the surgery.
Some emergency department physicians use controlled hypothermia to reduce damage to the heart in patients who have suffered a cardiac arrest. In the emergency department, the physician induces coma and lowers the patient’s body temperature to approximately 91 degrees. This condition, which is maintained for 24 hours, slows the patient’s metabolic rate. Because the patient’s organs require less blood to function, the heart’s workload is reduced.
Narrow Range of Atmospheric Pressure
Pressure is a force exerted by a substance that is in contact with another substance. Atmospheric pressure is pressure exerted by the mixture of gases (primarily nitrogen and oxygen) in the Earth’s atmosphere. Although you may not perceive it, atmospheric pressure is constantly pressing down on your body. This pressure keeps gases within your body, such as the gaseous nitrogen in body fluids, dissolved. If you were suddenly ejected from a space ship above Earth’s atmosphere, you would go from a situation of normal pressure to one of very low pressure. The pressure of the nitrogen gas in your blood would be much higher than the pressure of nitrogen in the space surrounding your body. As a result, the nitrogen gas in your blood would expand, forming bubbles that could block blood vessels and even cause cells to break apart.
Atmospheric pressure does more than just keep blood gases dissolved. Your ability to breathe—that is, to take in oxygen and release carbon dioxide—also depends upon a precise atmospheric pressure. Altitude sickness occurs in part because the atmosphere at high altitudes exerts less pressure, reducing the exchange of these gases, and causing shortness of breath, confusion, headache, lethargy, and nausea. Mountain climbers carry oxygen to reduce the effects of both low oxygen levels and low barometric pressure at higher altitudes (Figure \(\PageIndex{2}\)).
Decompression Sickness
Decompression sickness (DCS) is a condition in which gases dissolved in the blood or in other body tissues are no longer dissolved following a reduction in pressure on the body. This condition affects underwater divers who surface from a deep dive too quickly, and it can affect pilots flying at high altitudes in planes with unpressurized cabins.
In all cases, DCS is brought about by a reduction in barometric pressure. At high altitude, barometric pressure is much less than on Earth’s surface because pressure is produced by the weight of the column of air above the body pressing down on the body. The very great pressures on divers in deep water are likewise from the weight of a column of water pressing down on the body. For divers, DCS occurs at normal barometric pressure (at sea level), but it is brought on by the relatively rapid decrease of pressure as divers rise from the high pressure conditions of deep water to the now low, by comparison, pressure at sea level. Not surprisingly, diving in deep mountain lakes, where barometric pressure at the surface of the lake is less than that at sea level is more likely to result in DCS than diving in water at sea level.
In DCS, gases dissolved in the blood (primarily nitrogen) come rapidly out of solution, forming bubbles in the blood and in other body tissues. This occurs because when pressure of a gas over a liquid is decreased, the amount of gas that can remain dissolved in the liquid also is decreased. It is air pressure that keeps your normal blood gases dissolved in the blood. When pressure is reduced, less gas remains dissolved. You have seen this in effect when you open a carbonated drink. Removing the seal of the bottle reduces the pressure of the gas over the liquid. This in turn causes bubbles as dissolved gases (in this case, carbon dioxide) come out of solution in the liquid.
The most common symptoms of DCS are pain in the joints, with headache and disturbances of vision occurring in 10 percent to 15 percent of cases. Left untreated, very severe DCS can result in death. Immediate treatment is with pure oxygen. The affected person is then moved into a hyperbaric chamber. A hyperbaric chamber is a reinforced, closed chamber that is pressurized to greater than atmospheric pressure. It treats DCS by repressurizing the body so that pressure can then be removed much more gradually. Because the hyperbaric chamber introduces oxygen to the body at high pressure, it increases the concentration of oxygen in the blood. This has the effect of replacing some of the nitrogen in the blood with oxygen, which is easier to tolerate out of solution.
The dynamic pressure of body fluids is also important to human survival. For example, blood pressure, which is the pressure exerted by blood as it flows within blood vessels, must be great enough to enable blood to reach all body tissues, and yet low enough to ensure that the delicate blood vessels can withstand the friction and force of the pulsating flow of pressurized blood.
Chapter Review
Humans cannot survive for more than a few minutes without oxygen, for more than several days without water, and for more than several weeks without carbohydrates, lipids, proteins, vitamins, and minerals. Although the body can respond to high temperatures by sweating and to low temperatures by shivering and increased fuel consumption, long-term exposure to extreme heat and cold is not compatible with survival. The body requires a precise atmospheric pressure to maintain its gases in solution and to facilitate respiration—the intake of oxygen and the release of carbon dioxide. Humans also require blood pressure high enough to ensure that blood reaches all body tissues but low enough to avoid damage to blood vessels.
Review Questions
Q. Humans have the most urgent need for a continuous supply of ________.
A. food
B. nitrogen
C. oxygen
D. water
Answer: C
Q. Which of the following statements about nutrients is true?
A. All classes of nutrients are essential to human survival.
B. Because the body cannot store any micronutrients, they need to be consumed nearly every day.
C. Carbohydrates, lipids, and proteins are micronutrients.
D. Macronutrients are vitamins and minerals.
 Her body responds to the cold by ________.
A. increasing the blood to her hands and feet
B. becoming lethargic to conserve heat
C. breaking down stored energy
D. significantly increasing blood oxygen levels
Answer: C
Critical Thinking Questions
Q. When you open a bottle of sparkling water, the carbon dioxide gas in the bottle form bubbles. If the bottle is left open, the water will eventually “go flat.” Explain these phenomena in terms of atmospheric pressure.
A. In a sealed bottle of sparkling water, carbon dioxide gas is kept dissolved in the water under a very high pressure. When you open the bottle, the pressure of the gas above the liquid changes from artificially high to normal atmospheric pressure. The dissolved carbon dioxide gas expands, and rises in bubbles to the surface. When a bottle of sparkling water is left open, it eventually goes flat because its gases continue to move out of solution until the pressure in the water is approximately equal to atmospheric pressure.
Q. On his midsummer trek through the desert, Josh ran out of water. Why is this particularly dangerous?
A. The primary way that the body responds to high environmental heat is by sweating; however, sweating requires water, which comes from body fluids, including blood plasma. If Josh becomes dehydrated, he will be unable to sweat adequately to cool his body, and he will be at risk for heat stroke as his blood pressure drops too much from the loss of water from the blood plasma.
Glossary
- nutrient
- chemical obtained from foods and beverages that is critical to human survival
- pressure
- force exerted by a substance in contact with another substance


9.7: The Laplace Equation
The diffusion equation in two spatial dimensions is \[u_t=D(u_{xx}+u_{yy}).\nonumber\]
The steady-state solution, approached asymptotically in time, has \(u_t = 0\) so that the steady-state solution \(u = u(x, y)\) satisfies the two-dimensional Laplace equation \[\label{eq:1}u_{xx}+u_{yy}=0.\]
We will consider the mathematical problem of solving the two dimensional Laplace equation inside a rectangular or a circular boundary. The value of \(u(x, y)\) will be specified on the boundaries, defining this problem to be of Dirichlet type.
