5.1.2: Team Development Over Time
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Learning Objectives
- How do teams develop over time?
If you have been a part of a team—as most of us have—then you intuitively have felt that there are different “stages” of team development. Teams and team members often start from a position of friendliness and excitement about a project or endeavor, but the mood can sour and the team dynamics can go south very quickly once the real work begins. In 1965, educational psychologist Bruce Tuckman at Ohio State University developed a four-stage model to explain the complexities that he had witnessed in team development. The original model was called Tuckman’s Stages of Group Development, and he added the fifth stage of “Adjourning” in 1977 to explain the disbanding of a team at the end of a project. The four stages of the Tuckman model are: 3
- Forming
- Storming
- Norming
- Performing
- Adjourning
The forming stage begins with the introduction of team members. This is known as the “polite stage” in which the team is mainly focused on similarities and the group looks to the leader for structure and direction. The team members at this point are enthusiastic, and issues are still being discussed on a global, ambiguous level. This is when the informal pecking order begins to develop, but the team is still friendly.
The storming stage begins as team members begin vying for leadership and testing the group processes. This is known as the “win-lose” stage, as members clash for control of the group and people begin to choose sides. The attitude about the team and the project begins to shift to negative, and there is frustration around goals, tasks, and progress.
After what can be a very long and painful Storming process for the team, slowly the norming stage may start to take root. During Norming, the team is starting to work well together, and buy-in to group goals occurs. The team is establishing and maintaining ground rules and boundaries, and there is willingness to share responsibility and control. At this point in the team formation, members begin to value and respect each other and their contributions.
Finally, as the team builds momentum and starts to get results, it is entering the performing stage. The team is completely self-directed and requires little management direction. The team has confidence, pride, and enthusiasm, and there is a congruence of vision, team, and self. As the team continues to perform, it may even succeed in becoming a high-performing team. High-performing teams have optimized both task and people relationships—they are maximizing performance and team effectiveness. Katzenberg and Smith, in their study of teams, have created a “team performance curve” that graphs the journey of a team from a working group to a high-performing team. The team performance curve is illustrated in Figure 15.2.3.
The process of becoming a high-performance team is not a linear process. Similarly, the four stages of team development in the Tuckman model are not linear, and there are also factors that may cause the team to regress to an earlier stage of development. When a team member is added to the group, this may change the dynamic enough and be disruptive enough to cause a backwards slide to an earlier stage. Similarly, if a new project task is introduced that causes confusion or anxiety for the group, then this may also cause a backwards slide to an earlier stage of development. Think of your own experiences with project teams and the backslide that the group may have taken when another team member was introduced. You may have personally found the same to be true when a leader or project sponsor changes the scope or adds a new project task. The team has to re-group and will likely re-Storm and re-Form before getting back to Performing as a team.
CATCHING THE ENTREPRENEURIAL SPIRIT
Starting the Startup Team
Nothing is more exciting than a startup business. The enthusiasm is high, and people are excited about the new venture and the prospects that await. Depending on the situation, there may be funding that the startup has received from investors, or the startup could be growing and powering itself organically. Either way, the startup faces many different questions in the beginning, which will have a tremendous impact on its growth potential and performance down the road. One of the most critical questions that faces a startup —or any business for that matter—is the question of who should be on the team. Human capital is the greatest asset that any company can have, and it is an especially critical decision in a startup environment when you have limited resources and those resources will be responsible for building the company from ground up.
In Noam Wasserman’s January 2012 HBSP article “Assembling the Startup Team,” Wasserman asserts:
“ Nothing can bedevil a high-potential startup more than its people problems. In research on startup performance, venture capitalists attributed 65% of portfolio company failures to problems within the startup’s management team. Another study asked investors to identify problems that might occur at their portfolio companies; 61% of the problems involved team issues. These problems typically result from choices that founders make as they add team members…”
These statistics are based on people problems in startups, and it isn’t quite clear what percent of larger company failures could be directly or indirectly attributed to people and team issues. I would imagine that the percentage is also significant. The impact of people problems and team issues in a startup organization that is just getting its footing and trying to make the right connections and decisions can be very significant. If you know anyone who has a company in startup mode, you may have noticed that some of the early team members who are selected to join the team are trusted family members, friends, or former colleagues. Once a startup company grows to a certain level, then it may acquire an experienced CEO to take the helm. In any case, the startup is faced early on with important questions on how to build the team in a way that will maximize the chance of success.
In “Assembling the Startup Team,” the author refers to the three Rs: relationships, roles, and rewards as being key elements that must be managed effectively in order to avoid problems in the long term. Relationships refers to the actual team members that are chosen, and there are several caveats to keep in mind. Hiring relatives or close friends because they are trusted may seem like the right idea in the beginning, but the long-term hazards (per current research) outweigh the benefits. Family and friends may think too similarly, and the team misses the benefit of other perspectives and connections. Roles are important because you have to think about the division of labor and skills, as well as who is in the right roles for decision-making. The startup team needs to think through the implications of assigning people to specific roles, as that may dictate their decision power and status. Finally, defining the rewards can be difficult for the startup team because it essentially means that they are splitting the pie—i.e., both short-term and long-term compensation. For startup founders, this can be a very difficult decision when they have to weigh the balance of giving something away versus gaining human capital that may ultimately help the business to succeed. Thinking through the tradeoffs and keeping alignment between the “three Rs” is important because it challenges the startup team to think of the long-term consequences of some of their early decisions. It is easy to bring family and friends into the startup equation due to trust factors, but a careful analysis of the “three Rs” will help a startup leadership team make decisions that will pay off in the long term.
discussion questions
- Why might it be a bad decision to hire someone for a key startup role based only on the fact that the person is close family or a friend? What are the potential tradeoffs to the business?
- What does it mean for the “three Rs” to be in alignment? What is the potential risk of these not being in alignment? What could go wrong?
concept check
- What are the four stages of team development?
- What can cause a team to regress in its development?


4.3.1: Pyotr Ilyich Tchaikovsky - The Nutcracker
We’ve already encountered one ballet by a Russian composer: Igor Stravinsky’s The Rite of Spring. Now we will examine another. Stravinsky and Pyotr Ilyich Tchaikvosky (1840-1893) are, after all, two of the bestknown ballet composers. The most famous choreographers and dancers have been mostly Russian as well. Ballet, however, originated not in Russia but in France. We will begin, therefore, with the story of how this art form ended up thriving nearly two thousand miles away from its birthplace.
Ballet
Ballet’s origins can be traced to European courts of the 15th and 16th centuries, where dance became increasingly formalized and dramatic. Early ballets, however, were quite different from the art form we might be familiar with: Dancers wore regular shoes and clothing, the steps were taken from participatory court dances, and spectators usually joined in for the finale. Court ballet reached its pinnacle under Louis XIV, who ruled France from 1643 to 1715. He was an avid dancer, and frequently took leading roles in the productions. Louis XIV also sponsored the first professional ballet company, which was attached to the Paris Opera. The two genres—opera and ballet—were closely related in this period: French operas always contained extended dance numbers, while court ballets included singing.
Ballet as we know it today emerged in the second half of the 18th century, when a series of reforms were enacted by ballet master Jean-Georges Noverre. Noverre sought to make ballet more expressive and realistic by replacing the heavy costumes with light, form-fitting apparel, doing away with masks, and introducing the use of pantomime and facial expressions to communicate dramatic elements. The early 19th century saw the invention of the pointe shoe, which allows female dancers to balance on their toes, and the tutu, which accentuates their graceful movements and reveals the legs.
Soon, however, ballet in France was faltering. In seeking to compete with opera, ballet promoters were not successful. Ballet’s association with the aristocracy made it distasteful to French audiences of the late 19th century, and it was generally considered to be less expressive than opera—and therefore inferior. Ballet might have vanished completely were it not for Russian interest in the art form. Beginning in the late 18th century, Russian courts sought to establish their cultural credentials by importing European art forms. First, they brought in Italian opera. In the mid-19th century, they turned to French ballet.
Tchaikovsky’s Career
At first, the Russian ballet establishment was managed by French choreographers, scenarists, dancers, and composers. Gradually, however, Russians took over, and ballet became a distinctively Russian art form. Pyotr Ilyich Tchaikovsky was among the first Russian ballet composers, and his three masterpieces— Swan Lake (1877), Sleeping Beauty (1890), and The Nutcracker (1892)—are still frequently performed today.
Tchaikovsky, however, did not particularly care for his ballet scores, and he disdained their popularity. He would much prefer to have been remembered for his six symphonies, which he considered to be his greatest works. Tchaikovsky also wrote programmatic orchestral pieces, operas, and chamber music—in fact, he composed in all of the prominent European genres of the 19th century. Unlike Russian composers such as Modest Mussorgsky (discussed in Chapter 6), who rejected European influence, Tchaikovsky sought to follow in the European tradition.
Tchaikovsky was firmly entrenched in Russia’s European-style musical establishment. As a young man, he was at first frustrated in his desire to pursue a career in music by the fact that there were no opportunities to study music in Russia. He instead embarked upon a career in the civil service, but in 1862 was able to enroll in the first class at the newly-opened St. Petersburg Conservatory. He impressed his teachers, and upon graduating was offered a teaching position at the Moscow Conservatory, which opened in 1866. Tchaikovsky went on to establish an international reputation as a composer, although he faced criticism at home for not being “Russian enough” in his musical expression.
