4.4: Multiply and Divide Fractions (Part 2)
Find Reciprocals
The fractions \(\dfrac{2}{3}\) and \(\dfrac{3}{2}\) are related to each other in a special way. So are \(− \dfrac{10}{7}\) and \(− \dfrac{7}{10}\). Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be 1.
\[\dfrac{2}{3} \cdot \dfrac{3}{2} = 1 \quad and \quad - \dfrac{10}{7} \left(- \dfrac{7}{10}\right) = 1 \tag{4.2.53} \nonumber \]
Such pairs of numbers are called reciprocals.
The reciprocal of the fraction \(\dfrac{a}{b}\) is \(\dfrac{b}{a}\), where \(a ≠ 0\) and \(b ≠ 0\).
A number and its reciprocal have a product of \(1\).
\[\dfrac{a}{b} \cdot \dfrac{b}{a} = 1 \tag{4.2.54}\]
To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator.
To get a positive result when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.
To find the reciprocal, keep the same sign and invert the fraction. The number zero does not have a reciprocal. Why? A number and its reciprocal multiply to \(1\). Is there any number \(r\) so that \(0 • r = 1\)? No. So, the number \(0\) does not have a reciprocal.
Find the reciprocal of each number. Then check that the product of each number and its reciprocal is \(1\).
- \(\dfrac{4}{9}\)
- \(− \dfrac{1}{6}\)
- \(− \dfrac{14}{5}\)
- \(7\)
Solution
To find the reciprocals, we keep the sign and invert the fractions.
| Find the reciprocal of \(\dfrac{4}{9}\). | The reciprocal of \(\dfrac{4}{9}\) is \(\dfrac{9}{4}\). |
Check:
| Multiply the number and its reciprocal. | \(\dfrac{4}{9} \cdot \dfrac{9}{4}\) |
| Multiply numerators and denominators. | \(\dfrac{36}{36} \) |
| Simplify. | \(1 \; \checkmark \) |
| Find the reciprocal of \(- \dfrac{1}{6}\). | The reciprocal of \(- \dfrac{1}{6}\) is \(\dfrac{6}{1}\). |
| Simplify. | \(-6 \) |
| Check. | \(- \dfrac{1}{6} \cdot (-6) = 1 \; \checkmark \) |
| Find the reciprocal of \(- \dfrac{14}{5}\). | \(- \dfrac{5}{14} \) |
| Check. | \(- \dfrac{14}{5} \cdot \left(- \dfrac{5}{14}\right) = \dfrac{70}{70} = 1 \; \checkmark \) |
| Find the reciprocal of 7. | |
| Write 7 as a fraction. | \(\dfrac{7}{1}\) |
| Write the reciprocal of \(\dfrac{7}{1}\). | \(\dfrac{1}{7} \) |
| Check. | \(7 \cdot \left(\dfrac{1}{7}\right) = 1 \; \checkmark \) |
Find the reciprocal:
- \(\dfrac{5}{7}\)
- \(− \dfrac{1}{8}\)
- \(− \dfrac{11}{4}\)
- \(14\)
- Answer a
-
\(\dfrac{7}{5}\)
- Answer b
-
\(-8\)
- Answer c
-
\(-\dfrac{4}{11}\)
- Answer d
-
\(\dfrac{1}{14}\)
Find the reciprocal:
- \(\dfrac{3}{7}\)
- \(− \dfrac{1}{12}\)
- \(− \dfrac{14}{9}\)
- \(21\)
- Answer a
-
\(\dfrac{7}{3}\)
- Answer b
-
\(-12\)
- Answer c
-
\(-\dfrac{9}{14}\)
- Answer d
-
\(\dfrac{1}{21}\)
In a previous chapter, we worked with opposites and absolute values. Table \(\PageIndex{1}\) compares opposites, absolute values, and reciprocals.
| Opposite | Absolute Value | Reciprocal |
|---|---|---|
| has opposite sign | is never negative | has same sign, fraction inverts |
Fill in the chart for each fraction in the left column:
| Number | Opposite | Absolute Value | Reciprocal |
|---|---|---|---|
| \(- \dfrac{3}{8}\) | |||
| \(\dfrac{1}{2}\) | |||
| \(\dfrac{9}{5}\) | |||
| \(-5\) |
Solution
To find the opposite, change the sign. To find the absolute value, leave the positive numbers the same, but take the opposite of the negative numbers. To find the reciprocal, keep the sign the same and invert the fraction.
| Number | Opposite | Absolute Value | Reciprocal |
|---|---|---|---|
| \(- \dfrac{3}{8}\) | \(\dfrac{3}{8}\) | \(\dfrac{3}{8}\) | \(- \dfrac{8}{3}\) |
| \(\dfrac{1}{2}\) | \(- \dfrac{1}{2}\) | \(\dfrac{1}{2}\) | \(2\) |
| \(\dfrac{9}{5}\) | \(- \dfrac{9}{5}\) | \(\dfrac{9}{5}\) | \(\dfrac{5}{9}\) |
| \(-5\) | \(5\) | \(5\) | \(- \dfrac{1}{5}\) |
Fill in the chart for each number given:
| Number | Opposite | Absolute Value | Reciprocal |
|---|---|---|---|
| \(- \dfrac{5}{8}\) | |||
| \(\dfrac{1}{4}\) | |||
| \(\dfrac{8}{3}\) | |||
| \(-8\) |
- Answer
-
Number Opposite Absolute Value Reciprocal \(-\dfrac{5}{8}\) \(\dfrac{5}{8}\) \(\dfrac{5}{8}\) \(-\dfrac{8}{5}\) \(\dfrac{1}{4}\) \(-\dfrac{1}{4}\) \(\dfrac{1}{4}\) \(4\) \(\dfrac{8}{3}\) \(-\dfrac{8}{3}\) \(\dfrac{8}{3}\) \(\dfrac{3}{8}\) \(-8\) \(8\) \(8\) \(-\dfrac{1}{8}\)
Fill in the chart for each number given:
| Number | Opposite | Absolute Value | Reciprocal |
|---|---|---|---|
| \(- \dfrac{4}{7}\) | |||
| \(\dfrac{1}{8}\) | |||
| \(\dfrac{9}{4}\) | |||
| \(-1\) |
- Answer
-
Number Opposite Absolute Value Reciprocal \(-\dfrac{4}{7}\) \(\dfrac{4}{7}\) \(\dfrac{4}{7}\) \(- \dfrac{7}{4}\) \(\dfrac{1}{8}\) \(-\dfrac{1}{8}\) \(\dfrac{1}{8}\) \(8\) \(\dfrac{9}{4}\) \(-\dfrac{9}{4}\) \(\dfrac{9}{4}\) \(\dfrac{4}{9}\) \(-1\) \(1\) \(1\) \(-\dfrac{1}{1}\)
Divide Fractions
Why is \(12 ÷ 3 = 4\)? We previously modeled this with counters. How many groups of \(3\) counters can be made from a group of \(12\) counters?
Figure \(\PageIndex{2}\)
There are \(4\) groups of \(3\) counters. In other words, there are four \(3\)s in \(12\). So, \(12 ÷ 3 = 4\).
What about dividing fractions? Suppose we want to find the quotient: \(\dfrac{1}{2} \div \dfrac{1}{6}\). We need to figure out how many \(\dfrac{1}{6}\)s there are in \(\dfrac{1}{2}\). We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown in Figure \(\PageIndex{3}\). Notice, there are three \(\dfrac{1}{6}\) tiles in \(\dfrac{1}{2}\), so \(\dfrac{1}{2} \div \dfrac{1}{6} = 3\).
Figure \(\PageIndex{3}\)
Model: \(\dfrac{1}{4} \div \dfrac{1}{8}\).
Solution
We want to determine how many \(\dfrac{1}{8}\)s are in \(\dfrac{1}{4}\). Start with one \(\dfrac{1}{4}\) tile. Line up \(\dfrac{1}{8}\) tiles underneath the \(\dfrac{1}{4}\) tile.