Dirichlet Problem for a Rectangle
We consider the Laplace equation \(\eqref{eq:1}\) for the interior of a rectangle \(0 < x < a\), \(0 < y < b\), (see Fig. \(\PageIndex{1}\)), with boundary conditions \[\begin{array}{lllll}u(x,0)=0, && u(x,b)=0, && 0<x<a; \\ u(0,y)=0, && u(a,y)=f(y),&& 0\leq y\leq b.\end{array}\nonumber\]
More general boundary conditions can be solved by linear superposition of solutions.
We take our usual ansatz \[u(x,y)=X(x)Y(y),\nonumber\] and find after substitution into \(\eqref{eq:1}\), \[\frac{X''}{X}=-\frac{Y''}{Y}=\lambda,\nonumber\] with \(\lambda\) the separation constant. We thus obtain the two ordinary differential equations \[X''-\lambda X=0,\quad Y''+\lambda Y=0.\nonumber\]
The homogeneous boundary conditions are \(X(0) = 0,\: Y(0) = 0\) and \(Y(b) = 0\). We have already solved the equation for \(Y(y)\) in §9.5 , and the solution yields the eigenvalues \[\lambda_n=\left(\frac{n\pi}{b}\right)^2,\quad n=1,2,3,\ldots ,\nonumber\] with corresponding eigenfunctions \[Y_n(y)=\sin\frac{n\pi y}{b}.\nonumber\]
The remaining \(X\) equation and homogeneous boundary condition is therefore \[X''-\frac{n^2\pi^2}{b^2}X=0,\quad X(0)=0,\nonumber\] and the solution is the hyperbolic sine function \[X_n(x)=\sinh\frac{n\pi x}{b},\nonumber\] times a constant. Writing \(u_n = X_nY_n\), multiplying by a constant and summing over \(n\), yields the general solution \[u(x,y)=\sum\limits_{n=0}^\infty c_n\sinh\frac{n\pi x}{b}\sinh\frac{n\pi y}{b}.\nonumber\]
The remaining inhomogeneous boundary condition \(u(a, y) = f(y)\) results in \[f(y)=\sum\limits_{n=0}^\infty c_n\sinh\frac{n\pi a}{b}\sin\frac{n\pi y}{b},\nonumber\] which we recognize as a Fourier sine series for an odd function with period \(2b\), and coefficient \(c_n \sinh (n\pi a/b)\). The solution for the coefficient is given by \[c_n=\frac{2}{b\sinh\frac{n\pi a}{b}}\int_0^b f(y)\sin\frac{n\pi y}{b}dy.\nonumber\]
Dirichlet Problem for a Circle
The Laplace equation is commonly written symbolically as \[\label{eq:2}\nabla ^2u=0,\] where \(\nabla^2\) is called the Laplacian, sometimes denoted as \(\Delta\). The Laplacian can be written in various coordinate systems, and the choice of coordinate systems usually depends on the geometry of the boundaries. Indeed, the Laplace equation is known to be separable in \(13\) different coordinate systems! We have solved the Laplace equation in two dimensions, with boundary conditions specified on a rectangle, using \[\nabla ^2=\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}.\nonumber\]
Here we consider boundary conditions specified on a circle, and write the Laplacian in polar coordinates by changing variables from cartesian coordinates. Polar coordinates is defined by the transformation \((r,\theta )\to (x, y)\):
\[\label{eq:3}x=r\cos\theta,\quad y=r\sin\theta ;\] and the chain rule gives for the partial derivatives \[\label{eq:4}\frac{\partial u}{\partial r}=\frac{\partial u}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial r},\quad \frac{\partial u}{\partial\theta}=\frac{\partial u}{\partial x}\frac{\partial x}{\partial\theta}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial\theta}.\]
After taking the partial derivatives of \(x\) and \(y\) using \(\eqref{eq:3}\), we can write the transformation \(\eqref{eq:4}\) in matrix form as \[\label{eq:5}\left(\begin{array}{c}\partial u/\partial r \\ \partial u/\partial\theta\end{array}\right)=\left(\begin{array}{rr}\cos\theta &\sin\theta \\ -r\sin\theta & r\cos\theta\end{array}\right)\left(\begin{array}{c}\partial u/\partial x \\ \partial u/\partial y\end{array}\right).\]
Inversion of \(\eqref{eq:5}\) can be determined from the following result, commonly proved in a linear algebra class. If \[A=\left(\begin{array}{cc}a&b \\ c&d\end{array}\right),\quad\det A\neq 0,\nonumber\] then \[A^{-1}=\frac{1}{\det A}\left(\begin{array}{cc}d&-b \\ -c&a\end{array}\right).\nonumber\]
Therefore, since the determinant of the \(2\times 2\) matrix in \(\eqref{eq:5}\) is \(r\), we have \[\label{eq:6}\left(\begin{array}{c}\partial u/\partial x \\ \partial u/\partial y\end{array}\right)=\left(\begin{array}{rr}\cos\theta &-\sin\theta /r \\ \sin\theta &\cos\theta /r\end{array}\right)\left(\begin{array}{c}\partial u/\partial r \\ \partial u/\partial\theta\end{array}\right).\]
Rewriting \(\eqref{eq:6}\) in operator form, we have \[\label{eq:7}\frac{\partial}{\partial x}=\cos\theta\frac{\partial}{\partial r}-\frac{\sin\theta}{r}\frac{\partial}{\partial\theta},\quad\frac{\partial}{\partial y}=\sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial\theta}.\]
To find the Laplacian in polar coordinates with minimum algebra, we combine \(\eqref{eq:7}\) using complex variables as \[\label{eq:8}\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}=e^{i\theta}\left(\frac{\partial}{\partial r}+\frac{i}{r}\frac{\partial}{\partial\theta}\right),\] so that the Laplacian may be found by multiplying both sides of \(\eqref{eq:8}\) by its complex conjugate, taking care with the computation of the derivatives on the righthand-side:
\[\begin{aligned}\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}&=e^{i\theta}\left(\frac{\partial}{\partial r}+\frac{i}{r}\frac{\partial}{\partial\theta}\right)e^{-i\theta}\left(\frac{\partial}{\partial r}-\frac{1}{r}\frac{\partial}{\partial\theta}\right) \\ &=\frac{\partial ^2}{\partial r^2}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{\partial ^2}{\partial\theta ^2}.\end{aligned}\]
We have therefore determined that the Laplacian in polar coordinates is given by \[\label{eq:9}\nabla^2=\frac{\partial ^2}{\partial r^2}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{\partial ^2}{\partial\theta^2},\] which is sometimes written as \[\nabla^2 =\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial}{\partial r}\right)+\frac{1}{r^2}\frac{\partial ^2}{\partial\theta ^2}.\nonumber\]
We now consider the solution of the Laplace equation in a circle with radius \(r < a\) subject to the boundary condition \[\label{eq:10}u(a,\theta )=f(\theta),\quad 0\leq\theta\leq 2\pi.\]
An additional boundary condition due to the use of polar coordinates is that \(u(r, \theta )\) is periodic in \(\theta\) with period \(2\pi\). Furthermore, we will also assume that \(u(r, \theta )\) is finite within the circle.