It is no surprise that Tchaikovsky did not care for ballet work, for in the creation of a ballet, the composer found himself at the bottom of the hierarchy. Most of the creative work was completed by the scenarist (who outlined the dramatic contents of the ballet) and the choreographer (who designed the dance). The Nutcracker was conceived of by the renowned scenarist Marius Petipa, who chose and adapted the story, decided how it would be told through the ballet medium, and established a character for each of the dances. He went so far as to provide Tchaikovsky with the exact tempo and duration for each number, leaving the composer with little opportunity for creative expression. All the same, Tchaikovsky—who, while working on The Nutcracker , wrote to a friend that “I am daily becoming more and more attuned to my task”—was able to produce distinctive music that has charmed listeners for over a century.
The Nutcracker
The short story on which The Nutcracker was based required a great deal of alteration to become appropriate for the ballet stage. Indeed, the plot of The Nutcracker bears little resemblance to E.T.A. Hoffmann’s “The Nutcracker and the Mouse King” (1816), which can be categorized as a horror story. In Hoffmann’s account, a young girl, Marie (Clara in the ballet version), is subjected to terrifying nighttime encounters with the seven-headed Mouse King, who repeatedly threatens her. The tortuous plot hinges on a curse that transforms characters into hideous creatures with giant heads, gaping smiles, and long white beards. At the end of the story, Marie breaks the curse by pledging her love to the toy nutcracker that was given to her at Christmas by her godfather, the mysterious inventor Drosselmeyer.
Petipa (following an earlier adaptation by Alexandre Dumas) stripped this narrative of its horror elements, thereby transforming it into a family-friendly story about Christmas magic. The first act takes place at the home of Clara Stahlbaum, where guests have assembled for a Christmas Eve celebration. Drosselmeyer provides wonderful gifts for all of the children, including a nutcracker, to which Clara immediately becomes attached. After the party, Clara returns to the parlor to visit her nutcracker, where she witnesses a ferocious battle between the nutcracker—grown to life size and revealed to be a prince—and the Mouse King. She intercedes on the nutcracker’s behalf and he emerges victorious.
In the second act, the Nutcracker Prince takes Clara to his kingdom, the Land of the Sweets, where she is welcomed and celebrated. The courtiers put on a show for her to demonstrate their gratitude, presenting a series of dances while she and the Prince sit upon thrones. At the end of the ballet, Clara returns to her home—awakening, perhaps, from a fantastic dream.
The Nutcracker was premiered as part of a double-bill at the Imperial Mariinsky Theater in St. Petersburg. The other item on the program was Tchaikovsky’s newest opera, Iolanta, making for a complete evening of entertainment lasting about three hours. The premiere was not a success. Critics lambasted the dancing, the choreography, the adaption of the story, the sharp contrast between the acts, the prominence of children onstage, and the neglect of the principal ballerina, who, as the Sugar Plum Fairy, does not dance until nearly the end.
The music, on the other hand, was well-received, and Tchaikovsky was quick to salvage his work by transforming it into an orchestral suite that could be performed on concert programs. It was in this form that The Nutcracker first became popular. The Nutcracker was not staged again as a ballet until 1919, and did not enter the regular repertoire until 1934. A 1944 production by the San Francisco Ballet introduced it to American audiences. The New York City Ballet began offering annual performances in 1954, and the tradition of staging The Nutcracker during the Christmas season soon began to take hold across the United States. Today, The Nutcracker attracts millions of patrons every year, and is responsible for a large portion of the ticket sales by American ballet companies.
We will be taking a look at Act II of The Nutcracker . We will begin with the opening scene, in which Clara and the Prince arrive in the Land of the Sweets. Then we will examine part of the “Grand Divertissement” (that’s “grand entertainment” in English) that is staged for their amusement.
Act II: Introduction
Tchaikovsky opens Act II with harp arpeggios and a sweeping, romantic melody in the violins. Sustained pitches in the brass create a sense of calm and repose. When Clara and the Prince arrive onstage, they use pantomime and facial expressions to indicate their wonder at beholding the magical kingdom, while the music intensifies in excitement with the addition of ascending flourishes in the flutes and piccolo. Magical sounds are created by violin harmonics (a technique by which the player lightly touches the string to produce a high, wispy sound) and celesta, a keyboard instrument that produces bell-like sounds when hammers strike resonant metal bars. Tchaikovsky associated this instrument, invented in Paris in 1886, with the Sugar Plum Fairy, and he was the first to use it in a major work.
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Time |
Form |
What to listen for |
|---|---|---|
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45’30” |
Introduction to Act II |
Clara and the Prince are welcomed to the Land of the Sweets |
|
51’25” |
The Prince tells the story of his victory over the Mouse King; we hear music from Act I |
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52’51” |
The dancers for the “Grand Divertissement” are introduced to Clara and the Prince |
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53’38” |
Chocolate (Spanish dance) |
The melody is introduced by the trumpet, while castanets are heard throughout |
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55’04” |
Coffee (Arabian dance) |
A chromatic melody, heard first in the violins and later in the double reeds, floats above a droning rhythmic ostinato punctuated by tambourine strikes |
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59’00” |
Tea (Chinese dance) |
This dance pairs a flute/piccolo melody with pizzicato strings and a simple ostinato in the |
| 1:00’16” | Trepak |
Russian The fast-paced Trepak, which uses the entire orchestra, grows in intensity before culminating in a raucous final chord |
|
1:01’31” |
End of listening guide |
The entrance of the Sugar Plum Fairy, who has been ruling the Land of the Sweets in the Prince’s absence, is marked by another special effect (flutter tongue) in the flutes. She proceeds to greet Clara and the Prince, as do the subjects of the court. Next, the Prince tells the story of his battle with the Mouse King. He cannot use words, of course, so he reenacts the conflict in pantomime. He is aided by the orchestra, which repeats music from the battle scene—music that the audience heard only twenty or thirty minutes before and will easily recognize.
There has been dancing throughout the procedures thus far, of course, but nothing that could be described as a formal dance number. One of the challenges faced by any scenarist in designing a ballet is coming up with excuses for carefullychoreographed dance numbers. Ballet audiences enjoy the drama, but they want to see some good solo and ensemble dancing, not just pantomime. Petipa solved this problem by crafting a “Grand Divertissement” in which a series of dances are performed for Clara and the Prince. This “show within a show” is ostensibly put on for the benefit of the couple, but is in fact directed at the audience in the theater.
The “Grand Divertissement” consists of a diverse collection of themed dances. The first three dances are named after foods appropriate to the Land of the Sweets: chocolate, coffee, and tea. For each of these, Tchaikovsky drew inspiration from the lands from which these foods came: Spain, Arabia, and China. These are followed by a Russian dance, the “Dance of the Reed Flutes,” and a dance known as “Mother Ginger and the Little Clowns.” We will focus our attention on the first four dances for the purpose of considering how Tchaikovsky approached the task of representing national identity in music.
Act II: Chocolate
The Spanish dance is vibrant and exciting. The melody is first heard on the trumpet—an instrument that is not necessarily associated with Spanish music, but which introduces a bright timbre and sets the number apart from what has come before. The harmonies are simple and repetitive, suggesting a sort of generic “folk” style. What really marks the music as “Spanish” is the use of castanets , which are heard nowhere else in the ballet. Castanets are a simple percussion instrument
that consists of two concave pieces of wood. These are held in one hand and clapped together. Castanets are particularly associated with the Spanish tradition of flamenco , which encompasses unique forms of guitar playing, singing, and dancing. By prominently featuring castanets, Tchaikovsky clearly signalled to his audience that the music and dancing were meant to be Spanish.
11. In this example, a guitarist and castanet player perform “El Vito.”
12. In this example, the guitarist is accompanied by hand clapping, singing, and the rhythmic footwork of the dancer.
But how successful was he in replicating the sounds of flamenco music? We might compare Tchaikovsky’s music to a performance by flamenco musicians of the folk song “El Vito,” which dates to the 16th century. This rendition is typical of Spanish music in several ways. It is in a minor mode, with the melody supported by characteristics harmonies. The guitar player executes dissonant strums. And the castanet player provides complex accompanying rhythms. The performance of flamenco most often incorporates dance as well, and there is a rich vocabulary of movements and rhythmic steps that accompany and express the music. Next to these examples, Tchaikovsky’s Spanish dance sounds comically cheerful and simplistic.
Act II: Coffee
Next is the Arabian dance. This time, Tchaikovsky employs a variety of compositional techniques to signal to his listener that this is “Middle Eastern” music. The low strings play a repeating rhythmic ostinato that produces a hypnotic
effect. The ostinato features the interval of an open fifth, which leaves the question of mode open: we could be in major or minor. The ostinato also eliminates the possibility of complex harmonies, for we are to dwell on the same partial chord for the entire piece. Over the top of this, we hear a modal violin melody that incorporates both the raised and lowered seventh scale degrees and is characterized by unusual rhythms and phrasing. The melody prominently features the interval of an augmented second, which is seldom heard in European music, and it is scattered with trills. The melody is bookended by a repeated motif
from the clarinets and double reeds and punctuated by the jingle of a tambourine. Later, the melody is echoed in the oboe and the bassoon. A modal shift concludes the dance on a major harmony.
For an example of authentic Middle Eastern music, one can refer to the discussion of Turkish makam music in Chapter 8. Many of Tchaikovsky’s strategies for representing the Middle East in sound are indeed rooted in genuine practice. The tambourine, for example, features prominently in Persian and Turkish music. Likewise, Middle Eastern compositions use modes other than major and minor, and their melodies often feature augmented seconds. The trill is not an uncommon ornament in some instruments, such as the flute, and an ostinato sometimes provides a musical backdrop for improvisation. Finally, double reed instruments such as the sorna are native to the Middle East.