Model: \(\dfrac{1}{3} \div \dfrac{1}{6}\).
- Answer
-
Model: \(\dfrac{1}{2} \div \dfrac{1}{4}\).
- Answer
-
Model: \(2 ÷ \dfrac{1}{4}\).
Solution
We are trying to determine how many \(\dfrac{1}{4}\)s there are in \(2\). We can model this as shown.
Because there are eight \(\dfrac{1}{4}\)s in \(2\), \(2 ÷ \dfrac{1}{4} = 8\).
Model: \(2 ÷ \dfrac{1}{3}\)
- Answer
-
Model: \(3 ÷ \dfrac{1}{2}\)
- Answer
-
Let’s use money to model \(2 ÷ \dfrac{1}{4}\) in another way. We often read \(\dfrac{1}{4}\) as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown in Figure \(\PageIndex{4}\). So we can think of \(2 ÷ \dfrac{1}{4}\) as, “How many quarters are there in two dollars?” One dollar is \(4\) quarters, so \(2\) dollars would be \(8\) quarters. So again, \(2 ÷ \dfrac{1}{4} = 8\).
Figure \(\PageIndex{4}\):The U.S. coin called a quarter is worth one-fourth of a dollar.
Using fraction tiles, we showed that \(\dfrac{1}{2} \div \dfrac{1}{6} = 3\). Notice that \(\dfrac{1}{2} \cdot \dfrac{6}{1} = 3\) also. How are \(\dfrac{1}{6}\) and \(\dfrac{6}{1}\) related? They are reciprocals. This leads us to the procedure for fraction division.
If \(a, b, c,\) and \(d\) are numbers where \(b ≠ 0\), \(c ≠ 0\), and \(d ≠ 0\), then
\[\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \cdot \dfrac{d}{c} \]
To divide fractions, multiply the first fraction by the reciprocal of the second.
We need to say \(b ≠ 0\), \(c ≠ 0\) and \(d ≠ 0\) to be sure we don’t divide by zero.
Divide, and write the answer in simplified form: \(\dfrac{2}{5} \div \left(- \dfrac{3}{7}\right).
Solution
| Multiply the first fraction by the reciprocal of the second. | \(\dfrac{2}{5} \left(- \dfrac{7}{3}\right) \) |
| Multiply. The product is negative. | \(- \dfrac{14}{15}\) |
Divide, and write the answer in simplified form: \(\dfrac{3}{7} \div \left(− \dfrac{2}{3}\right)\).
- Answer
-
\(-\dfrac{9}{14}\)
Divide, and write the answer in simplified form: \(\dfrac{2}{3} \div \left(− \dfrac{7}{5}\right)\).
- Answer
-
\(-\dfrac{10}{21}\)
Divide, and write the answer in simplified form: \(\dfrac{2}{3} \div \dfrac{n}{5}\).
Solution
| Multiply the first fraction by the reciprocal of the second. | \(\dfrac{2}{3} \div \dfrac{5}{n} \) |
| Multiply. | \(\dfrac{10}{3n}\) |
Divide, and write the answer in simplified form: \(\dfrac{3}{5} \div \dfrac{p}{7}\).
- Answer
-
\(\dfrac{21}{5p}\)
Divide, and write the answer in simplified form: \(\dfrac{5}{8} \div \dfrac{q}{3}\).
- Answer
-
\(\dfrac{15}{8q}\)
Divide, and write the answer in simplified form: \(− \dfrac{3}{4} \div \left(− \dfrac{7}{8}\right)\).
Solution
| Multiply the first fraction by the reciprocal of the second. | \(- \dfrac{3}{4} \cdot \left(- \dfrac{8}{7}\right) \) |
| Multiply. Remember to determine the sign first. | \(\dfrac{3 \cdot 8}{4 \cdot 7}\) |
| Rewrite to show common factors. | \(\dfrac{3 \cdot \cancel{4} \cdot 2}{\cancel{4} \cdot 7} \) |
| Remove common factors and simplify. | \(\dfrac{6}{7} \) |
Divide, and write the answer in simplified form: \(− \dfrac{2}{3} \div \left(− \dfrac{5}{6}\right)\).
- Answer
-
\(\dfrac{4}{5}\)
Divide, and write the answer in simplified form: \(− \dfrac{5}{6} \div \left(− \dfrac{2}{3}\right)\).
- Answer
-
\(\dfrac{5}{4}\)
Divide, and write the answer in simplified form: \(\dfrac{7}{18} \div \dfrac{14}{27}\).
Solution
| Multiply the first fraction by the reciprocal of the second. | \(\dfrac{7}{18} \cdot \dfrac{27}{14} \) |
| Multiply. | \(\dfrac{7 \cdot 27}{18 \cdot 14} \) |
| Rewrite showing common factors. | \(\dfrac{\cancel{\textcolor{red}{7}} \cdot \cancel{\textcolor{red}{9}} \cdot 3}{\cancel{\textcolor{red}{9}} \cdot \cancel{\textcolor{red}{7}} \cdot 2}\) |
| Remove common factors. | \(\dfrac{3}{2 \cdot 2} \) |
| Simplify. | \(\dfrac{3}{4} \) |
Divide, and write the answer in simplified form: \(\dfrac{7}{27} \div \dfrac{35}{36}\).
- Answer
-
\(\dfrac{4}{15}\)
Divide, and write the answer in simplified form: \(\dfrac{5}{14} \div \dfrac{15}{28}\).
- Answer
-
\(\dfrac{2}{3}\)
Access Additional Online Resources
- Simplifying Fractions
- Multiplying Fractions (Positive Only)
- Multiplying Signed Fractions
- Dividing Fractions (Positive Only)
- Dividing Signed Fractions
Key Concepts
- Equivalent Fractions Property
- If \(a, b, c\) are numbers where \(b\neq 0, c\neq 0\), then \(\dfrac{a}{b} = \dfrac{a\cdot c}{b\cdot c}\) and \(\dfrac{a\cdot c}{b\cdot c} = \dfrac{a}{b}\)
-
Simplify a fraction.
- Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
- Simplify, using the equivalent fractions property, by removing common factors.
- Multiply any remaining factors.
-
Fraction Multiplication
- If \(a, b, c,\) and
-
Reciprocal
- A number and its reciprocal have a product of 1. \(\frac{a}{b} \cdot \frac{b}{a} = 1\)
-
Opposite Absolute Value Reciprocal has opposite sign is never negative has same sign, fraction inverts
-
Fraction Division
- If \(a, b, c,\) and \(d\) are numbers where \(b\neq 0\), \(c\neq 0\), and \(d\neq 0\), then \(\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b}\cdot \dfrac{d}{c}\)
- To divide fractions, multiply the first fraction by the reciprocal of the second.
Glossary
- reciprocal
-
The reciprocal of the fraction \(\dfrac{a}{b}\) is \(\dfrac{b}{a}\) where \(a\neq 0\) and \(b\neq 0\).
- simplified fraction
-
A fraction is considered simplified if there are no common factors in the numerator and denominator.
Practice Makes Perfect
Simplify Fractions
In the following exercises, simplify each fraction. Do not convert any improper fractions to mixed numbers.
- \(\dfrac{7}{21}\)
- \(\dfrac{8}{24}\)
- \(\dfrac{15}{20}\)
- \(\dfrac{12}{18}\)
- \(- \dfrac{40}{88}\)
- \(- \dfrac{63}{99}\)
- \(- \dfrac{108}{63}\)
- \(- \dfrac{104}{48}\)
- \(\dfrac{120}{252}\)
- \(\dfrac{182}{294}\)
- \(- \dfrac{168}{192}\)
- \(- \dfrac{140}{224}\)
- \(\dfrac{11x}{11y}\)
- \(\dfrac{15a}{15b}\)
- \(− \dfrac{3x}{12y}\)
- \(− \dfrac{4x}{32y}\)
- \(\dfrac{14x^{2}}{21y}\)
- \(\dfrac{24a}{32b^{2}}\)
Multiply Fractions
In the following exercises, use a diagram to model.