The method of separation of variables takes as our ansatz \[u(r,\theta )=R(r)\Theta (\theta ),\nonumber\] and substitution into the Laplace equation \(\eqref{eq:2}\) using \(\eqref{eq:9}\) yields \[R''\Theta +\frac{1}{r}R'\Theta +\frac{1}{r^2}R\Theta ''=0,\nonumber\] or \[r^2\frac{R''}{R}+r\frac{R'}{R}=-\frac{\Theta ''}{\Theta }=\lambda,\nonumber\] where \(\lambda\) is the separation constant. We thus obtain the two ordinary differential equations \[r^2R''+rR'-\lambda R=0,\quad \Theta ''+\lambda\Theta =0.\nonumber\]
The \(\Theta\) equation is solved assuming periodic boundary conditions with period \(2\pi\). If \(\lambda < 0\), then no periodic solution exists. If \(\lambda = 0\), then \(\Theta\) can be constant. If \(\lambda = \mu^2 > 0\), then \[\Theta (\theta )=A\cos\mu\theta +B\sin\mu\theta ,\nonumber\] and the requirement that \(\Theta\) is periodic with period \(2\pi\) forces \(\mu\) to be an integer. Therefore, \[\lambda_n=n^2,\quad n=0,1,2,\ldots ,\nonumber\] with corresponding eigenfunctions \[\Theta_n(\theta )=A_n\cos n\theta +B_n\sin n\theta.\nonumber\]
The \(R\) equation for each eigenvalue \(\lambda_n\) then becomes \[\label{eq:11}r^2R''+rR'-n^2R=0,\] which is an Euler equation. With the ansatz \(R = r^s\), \(\eqref{eq:11}\) reduces to the algebraic equation \(s(s − 1) + s − n^2 = 0\), or \(s^2 = n^2\). Therefore, \(s = ±n\), and there are two real solutions when \(n > 0\) and degenerate solutions when \(n = 0\). When \(n > 0\), the solution for \(R(r)\) is \[R_n(r)=Ar^n+Br^{-n}.\nonumber\]
The requirement that \(u(r,\theta )\) is finite in the circle forces \(B = 0\) since \(r^{−n}\) becomes unbounded as \(r\to 0\). When \(n = 0\), the solution for \(R(r)\) is \[R_n(r)=A+B\ln r,\nonumber\] and again finite \(u\) in the circle forces \(B = 0\). Therefore, the solution for \(n = 0,\: 1,\: 2,\ldots\) is \(R_n = r^n\). Thus the general solution for \(u(r,\theta )\) may be written as \[\label{eq:12} u(r,\theta )=\frac{A_0}{2}+\sum\limits_{n=1}^\infty r^n (A_n\cos n\theta +B_n\sin n\theta ),\] where we have separated out the \(n = 0\) solution to write our solution in a form similar to the standard Fourier series given by (9.3.1) . The remaining boundary condition \(\eqref{eq:10}\) specifies the values of u on the circle of radius a, and imposition of this boundary condition results in \[\label{eq:13} f(\theta )=\frac{A_0}{2}+\sum\limits_{n=1}^\infty a^n (A_n\cos n\theta +B_n\sin n\theta ).\]
Equation \(\eqref{eq:13}\) is a Fourier series for the periodic function \(f(\theta )\) with period \(2\pi\), i.e., \(L = \pi\) in (9.3.1) . The Fourier coefficients \(a^nA_n\) and \(a^nB_n\) are therefore given by (9.3.5) and (9.3.6) to be \[\begin{align} a^nA_n&=\frac{1}{\pi}\int_0^{2\pi} f(\phi )\cos n\phi d\phi, \quad n=0,1,2,\ldots ; \nonumber \\ a^nB_n&=\frac{1}{\pi}\int_0^{2\pi}f(\phi )\sin n\phi d\phi ,\quad n=1,2,3,\ldots ,\label{eq:14}\end{align}\] where we have used \(\phi\) for the dummy variable of integration.
A remarkable fact is that the infinite series solution for \(u(r, \theta )\) can be summed explicitly. Substituting \(\eqref{eq:14}\) into \(\eqref{eq:12}\), we obtain \[\begin{aligned}u(r,\theta )&=\frac{1}{2\pi}\int_0^{2\pi}d\phi f(\phi )\left[ 1+2\sum\limits_{n=1}^\infty \left(\frac{r}{a}\right)^n (\cos n\theta\cos n\phi +\sin n\theta\sin n\phi )\right] \\ &=\frac{1}{2\pi}\int_0^{2\pi} d\phi f(\phi)\left[1+2\sum_{n=1}^\infty \left(\frac{r}{a}\right)^n\cos n(\theta -\phi)\right].\end{aligned}\]
We can sum the infinite series by writing \(2 cos n(\theta − \phi ) = e^{in(\theta−\phi)} + e^{−in(\theta −\phi)}\), and using the sum of the geometric series \(\sum_{n=1}^\infty z^n = z/(1 − z)\) to obtain \[\begin{aligned} 1+2\sum\limits_{n=1}^\infty \left(\frac{r}{a}\right)^n\cos n(\theta -\phi)&=1+\sum\limits_{n=1}^\infty \left(\frac{re^{i(\theta -\phi)}}{a}\right)^n +\sum\limits_{n=1}^\infty \left(\frac{re^{-i(\theta -\phi )}}{a}\right)^n \\ &=1+\left(\frac{re^{i(\theta -\phi)}}{a-re^{i(\theta -\phi )}}+c.c.\right) \\ &=\frac{a^2-r^2}{a^2-2ar\cos (\theta -\phi )+r^2}.\end{aligned}\]
Therefore, \[u(r,\theta )=\frac{a^2-r^2}{2\pi}\int_0^{2\pi}\frac{f(\phi )}{a^2-2ar\cos (\theta -\phi )+r^2}d\phi,\nonumber\] an integral result for \(u(r, \theta )\) known as Poisson’s formula. As a trivial example, consider the solution for \(u(r, \theta )\) if \(f(\theta ) = F\), a constant. Clearly, \(u(r, \theta ) = F\) satisfies both the Laplace equation and the boundary conditions so must be the solution. You can verify that \(u(r, \theta ) = F\) is indeed the solution by showing that \[\int_0^{2\pi}\frac{d\phi}{a^2-2ar\cos (\theta -\phi )+r^2}=\frac{2\pi }{a^2-r^2}.\nonumber\]


3.7E: Inverse Functions (Exercises)
For the following exercises, find \(f^{-1}(x)\) for each function.