In short, Tchaikovsky captures the sound of Middle Eastern music with considerable success. The main differences between the genuine article and Tchaikovsky’s imitation are the different timbres of the instruments, the inauthentic complexity of Tchaikovsky’s orchestration, and the Western intonation of the orchestral players, who do not tune their pitches in the same way as members of a takht ensemble.
All the same, Tchaikovsky contributes to a musical stereotype that casts Middle Eastern music as static and hypnotic. While it can have these characteristics, it usually does not. Unfortunately, these have become the hallmarks of Western imitations, with the result that a rich music tradition is reduced to a handful of cliches.
Act II: Tea
Next up is the Chinese dance, representative of tea. Tchaikovsky again makes use of an ostinato—this time, a rapid oscillation between the first and fifth scale degrees by the bassoon player. Above this we hear a high melody in the flutes and piccolos, punctuated by pizzicato from the strings. As the music grows in intensity, clarinets provide an arpeggiated accompaniment while bells sparkle alongside the flutes.
For an example of authentic Chinese music, we need only look to the previous example in this chapter. Any listener will immediately note that the Tchaikovsky’s dance has little to do with actual Chinese music. His choice of flutes for the melody might bring to mind the bright timbres often favored in Chinese music, but the similarity ends there. The steady rhythm, choice of scale, texture, and repetitive form all suggest that he had never actually heard Chinese music at all—or at least had no interest in creating a faithful reproduction.
Creating a faithful reproduction, of course, was never Tchaikovsky’s goal in any of these cases. Whether or not he accurately reflected the music of the cultures he parodied was purely incidental. Tchaikovsky’s only task was to entertain the Moscow audience members that purchased tickets to see the ballet. His audience was Russian, and he knew that they enjoyed exotic escapism as part of their theatrical entertainment. They were not alone.
Exoticism— the use of stereotypes to portray other cultures as exciting or mysterious—has a long history in European music, and especially in music for the opera stage. European audiences of the 18th and 19th centuries were intrigued by the cultural practices of distant lands, and they had a limitless appetite for their representation in the arts. The East held particular fascination, such that the term orientalism has been coined to describe the stereotyped representation of Eastern cultures. Such representations, however, are seldom accurate or flattering. Instead, exoticism dehumanizes its subject so as to provide an escapist experience to the consumer. Europeans often perceived exoticized subjects as sexually licentious, primitive, and driven by their emotions— or, to put it another way, free from the strictures of society. In this way, exoticized subjects became an object of both adoration and loathing. They could give in to temptations that were denied to European viewers, but only because they were less than human.
All of this might seem a bit tangential to The Nutcracker , which, after all, tells a charming story set in an imagined candy land, but it is not. The ballet’s representations of exoticized others—whether Romani people of Spain, or Arabs, or Chinese—contributes to a centuries-long practice that can still dehumanize these people today. This is best exemplified by a current controversy surrounding the performance of Tchaikovsky’s Chinese dance, which usually relies on stereotyped costumes, makeup, and choreography that many people find offensive.
In 2017, yellowface.org was founded specifically to advocate for changes in how the Chinese dance is presented in productions of The Nutcracker . The organization encourages arts leaders to sign the “Final Bow for Yellowface” pledge, which is a commitment to end racist representations of the Chinese characters. It also provides resources for creating new costumes, makeup, and choreography for use in Nutcracker productions that reflect genuine Chinese cultural practices instead of racist stereotypes. The movement has gained support, but most productions— including that associated with this text—continue to present a stereotyped visual representation of Chinese culture alongside Tchaikovky’s musical one.
Act II: Trepak
In this video, you can see the kind of folk dance that inspired Tchaikovsky’s Trepak.
The final selection from the “Grand Divertissement” that we will examine here is the Russian dance, or Trepak. This time, Tchaikovsky took a model closer to home, for this dance is based on local Russian and Ukrainian folk practices.13 Unsurprisingly, the Russian dance is Tchaikovsky’s best imitation of the “real thing.” Both his Russian dance and the authentic trepak are fast-paced and in duple meter, with a driving rhythm suited to high-energy dancing. We again hear the tambourine—now a symbol not of the Middle East but of native folk culture.


17.3: Examples
A bartender slides a bottle down his bar to a customer. Unfortunately, he pushes it too hard and the bottle flies off the end of the bar!
- The force of friction between the bar and the bottle is 0.150 N, and the bottle has a mass of 0.520 kg. What is the acceleration of the bottle as it travels down the bar?
- If it started with a speed of 2.97 m/s, what is the speed of the bottle as it leaves the bar, 2.80 m away from where the bartender pushed it?
- Black Hat from the webcomic XKCD is standing nearby, a distance 1.25 m from the end of the bar (which is 1.15 m tall). Does the bottle hit him after it slides off the end of the bar?
Consider two blocks which are in contact with each other on a horizontal frictionless surface. The masses of the blocks are 5 kg and 10 kg. The 5 kg block is being pushed with an unknown force \(F_p\), which then pushes on the 10 kg block, making them both accelerate to the right at 2.00 m/s\(^2\).
- How big is the force pushing on the 10 kg block from the 5 kg block?
- What is the magnitude of the pushing force \(F_p\)?
Consider three blocks which are in contact with each other on a horizontal frictionless surface. From left to right, the masses of the blocks are 5 kg, 10 kg, and 25 kg. The leftmost block is being pushed to the right with an unknown force \(F_p\), and the blocks are accelerating to the right at 2.00 m/s\(^2\).
- What is the magnitude and direction of the forces the blocks are exerting on each other?
- What is the magnitude of the pushing force \(F_p\)?
A 40.0 kg crate is sitting on top of a 60.0 kg crate on the floor of an elevator. The elevator floor is exerting an upwards force of 1050 N to the 60.0 kg crate.
- Determine the magnitude and direction of the acceleration of the crates in the elevator.
- Determine the magnitude and direction of the contact force that the 40.0 kg crate exerts on the 60.0 kg crate.
- Determine the magnitude and direction of the contact force that the 60.0 kg crate exerts on the 40.0 kg crate.


3.3.11: Jamaica Kincaid (1949- )
Girl Antiguan-American Contemporary Literature Jamaica Kincaid, whose original name is Elaine Potter Richardson, was born in Antigua in 1949. She is currently Professor of African and African-American Studies at Harvard University. She grew up in Antigua in poverty, and she had a troubled relationship with her mother, whom she believed neglected her in favor of her brothers. She has said that this relationship shaped her as a writer. As a teenager, she moved to New York city, where she began her career as a writer in her twenties publishing short stories in teen magazines but eventually publishing short fiction in The Village Voice , The Paris Review , and The New Yorker . While she has no college degree, Kincaid wrote for The New Yorker for nearly 20 years. Kincaid's work is often semi-autobiographical; she explores themes of race and gender, particularly in a neo-colonial setting. "Girl" was originally published in the New Yorker magazine in 1978. Written as a dispute between a mother and a daughter, it is a powerful illustration of the limits of a young woman's life in the Caribbean culture of the time. Consider while reading:
- Why is the story written in second person?
- What does the mother tell the young girl about her role as a woman?
- How does the tone change as the conversation progresses?
- How does this very short story reflect the author's experiences?
Written by Anita Turlington


7.10.2: Risk Factors/Protective Factors
Risk factors for diabetes may include biological or disease factors outside of a person’s control such as:
- Age (the risk of type II diabetes increases with age)
- Genetics (as in monogenic diabetes syndromes)
- Other diseases (cystic fibrosis, polycystic ovarian syndrome, pancreatitis)
- Treatments for other diseases (such as medicines for HIV) (ElSayed et al., 2022)
Particularly for type II diabetes however, there are several risk factors that are related to lifestyle behaviors:
- Obesity
- Poor nutrition (overconsumption of calories)
- Lack of physical activity
Conversely, protective factors against type II diabetes, and strategies used to reverse prediabetes include nutrition, exercise, and weight loss (if obesity is a factor). Exercise alone can increase insulin sensitivity both acutely and over the long term. When skeletal muscles contract, they have an “insulin-like effect” which draws glucose into the muscle cell, while becoming more sensitized to any existing insulin in the bloodstream. This allows them to take up glucose to use for the energy they need. Insulin sensitivity is increased for at least 16 hours after a workout, and if a person exercises regularly (and perhaps also decreases body fat) this can have lasting effects on their ability to manage blood sugar. Physical activity can also reduce the chronic inflammation associated with type II diabetes, and helps those with type I diabetes also manage the disease (Venkatasamy et al., 2013). For people who have obesity, losing just 5-7% of their body weight and getting 150 minutes of moderate physical activity per week reduces the risk of type II diabetes by 58%-71% (CDC, 2023b).


1.6: Research Designs
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We have just been looking at models of the research process and goals of research. The following is a comparison of research methods or techniques used to describe, explain, or evaluate. Each of these designs has strengths and weaknesses and is sometimes used in combination with other designs within a single study.
Observational studies involve watching and recording the actions of participants. This may take place in the natural setting, such as observing children at play at a park, or behind a one-way glass while children are at play in a laboratory playroom. The researcher may follow a check list and record the frequency and duration of events (perhaps how many conflicts occur among 2 year olds) or may observe and record as much as possible about an event as a participant (such as attending an Alcoholics Anonymous meeting and recording the slogans on the walls, the structure of the meeting, the expressions commonly used, etc.). The researcher may be a participant or a non-participant. What would be the strengths of being a participant? What would be the weaknesses? Consider the strengths and weaknesses of not participating. In general, observational studies have the strength of allowing the researcher to see how people behave rather than relying on self-report. What people do and what they say they do are often very different. A major weakness of observational studies is that they do not allow the researcher to explain causal relationships. Yet, observational studies are useful and widely used when studying children. Children tend to change their behavior when they know they are being watched (known as the Hawthorne effect ) and may not survey well.