- \(\dfrac{1}{2} \cdot \dfrac{2}{3}\)
- \(\dfrac{1}{2} \cdot \dfrac{5}{8}\)
- \(\dfrac{1}{3} \cdot \dfrac{5}{6}\)
- \(\dfrac{1}{3} \cdot \dfrac{2}{5}\)
In the following exercises, multiply, and write the answer in simplified form.
- \(\dfrac{2}{5} \cdot \dfrac{1}{3}\)
- \(\dfrac{1}{2} \cdot \dfrac{3}{8}\)
- \(\dfrac{3}{4} \cdot \dfrac{9}{10}\)
- \(\dfrac{4}{5} \cdot \dfrac{2}{7}\)
- \(− \dfrac{2}{3} \left(− \dfrac{3}{8}\right)\)
- \(− \dfrac{3}{4} \left(− \dfrac{4}{9}\right)\)
- \(- \dfrac{5}{9} \cdot \dfrac{3}{10}\)
- \(- \dfrac{3}{8} \cdot \dfrac{4}{15}\)
- \(− \dfrac{7}{12} \left(− \dfrac{8}{21}\right)\)
- \(\dfrac{5}{12} \left(− \dfrac{8}{15}\right)\)
- \(\left(− \dfrac{14}{15}\right) \left(\dfrac{9}{20}\right)\)
- \(\left(− \dfrac{9}{10}\right) \left(\dfrac{25}{33}\right)\)
- \(\left(− \dfrac{63}{84}\right) \left(- \dfrac{44}{90}\right)\)
- \(\left(− \dfrac{33}{60}\right) \left(- \dfrac{40}{88}\right)\)
- \(4 \cdot \dfrac{5}{11}\)
- \(5 \cdot \dfrac{8}{3}\)
- \(\dfrac{3}{7} \cdot 21n\)
- \(\dfrac{5}{6} \cdot 30m\)
- \(−28p \left(− \dfrac{1}{4}\right)\)
- \(−51q \left(− \dfrac{1}{3}\right)\)
- \(−8 \left(\dfrac{17}{4}\right)\)
- \(\dfrac{14}{5} (−15)\)
- \(−1 \left(− \dfrac{3}{8}\right)\)
- \((−1) \left(- \dfrac{6}{7}\right)\)
- \(\left(\dfrac{2}{3}\right)^{3}\)
- \(\left(\dfrac{4}{5}\right)^{2}\)
- \(\left(\dfrac{6}{5}\right)^{4}\)
- \(\left(\dfrac{4}{7}\right)^{4}\)
Find Reciprocals In the following exercises, find the reciprocal.
- \(\dfrac{3}{4}\)
- \(\dfrac{2}{3}\)
- \(− \dfrac{5}{17}\)
- \(− \dfrac{6}{19}\)
- \(\dfrac{11}{8}\)
- −13
- −19
- −1
- 1
-
Fill in the chart.
Opposite Absolute Value Reciprocal \(- \dfrac{7}{11}\) \(\dfrac{4}{5}\) \(\dfrac{10}{7}\) \(-8\) -
Fill in the chart.
Opposite Absolute Value Reciprocal \(- \dfrac{3}{13}\) \(\dfrac{9}{14}\) \(\dfrac{15}{7}\) \(-9\)
Divide Fractions
In the following exercises, model each fraction division.
- \(\dfrac{1}{2} \div \dfrac{1}{4}\)
- \(\dfrac{1}{2} \div \dfrac{1}{8}\)
- \(2 \div \dfrac{1}{5}\)
- \(3 \div \dfrac{1}{4}\)
In the following exercises, divide, and write the answer in simplified form.
- \(\dfrac{1}{2} \div \dfrac{1}{4}\)
- \(\dfrac{1}{2} \div \dfrac{1}{8}\)
- \(\dfrac{3}{4} \div \dfrac{2}{3}\)
- \(\dfrac{4}{5} \div \dfrac{3}{4}\)
- \(- \dfrac{4}{5} \div \dfrac{4}{7}\)
- \(- \dfrac{3}{4} \div \dfrac{3}{5}\)
- \(− \dfrac{7}{9} \div \left(- \dfrac{7}{9}\right)\)
- \(− \dfrac{5}{6} \div \left(- \dfrac{5}{6}\right)\)
- \(\dfrac{3}{4} \div \dfrac{x}{11}\)
- \(\dfrac{2}{5} \div \dfrac{y}{9}\)
- \(\dfrac{5}{8} \div \dfrac{a}{10}\)
- \(\dfrac{5}{6} \div \dfrac{c}{15}\)
- \(\dfrac{5}{18} \div \left(- \dfrac{15}{24}\right)\)
- \(\dfrac{7}{18} \div \left(- \dfrac{14}{27}\right)\)
- \(\dfrac{7p}{12} \div \dfrac{21p}{8}\)
- \(\dfrac{5q}{12} \div \dfrac{15q}{8}\)
- \(\dfrac{8u}{15} \div \dfrac{12v}{25}\)
- \(\dfrac{12r}{25} \div \dfrac{18s}{35}\)
- \(-5 \div \dfrac{1}{2}\)
- \(-3 \div \dfrac{1}{4}\)
- \(\dfrac{3}{4} \div (-12)\)
- \(\dfrac{2}{5} \div (-10)\)
- \(−18 \div \left(− \dfrac{9}{2}\right)\)
- \(−15 \div \left(− \dfrac{5}{3}\right)\)
- \(\dfrac{1}{2} \div \left(- \dfrac{3}{4}\right) \div \dfrac{7}{8}\)
- \(\dfrac{11}{2} \div \dfrac{7}{8} \cdot \dfrac{2}{11}\)
Everyday Math
-
Baking
A recipe for chocolate chip cookies calls for 3 4 cup brown sugar. Imelda wants to double the recipe.
- How much brown sugar will Imelda need? Show your calculation. Write your result as an improper fraction and as a mixed number.
- Measuring cups usually come in sets of \(\dfrac{1}{8}, \dfrac{1}{4}, \dfrac{1}{3}, \dfrac{1}{2}\), and 1 cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the recipe.
-
Baking
Nina is making 4 pans of fudge to serve after a music recital. For each pan, she needs 2 3 cup of condensed milk.
- How much condensed milk will Nina need? Show your calculation. Write your result as an improper fraction and as a mixed number.
- Measuring cups usually come in sets of \(\dfrac{1}{8}, \dfrac{1}{4}, \dfrac{1}{3}, \dfrac{1}{2}\), and 1 cup. Draw a diagram to show two different ways that Nina could measure the condensed milk she needs.
- Portions Don purchased a bulk package of candy that weighs 5 pounds. He wants to sell the candy in little bags that hold \(\dfrac{1}{4}\) pound. How many little bags of candy can he fill from the bulk package?
- Portions Kristen has \(\dfrac{3}{4}\) yards of ribbon. She wants to cut it into equal parts to make hair ribbons for her daughter’s 6 dolls. How long will each doll’s hair ribbon be?
Writing Exercises
- Explain how you find the reciprocal of a fraction.
- Explain how you find the reciprocal of a negative fraction.
- Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into 6 or 8 slices. Would he prefer 3 out of 6 slices or 4 out of 8 slices? Rafael replied that since he wasn’t very hungry, he would prefer 3 out of 6 slices. Explain what is wrong with Rafael’s reasoning.
- Give an example from everyday life that demonstrates how \(\dfrac{1}{2} \cdot \dfrac{2}{3}\) is \(\dfrac{1}{3}\).
Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) After reviewing this checklist, what will you do to become confident for all objectives?
Contributors and Attributions
-
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at<EMAIL_ADDRESS>."