69. \(f(x)=9+10 x\)
70. \(f(x)=\frac{x}{x+2}\)
For the following exercise, find a domain on which the function \(f\) is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of \(f\) restricted to that domain.
- \(f(x)=x^{2}+1\)
-
Given \(f(x)=x^{3}-5\) and \(g(x)=\sqrt[3]{x+5}\) :
- Find \(f(g(x))\) and \(g(f(x))\).
- What does the answer tell us about the relationship between \(f(x)\) and \(g(x)\) ?
For the following exercises, use a graphing utility to determine whether each function is one-to-one.
73. \(f(x)=\frac{1}{x}\)
74. \(f(x)=-3 x^{2}+x\)
Practice with function notation:
75. If \(f(5)=2,\) find \(f^{-1}(2)\).
76. If \(f(1)=4,\) find \(f^{-1}(4)\).


11.1: Introduction to Light
Light from this page or screen is formed into an image by the lens of your eye, much as the lens of the camera that made this photograph. Mirrors, like lenses, can also form images that in turn are captured by your eye.
Our lives are filled with light. Through vision, the most valued of our senses, light can evoke spiritual emotions, such as when we view a magnificent sunset or glimpse a rainbow breaking through the clouds. Light can also simply amuse us in a theater, or warn us to stop at an intersection. It has innumerable uses beyond vision. Light can carry telephone signals through glass fibers or cook a meal in a solar oven. Life itself could not exist without light’s energy. From photosynthesis in plants to the sun warming a cold-blooded animal, its supply of energy is vital.
We will start our discussion of visible light as a type of electromagnetic wave. This knowledge will help us answer questions regarding the nature of light and vision. What is color, and how do our eyes detect it? Why do diamonds sparkle? How does light travel? How do lenses and mirrors form images? These are but a few of the questions that are answered by the study of optics. Optics is the branch of physics that deals with the behavior of visible light and other electromagnetic waves.
It is convenient to divide optics into two major parts based on the features we are interested in. The wave characteristics of light, such as frequency and wavelength, relates to the colors we perceive and to how we characterize different types of electromagnetic wave along the electromagnetic spectrum. The wave nature of light is also responsible for phenomena such as diffraction and interference. We call this part of optics "wave optics" or "physical optics." But when light interacts with an object that is several times as large as the light’s wavelength, its observable behavior is like that of a ray; it does not prominently display its wave characteristics, and we can look at primarily how it refracts and reflects. We call this part of optics “geometric optics” or "ray optics."


9.1.1: The Home Health Aide/Personal Care Aide’s Role in Family Spending and Budgeting
Home Health Aides/Personal Care Aides may be asked, as directed in the Care Plan, to assist their patient and their family with developing and sticking to a budget . They may also be asked to assist the patient with paying bills and running errands, such as grocery shopping or banking for their patient. It is important to remember that assisting someone with their money is a task in which accurate record keeping is paramount (very important). Home Health Aides/Personal Care Aides must always carefully and immediately document how much of the patient’s money they spent and what they spent it on.
Home Health Aides/Personal Care Aides should only provide assistance with managing and handling a patient’s money and budget as directed by the Care Plan. Their employer will tell them the policies and procedures for the agency they work regarding whether or not they will help with handling a patient’s money and what the rules are for doing so. If Home Health Aides/Personal Care Aides are not allowed to handle money, they must NEVER agree to do so, as they could get into a lot of trouble and potentially lose their job. If a patient or family member ever asks a HHA/PCA to handle their money and they are not allowed to do so or do not feel comfortable doing so, they should discuss this with a supervisor and seek their guidance.
Even if part of their job is to handle a patient’s money, Home Health Aides/Personal Care Aides should NEVER use the patient’s money for their own needs, even if they are planning to pay them back (Leahy, Fuzy & Grafe, 2013). This is considered to be stealing . They could lose their job, get their agency into a lot of trouble, and face legal and criminal action, such as getting arrested. Document all use of the patient’s money as directed by the Care Plan and the agency’s policies and procedures. Use checks instead of cash whenever possible. This is the easiest way to accurately keep track of how money was spent. It is much more difficult to keep track of cash. The patient must sign the checks, which demonstrates that they intended the money to be used for the purpose they assigned.
1. It is okay to buy yourself something at the store with your patient’s money. True or False? ___________
2. It is okay to borrow money from your patient, as long as you promise to pay it back. True or False? ___________
3. Using a patient’s money in any way other than that directed in the Care Plan or for which they have specifically instructed you to do so is considered stealing and could result in losing your job or going to jail. True or False? ___________
- Answer
-
1. False
2. False
3. True
FEEDBACK :
1. It is never okay to use a patient’s money for any purpose other than that directed within the Care Plan and by the patient.
2. It is never okay to borrow money from a patient or their family, even if you intend to pay it back.
3. Using a patient’s money in any way than that directed in the Care Plan could be considered stealing and is punishable by loss of job, fines, and/or jail time. The agency for which the HHA/PCA works can also face serious penalty.
Income
Many people, including patients and their families may live on a fixed income . A fixed income means that a person lives on a set income and must plan their expenses in order to fit into the amount of money that they have coming in. People may have income from a variety of sources. Income is the amount of money that enters a household. Some people get a paycheck once a week or biweekly . Biweekly paychecks are usually paychecks that come in twice per month, or every other week. Others receive a pension from a retirement plan. Some people may receive financial assistance from state or government sources.
If Home Health Aides/Personal Care Aides will be helping their patient plan a budget , they must first know the amount of money that is coming in to the household in order to help make a plan. The patient may be sensitive about the amount of money they make. Never be judgmental about the amount of money or the source of a patient’s income . Allow a patient to manage their budget independently if they prefer, but offer assistance when needed. Some patients may require complete assistance with planning for their budget. The Care Plan will direct the HHA/PCA as to how much assistance they will provide their patient and their family with managing expenses.
1. Income is the amount of money coming in to a household, regardless of source. True or False? _________
2. A fixed income means a person can spend as much as they want regardless of how much money they make. True or False? _________
- Answer
-
1. True
2. False
FEEDBACK :
1. The amount of money entering a household is known as income. Income can be from a variety of sources such as pension plans, employment, and financial assistance from a government.