Experiments are designed to test hypotheses (or specific statements about the relationship between variables ) in a controlled setting in efforts to explain how certain factors or events produce outcomes. A variable is anything that changes in value. Concepts are operationalized or transformed into variables in research which means that the researcher must specify exactly what is going to be measured in the study. For example, if we are interested in studying marital satisfaction, we have to specify what marital satisfaction really means or what we are going to use as an indicator of marital satisfaction. What is something measurable that would indicate some level of marital satisfaction? Would it be the amount of time couples spend together each day? Or eye contact during a discussion about money? Or maybe a subject’s score on a marital satisfaction scale. Each of these is measurable but these may not be equally valid or accurate indicators of marital satisfaction. What do you think? These are the kinds of considerations researchers must make when working through the design.
Three conditions must be met in order to establish cause and effect. Experimental designs are useful in meeting these conditions.
The independent and dependent variables must be related . In other words, when one is altered, the other changes in response. (The independent variable is something altered or introduced by the researcher. The dependent variable is the outcome or the factor affected by the introduction of the independent variable. For example, if we are looking at the impact of exercise on stress levels, the independent variable would be exercise; the dependent variable would be stress.)
The cause must come before the effect. Experiments involve measuring subjects on the dependent variable before exposing them to the independent variable (establishing a baseline). So we would measure the subjects’ level of stress before introducing exercise and then again after the exercise to see if there has been a change in stress levels. (Observational and survey research does not always allow us to look at the timing of these events which makes understanding causality problematic with these designs.)
The cause must be isolated. The researcher must ensure that no outside, perhaps unknown variables are actually causing the effect we see. The experimental design helps make this possible. In an experiment, we would make sure that our subjects’ diets were held constant throughout the exercise program. Otherwise, diet might really be creating the change in stress level rather than exercise.
A basic experimental design involves beginning with a sample (or subset of a population) and randomly assigning subjects to one of two groups: the experimental group or the control group . The experimental group is the group that is going to be exposed to an independent variable or condition the researcher is introducing as a potential cause of an event. The control group is going to be used for comparison and is going to have the same experience as the experimental group but will not be exposed to the independent variable. After exposing the experimental group to the independent variable, the two groups are measured again to see if a change has occurred. If so, we are in a better position to suggest that the independent variable caused the change in the dependent variable . The basic experimental model looks like this:
|
Sample is randomly assigned to one of the groups below: |
Measure DV |
Introduce IV |
Measure DV |
|---|---|---|---|
|
Experimental Group |
X |
X |
X |
|
Control Group |
X |
– |
X |
The major advantage of the experimental design is that of helping to establish cause and effect relationships. A disadvantage of this design is the difficulty of translating much of what concerns us about human behavior into a laboratory setting. I hope this brief description of experimental design helps you appreciate both the difficulty and the rigor of conducting an experiment.
Case studies involve exploring a single case or situation in great detail. Information may be gathered with the use of observation, interviews, testing, or other methods to uncover as much as possible about a person or situation. Case studies are helpful when investigating unusual situations such as brain trauma or children reared in isolation. And they often used by clinicians who conduct case studies as part of their normal practice when gathering information about a client or patient coming in for treatment. Case studies can be used to explore areas about which little is known and can provide rich detail about situations or conditions. However, the findings from case studies cannot be generalized or applied to larger populations; this is because cases are not randomly selected and no control group is used for comparison. (Read “The Man Who Mistook His Wife for a Hat” by Dr. Oliver Sacks as a good example of the case study approach.)
Surveys are familiar to most people because they are so widely used. Surveys enhance accessibility to subjects because they can be conducted in person, over the phone, through the mail, or online. A survey involves asking a standard set of questions to a group of subjects. In a highly structured survey, subjects are forced to choose from a response set such as “strongly disagree, disagree, undecided, agree, strongly agree”; or “0, 1-5, 6-10, etc.” Surveys are commonly used by sociologists, marketing researchers, political scientists, therapists, and others to gather information on many independent and dependent variables in a relatively short period of time. Surveys typically yield surface information on a wide variety of factors, but may not allow for in-depth understanding of human behavior. Of course, surveys can be designed in a number of ways. They may include forced choice questions and semi-structured questions in which the researcher allows the respondent to describe or give details about certain events. One of the most difficult aspects of designing a good survey is wording questions in an unbiased way and asking the right questions so that respondents can give a clear response rather that choosing “undecided” each time. Knowing that 30% of respondents are undecided is of little use! So a lot of time and effort should be placed on the construction of survey items. One of the benefits of having forced choice items is that each response is coded so that the results can be quickly entered and analyzed using statistical software. Analysis takes much longer when respondents give lengthy responses that must be analyzed in a different way. Surveys are useful in examining stated values, attitudes, opinions, and reporting on practices. However, they are based on self-report or what people say they do rather than on observation and this can limit accuracy.
Secondary/Content analysis involves analyzing information that has already been collected or examining documents or media to uncover attitudes, practices or preferences. There are a number of data sets available to those who wish to conduct this type of research. For example, the U. S. Census Data is available and widely used to look at trends and changes taking place in the United States (go to http://www.census.gov/ and check it out). There are a number of other agencies that collect data on family life, sexuality, and many other areas of interest in human development (go to http://www.norc.uchicago.edu/ or http://www.kff.org/ and see what you find.). The researcher conducting secondary analysis does not have to recruit subjects but does need to know the quality of the information collected in the original study.
Content analysis involves looking at media such as old texts, pictures, commercials, lyrics or other materials to explore patterns or themes in culture. An example of content analysis is the classic history of childhood by Aries (1962) called “Centuries of Childhood” or the analysis of television commercials for sexual or violent content. Passages in text or programs that air can be randomly selected for analysis as well. Again, one advantage of analyzing work such as this is that the researcher does not have to go through the time and expense of finding respondents, but the researcher cannot know how accurately the media reflects the actions and sentiments of the population.
Developmental designs are techniques used in life span research (and other areas as well). These techniques try to examine how age, cohort, gender, and social class impact development. Cross-sectional research involves beginning with a sample that represents a cross-section of the population. Respondents who vary in age, gender, ethnicity, and social class might be asked to complete a survey about television program preferences or attitudes toward the use of the Internet. The attitudes of males and females could then be compared as could attitudes based on age. In cross-sectional research, respondents are measured only once. This method is much less expensive than longitudinal research but does not allow the researcher to distinguish between the impact of age and the cohort effect. Different attitudes about the Internet, for example, might not be altered by a person’s biological age as much as their life experiences as members of a cohort.
Longitudinal research involves beginning with a group of people who may be of the same age and background, and measuring them repeatedly over a long period of time. One of the benefits of this type of research is that people can be followed through time and be compared with them when they were younger. A problem with this type of research is that it is very expensive and subjects may drop out over time. (The film 49 Up is a example of following individuals over time. You see how people change physically, emotionally, and socially through time.) What would be the drawbacks of being in a longitudinal study? What about 49 Up? Would you want to be filmed every 7 years? What would be the advantages and disadvantages? Can you imagine why some would continue and others drop out of the project?
Cross-sequential research involves combining aspects of the previous two techniques; beginning with a cross-sectional sample and measuring them through time. This is the perfect model for looking at age, gender, social class, and ethnicity. But it the drawbacks of high costs and attrition are here as well.


39.6: Review Questions
The nurse teaches parents of school-age children that which structure protects the middle ear from infection?
- Answer
-
Eustachian tube
The nurse is caring for a client who has an upper respiratory illness but is getting ready to travel by plane. Which medication should the nurse anticipate being prescribed to minimize ear-related problems during the flight?
- Answer
-
Pseudoephedrine
Which instruction should the nurse give to a client to reduce tinnitus?
- Answer
-
Decrease sodium intake.
Which communication technique is most effective for the nurse to use when communicating with a client with hearing impairment?
- Answer
-
Avoid interrupting the client with a hearing impairment.
Which of the following is a proper technique regarding otic antibiotics?
- Answer
-
Avoid touching the tip of the dropper to any surface.
Which assessment finding indicates to the nurse that a client receiving an otic antibiotic may have a resistant infection, also known as a superinfection?
- Answer
-
Pain and ear itching
Which adverse effect should the nurse caution the client taking amoxicillin-clavulanate to expect?
- Answer
-
The client could experience diarrhea.
Which instruction should the nurse give to a client with diabetes who is taking a decongestant for relief from sinus and ear congestion caused by an upper respiratory infection?
- Answer
-
Regularly check serum glucose while taking the decongestant.
Which action should the nurse take to evaluate the effectiveness of carbamide peroxide administered to a client’s ear?
- Answer
-
Examine the ear canal.
Which diagnosis does the nurse anticipate in the client who reports hearing loss, vertigo, nausea, and vomiting?