12.2: Interpersonal Communication and the Workplace
Making the Connection
People do not always associate interpersonal communication and organizational communication. However, we do not live in silos, nor are we robots. The workplace is full of interpersonal interactions. According to McKeown & Ayoko:
Often it is the people who make or break a job for us—getting on with those we work with impacts on our happiness is not just work but in our wider life as well as on our physical and mental health. Some help us grow personally and professionally. Some serve our need for social interaction in our daily lives. Some can be harmful and inappropriate, for which we should be empowered to stand up against. So let’s look at the basics of why we are inclined to engage in interpersonal communication at work. (2020, para. 1)
Why Learn Professional Interpersonal Communication Skills?
In the pie chart in Figure \(\PageIndex{1}\) shows that work and work-related activities take up the bulk of most Americans' days with 8.9 hours or 37% of our total day. Sleeping and personal care follow closely at 8.64 hours or 35%. Leisure and sports account for 3.01 hours or 13%. We spend about 1 hour each day in the following activities, which each account for about 4% of our day: eating and drinking, household activities, and a category labeled "Other." Caring for others came in at 0.59 hours, or 2%. of our time.
A survey from the Bureau of Labor Statistics shows that most people spend more time working and/or spending time with people from work than they do on any other activity on a weekday, even more than time spent sleeping. In fact, given this information, consider the time spent with family, friends, children, or any other person with which you have an interpersonal relationship. You might see that those working relationships take up a significant amount of your day.
However, learning about effective interpersonal communication skills is more than just a numbers game of hours spent in a day. If you recall from Chapter 1, many researchers have pointed to how interpersonal communication can help meet our needs as humans (Maslow, 1987; Schutz, 1958). Being that many of us spend so much of our time at our workplace, it could be advantageous to utilize this working time to help meet more of our interpersonal needs. In addition, if you remember our previous chapters on the self, verbal and nonverbal communication, and perception (Chapters 2 through 5), the way we communicate with others has an impact on how we view ourselves and those around us. Practicing our interpersonal communication skills at work further reinforces things like our identity, our observations, and our relationships. Let’s review these concepts and discuss some specific examples present in the workplace.
Interpersonal Communication Meets Our Basic Needs
In Chapter 1 you learned that communication helps to meet our basic needs. When it comes to the workplace, research from Cheung (as cited in Morgan, 2020) shows us that many people gain a sense of belonging from their jobs, and that even employee happiness is related to their social interactions at work. Psychologists have cited interpersonal relationships having a considerable impact on our health, to include mental, behavioral, and physical health (Umberson & Montez, 2010). Probably best related to interpersonal communication are esteem needs , for which reputation recognition, attention, or appreciation (as defined by others) (Maslow, 1943) are met. Our workplace is a common venue through which we gain a reputation, attention, appreciation or recognition. Therefore, our workspaces often serve as a significant source for meeting our esteem needs.
Interpersonal Communication Influences Your Thinking about Yourself
As you learned in Chapter 2, research by George Herbert Mead (1934) shows that our understanding of ourselves is shaped by interacting with others. Because we learn, grow, and develop at our places of work, our interactions with our co-workers help to shape how we see ourselves through theories like Reflective Appraisal (Mead, 1934; Cooley, 1902; Sullivan, 1947) and Social Comparison (Festinger, 1954). From how you dress to the praise or criticism you receive on a project, our interpersonal interactions in the workplace continuously shape our self-concept. And since we have already reviewed the data about the amount of waking hours spent at our workplace, consider just how much impact your workplace relationships have on your self-concept.
Interpersonal Communication Influences Your Thinking about Others
As we learned in Chapter 3, our understanding of others is highly influenced by our perceptions (Knudsen, et al., 202). At the workplace, this may look like past experiences with bosses, colleagues, customers, and meeting dynamics, which could have shaped how we expect others to interact or behave. Perhaps by starting a new job or working with a new colleague, you might notice that your understanding of others’ roles at work is contextual, or learned through past experiences. The competent communicator could recognize how perception can create biases in the workplace that could influence pay, opportunities, leadership roles, and even harassment in the workplace. For example, say that your last employer was very strict regarding punctuality. Being late was frowned upon and happened rarely. At your new place of work, your colleagues seem to be very relaxed in their approach to punctuality. Arriving to a meeting with a coffee in hand, 5–10 minutes late, is not uncommon. However, you find yourself judging those who are late, perhaps even labeling them as unprofessional or disrespectful. Your past working norms have influenced your understanding of what is, and is not, an appropriate way to interact with others.
Employers Look for Interpersonal Communication Skills
Have you ever been interested in a job as a brand strategist, social media manager, content marketing manager, web producer, social media planner, or public relations specialist? If so, these were reported as some of the top Communications degree jobs in 2021 (Keiling, 2021). But it is not just for Communication-specific titles that employers look for strong communication skills. In Figure \(\PageIndex{3}\) we show the communication-specific skills that were cited in the Job Outlook 2021 Spring Update by the National Association of Colleges and Employers—skills in which communication, working with others, and/or the combination of the two, essentially dominate the attributes employers report seeking out in candidates.
When you look at the various jobs you’ve had throughout your life, whether they be typical nine-to-five office positions, part-time work, or chores around your living space, what made your work more effective and enjoyable? Did you have a co-worker that would make you laugh to get you through a long shift? A boss that was especially kind when giving praise? Did you have a parent or roommate that set clear expectations about when things needed to be done or offered help when you could not carry your own weight? It may seem obvious now that you look back, but those interpersonal communication dynamics in a workspace are particularly impactful to how effective and enjoyable our work is. In fact, they could be the difference between loving or hating the work, and motivation to get the work done and at what quality.
Read this article: Interpersonal Skills: Definitions and Examples .
Reflection Questions
- Out of the nine interpersonal skills listed, which do you think are your strengths?
- The article mentions using your interpersonal skills from the job interview to the daily interactions at work. How have you seen these show up effectively in your own experience?
- The video embedded in the article talks about EQ. What is EQ and how is it related to workplace communication?
Why Do We Seek Out Relationships at Work?
If you take stock of your own workplace experiences, or those of people you know, you might notice co-workers at children's birthday parties, receiving invites to a colleague’s wedding, or meeting work-friends for drinks. You may know past or present co-workers who are dating, or even getting married themselves.
We know from interpersonal communication research that we seek out relationships with others based on proximity, similarity, and physical attractiveness. When you consider how physically close a co-worker is to us for much of our waking hours, along with all we might have in common with that person (we work on the same industry, we possibly have the same boss and co-workers, we may have a similar commute, and we might be close in socioeconomic class, etc.), you could see similarities emerging quickly. Therefore, relationships at work are very convenient for us.
In addition to ease and accessibility of relationships, there are emotional and physical benefits to workplace relationships, as many researchers point out. An article from the Harvard Business Review cites studies showing that “teams of friends perform better; that people with supportive co-workers have more work/life balance and are less stressed; that strong personal ties increase information- and idea-sharing, self-confidence” (Beard, 2020, para. 3). Furthermore, this article cites a book written by former US surgeon general Vivek Murthy, who refers to friendship as “fundamental to successful professional relationships” and says that by just having a friend around significantly impacts our blood pressure and immune cells" (para. 4).
In most workplaces, the most basic level of communication that we will have is in our interpersonal interactions—in the form of interactions with co-workers or customers/clients, organizational leaders, or subordinates. Therefore, most communication will flow either vertically or horizontally. Communication amongst superiors and subordinates is considered vertical communication , which flows either upward or downward, while communication with our colleagues and customers or clients is considered horizontal communication . Communication across these channels can be structured formally, due to organizational processes such as with job instructions or contracts or performance reviews, or can occur more informally through networks that are created by the types of relationships established in the workplace. In the next section, we describe some of the types of workplace relationships where dyadic interpersonal communication plays an important role.