2. When people are on fixed incomes, meaning they have a set amount of money coming in to the household, they must be watchful of how much they spend. They should not spend more than the amount of money earned from income.
Debt and Stress
Money can be a real source of stress for many people. If we are unable to buy things we need or want, it can cause great frustration, anger, or depression. When people spend more than the amount of money they have, they can incur debt. Debt is any money that is owed. Some people incur debt by spending more money than their income (money they earn). They may borrow money from family, friends, or lending agencies, such as banks. They may also use credit cards to purchase items they cannot afford. If a person does not pay their credit card bills on time or in the full amount due, over a relatively short period of time, a higher amount of money than what the person borrowed becomes due. This is because credit card and lending agencies charge what is called interest. Interest is the amount of money a person pays on top of what they borrowed. When we borrow money from lending agencies or credit card companies, we are charged interest. This is the cost of allowing us to borrow money that we do not currently have.
If a patient has a high amount of debt and is having difficulty paying their bills or loans, and would like assistance with this aspect of their life, Home Health Aides/Personal Care Aides could speak with their supervisor about what options their patient has. Many communities may have credit counseling services available to help patients and families manage their debts and plan for living expenses appropriately. If Home Health Aides/Personal Care Aides think their patient could benefit from this type of service, they should discuss their concerns with their supervisor. The home care team may be able to provide budgeting assistance and referrals to community agencies to help families with this type of need. The best way to reduce stress about money is to budget and plan for expenses.
To help your patient and family determine their debt to income ratio, a Debt to Income Ratio Calculator Worksheet can be downloaded and copied from American Consumer Credit Counseling at: http :// www . consumercredit . com / media /11405/ debt – to – income – ratio . pdf
1. Which of the following best describes debt?
a). The amount of money coming in to the household.
b). The expenses that a family has.
c). The amount of money in a person’s savings account.
d). The amount of money a person owes after borrowing.
- Answer
-
1. D
FEEDBACK :
Debt is the amount of money that a person owes after borrowing. Income is the amount of money coming in to a household.
Daily Expense Tracker
One way to determine where money is going in the household and to properly set up a budget is to use a daily expense tracker which helps to keep track of daily expenses for a period of time. This can give a patient and their HHA/PCA useful information where money needs to be allocated (distributed or assigned to a particular category) and where spending cuts can be made. When doing this, it is important to keep track of every item that is purchased, even if it seems small or insignificant. Oftentimes, people may not even realize the amount of money they spend on unnecessary items. These can be helpful places to cut back on expenses without too much hardship.
You can make a daily expense tracker in a software program such as Excel or using a word document. You can even keep track on a piece of paper with headings indicating the date, type of expense, and amount spent. Alternatively, a Daily Expense Tracker Worksheet can be downloaded from American Consumer Credit Counseling at: http :// www . consumercredit . com / media /11395/ daily – expense – tracker . pdf
A Daily Expense Tracker Worksheet Home Health Aides/Personal Care Aides might look like the following example.
Daily Expense Tracker Worksheet
| Date | Type of Expense (What and where was the money spent on) | Amount Spent |
Let’s look at a sample daily expense tracker that has been filled out:
Daily Expense Tracker Worksheet
| Date | Type of Expense (What and where was the money spent on) | Amount Spent |
| 01/15/15 | Gas for car-gas station | $20.00 |
| 01/16/15 | Groceries-supermarket | $27.21 |
| 01/16/15 | Lunch-Fast food restaurant | $5.99 |
| 01/16/15 | Gum & soda-gas station | $4.51 |
| 01/17/15 | Cleaning products-grocery store | $15.61 |
As you can see from the previous example, expenses were tracked for 3 days. Money was spent on gas, groceries, lunch, gum & soda, and cleaning products. Using a daily expense tracker such as this may be a helpful way for a person to recognize that they are spending some money on unnecessary items such as eating out or purchasing gum or soda. If they find their expenses exceed their income, eliminating unnecessary items is one way to help income meet their expenses.
1. Which of the items from the sample daily expense tracker worksheet are probably unnecessary? Select all that apply .
a). Gas for car
b). Groceries
c). Fast food
d). Gum and soda
- Answer
-
1. C and D
FEEDBACK :
While some of these items are necessary, such as gas for the car, or groceries, other items such as gum and a lunch from a fast food restaurant are not necessary. These types of items can add up in the long run and are often items we forget about spending money on when we try to figure out where our money has gone!


KEY TAKEAWAYs
- The General Agreement on Tariffs and Trade (GATT) is a series of rules governing trade that were first created in 1947 by twenty-three countries. It remained in force until 1995, when it was replaced by the WTO.
- The World Trade Organization (WTO) is the only global, international organization dealing with the rules of trade between nations. The WTO agreements that have been negotiated and signed by the organization’s 153 member nations and ratified in their parliaments are the heart of the organization. Its goal is to help the producers, exporters, and importers of goods and services conduct business. The latest round of the WTO trade negotiations is called the Doha Round.
EXERCISES
(AACSB: Reflective Thinking, Analytical Skills)
- Define GATT and discuss the importance of the successive rounds. Do you think that GATT was essential to promoting world trade or would we be in the same place today without it? Why or why not?
- Define WTO. In what ways do you think the WTO is still essential to global trade? Discuss how a private-sector firm would use the WTO to protect its business interests.
- Read the following excerpt from a 2010 Wall Street Journal article about the WTO:
The World Trade Organization formally condemned European subsidies to civil-aircraft maker Airbus, concluding the first half of the most expensive trade dispute in WTO history.
Its main finding was that more than $20 billion in low-interest government loans used to develop six models of passenger jet constituted prohibited export subsidies.
The ruling could force the parent company of Airbus, European Aeronautic Defence & Space Co., to repay some aid money or risk giving the U.S. the right to raise import tariffs in retaliation on goods imported from Europe, such as cars, wines and cheese.
Source:
John W. Miller and Daniel Michaels, “WTO Condemns Airbus Subsidies,” Wall Street Journal, July 1, 2010, accessed December 29, 2010,
http://online.wsj.com/article/SB10001424052748703426004575338773153793294.html
.
Do you agree with the WTO’s assessment? Is it fair for the United States to retaliate against the airplane manufacturer with tariffs on other imported products? How might US consumers react to additional taxes imposed on popular imported products such as cars, wine, and cheese?


Definition of trusses, and their classification into plane and space trusses. Overview of the methods for analyzing trusses: the method of joints and method of sections.
Using the method of joints, a process centering on the joints or connection points between the members, to solve for the unknown forces acting on members of a truss. Includes several worked examples.
Using the method of sections, a process that involves breaking the truss down into sections to each be analyzed as a separate rigid body, to solve for unknown forces acting on members of a truss. Includes several worked examples.