- Answer
-
Ménière’s disease


6.8: Imperative Construction
Imperative Construction Video
Instructor Video: Imperative Construction
Imperative sentences are sentences that express request, commands, order, instruction, etc. Like in English, Subject “You” is dropped. However, unlike English, Hindi has three different forms of the pronoun ‘You,’ which are used in three different sociocultural contexts, as explained below.
| English | Hindi | Uses |
| You | तू | तू is used for someone who is younger in age, lower socio-economic class. It is also used with someone who is very intimate to the speaker such as friend, mother, and God. |
| You | तुम | तुम is used for someone who is of the same age group, and same socio-economic class. |
| You | आप | आप is used for someone who is older in age, and higher socio-economic class. It is used for anyone who one wants to show respect. |
Forms of Imperative Construction Based on Three Different Forms of “ You “
The forms for “तू” and “तुम” are used for instruction, and command. Whereas, the form for “आप” is used for request.
| Verbs |
“तू “
Drop the -ना part of the verb |
“तुम”
Drop the -ना part and add “ओ” |
“आप”
Drop the -ना part and add “इये” |
| खाना “To eat” | खा | खाओ | खाइये |
| जाना “To go” | जा | जाओ | जाइये |
| आना “To come” | आ | आओ | आइये |
| पढ़ना “To read/study” | पढ़ | पढ़ो | पढ़िये |
| लिखना “To write” | लिख | लिखो | लिखिए |
| सोना “To sleep” | सो | सोओ | सोइये |
| बोलना “To speak” | बोल | बोलो | बोलिये |
| सुनना “To listen” | सुन | सुनो | सुनिये |
| बैठना “To sit” | बैठ | बैठो | बैठिये |
| बनाना “To make” | बना | बनाओ | बनाइये |
Some Irregular Imperative Forms
| Verbs | “तू “ | “तुम” | “आप” |
| करना “To do” | कर | करो | कीजिये |
| लेना “To take” | ले | लो | लीजिये |
| देना “To give” | दे | दो | दीजिये |
| पीना | पी | पियो | पीजिये |
Giving Instruction and Command
To give instruction and command, in Hindi, “तू” and “तुम” forms of the verb are used.
For example:
- यहाँ आ/आओ “Come here.”
- घर जा/जाओ “Go home.”
- मेरी किताब ला/लाओ “Bring my book.”
- मुझ से बात कर/करो “Talk to me.”
- जॉन को अपनी कार बेच/बेचो “Sell your car to John.”
Making Requests
To make request, in Hindi, “आप” form of the verb is used.
For example:
- यहाँ आइये “Please come here.”
- घर जाइये “Please go home.”
- मेरी किताब लाइये “Please bring my book.”
- मुझ से बात कीजिये “Please talk to me.”
- जॉन को अपनी कार बेचिये “Please sell your car to John.”
Use of ज़रा/कृपया
In Hindi, formal imperatives do not necessarily imply “ please. ” Just as in any language, the tone of voice, context, and personal relationships affect politeness.
The adverb कृपया (please), ज़रा ( just, a little ) is used to make a command milder/extra polite.
For example:
- कृपया/ज़रा यहाँ आइये “Please come here.”
- कृपया/ज़रा घर जाइये “Please go home.”
- कृपया/ज़रा मेरी किताब लाइये “Please bring my book.”
- कृपया/ज़रा मुझ से बात कीजिये “Please talk to me.”
- कृपया/ज़रा जॉन को अपनी कार बेचिये “Please sell your car to John.”
Negation of Imperative Construction
In order to make negative imperative, command, and making negative request, in Hindi, words like “न” and “मत” can be used.
For example
- यहाँ न/मत आ/आओ “Do not come here.”
- घर न/मत जा/जाओ “Do not go home.”
- मेरी किताब न/मत लाइये “Please do not bring my book.”
- मुझ से बात न/मत कीजिये “Please do not talk to me.”
- कृपया/ज़रा जॉन को अपनी कार न/मत बेचिये “Please do not sell your car to John.
Note: Use of “न” post verb place in an imperative construction will make it request, and not a negative sentence.
For example:
| “न” before verb for negation | “न” before verb for request |
| यहाँ न आइये “Please do not come here.” | यहाँ आइये न “Please come here.” |
| मुझ से बात न कीजिये “Please do not talk to me.” | मुझ से बात कीजिये न “Please talk to me.” |
| मेरी बात न सुनिये “Please do not listen to me.” | मेरी बात सुनिये न “Please listen to me” |
Activities
a) Please write ten sentences of instructions and ask your friend to enact them, and vice-versa.
b) Make a list of ten things that you want your parents to do for you. Please use “आप” form of the imperative construction.
c) Please choose the correct form of the pronoun “You.”
https://openbooks.lib.msu.edu/ll151/?p=1152
धन्यवाद !


6.3: Coding
What is Coding?
Coding is step-by-step instructions that a computer understands for programs to work. Our cars, tablets, and cell phones all use coding to work properly. However, children don't have to start out making coding instructions for computers. There are many activities children can participate in called "pre-coding" activities that don't involve using a computer.
Coding Stories and Games for Preschool
Pre-coding activities for preschoolers involve directional language, ordinal numbers, and one-to-one correspondence. Directional language involves understanding words such as up and down, backward and forward, left and right. Children must also know how to count using ordinal numbers, such as first, second, and third. Children must also understand one-to-one correspondence meaning they only count each item once. All children will be at different places in their understanding of these concepts. Some children will have mastered these skills while others are still learning them. That is ok because pre-coding activities help to develop these skills.
See Creating Coding Stories and Games to learn more about the benefits of coding with young children and for ideas for pre-coding activities.


7.8: Summary
Parenteral medication administration is an effective method of delivering medication to patients, and can be safely accomplished by utilizing the appropriate guidelines and policies in place to keep patients safe from harm. IV medications have a higher risk of harm than non-parenteral medication. The ever-increasing complexity of the health care environment increases the risk of a medication error with parenteral medications. The key takeaways below provide advice for preventing errors with parenteral medications.
Key Takeaways
- Parenteral medications have a quicker onset of therapeutic effects. Be aware of the onset, peak, and duration of all parenteral medications.
- Know which medications are considered high-alert medications and perform independent double checks to minimize errors.
- Always consider the therapeutic effects and adverse effects when administering parenteral medications.
- Safeguards for medication administration exist in most hospitals. Make use of all safety strategies (such as smart IV pumps, no-interruption zones, two patient identifiers, and checklists) to administer medications safely.
- Quality and safety controls for safe medication administration must be considered along the entire process, not just at the point of administration.
- It is human nature to look for quick and easy ways to perform a task, but doing so may lead to errors. Avoid workarounds: Most hospitals have operational failures that lead to front line health care providers finding ways to manage deficiency in hospital operating systems. Rather than creating workarounds, engage in the additional steps to prevent re-occurrence of issues.
- Stay current with evidence-based research regarding potential system errors in health care. Commit to improving patient safety with medication administration.
- Report all errors, near misses, and adverse reactions to ensure knowledge is shared, and to prevent further errors from occurring.
Suggested Online Resources
1. Agency for Healthcare Research and Quality (AHRQ). This website provides evidence-based research, guidelines, recommendations, and resources on improving patient safety .
2. Canadian Patient Safety Institute (CPSI). This organization’s website provides guidelines, research, and recommendations for improving patient safety in Canada.
3. Drug calculations. This medication calculation website reviews how to calculate the dosages for parenteral and non-parenteral medications, and IV fluids. It also includes metric conversions and IV drop rate calculations.
4. Institute for Healthcare Improvement (IHI). This group’s website provides educational resources, webinars, publications, and improvement stories and tools to enhance patient safety.
5. Institute for Safe Medication Practices (ISMP). This organization focuses on improving medication administration. Its website lists high-alert medications, offers newsletters and webinars, and provides a system for reporting medication errors, guidelines, and policies on safe practices.
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Prakash, V., Koczmara, C., Saveage, P., Trip, K., Stewart, J., Mccurdie, T., Cafazzo, J., & Trbovich, P. (2014). Mitigating errors caused by interruptions during medication verification and administration: Interventions in a simulated ambulatory chemotherapy setting. BMJ Quality and Safety, 0 (1-10). doi:10.1136/bmjqs-2013-002484
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Provincial Infectious Disease Advisory Committee. (2014). Best practices for hand hygiene in all health care settings (4th ed.). Retrieved on June 4, 2015, from http://www.publichealthontario.ca/en...%20Hygiene.pdf
Rodgers, M.A., & King, L. (2000). Drawing up and administering intramuscular injections: A review of the literature. Journal of Advanced Nursing , 31 (3), 574–582. doi: 10.1046/j.1365-2648.2000.01312.x
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1.E: Units and Measurement (Exercises)
Conceptual Questions
1.1 The Scope and Scale of Physics
- What is physics?
- Some have described physics as a “search for simplicity.” Explain why this might be an appropriate description.
- If two different theories describe experimental observations equally well, can one be said to be more valid than the other (assuming both use accepted rules of logic)?
- What determines the validity of a theory?
- Certain criteria must be satisfied if a measurement or observation is to be believed. Will the criteria necessarily be as strict for an expected result as for an unexpected result?
- Can the validity of a model be limited or must it be universally valid? How does this compare with the required validity of a theory or a law?
1.2 Units and Standards
- Identify some advantages of metric units.
- What are the SI base units of length, mass, and time?
- What is the difference between a base unit and a derived unit? (b) What is the difference between a base quantity and a derived quantity? (c) What is the difference between a base quantity and a base unit?
- For each of the following scenarios, refer to Figure 1.4 and Table 1.2 to determine which metric prefix on the meter is most appropriate for each of the following scenarios. (a) You want to tabulate the mean distance from the Sun for each planet in the solar system. (b) You want to compare the sizes of some common viruses to design a mechanical filter capable of blocking the pathogenic ones. (c) You want to list the diameters of all the elements on the periodic table. (d) You want to list the distances to all the stars that have now received any radio broadcasts sent from Earth 10 years ago.
1.6 Significant Figures
- (a) What is the relationship between the precision and the uncertainty of a measurement? (b) What is the relationship between the accuracy and the discrepancy of a measurement?
1.7 Solving Problems in Physics
- What information do you need to choose which equation or equations to use to solve a problem?
- What should you do after obtaining a numerical answer when solving a problem?