3.1.14: William Faulkner (1897-1962)
Selected Stories American Modernism Considered by many scholars to be the most distinguished writer of the 20th century, Faulkner was born in Oxford, Mississippi in 1897. He is mostly known for his novels and short stories, set in the fictional Yoknapatawpha County in Mississippi. Both his novels and short stories confront the complexities of Southern culture, shaped by its heritage of slavery, the loss of the American Civil War, and continued struggles with racism through Jim Crow laws and the atrocities committed by the Ku Klux Klan. He is also keenly aware of the close ties of Southerners to the land and the ways that the old agrarian values continued to shape ideas about class in the South well into the 20th century. Faulkner draws on family histories as well as aspects of Southern gothic ghost stories in his novels, and most of his works explore the complex and troubled mix of race and sexuality in the South. Most of his works were published in the 1920s and 30s, among which "A Rose for Emily" (1930) and "Barn Burning" (1939) are included here, but he was primarily known in America as a Southern writer until he won the Nobel Prize for Literature in 1942. His Nobel Prize acceptance speech is often reprinted for his optimistic declaration of the importance of art. After decrying the anxiety and pessimism that he felt characterized the literature of the period, Faulkner declared that humanity would prevail because of the strength of the human spirit: "The poet's, the writer's, duty is to write about these things. It is his privilege to help man endure by . . . reminding him of the courage and honor and hope and pride and compassion and pity and sacrifice which have been the glory of his past. The poet's voice need not merely be the record of man, it can be one of the props, the pillars to help him endure and prevail." Consider while reading:
- In "A Rose for Emily," the story of Emily's life and death is told through the voice of the townspeople. How does this technique affect the story and what we know and don't know about Emily? How does it affect the timeline of the story, which is not told in chronological order?
- What elements in the story can be considered "gothic"?
- What is the significance of the "rose" in the title of the story? What connotations of the word are meaningful in the context of Emily's life?
- In "Barn Burning," why is burning a barn such a serious crime?
- How would you characterize Abner? Sarty? What is the primary problem between the two of them?
- Why does Sarty ultimately betray his father at the end of the story?
Written by Anita Turlington


3: Introduction to Brownian Motion
This chapter introduces Brownian motion as a model of trait evolution. I first connected Brownian motion to a model of neutral genetic drift for traits that have no effect on fitness. However, as I demonstrated, Brownian motion can result from a variety of other models, some of which include natural selection. For example, traits will follow Brownian motion under selection is if the strength and direction of selection varies randomly through time. In other words, testing for a Brownian motion model with your data tells you nothing about whether or not the trait is under selection.
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- 3.1: Introduction to Brownian Motion
- Imagine that you want to use statistical approaches to understand how traits change through time. This requires an exact mathematical specification of how evolution takes place. Obviously there are a wide variety of models of trait evolution, from simple to complex. e.g., creating a model where a trait starts with a certain value and has some constant probability of changing in any unit of time or an alternative model that is more detailed and explicit and considers a large set of individuals.
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- 3.2: Properties of Brownian Motion
- We can use Brownian motion to model the evolution of a continuously valued trait through time. Brownian motion is an example of a “random walk” model because the trait value changes randomly, in both direction and distance, over any time interval. The statistical process of Brownian motion was originally invented to describe the motion of particles suspended in a fluid.
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- 3.5: Multivariate Brownian motion
- The Brownian motion model we described above was for a single character. However, we often want to consider more than one character at once. This requires the use of multivariate models. The situation is more complex than the univariate case – but not much! In this section I will derive the expectation for a set of (potentially correlated) traits evolving together under a multivariate Brownian motion model.
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- 3.6: Simulating Brownian motion on trees
- To simulate Brownian motion evolution on trees, we use the three properties of the model described above. For each branch on the tree, we can draw from a normal distribution (for a single trait) or a multivariate normal distribution (for more than one trait) to determine the evolution that occurs on that branch. We can then add these evolutionary changes together to obtain character states at every node and tip of the tree.


13.1.3: Observatories and Early Astronomy Clubs
Observatories and Early Astronomy Clubs
Interest in astronomy was strong, so setting up telescopes in public was regularly done. Today, these events are usually called star parties, where the public is invited to come to a park, museum, or other site to look through telescopes often staffed by amateur astronomers. Perhaps, one of the best known of these groups is the San Francisco Sidewalk Astronomers, started by the late John Dobson.
Many astronomy clubs, colleges/universities, museums/science centers, and even private individuals have observatories: a permanent housing for their telescope or telescopes. A number of these open for the public on regular occasions, or for special astronomical events like an eclipse or when a planet is in the position for optimum viewing.


15.6: Free Energy and Equilibrium
You are expected to be able to define and explain the significance of terms identified in italic type .
- As a homogeneous chemical reaction proceeds, the Gibbs energies of the reactants become more negative and those of the products more positive as the composition of the system changes.
- The total Gibbs energy of the system (reactants + products) always becomes more negative as the reaction proceeds. Eventually it reaches a minimum value at a system composition that defines the equilibrium composition of the system, after which time no further net change will occur.
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The
equilibrium constant
for the reaction is determined the standard Gibbs energy change:
ΔG° = - RT ln K p - The sign of the temperature dependence of the equilibrium constant is governed by the sign of Δ H °. This is the basis of the Le Chatelier Principle .
- The Gibbs energies of solid and liquid components are constants that do not change with composition. Thus in heterogeneous reactions such as phase changes, the total Gibbs energy does not pass through a minimum and when the system is not at equilibrium only all-products or all-reactants will be stable.
- Two reactions are coupled when the product of one reaction is consumed in the other. If Δ G ° for the first reaction is positive, the overall process can still be spontaneous if Δ G ° for the second reaction is sufficiently negative— in which case the second reaction is said to "drive" the first reaction.
Under conditions of constant temperature and pressure, chemical change will tend to occur in whatever direction leads to a decrease in the value of the Gibbs energy . In this lesson we will see how G varies with the composition of the system as reactants change into products. When G falls as far as it can, all net change comes to a stop. The equilibrium composition of the mixture is determined by ΔG° which also defines the equilibrium constant K.
The Road to Equilibrium is Down the Gibbs Energy Hill
This means, of course, that if the total Gibbs energy \(G\) of a mixture of reactants and products goes through a minimum value as the composition changes, then all net change will cease— the reaction system will be in a state of chemical equilibrium . You will recall that the relative concentrations of reactants and products in the equilibrium state is expressed by the equilibrium constant . In this lesson we will examine the relation between the Gibbs energy change for a reaction and the equilibrium constant.
To keep things as simple as possible, we will consider a homogeneous chemical reaction of the form
\[A + B \rightleftharpoons C + D\]
in which all components are gases at the temperature of interest. If the sum of the standard Gibbs energies of the products is less than that of the reactants, Δ G ° for the reaction will be negative and the reaction will proceed to the right. But how far? If the reactants are completely transformed into products, the equilibrium constant would be infinity. The equilibrium constants we actually observe all have finite values, implying that even if the products have a lower Gibbs energy than the reactants, some of the latter will always remain when the process comes to equilibrium.
A homogeneous reaction is one in which everything takes place in a single gas or liquid phase.
To understand how equilibrium constants relate to Δ G ° values, assume that all of the reactants are gases, so that the Gibbs energy of gas A, for example, is given at all times by
\[G_A = G_A^° + RT \ln P_A \label{5-1}\]
The Gibbs energy change for the reaction is sum of the Gibbs energies of the products, minus the sum of Gibbs energies of the reactants:
\[\Delta G = \underbrace{G_C + G_D}_{\text{products}} \underbrace{– G_A – G_B}_{\text{reactants}} \label{5-2}\]
Using Equation \(\ref{5-1}\) to expand each term on the right of Equation \ref{5-2}, we have
\[\Delta G = (G^°_C + RT \ln P_C) + (G^°_D + RT \ln P_D) – (G^°_B + RT \ln P_B) – (G^°_A + RT \ln P+A) \label{5-3}\]
We can now express the \(G^°\) terms collectively as \(\Delta G^°\), and combine the logarithmic pressure terms into a single fraction
\[ \Delta G = \Delta G° + RT \ln \left( \dfrac{P_CP_D}{P_AP_B} \right) \label{5-4}\]
which is more conveniently expressed in terms of the reaction quotient \(Q\).
\[\Delta{G} = \Delta G^° + RT \ln Q \label{5-5}\]
The Gibbs energy \(G\) is a quantity that becomes more negative during the course of any natural process. Thus as a chemical reaction takes plac e, \(G\) only falls and will never become more positive. Eventually a point is reached where any further transformation of reactants into products would cause \(G\) to increase. At this point \(G\) is at a minimum (see below), and no further net change can take place; the reaction is then at equilibrium.