Definition of a frame or machine as a structure with at least one non-two-force member. Distinguishing between frames and machines by structure rigidity.


2.1: Early Ideas in Atomic Theory
Skills to Develop
By the end of this section, you will be able to:
- State the postulates of Dalton’s atomic theory
- Use postulates of Dalton’s atomic theory to explain the laws of definite and multiple proportions
The language used in chemistry is seen and heard in many disciplines, ranging from medicine to engineering to forensics to art. The language of chemistry includes its own vocabulary as well as its own form of shorthand. Chemical symbols are used to represent atoms and elements. Chemical formulas depict molecules as well as the composition of compounds. Chemical equations provide information about the quality and quantity of the changes associated with chemical reactions.
This chapter will lay the foundation for our study of the language of chemistry. The concepts of this foundation include the atomic theory, the composition and mass of an atom, the variability of the composition of isotopes, ion formation, chemical bonds in ionic and covalent compounds, the types of chemical reactions, and the naming of compounds. We will also introduce one of the most powerful tools for organizing chemical knowledge: the periodic table.
Atomic Theory through the Nineteenth Century
The earliest recorded discussion of the basic structure of matter comes from ancient Greek philosophers, the scientists of their day. In the fifth century BC, Leucippus and Democritus argued that all matter was composed of small, finite particles that they called atomos , a term derived from the Greek word for “indivisible.” They thought of atoms as moving particles that differed in shape and size, and which could join together. Later, Aristotle and others came to the conclusion that matter consisted of various combinations of the four “elements”—fire, earth, air, and water—and could be infinitely divided. Interestingly, these philosophers thought about atoms and “elements” as philosophical concepts, but apparently never considered performing experiments to test their ideas.
The Aristotelian view of the composition of matter held sway for over two thousand years, until English schoolteacher John Dalton helped to revolutionize chemistry with his hypothesis that the behavior of matter could be explained using an atomic theory. First published in 1807, many of Dalton’s hypotheses about the microscopic features of matter are still valid in modern atomic theory. Here are the postulates of Dalton’s atomic theory .
- Matter is composed of exceedingly small particles called atoms. An atom is the smallest unit of an element that can participate in a chemical change.
- An element consists of only one type of atom, which has a mass that is characteristic of the element and is the same for all atoms of that element (Figure \(\PageIndex{1}\)). A macroscopic sample of an element contains an incredibly large number of atoms, all of which have identical chemical properties.
- Atoms of one element differ in properties from atoms of all other elements.
- A compound consists of atoms of two or more elements combined in a small, whole-number ratio. In a given compound, the numbers of atoms of each of its elements are always present in the same ratio (Figure \(\PageIndex{2}\)).
- Atoms are neither created nor destroyed during a chemical change, but are instead rearranged to yield substances that are different from those present before the change (Figure \(\PageIndex{3}\)).
Dalton’s atomic theory provides a microscopic explanation of the many macroscopic properties of matter that you’ve learned about. For example, if an element such as copper consists of only one kind of atom, then it cannot be broken down into simpler substances, that is, into substances composed of fewer types of atoms. And if atoms are neither created nor destroyed during a chemical change, then the total mass of matter present when matter changes from one type to another will remain constant (the law of conservation of matter).
Example \(\PageIndex{1}\): Testing Dalton’s Atomic Theory
In the following drawing, the green spheres represent atoms of a certain element. The purple spheres represent atoms of another element. If the spheres touch, they are part of a single unit of a compound. Does the following chemical change represented by these symbols violate any of the ideas of Dalton’s atomic theory? If so, which one?
Solution The starting materials consist of two green spheres and two purple spheres. The products consist of only one green sphere and one purple sphere. This violates Dalton’s postulate that atoms are neither created nor destroyed during a chemical change, but are merely redistributed. (In this case, atoms appear to have been destroyed.)
Exercise \(\PageIndex{1}\)
In the following drawing, the green spheres represent atoms of a certain element. The purple spheres represent atoms of another element. If the spheres touch, they are part of a single unit of a compound. Does the following chemical change represented by these symbols violate any of the ideas of Dalton’s atomic theory? If so, which one
Answer:
The starting materials consist of four green spheres and two purple spheres. The products consist of four green spheres and two purple spheres. This does not violate any of Dalton’s postulates: Atoms are neither created nor destroyed, but are redistributed in small, whole-number ratios.
Dalton knew of the experiments of French chemist Joseph Proust, who demonstrated that all samples of a pure compound contain the same elements in the same proportion by mass . This statement is known as the law of definite proportions or the law of constant composition . The suggestion that the numbers of atoms of the elements in a given compound always exist in the same ratio is consistent with these observations. For example, when different samples of isooctane (a component of gasoline and one of the standards used in the octane rating system) are analyzed, they are found to have a carbon-to-hydrogen mass ratio of 5.33:1, as shown in Table \(\PageIndex{1}\).
| Sample | Carbon | Hydrogen | Mass Ratio |
|---|---|---|---|
| A | 14.82 g | 2.78 g | \(\mathrm{\dfrac{14.82\: g\: carbon}{2.78\: g\: hydrogen}=\dfrac{5.33\: g\: carbon}{1.00\: g\: hydrogen}}\) |
| B | 22.33 g | 4.19 g | \(\mathrm{\dfrac{22.33\: g\: carbon}{4.19\: g\: hydrogen}=\dfrac{5.33\: g\: carbon}{1.00\: g\: hydrogen}}\) |
| C | 19.40 g | 3.64 g | \(\mathrm{\dfrac{19.40\: g\: carbon}{3.63\: g\: hydrogen}=\dfrac{5.33\: g\: carbon}{1.00\: g\: hydrogen}}\) |
It is worth noting that although all samples of a particular compound have the same mass ratio, the converse is not true in general. That is, samples that have the same mass ratio are not necessarily the same substance. For example, there are many compounds other than isooctane that also have a carbon-to-hydrogen mass ratio of 5.33:1.00.
Dalton also used data from Proust, as well as results from his own experiments, to formulate another interesting law. The law of multiple proportions states that when two elements react to form more than one compound, a fixed mass of one element will react with masses of the other element in a ratio of small, whole numbers . For example, copper and chlorine can form a green, crystalline solid with a mass ratio of 0.558 g chlorine to 1 g copper, as well as a brown crystalline solid with a mass ratio of 1.116 g chlorine to 1 g copper. These ratios by themselves may not seem particularly interesting or informative; however, if we take a ratio of these ratios, we obtain a useful and possibly surprising result: a small, whole-number ratio.
\[\mathrm{\dfrac{\dfrac{1.116\: g\: Cl}{1\: g\: Cu}}{\dfrac{0.558\: g\: Cl}{1\: g\: Cu}}=\dfrac{2}{1}}\]
This 2-to-1 ratio means that the brown compound has twice the amount of chlorine per amount of copper as the green compound.