Problems
1.1 The Scope and Scale of Physics
-
Find the order of magnitude of the following physical quantities.
- The mass of Earth’s atmosphere: 5.1 × 10 18 kg;
- The mass of the Moon’s atmosphere: 25,000 kg;
- The mass of Earth’s hydrosphere: 1.4 × 10 21 kg;
- The mass of Earth: 5.97 × 10 24 kg;
- The mass of the Moon: 7.34 × 10 22 kg;
- The Earth–Moon distance (semi-major axis): 3.84 × 10 8 m;
- The mean Earth–Sun distance: 1.5 × 10 11 m;
- The equatorial radius of Earth: 6.38 × 10 6 m;
- The mass of an electron: 9.11 × 10 −31 kg;
- The mass of a proton: 1.67 × 10 −27 kg;
- The mass of the Sun: 1.99 × 10 30 kg.
-
Use the orders of magnitude you found in the previous problem to answer the following questions to within an order of magnitude.
- How many electrons would it take to equal the mass of a proton?
- How many Earths would it take to equal the mass of the Sun?
- How many Earth–Moon distances would it take to cover the distance from Earth to the Sun?
- How many Moon atmospheres would it take to equal the mass of Earth’s atmosphere?
- How many moons would it take to equal the mass of Earth?
- How many protons would it take to equal the mass of the Sun?
For the remaining questions, you need to use Figure 1.4 to obtain the necessary orders of magnitude of lengths, masses, and times.
- Roughly how many heartbeats are there in a lifetime?
- A generation is about one-third of a lifetime. Approximately how many generations have passed since the year 0 AD?
- Roughly how many times longer than the mean life of an extremely unstable atomic nucleus is the lifetime of a human?
- Calculate the approximate number of atoms in a bacterium. Assume the average mass of an atom in the bacterium is 10 times the mass of a proton.
- (a) Calculate the number of cells in a hummingbird assuming the mass of an average cell is 10 times the mass of a bacterium. (b) Making the same assumption, how many cells are there in a human?
- Assuming one nerve impulse must end before another can begin, what is the maximum firing rate of a nerve in impulses per second?
- About how many floating-point operations can a supercomputer perform each year?
- Roughly how many floating-point operations can a supercomputer perform in a human lifetime?
1.2 Units and Standards
-
The following times are given using metric prefixes on the base SI unit of time: the second. Rewrite them in scientific notation without the prefix. For example, 47 Ts would be rewritten as 4.7 × 10
13
s.
- 980 Ps;
- 980 fs;
- 17 ns;
- 577 µs.
-
The following times are given in seconds. Use metric prefixes to rewrite them so the numerical value is greater than one but less than 1000. For example, 7.9 × 10
−2
s could be written as either 7.9 cs or 79 ms.
- 9.57 × 10 5 s;
- 0.045 s;
- 5.5 × 10 −7 s;
- 3.16 × 10 7 s.
-
The following lengths are given using metric prefixes on the base SI unit of length: the meter. Rewrite them in scientific notation without the prefix. For example, 4.2 Pm would be rewritten as 4.2 × 10
15
m.
- 89 Tm;
- 89 pm;
- 711 mm;
- 0.45 µm.
-
The following lengths are given in meters. Use metric prefixes to rewrite them so the numerical value is bigger than one but less than 1000. For example, 7.9 × 10
−2
m could be written either as 7.9 cm or 79 mm.
- 7.59 × 10 7 m;
- 0.0074 m;
- 8.8 × 10 −11 m;
- 1.63 × 10 13 m.
-
The following masses are written using metric prefixes on the gram. Rewrite them in scientific notation in terms of the SI base unit of mass: the kilogram. For example, 40 Mg would be written as 4 × 10
4
kg.
- 23 mg;
- 320 Tg;
- 42 ng;
- 7 g;
- 9 Pg.
-
The following masses are given in kilograms. Use metric prefixes on the gram to rewrite them so the numerical value is bigger than one but less than 1000. For example, 7 × 10−4 kg could be written as 70 cg or 700 mg.
- 3.8 × 10−5 kg;
- 2.3 × 1017 kg;
- 2.4 × 10−11 kg;
- 8 × 1015 kg;
- 4.2 × 10−3 kg.
1.3 Unit Conversion
- The volume of Earth is on the order of 10 21 m 3 . (a) What is this in cubic kilometers (km 3 )? (b) What is it in cubic miles (mi 3 )? (c) What is it in cubic centimeters (cm 3 )?
- The speed limit on some interstate highways is roughly 100 km/h. (a) What is this in meters per second? (b) How many miles per hour is this?
- A car is traveling at a speed of 33 m/s. (a) What is its speed in kilometers per hour? (b) Is it exceeding the 90 km/ h speed limit?
- In SI units, speeds are measured in meters per second (m/s). But, depending on where you live, you’re probably more comfortable of thinking of speeds in terms of either kilometers per hour (km/h) or miles per hour (mi/h). In this problem, you will see that 1 m/s is roughly 4 km/h or 2 mi/h, which is handy to use when developing your physical intuition. More precisely, show that (a) 1.0 m/s = 3.6 km/h and (b) 1.0 m/s = 2.2 mi/h.
- American football is played on a 100-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 m = 3.281 ft.)
- Soccer fields vary in size. A large soccer field is 115 m long and 85.0 m wide. What is its area in square feet? (Assume that 1 m = 3.281 ft.)
- What is the height in meters of a person who is 6 ft 1.0 in. tall?
- Mount Everest, at 29,028 ft, is the tallest mountain on Earth. What is its height in kilometers? (Assume that 1 m = 3.281 ft.)
- The speed of sound is measured to be 342 m/s on a certain day. What is this measurement in kilometers per hour?
- Tectonic plates are large segments of Earth’s crust that move slowly. Suppose one such plate has an average speed of 4.0 cm/yr. (a) What distance does it move in 1.0 s at this speed? (b) What is its speed in kilometers per million years?
- The average distance between Earth and the Sun is 1.5 × 10 11 m. (a) Calculate the average speed of Earth in its orbit (assumed to be circular) in meters per second. (b) What is this speed in miles per hour?
- The density of nuclear matter is about 10 18 kg/m 3 . Given that 1 mL is equal in volume to cm 3 , what is the density of nuclear matter in megagrams per microliter (that is, Mg/µL)?
- The density of aluminum is 2.7 g/cm 3 . What is the density in kilograms per cubic meter?
- A commonly used unit of mass in the English system is the pound-mass, abbreviated lbm, where 1 lbm = 0.454 kg. What is the density of water in pound-mass per cubic foot?
- A furlong is 220 yd. A fortnight is 2 weeks. Convert a speed of one furlong per fortnight to millimeters per second.
- It takes \(2 \pi\) radians (rad) to get around a circle, which is the same as 360°. How many radians are in 1°?
- Light travels a distance of about 3 × 10 8 m/s. A light-minute is the distance light travels in 1 min. If the Sun is 1.5 × 10 11 m from Earth, how far away is it in lightminutes?
- A light-nanosecond is the distance light travels in 1 ns. Convert 1 ft to light-nanoseconds.
- An electron has a mass of 9.11 × 10 −31 kg. A proton has a mass of 1.67 × 10 −27 kg. What is the mass of a proton in electron-masses?
- A fluid ounce is about 30 mL. What is the volume of a 12 fl-oz can of soda pop in cubic meters?
1.4 Dimensional Analysis
- A student is trying to remember some formulas from geometry. In what follows, assume A is area, V is volume, and all other variables are lengths. Determine which formulas are dimensionally consistent. (a) V = \(\pi r^{2} h\); (b) A = \(2 \pi r^{2} + 2 \pi r h\); (c) V = 0.5bh; (d) V = \(\pi d^{2}\) ; (e) V = \(\frac{\pi d^{3}}{6}\)
- Consider the physical quantities s, v, a, and t with dimensions [s] = L, [v] = LT −1 , [a] = LT −2 , and [t] = T. Determine whether each of the following equations is dimensionally consistent. (a) v 2 = 2as; (b) s = vt 2 + 0.5at 2 ; (c) v = s/t; (d) a = v/t.
- Consider the physical quantities m, s, v, a, and t with dimensions [m] = M, [s] = L, [v] = LT –1 , [a] = LT –2 , and [t] = T. Assuming each of the following equations is dimensionally consistent, find the dimension of the quantity on the left-hand side of the equation: (a) F = ma; (b) K = 0.5mv 2 ; (c) p = mv; (d) W = mas; (e) L = mvr.
- Suppose quantity s is a length and quantity t is a time. Suppose the quantities v and a are defined by v = ds/dt and a = dv/dt. (a) What is the dimension of v? (b) What is the dimension of the quantity a? What are the dimensions of (c) \(\int\)vdt, (d) \(\int\)adt, and (e) da/dt?
- Suppose [V] = L3 , [ρ] = ML –3 , and [t] = T. (a) What is the dimension of \(\int \rho\)dV ? (b) What is the dimension of dV/dt? (c) What is the dimension of \(\rho\)(dV/dt)?
- The arc length formula says the length s of arc subtended by angle \(\Theta\) in a circle of radius r is given by the equation s = r\(\Theta\). What are the dimensions of (a) s, (b) r, and (c) \(\Theta\)?
1.5 Estimates and Fermi Calculations
- Assuming the human body is made primarily of water, estimate the volume of a person.
- Assuming the human body is primarily made of water, estimate the number of molecules in it. (Note that water has a molecular mass of 18 g/mol and there are roughly 10 24 atoms in a mole.)
- Estimate the mass of air in a classroom.
- Estimate the number of molecules that make up Earth, assuming an average molecular mass of 30 g/mol. (Note there are on the order of 10 24 objects per mole.)