Although Equations \ref{5-1}-\ref{5-5} are strictly correct only for perfect gases, we will see later that equations of similar form can be applied to many liquid solutions by substituting concentrations for pressures.
Consider the gas-phase dissociation reaction
\[\ce{N_2O_4 \rightarrow 2 NO_2 } \nonumber\]
which is a simple example of the Gibbs energy relationships in a homogeneous reaction.
The Gibbs energy of 1 mole of N 2 O 4 (1) is smaller than that of 2 moles of NO 2 (2) by 5.3 kJ; thu s \( \Delta G^o = +5.3\, \text{kJ}\) fo r the complete transformation of reactants into products. The straight diagonal line shows the Gibbs energy of all possible compositions if the two gases were prevented from mixing. The red curved line show the Gibbs energy of the actual reaction mixture. This passes through a minimum at (3) where 0.814 mol of \(N_2O_4\) are in equilibrium with 0.372 mol of \(NO_2\). The difference (4) corresponds to the Gibbs energy of mixing of reactants and products which always results in an equilibrium mixture whose Gibbs energy is lower than that of either pure reactants or pure products. Thus some amount of reaction will occur even if Δ G ° for the process is positive.
What’s the difference between Δ G and Δ G °?
It’s very important to be aware of this distinction; that little ° symbol makes a world of difference! First, the standard Gibbs energy change Δ G ° has a single value for a particular reaction at a given temperature and pressure; this is the difference
\[ \sum G^°_{f} (\text{products}) – \sum G^°_{f}(\text{reactants}) \]
that are tabulated in thermodynamic tables . It corresponds to the Gibbs energy change for a process that never really happens: the complete transformation of pure N 2 O 4 into pure NO 2 at a constant pressure of 1 atm.
The other quantity \(\Delta G\), defined by Equation \(\ref{5-5}\), represents the total Gibbs energies of all substances in the reaction mixture at any particular system composition. In contrast to \(\Delta G^°\) which is a constant for a given reaction, \(\Delta G\) varies continuously as the composition changes, finally reaching zero at equilibrium. \(\Delta G\) is the “distance” (in Gibbs energy) from the equilibrium state of a given reaction. Thus for the limiting cases of pure \(\ce{N_2O_4}\) or \(\ce{NO_2}\) (a s far from the equilibrium state as the system can be!),
\[Q = \dfrac{[NO_2]^2}{[N_2O_4]} = \pm\infty\]
which makes the logarithm in Equation \(\ref{5-5}\), and thus the value o f \(\Delta G\), approach the same asymptotic limits (1) or (2) . As the reaction proceeds in the appropriate direction \(\Delta G\) approaches zero; once there (3) , the system is at its equilibrium composition and no further net change will occur.
The standard molar Gibbs energy change for this very simple reaction is –2.26 kJ, but mixing of the unreacted butane with the product brings the Gibbs energy of the equilibrium mixture down to about –3.1 kJ mol –1 at the equilibrium composition corresponding to 77 percent conversion.
Notice particularly that
- The sum of the Gibbs energies of the two gases ( n -butane and iso -butane) separately varies linearly with the composition of the mixture (red line ).
- The green curve adds the Gibbs energy of mixing to the above sum; its minimum defines the equilibrium composition.
- As the composition approaches the equilibrium value , \(ΔG\) (which denotes how much farther the Gibbs energy of the system can fall) approaches zero.
The detailed calculations that lead to the values shown above can be found here .
Why reactions lead to mixtures of reactants and products
We are now in a position to answer the question posed earlier: if Δ G ° for a reaction is negative, meaning that the Gibbs energies of the products are more negative than those of the reactants, why will some of the latter remain after equilibrium is reached? The answer is that no matter how low the Gibbs energy of the products, the Gibbs energy of the system can be reduced even more by allowing some of the products to be "contaminated" (i.e., diluted) by some reactants. Owing to the entropy associated with mixing of reactants and products, no homogeneous reaction will be 100% complete. An interesting corollary of this is that any reaction for which a balanced chemical equation can be written can in principle take place to some extent, however minute that might be.
Gibbs energies of mixing of products with reactants tend to be rather small, so for reactions having Δ G ° values that are highly negative or positive (±20 kJ mol –1 , say), the equilibrium mixture will, for all practical purposes, be either [almost] "pure" reactants or products.
The Equilibrium Constant
Now let us return to Equation \(\ref{5-5}\) which we reproduce here:
\[\Delta{G} = \Delta{G^°} + RT \ln Q \]
As the reaction approaches equilibrium, \(\Delta G\) becomes less negative and finally reaches zero. At equilibrium \(\Delta{G} = 0\) and \(Q = K\), so we can write ( must know this!)
\[\Delta{G^°} = –RT \ln K_p \label{5-6}\]
in which \(K_p\), the equilibrium constant expressed in pressure units, is the special value of \(Q\) that corresponds to the equilibrium composition.
This equation is one of the most important in chemistry because it relates the equilibrium composition of a chemical reaction system to measurable physical properties of the reactants and products. If you know the entropies and the enthalpies of formation of a set of substances, you can predict the equilibrium constant of any reaction involving these substances without the need to know anything about the mechanism of the reaction.
Instead of writing Equation \(\ref{5-6}\) in terms of K p , we can use any of the other forms of the equilibrium constant such as K c (concentrations), K x (mole fractions), K n (numbers of moles), etc. Remember, however, that for ionic solutions especially, only the K a , in which activities are used, will be strictly valid.
It is often useful to solve Equation \(\ref{5-6}\) for the equilibrium constant, yielding
\[ K = \exp {\left ( {-\Delta G \over RT} \right )} \label{5-7}\]
This relation is most conveniently plotted against the logarithm of \(K\) as shown in Figure \(\PageIndex{3}\), where it can be represented as a straight line that passes through the point (0,0).
Calculate the equilibrium constant for the reaction from the following thermodynamic data :
\[\ce{H^{+}(aq) + OH^{–}(aq) <=> H_2O(l)} \nonumber\]
|
\(H^+(aq)\)
|
|---|
* Note that the standard entropy of the hydrogen ion is zero by definition. This reflects the fact that it is impossible to carry out thermodynamic studies on a single charged species. All ionic entropies are relative to that of \(\ce{H^{+}(aq)}\), which explains why some values (as for aqueous hydroxide ion) are negative.
Solution
From the above data, we can evaluate the following quantities:
\[\begin{align*} \Delta{H}^o &= \sum \Delta H^o_{f}(\text{products}) - \sum \Delta H^o_{f}(\text{reactants}) \\[4pt] &= (–285.8) - (-230) \\[4pt] &= –55.8\, kJ \; mol^{-1} \end{align*}\]
\[\begin{align*}\Delta{S}^o &= \sum \Delta S^o (\text{products}) - \sum \Delta S° (\text{reactants}) \\[4pt] &= (70.0) – (–10.9) \\[4pt] &= +80.8\, J \; K^{-1}\; mol^{-1} \end{align*}\]
The value of \(\Delta{G}°\) at 298 K is
\[\begin{align*} \Delta H^o – T\Delta S^o &= (–55800) – (298)(80.8) \\[4pt] &= –79900\, J\, mol^{–1} \end{align*}\]
From Equation \(\ref{5-7}\) we have
\[\begin{align*} K &= \exp\left(\dfrac{–79900}{8.314 \times 298}\right) \\[4pt] &= e^{32.2} = 1.01 \times 10^{–14} \end{align*}\]
Equilibrium and Temperature
We have already discussed how changing the temperature will increase or decrease the tendency for a process to take place, depending on the sign of Δ S °. This relation can be developed formally by differentiating the relation
\[ \Delta G^° = \Delta H^° – T\Delta S^° \label{5-8}\]
with respect to the temperature:
\[ \dfrac{d(-\Delta G^°)}{dT} = -\Delta S^° \label{5-9}\]
Hence, the sign of the entropy change determines whether the reaction becomes more or less allowed as the temperature increases.