This can be explained by atomic theory if the copper-to-chlorine ratio in the brown compound is 1 copper atom to 2 chlorine atoms, and the ratio in the green compound is 1 copper atom to 1 chlorine atom. The ratio of chlorine atoms (and thus the ratio of their masses) is therefore 2 to 1 (Figure \(\PageIndex{4}\)).
Example \(\PageIndex{2}\): Laws of Definite and Multiple Proportions
A sample of compound A (a clear, colorless gas) is analyzed and found to contain 4.27 g carbon and 5.69 g oxygen. A sample of compound B (also a clear, colorless gas) is analyzed and found to contain 5.19 g carbon and 13.84 g oxygen. Are these data an example of the law of definite proportions, the law of multiple proportions, or neither? What do these data tell you about substances A and B?
Solution
In compound A, the mass ratio of carbon to oxygen is:
\[\mathrm{\dfrac{1.33\: g\: O}{1\: g\: C}}\]
In compound B, the mass ratio of carbon to oxygen is:
\[\mathrm{\dfrac{2.67\: g\: O}{1\: g\: C}}\]
The ratio of these ratios is:
\[\mathrm{\dfrac{\dfrac{1.33\: g\: O}{1\: g\: C}}{\dfrac{2.67\: g\: O}{1\: g\: C}}=\dfrac{1}{2}}\]
This supports the law of multiple proportions. This means that A and B are different compounds, with A having one-half as much carbon per amount of oxygen (or twice as much oxygen per amount of carbon) as B. A possible pair of compounds that would fit this relationship would be A = CO 2 and B = CO.
Exercise \(\PageIndex{2}\)
A sample of compound X (a clear, colorless, combustible liquid with a noticeable odor) is analyzed and found to contain 14.13 g carbon and 2.96 g hydrogen. A sample of compound Y (a clear, colorless, combustible liquid with a noticeable odor that is slightly different from X’s odor) is analyzed and found to contain 19.91 g carbon and 3.34 g hydrogen. Are these data an example of the law of definite proportions, the law of multiple proportions, or neither? What do these data tell you about substances X and Y?
Answer :
In compound X, the mass ratio of carbon to hydrogen is \(\mathrm{\dfrac{14.13\: g\: C}{2.96\: g\: H}}\). In compound Y, the mass ratio of carbon to oxygen is \(\mathrm{\dfrac{19.91\: g\: C}{3.34\: g\: H}}\). The ratio of these ratios is \(\mathrm{\dfrac{\dfrac{14.13\: g\: C}{2.96\: g\: H}}{\dfrac{19.91\: g\: C}{3.34\: g\: H}}=\dfrac{4.77\: g\: C/g\: H}{5.96\: g\: C/g\: H}=0.800=\dfrac{4}{5}}\). This small, whole-number ratio supports the law of multiple proportions. This means that X and Y are different compounds.
Summary
The ancient Greeks proposed that matter consists of extremely small particles called atoms. Dalton postulated that each element has a characteristic type of atom that differs in properties from atoms of all other elements, and that atoms of different elements can combine in fixed, small, whole-number ratios to form compounds. Samples of a particular compound all have the same elemental proportions by mass. When two elements form different compounds, a given mass of one element will combine with masses of the other element in a small, whole-number ratio. During any chemical change, atoms are neither created nor destroyed.
Glossary
- Dalton’s atomic theory
- set of postulates that established the fundamental properties of atoms
- law of constant composition
- (also, law of definite proportions) all samples of a pure compound contain the same elements in the same proportions by mass
- law of definite proportions
- (also, law of constant composition) all samples of a pure compound contain the same elements in the same proportions by mass
- law of multiple proportions
- when two elements react to form more than one compound, a fixed mass of one element will react with masses of the other element in a ratio of small whole numbers
Contributors
-
Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).


9.1: Valence Bond Theory
- Describe the formation of covalent bonds in terms of atomic orbital overlap
- Define and give examples of σ and π bonds
We have examined the basic ideas of bonding, showing that atoms share electrons to form molecules with stable Lewis structures and that we can predict the shapes of those molecules by valence shell electron pair repulsion (VSEPR) theory. These ideas provide an important starting point for understanding chemical bonding. But these models sometimes fall short in their abilities to predict the behavior of real substances. How can we reconcile the geometries of s, p, and d atomic orbitals with molecular shapes that show angles like 120° and 109.5°?
As we know, a scientific theory is a strongly supported explanation for observed natural laws or large bodies of experimental data. For a theory to be accepted, it must explain experimental data and be able to predict behavior. For example, VSEPR theory has gained widespread acceptance because it predicts three-dimensional molecular shapes that are consistent with experimental data collected for thousands of different molecules. However, VSEPR theory does not provide an explanation of chemical bonding.
There are successful theories that describe the electronic structure of atoms. We can use quantum mechanics to predict the specific regions around an atom where electrons are likely to be located: A spherical shape for an s orbital, a dumbbell shape for a p orbital, and so forth. However, these predictions only describe the orbitals around free atoms. When atoms bond to form molecules, atomic orbitals are not sufficient to describe the regions where electrons will be located in the molecule. A more complete understanding of electron distributions requires a model that can account for the electronic structure of molecules. One popular theory holds that a covalent bond forms when a pair of electrons is shared by two atoms and is simultaneously attracted by the nuclei of both atoms. In the following sections, we will discuss how such bonds are described by valence bond theory and hybridization.
Valence bond theory describes a covalent bond as the overlap of half-filled atomic orbitals (each containing a single electron) that yield a pair of electrons shared between the two bonded atoms. We say that orbitals on two different atoms overlap when a portion of one orbital and a portion of a second orbital occupy the same region of space. According to valence bond theory, a covalent bond results when two conditions are met:
- an orbital on one atom overlaps an orbital on a second atom and
- the single electrons in each orbital combine to form an electron pair.
The mutual attraction between this negatively charged electron pair and the two atoms’ positively charged nuclei serves to physically link the two atoms through a force we define as a covalent bond. The strength of a covalent bond depends on the extent of overlap of the orbitals involved. Orbitals that overlap extensively form bonds that are stronger than those that have less overlap.