- Estimate the surface area of a person.
- Roughly how many solar systems would it take to tile the disk of the Milky Way?
- (a) Estimate the density of the Moon. (b) Estimate the diameter of the Moon. (c) Given that the Moon subtends at an angle of about half a degree in the sky, estimate its distance from Earth.
- The average density of the Sun is on the order 10 3 kg/ m 3 . (a) Estimate the diameter of the Sun. (b) Given that the Sun subtends at an angle of about half a degree in the sky, estimate its distance from Earth. 64. Estimate the mass of a virus.
- A floating-point operation is a single arithmetic operation such as addition, subtraction, multiplication, or division. (a) Estimate the maximum number of floating-point operations a human being could possibly perform in a lifetime. (b) How long would it take a supercomputer to perform that many floating-point operations?
1.6 Significant Figures
- Consider the equation 4000/400 = 10.0. Assuming the number of significant figures in the answer is correct, what can you say about the number of significant figures in 4000 and 400?
- Suppose your bathroom scale reads your mass as 65 kg with a 3% uncertainty. What is the uncertainty in your mass (in kilograms)?
- A good-quality measuring tape can be off by 0.50 cm over a distance of 20 m. What is its percent uncertainty?
- An infant’s pulse rate is measured to be 130 ± 5 beats/ min. What is the percent uncertainty in this measurement?
- (a) Suppose that a person has an average heart rate of 72.0 beats/min. How many beats does he or she have in 2.0 years? (b) In 2.00 years? (c) In 2.000 years?
- A can contains 375 mL of soda. How much is left after 308 mL is removed?
- State how many significant figures are proper in the results of the following calculations: (a) (106.7)(98.2) / (46.210)(1.01); (b) (18.7) 2 ; (c) (1.60 × 10 −19 )(3712)
- (a) How many significant figures are in the numbers 99 and 100.? (b) If the uncertainty in each number is 1, what is the percent uncertainty in each? (c) Which is a more meaningful way to express the accuracy of these two numbers: significant figures or percent uncertainties?
- (a) If your speedometer has an uncertainty of 2.0 km/h at a speed of 90 km/h, what is the percent uncertainty? (b) If it has the same percent uncertainty when it reads 60 km/ h, what is the range of speeds you could be going?
- (a) A person’s blood pressure is measured to be 120 ± 2 mm Hg. What is its percent uncertainty? (b) Assuming the same percent uncertainty, what is the uncertainty in a blood pressure measurement of 80 mm Hg?
- A person measures his or her heart rate by counting the number of beats in 30 s. If 40 ± 1 beats are counted in 30.0 ± 0.5 s, what is the heart rate and its uncertainty in beats per minute?
- What is the area of a circle 3.102 cm in diameter?
- Determine the number of significant figures in the following measurements: (a) 0.0009, (b) 15,450.0, (c) 6×103 , (d) 87.990, and (e) 30.42.
- Perform the following calculations and express your answer using the correct number of significant digits. (a) A woman has two bags weighing 13.5 lb and one bag with a weight of 10.2 lb. What is the total weight of the bags? (b) The force F on an object is equal to its mass m multiplied by its acceleration a. If a wagon with mass 55 kg accelerates at a rate of 0.0255 m/s 2 , what is the force on the wagon? (The unit of force is called the newton and it is expressed with the symbol N.)
Additional Problems
- Consider the equation y = mt +b, where the dimension of y is length and the dimension of t is time, and m and b are constants. What are the dimensions and SI units of (a) m and (b) b?
- Consider the equation \(s = s_{0} + v_{0} t + \frac{a_{0} t^{2}}{2} + \frac{j_{0} t^{3}}{6} + \frac{S_{0} t^{4}}{24} + \frac{ct^{5}}{120}\), where s is a length and t is a time. What are the dimensions and SI units of (a) s 0 , (b) v 0 , (c) a 0 , (d) j 0 , (e) S 0 , and (f) c?
- (a) A car speedometer has a 5% uncertainty. What is the range of possible speeds when it reads 90 km/h? (b) Convert this range to miles per hour. Note 1 km = 0.6214 mi.
- A marathon runner completes a 42.188-km course in 2 h, 30 min, and 12 s. There is an uncertainty of 25 m in the distance traveled and an uncertainty of 1 s in the elapsed time. (a) Calculate the percent uncertainty in the distance. (b) Calculate the percent uncertainty in the elapsed time. (c) What is the average speed in meters per second? (d) What is the uncertainty in the average speed?
- The sides of a small rectangular box are measured to be 1.80 ± 0.1 cm, 2.05 ± 0.02 cm, and 3.1 ± 0.1 cm long. Calculate its volume and uncertainty in cubic centimeters.
- When nonmetric units were used in the United Kingdom, a unit of mass called the pound-mass (lbm) was used, where 1 lbm = 0.4539 kg. (a) If there is an uncertainty of 0.0001 kg in the pound-mass unit, what is its percent uncertainty? (b) Based on that percent uncertainty, what mass in pound-mass has an uncertainty of 1 kg when converted to kilograms?
- The length and width of a rectangular room are measured to be 3.955 ± 0.005 m and 3.050 ± 0.005 m. Calculate the area of the room and its uncertainty in square meters.
- A car engine moves a piston with a circular cross-section of 7.500 ± 0.002 cm in diameter a distance of 3.250 ± 0.001 cm to compress the gas in the cylinder. (a) By what amount is the gas decreased in volume in cubic centimeters? (b) Find the uncertainty in this volume.
Challenge Problems
-
The first atomic bomb was detonated on July 16, 1945, at the Trinity test site about 200 mi south of Los Alamos. In 1947, the U.S. government declassified a film reel of the explosion. From this film reel, British physicist G. I. Taylor was able to determine the rate at which the radius of the fireball from the blast grew. Using dimensional analysis, he was then able to deduce the amount of energy released in the explosion, which was a closely guarded secret at the time. Because of this, Taylor did not publish his results until 1950. This problem challenges you to recreate this famous calculation.
- Using keen physical insight developed from years of experience, Taylor decided the radius r of the fireball should depend only on time since the explosion, t, the density of the air, ρ, and the energy of the initial explosion, E. Thus, he made the educated guess that \(r = kE^{a} \rho^{b} t^{c}\) for some dimensionless constant k and some unknown exponents a, b, and c. Given that [E] = ML 2 T –2 , determine the values of the exponents necessary to make this equation dimensionally consistent. (Hint: Notice the equation implies that \(k = rE^{-a} \rho^{-b} t^{-c}\) and that [k] = 1.)
- By analyzing data from high-energy conventional explosives, Taylor found the formula he derived seemed to be valid as long as the constant k had the value 1.03. From the film reel, he was able to determine many values of r and the corresponding values of t. For example, he found that after 25.0 ms, the fireball had a radius of 130.0 m. Use these values, along with an average air density of 1.25 kg/m 3 , to calculate the initial energy release of the Trinity detonation in joules (J). (Hint: To get energy in joules, you need to make sure all the numbers you substitute in are expressed in terms of SI base units.) (c) The energy released in large explosions is often cited in units of “tons of TNT” (abbreviated “t TNT”), where 1 t TNT is about 4.2 GJ. Convert your answer to (b) into kilotons of TNT (that is, kt TNT). Compare your answer with the quick-and-dirty estimate of 10 kt TNT made by physicist Enrico Fermi shortly after witnessing the explosion from what was thought to be a safe distance. (Reportedly, Fermi made his estimate by dropping some shredded bits of paper right before the remnants of the shock wave hit him and looked to see how far they were carried by it.)
- The purpose of this problem is to show the entire concept of dimensional consistency can be summarized by the old saying “You can’t add apples and oranges.” If you have studied power series expansions in a calculus course, you know the standard mathematical functions such as trigonometric functions, logarithms, and exponential functions can be expressed as infinite sums of the form \(\sum_{n=0}^{\infty} a_{n} x^{n} = a_{0} + a_{1} x + a_{2} x^{2} + a^{3} x^{3} + \cdotp \cdotp \cdotp,\) where the a n are dimensionless constants for all n = 0, 1, 2, ⋯ and x is the argument of the function. (If you have not studied power series in calculus yet, just trust us.) Use this fact to explain why the requirement that all terms in an equation have the same dimensions is sufficient as a definition of dimensional consistency. That is, it actually implies the arguments of standard mathematical functions must be dimensionless, so it is not really necessary to make this latter condition a separate requirement of the definition of dimensional consistency as we have done in this section.
Contributors and Attributions
-
Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0) .


10.A: Review Questions
Complete as directed.
Fill in the Blanks:
- In general, each scaffold and scaffold component shall be capable supporting, without failure, its own weight and at least times the maximum intended load applied or transmitted to it.
- In general, each platform on all working levels of scaffolds shall be or decked between the front uprights in the guardrail supports.
- The front edge of all platforms shall not be more than ________ inches from the face of the work, unless guardrail systems are erected along the front edge and/or personal fall arrest systems are used.
- Each platform greater than 10 feet in length shall not extend over its support more than ________ inches, unless it is designed and installed so that the cantilevered portion of the platform is able to support employees without tipping, or has guardrails which block employee access to the cantilevered end.
- Scaffold components manufactured by different manufactures shall not be ________ unless the components fit together without force and the scaffold's structural integrity is maintained by the user.
- Guardrail systems shall be capable of withstanding, without failure, a force of at least ________ pounds (890 N) applied within ________ i nches (5.1 cm) of the top edge, in any outward or downward direction, at any point along the top edge.
- Cross braces on tubular Weldon frame scaffold ________ be used as a means of access or egress.