We often want to know how a change in the temperature will affect the value of an equilibrium constant whose value is known at some fixed temperature. Suppose that the equilibrium constant has the value \(K_1\) at temperature \(T_1\) and we wish to estimate \(K_2\) at temperature \(T_2\). Expanding Equation \(\ref{5-7}\) in terms of \(\Delta H^°\) and \(\Delta S^°\), we obtain
\[–RT_1 \ln K_1 = \Delta H^ ° – T_1 \Delta S^° \]
and
\[–RT_2 \ln K_2 = \Delta H ^° – T_2 \Delta S^°\]
Dividing both sides by RT and subtracting, we obtain
\[ \ln K_1 - \ln K_2 = - \left( \dfrac{\Delta H^°}{RT_1} -\dfrac{\Delta H^°}{RT_2} \right) \label{5-10}\]
Which is most conveniently expressed as the ratio
\[ \ln \dfrac{K_1}{K_2} = - \dfrac{\Delta H^°}{R} \left( \dfrac{1}{T_1} -\dfrac{1}{T_2} \right) \label{5-11}\]
This is its theoretical foundation of Le Ch a telier's Principle with respect to the effect of the temperature on equilibrium:
- if the reaction is exothermic \(\Delta H^° < 0\), then increasing temperature will make the second exponential term smaller and \(K\) will decrease. The equilibrium will then “shift to the left”.
- If \(\Delta H^° > 0\), then increasing T will make the exponent less negative and \(K\) will increase and the equilibrium will “shift to the right”.
This is an extremely important relationship, but not just because of its use in calculating the temperature dependence of an equilibrium constant. Even more important is its application in the “reverse” direction to experimentally determine Δ H ° from two values of the equilibrium constant measured at different temperatures. Direct calorimetric determinations of heats of reaction are not easy to make; relatively few chemists have the equipment and experience required for this rather exacting task. Measurement of an equilibrium constant is generally much easier, and often well within the capabilities of anyone who has had an introductory Chemistry course. Once the value of ΔH° is determined it can be combined with the Gibbs energy change (from a single observation of K , through Equation \(\ref{5-7}\)) to allow Δ S ° to be calculated through Equation \(\ref{5-9}\).
Equilibrium Without Mixing: it's all or nothing
You should now understand that for homogeneous reactions (those that take place entirely in the gas phase or in solution) the equilibrium composition will never be 100% products, no matter how much lower their Gibbs energy relative to the reactants. As was summarized in the N 2 O 4 -dissociation example discussed previously. This is due to "dilution" of the products by the reactants. In heterogeneous reactions (those which involve more than one phase) this dilution, and the effects that flow from it, may not be possible.
A particularly simple but important type of a heterogeneous process is phase change . Consider, for example, an equilibrium mixture of ice and liquid water. The concentration of H 2 O in each phase is dependent only on the density of the phase; there is no way that ice can be “diluted” with water, or vice versa. This means that at all temperatures other than the freezing point, the lowest Gibbs energy state will be that corresponding to pure ice or pure liquid. Only at the freezing point, where the Gibbs energies of water and ice are identical, can both phases coexist, and they may do so in any proportion.
Gibbs energy of the ice-water system
Only at 0°C can ice and liquid water coexist in any proportion. Note that in contrast to the homogeneous N 2 O 4 example, there is no Gibbs energy minimum at intermediate compositions.
Coupled Reactions
Two reactions are said to be coupled when the product of one of them is the reactant in the other:
\[A \rightarrow B \nonumber\]
and
\[B \rightarrow C \nonumber\]
If the standard Gibbs energy of the first reaction is positive but that of the second reaction is sufficiently negative, then for the overall process will be negative and we say that the first reaction is “driven” by the second one. This, of course, is just another way of describing an effect that you already know as the Le Chatelier principle: the removal of substance B by the second reaction causes the equilibrium of the first to “shift to the right”. Similarly, the equilibrium constant of the overall reaction is the product of the equilibrium constants of the two steps.
| 1 Cu 2 S(s) → 2 Cu(s) + S(s) | Δ G ° = + 86.2 kJ | Δ H ° = + 76.3 kJ |
| 2 S(s) + O 2 (g)→ SO 2 (g) | Δ G ° = –300.1 kJ | Δ H ° = + 296.8 kJ |
| 3 Cu 2 S(s)→ 2 Cu(s) + SO 2 (g) | Δ G ° = –213.9 kJ | Δ H ° = – 217.3 kJ |
In the above example, reaction 1 is the first step in obtaining metallic copper from one of its principal ores. This reaction is endothermic and it has a positive Gibbs energy change, so it will not proceed spontaneously at any temperature. If Cu 2 S is heated in the air, however, the sulfur is removed as rapidly as it is formed by oxidation in the highly spontaneous reaction 2 , which supplies the Gibbs energy required to drive 1 . The combined process, known as roasting , is of considerable industrial importance and is one of a large class of processes employed for winning metals from their ores.


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5: Nonverbal Processes in Intercultural Communication
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5.1: Principles of Nonverbal Communication
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In 2015, Dzhokhar Tsarnaev, involved in the Boston Marathon bombing, was sentenced to death after a trial focused on whether he should face life imprisonment or death. The jury's decision was influenced by character testimonies and Tsarnaev's demeanor, which appeared detached and unremorseful. The page further explores nonverbal communication, illustrating its importance and complexity across cultures.
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5.2: Types of Nonverbal Communication
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The page discusses various types of nonverbal communication and highlights their cultural variations. It covers nonverbal codes such as kinesics (including gestures, facial expressions, and posture), vocalics, proxemics, haptics, appearance and artifacts, olfactics, and music. Each of these codes communicates different aspects like emotions, power, or status and can differ greatly across cultures.
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5.3: Time and Space
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Chronemics refers to the study of time perception and orientation, distinguishing between monochronic and polychronic cultures. Monochronic cultures like the U.S. and Germany view time as linear and emphasize punctuality, while polychronic cultures such as those in Latin America are more fluid with time. Cultural spaces, influenced by architecture, also impact communication.
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5.4: Nonverbal Communication Challenges
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The Nonverbal Expectancy Violation Theory, developed by Judee Burgoon, suggests that individuals have culturally driven expectations about appropriate nonverbal behavior. When these norms are violated, it triggers reactions ranging from physiological to cognitive responses, with outcomes varying depending on factors like the severity of the breach and perceptions of the individual involved. These responses can differ across cultures, influenced by the context and local norms.


5.15: Outcome- Social Responsibility Marketing Impact
What you’ll learn to do: explain how demonstrating corporate social responsibility can impact marketing
We have reviewed many ethical challenges and potential traps for marketers. How can a marketer win? Actually, in lots of ways. Increasingly, marketers are doing more than just trying to avoid doing harm; as you’ll see, they’re taking on important issues and are making a difference, actively doing good.
Earlier in this module we discussed what corporate social responsibility is and how social responsibility programs impact many different stakeholders in a business. In this section we focus on the role of corporate social responsibility in marketing. We will look at the marketing mix—product, price, promotion, and distribution—and see how companies are changing their marketing strategies to visibly contribute to their communities.
Finally, we’ll talk about the results that companies achieve when social responsibility is part of the marketing strategy.