The energy of the system depends on how much the orbitals overlap. Figure \(\PageIndex{1}\) illustrates how the sum of the energies of two hydrogen atoms (the colored curve) changes as they approach each other. When the atoms are far apart there is no overlap, and by convention we set the sum of the energies at zero. As the atoms move together, their orbitals begin to overlap. Each electron begins to feel the attraction of the nucleus in the other atom. In addition, the electrons begin to repel each other, as do the nuclei. While the atoms are still widely separated, the attractions are slightly stronger than the repulsions, and the energy of the system decreases. (A bond begins to form.) As the atoms move closer together, the overlap increases, so the attraction of the nuclei for the electrons continues to increase (as do the repulsions among electrons and between the nuclei). At some specific distance between the atoms, which varies depending on the atoms involved, the energy reaches its lowest (most stable) value. This optimum distance between the two bonded nuclei is the bond distance between the two atoms. The bond is stable because at this point, the attractive and repulsive forces combine to create the lowest possible energy configuration. If the distance between the nuclei were to decrease further, the repulsions between nuclei and the repulsions as electrons are confined in closer proximity to each other would become stronger than the attractive forces. The energy of the system would then rise (making the system destabilized), as shown at the far left of Figure \(\PageIndex{1}\).
The bond energy is the difference between the energy minimum (which occurs at the bond distance) and the energy of the two separated atoms. This is the quantity of energy released when the bond is formed. Conversely, the same amount of energy is required to break the bond. For the \(H_2\) molecule shown in Figure \(\PageIndex{1}\), at the bond distance of 74 pm the system is \(7.24 \times 10^{−19}\, J\) lower in energy than the two separated hydrogen atoms. This may seem like a small number. However, we know from our earlier description of thermochemistry that bond energies are often discussed on a per-mole basis. For example, it requires \(7.24 \times 10^{−19}\; J\) to break one H–H bond, but it takes \(4.36 \times 10^5\; J\) to break 1 mole of H–H bonds. A comparison of some bond lengths and energies is shown in Table \(\PageIndex{1}\). We can find many of these bonds in a variety of molecules, and this table provides average values. For example, breaking the first C–H bond in CH4 requires 439.3 kJ/mol, while breaking the first C–H bond in \(\ce{H–CH2C6H5}\) (a common paint thinner) requires 375.5 kJ/mol.
| Bond | Length (pm) | Energy (kJ/mol) | Bond | Length (pm) | Energy (kJ/mol) | |
|---|---|---|---|---|---|---|
| H–H | 74 | 436 | C–O | 140.1 | 358 | |
| H–C | 106.8 | 413 | \(\mathrm{C=O}\) | 119.7 | 745 | |
| H–N | 101.5 | 391 | \(\mathrm{C≡O}\) | 113.7 | 1072 | |
| H–O | 97.5 | 467 | H–Cl | 127.5 | 431 | |
| C–C | 150.6 | 347 | H–Br | 141.4 | 366 | |
| \(\mathrm{C=C}\) | 133.5 | 614 | H–I | 160.9 | 298 | |
| \(\mathrm{C≡C}\) | 120.8 | 839 | O–O | 148 | 146 | |
| C–N | 142.1 | 305 | \(\mathrm{O=O}\) | 120.8 | 498 | |
| \(\mathrm{C=N}\) | 130.0 | 615 | F–F | 141.2 | 159 | |
| \(\mathrm{C≡N}\) | 116.1 | 891 | Cl–Cl | 198.8 | 243 |
In addition to the distance between two orbitals, the orientation of orbitals also affects their overlap (other than for two s orbitals, which are spherically symmetric). Greater overlap is possible when orbitals are oriented such that they overlap on a direct line between the two nuclei. Figure \(\PageIndex{2}\) illustrates this for two p orbitals from different atoms; the overlap is greater when the orbitals overlap end to end rather than at an angle.
The overlap of two s orbitals (as in H 2 ), the overlap of an s orbital and a p orbital (as in HCl), and the end-to-end overlap of two p orbitals (as in Cl 2 ) all produce sigma bonds (σ bonds) , as illustrated in Figure \(\PageIndex{3}\). A σ bond is a covalent bond in which the electron density is concentrated in the region along the internuclear axis; that is, a line between the nuclei would pass through the center of the overlap region. Single bonds in Lewis structures are described as σ bonds in valence bond theory.
A pi bond (π bond) is a type of covalent bond that results from the side-by-side overlap of two p orbitals, as illustrated in Figure \(\PageIndex{4}\). In a π bond, the regions of orbital overlap lie on opposite sides of the internuclear axis. Along the axis itself, there is a node , that is, a plane with no probability of finding an electron.
While all single bonds are σ bonds, multiple bonds consist of both σ and π bonds. As the Lewis structures suggest, O 2 contains a double bond, and N 2 contains a triple bond. The double bond consists of one σ bond and one π bond, and the triple bond consists of one σ bond and two π bonds. Between any two atoms, the first bond formed will always be a σ bond, but there can only be one σ bond in any one location. In any multiple bond, there will be one σ bond, and the remaining one or two bonds will be π bonds. These bonds are described in more detail later in this chapter.
As seen in Table \(\PageIndex{1}\), an average carbon-carbon single bond is 347 kJ/mol, while in a carbon-carbon double bond, the π bond increases the bond strength by 267 kJ/mol. Adding an additional π bond causes a further increase of 225 kJ/mol. We can see a similar pattern when we compare other σ and π bonds. Thus, each individual π bond is generally weaker than a corresponding σ bond between the same two atoms. In a σ bond, there is a greater degree of orbital overlap than in a π bond.
Butadiene, C 4 H 6 , is used to make synthetic rubber. Identify the number of σ and π bonds contained in this molecule.
Solution
There are six σ C–H bonds and one σ C–C bond, for a total of seven from the single bonds. There are two double bonds that each have a π bond in addition to the σ bond. This gives a total nine σ and two π bonds overall.
Identify each illustration as depicting a σ or π bond:
- side-by-side overlap of a 4 p and a 2 p orbital
- end-to-end overlap of a 4 p and 4 p orbital
- end-to-end overlap of a 4 p and a 2 p orbital
- Answer
-
(a) is a π bond with a node along the axis connecting the nuclei while (b) and (c) are σ bonds that overlap along the axis.
Summary
Contributors and Attributions
Valence bond theory describes bonding as a consequence of the overlap of two separate atomic orbitals on different atoms that creates a region with one pair of electrons shared between the two atoms. When the orbitals overlap along an axis containing the nuclei, they form a σ bond. When they overlap in a fashion that creates a node along this axis, they form a π bond.
Glossary
- overlap
- coexistence of orbitals from two different atoms sharing the same region of space, leading to the formation of a covalent bond
- node
- plane separating different lobes of orbitals, where the probability of finding an electron is zero
- pi bond (π bond)
- covalent bond formed by side-by-side overlap of atomic orbitals; the electron density is found on opposite sides of the internuclear axis
- sigma bond (σ bond)
- covalent bond formed by overlap of atomic orbitals along the internuclear axis
- valence bond theory
- description of bonding that involves atomic orbitals overlapping to form σ or π bonds, within which pairs of electrons are shared
Contributors and Attributions
-
Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).