- Only ________ person(s) shall operate an aerial lift.
- In addition to wearing ________ each employee on a scaffold shall be provided with additional protection from falling hand tools, debris, and other small objects through the installation of toeboards, screens, or guardrail systems, or through the erection of debris nets, catch platforms, or canopy structures that contain or deflect the falling objects.
- Scaffolds shall be erected, moved, dismantled, or altered only under the supervision and direction of a ________ qualified in scaffold erection, moving, dismantling or alteration. Such activity shall be performed only by experienced and trained employees selected for such work by the competent person.


7.8: Chemical Potential and Fugacity
The chemical potential , \(\mu\), of a pure substance has as one of its definitions (Sec. 5.5) \begin{gather} \s{ \mu \defn G\m = \frac{G}{n} } \tag{7.8.1} \cond{(pure substance)} \end{gather} That is, \(\mu\) is equal to the molar Gibbs energy of the substance at a given temperature and pressure. (Section 9.2.6 will introduce a more general definition of chemical potential that applies also to a constituent of a mixture.) The chemical potential is an intensive state function.
The total differential of the Gibbs energy of a fixed amount of a pure substance in a single phase, with \(T\) and \(p\) as independent variables, is \(\dif G = -S\dif T + V\difp\) (Eq. 5.4.4). Dividing both sides of this equation by \(n\) gives the total differential of the chemical potential with these same independent variables: \begin{gather} \s{ \dif \mu = -S\m \dif T + V\m \difp } \tag{7.8.2} \cond{(pure substance, \(P{=}1\))} \end{gather} (Since all quantities in this equation are intensive, it is not necessary to specify a closed system; the amount of the substance in the system is irrelevant.)
We identify the coefficients of the terms on the right side of Eq. 7.8.2 as the partial derivatives \begin{gather} \s{ \Pd{\mu}{T}{\!p} = -S\m } \tag{7.8.3} \cond{(pure substance, \(P{=}1\))} \end{gather} and \begin{gather} \s{ \Pd{\mu}{p}{T} = V\m } \tag{7.8.4} \cond{(pure substance, \(P{=}1\))} \end{gather} Since \(V\m\) is positive, Eq. 7.8.4 shows that the chemical potential increases with increasing pressure in an isothermal process.
The standard chemical potential , \(\mu\st\), of a pure substance in a given phase and at a given temperature is the chemical potential of the substance when it is in the standard state of the phase at this temperature and the standard pressure \(p\st\).
There is no way we can evaluate the absolute value of \(\mu\) at a given temperature and pressure, or of \(\mu\st\) at the same temperature—at least not to any useful degree of precision. The values of \(\mu\) and \(\mu\st\) include the molar internal energy whose absolute value can only be calculated from the Einstein relation; see Sec. 2.6.2. We can however measure or calculate the difference \(\mu - \mu\st\). The general procedure is to integrate \(\dif\mu=V\m\difp\) (Eq. 7.8.2 with \(\dif T\) set equal to zero) from the standard state at pressure \(p\st\) to the experimental state at pressure \(p'\): \begin{gather} \s{ \mu(p') - \mu\st = \int_{p\st}^{p'} V\m \difp} \tag{7.8.5} \cond{(constant \(T\))} \end{gather}
7.8.1 Gases
For the standard chemical potential of a gas, this e-book will usually use the notation \(\mu\st\gas\) to emphasize the choice of a gas standard state.
An ideal gas is in its standard state at a given temperature when its pressure is the standard pressure. We find the relation of the chemical potential of an ideal gas to its pressure and its standard chemical potential at the same temperature by setting \(V\m\) equal to \(RT/p\) in Eq. 7.8.5: \(\mu(p') - \mu\st = \int_{p\st}^{p'} (RT/p) \difp = RT\ln(p'/p\st)\). The general relation for \(\mu\) as a function of \(p\), then, is \begin{gather} \s{ \mu = \mu\st\gas + RT\ln\frac{p}{p\st} } \tag{7.8.6} \cond{(pure ideal gas, constant \(T\))} \end{gather} This function is shown as the dashed curve in Fig. 7.6.
If a gas is not an ideal gas, its standard state is a hypothetical state. The fugacity , \(\fug\), of a real gas (a gas that is not necessarily an ideal gas) is defined by an equation with the same form as Eq. 7.8.6: \begin{gather} \s{ \mu = \mu\st\gas + RT \ln \frac{\fug}{p\st}} \tag{7.8.7} \cond{(pure gas)} \end{gather} or \begin{gather} \s{\fug \defn p\st \exp\left[ \frac{\mu-\mu\st\gas }{RT} \right]} \tag{7.8.8} \cond{(pure gas)} \end{gather} Note that fugacity has the dimensions of pressure. Fugacity is a kind of effective pressure. Specifically, it is the pressure that the hypothetical ideal gas (the gas with intermolecular forces “turned off” ) would need to have in order for its chemical potential at the given temperature to be the same as the chemical potential of the real gas (see point C in Fig. 7.6). If the gas is an ideal gas, its fugacity is equal to its pressure.
To evaluate the fugacity of a real gas at a given \(T\) and \(p\), we must relate the chemical potential to the pressure–volume behavior. Let \(\mu'\) be the chemical potential and \(\fug'\) be the fugacity at the pressure \(p'\) of interest; let \(\mu''\) be the chemical potential and \(\fug''\) be the fugacity of the same gas at some low pressure \(p''\) (all at the same temperature). Then we use Eq. 7.8.5 to write \(\mu'-\mu\st\gas = RT\ln(\fug'/p\st)\) and \(\mu''-\mu\st\gas =RT\ln(\fug''/p\st)\), from which we obtain \begin{equation} \mu' - \mu'' = RT\ln\frac{\fug'}{\fug''} \tag{7.8.9} \end{equation} By integrating \(\dif\mu = V\m\difp\) from pressure \(p''\) to pressure \(p'\), we obtain \begin{equation} \mu' - \mu'' = \int_{\mu''}^{\mu'}\dif \mu = \int_{p''}^{p'}V\m\difp \tag{7.8.10} \end{equation} Equating the two expressions for \(\mu' - \mu''\) and dividing by \(RT\) gives \begin{equation} \ln\frac{\fug'}{\fug''} = \int_{p''}^{p'}\frac{V\m}{RT} \difp \tag{7.8.11} \end{equation}
In principle, we could use the integral on the right side of Eq. 7.8.11 to evaluate \(\fug'\) by choosing the lower integration limit \(p''\) to be such a low pressure that the gas behaves as an ideal gas and replacing \(\fug''\) by \(p''\). However, because the integrand \(V\m/RT\) becomes very large at low pressure, the integral is difficult to evaluate. We avoid this difficulty by subtracting from the preceding equation the identity \begin{equation} \ln\frac{p'}{p''} = \int_{p''}^{p'}\frac{\difp}{p} \tag{7.8.12} \end{equation} which is simply the result of integrating the function \(1/p\) from \(p''\) to \(p'\). The result is \begin{equation} \ln\frac{\fug' p''}{\fug'' p'} = \int_{p''}^{p'} \left( \frac{V\m}{RT} - \frac{1}{p} \right) \difp \tag{7.8.13} \end{equation} Now we take the limit of both sides of Eq. 7.8.13 as \(p''\) approaches zero. In this limit, the gas at pressure \(p''\) approaches ideal-gas behavior, \(\fug''\) approaches \(p''\), and the ratio \(\fug' p''/\fug'' p'\) approaches \(\fug'/p'\): \begin{equation} \ln\frac{\fug'}{p'} = \int_{0}^{p'} \left( \frac{V\m}{RT} - \frac{1}{p} \right) \difp \tag{7.8.14} \end{equation} The integrand \((V\m/RT-1/p)\) of this integral approaches zero at low pressure, making it feasible to evaluate the integral from experimental data.
The fugacity coefficient \(\phi\) of a gas is defined by \begin{gather} \s{\phi\defn \frac{\fug}{p} \quad \tx{or} \quad \fug = \phi p } \tag{7.8.15} \cond{(pure gas)} \end{gather} The fugacity coefficient at pressure \(p'\) is then given by Eq. 7.8.14: \begin{gather} \s{ \ln \phi(p') = \int_{0}^{p'} \left( \frac{V\m}{RT} - \frac{1}{p} \right) \difp } \tag{7.8.16} \cond{(pure gas, constant \(T\))} \end{gather}
The isothermal behavior of real gases at low to moderate pressures (up to at least \(1\br\)) is usually adequately described by a two-term equation of state of the form given in Eq. 2.2.8: \begin{equation} V\m \approx \frac{RT}{p} + B \tag{7.8.17} \end{equation} Here \(B\) is the second virial coefficient, a function of \(T\). With this equation of state, Eq. 7.8.16 becomes \begin{equation} \ln\phi \approx \frac{Bp}{RT} \tag{7.8.18} \end{equation}
For a real gas at temperature \(T\) and pressure \(p\), Eq. 7.8.16 or 7.8.18 allows us to evaluate the fugacity coefficient from an experimental equation of state or a second virial coefficient. We can then find the fugacity from \(\fug=\phi p\).
As we will see in Sec. 9.7, the dimensionless ratio \(\phi = \fug /p\) is an example of an activity coefficient and the dimensionless ratio \(\fug /p\st\) is an example of an activity .
7.8.2 Liquids and solids
The dependence of the chemical potential on pressure at constant temperature is given by Eq. 7.8.5. With an approximation of zero compressibility, this becomes \begin{gather} \s{ \mu \approx \mu\st + V\m (p - p\st) } \tag{7.8.19} \cond{(pure liquid or solid,} \nextcond{constant \(T\))} \end{gather}