The specific things you’ll learn in this section include:
- Define social responsibility
- Identify examples of social responsibility that create value for customers
- Explain the impact of social responsibility on marketing results
Learning Activities
The learning activities for this section include the following:
- Reading: Social Responsibility in the Marketing Strategy
- Reading: Social Responsibility Initiatives
- Self Check: Social Responsibility Marketing Impact
Contributors and Attributions
- Outcome: Social Responsibility Marketing Impact. Provided by : Lumen Learning. License : CC BY: Attribution


10.8E: Vectors (Exercises)
For the following exercises, determine whether the two vectors, \(\mathbf{u}\) and \(\mathbf{v},\) are equal, where \(\mathbf{u}\) has an initial point \(P_{1}\) and a terminal point \(P_{2},\) and \(\mathbf{v}\) has an initial point \(P_{3}\) and a terminal point \(P_{4}\).
52. \(P_{1}=(-1,4), P_{2}=(3,1), P_{3}=(5,5)\) and \(P_{4}=(9,2)\)
53. \(P_{1}=(6,11), P_{2}=(-2,8), P_{3}=(0,-1)\) and \(P_{4}=(-8,2)\)
For the following exercises, use the vectors \(\mathbf{u}=2 \mathbf{i}-\mathbf{j}, \mathbf{v}=4 \mathbf{i}-3 \mathbf{j},\) and \(w=-2 \mathbf{i}+5 \mathbf{j}\) to evaluate the expression.
54. \(u-v\)
55. \(2 v-u+w\)
For the following exercises, find a unit vector in the same direction as the given vector.
56. \(a=8 i-6 j\)
57. \(b=-3 i-j\)
For the following exercises, find the magnitude and direction of the vector.
58. \(\langle 6,-2\rangle\)
59. \(\langle-3,-3\rangle\)
For the following exercises, calculate \(\mathbf{u} \cdot \mathbf{v}\).
60. \(u=-2 i+j\) and \(v=3 i+7 j\)
61. \(u=i+4 j\) and \(v=4 i+3 j\)
62. Given \(\boldsymbol{v}=\langle-3,4\rangle\) draw \(\boldsymbol{v}, 2 \boldsymbol{v},\) and \(\frac{1}{2} \boldsymbol{v}\)
63. Given the vectors shown in Figure 4 , sketch \(\boldsymbol{u}+\boldsymbol{v}, \boldsymbol{u}-\boldsymbol{v}\) and \(3 \boldsymbol{v}\).
Figure 4
64. Given initial point \(P_{1}=(3,2)\) and terminal point \(P_{2}=(-5,-1),\) write the vector \(\mathbf{v}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\). Draw the points and the vector on the graph.


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Study Questions
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What are the steps in the SDLC methodology?
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What is RAD software development?
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What is the Waterfall model?
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What makes the lean methodology unique?
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What is the difference between the Waterfall and Agile models?
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What is a sprint?
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What are three differences between second-generation and third-generation languages?
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Why would an organization consider building its own software application if it is cheaper to buy one?
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What is the difference between the pilot implementation methodology and the parallel implementation methodology?
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What is change management?
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What are the four different implementation methodologies?
Exercises
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Which software-development methodology would be best if an organization needed to develop a software tool for a small group of users in the marketing department? Why? Which implementation methodology should they use? Why?
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Doing your own research, find three programming languages and categorize them in these areas: generation, compiled vs. interpreted, procedural vs. object-oriented.
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Some argue that HTML is not a programming language. Doing your own research, find three arguments for why it is not a programming language and three arguments for why it is.
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Read more about responsive design using the link given in the text. Provide the links to three websites that use responsive design and explain how they demonstrate responsive-design behavior.
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Research the criteria and cost to put a mobile app into Apple’s App Store. Write a report.
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Research to find out what elements to use to estimate the cost to build an app. Write a report.


10.9: Returning to Rehabilitation in the Contemporary Juvenile Justice System
Empirical research drives recent reform efforts. The past decade has witnessed the identification of key developmental processes associated with delinquent behavior, such as brain development research. Ergo, evidence-based practices , which utilize the scientific method to assess the effectiveness of interventions, policies, and programs. In looking at what works, what doesn’t, and what is promising, researchers and policymakers alike assess the implementation of interventions to best meet the needs of the individual youth.
Additionally, several noteworthy Supreme Court cases exemplify society’s evolving standards of decency and the treatment of youth. These key cases demonstrate a move back to rehabilitation and acknowledge the fundamental differences between children and adults.
Key Supreme Court Cases
Roper v Simmons (2005)
Roper v. Simmons). Although Christopher Simmons planned and committed a capital offense (he murdered his neighbor, Shirley Cook), the court ruled that 18 years of age is where criminal responsibility should rest. That is to say, if a child is too young to vote, sign contracts, or do a number of other things (because society deems them not responsible enough), then they are too young to receive the death penalty. The court stated,“the evolving standards of decency that mark the progress of a maturing society” to determine which punishments are so disproportionate as to be “cruel and unusual.” Simmons received life in prison. It was ruled that imposing the death penalty on a person who was under the age of 18 at the time of the crime constituted cruel and unusual punishment. At the time of the Roper v Simmons verdict, the U.S. was only one out of a handful of countries that still imposed the death penalty on juveniles (among other countries were Yemen, Saudi Arabia, and Iran).
Graham v Florida (2010)
life in prison without the possibility of parole (LWOP ). This was until the 2010 case of Graham v. Florida . Terrance Graham received life in prison for a felony offense (armed burglary) when he was only 16 years old. Since Florida does not have parole, his sentence de facto became a life without the possibility of parole. The Supreme Court heard his case and ruled that it was unconstitutional to sentence a minor to life without the possibility of parole for a non-homicide offense.
Miller v Alabama (2012)
Graham v. Florida abolished life without the possibility of parole for non-homicide offenses, youth under the age of 18 were still receiving that sentence for crimes of murder. In 2012, Evan Miller was 14 years old when he killed his neighbor by severely beating him with a baseball bat while attempting to rob him.  In the majority opinion, Justice Elena Kagan wrote that “mandatory life without parole for those under age of 18 at the time of their crime violates the 8th Amendment ’s prohibition on cruel and unusual punishments. ” “Mandatory life without parole for a juvenile precludes consideration of his chronological age and its hallmark features – among them, immaturity, impetuosity, and failure to appreciate risks and consequences,” Justice Kagan said. “It prevents taking into account the family and home environment that surrounds him – and from which he cannot usually extricate himself – no matter how brutal or dysfunctional.”
Montgomery v Louisiana (2016)
Montgomery v. Alabama barred mandatory life without parole sentences retroactively. This meant that all youth sentenced prior to 2012 with LWOP sentences needed to be retried.


6.3: Diversity and Inclusion in the Workplace
Diversity and inclusion in the workforce are important to understand as you prepare for your future career. Diversity is not simply a box to be checked but rather is an approach to business that unites ethical management and high performance. Business leaders in the global economy recognize the benefits of a diverse workforce as an organizational strength, not as a mere slogan or a form of regulatory compliance with the law. They recognize that diversity can enhance performance and drive innovation; conversely, adhering to the traditional business practices of the past can cost them talented employees and loyal customers (Byers & Stanberry, 2020).
A study by global management consulting firm McKinsey & Company indicates that businesses with gender and ethnic diversity outperform others. According to Mike Dillon, chief diversity and inclusion officer for PwC in San Francisco, “attracting, retaining and developing a diverse group of professionals stirs innovation and drives growth.”
Living this goal means not only recruiting, hiring, and training talent from a wide demographic spectrum but also including all employees in every aspect of the organization (Byers & Stanberry, 2020).
Workplace Diversity
The twenty-first century workplace features much greater diversity than was common even a couple of generations ago. Individuals who might once have faced employment challenges because of religious beliefs, ability differences, or sexual orientation now regularly join their peers in interview pools and on the job. Each may bring a new outlook and different information to the table, and employees no longer can take for granted that their coworkers think the same way they do. This change pushes employers to question their own assumptions, expand their understanding and appreciate alternate viewpoints. The result is more creative ideas, approaches, and solutions. Thus, diversity may also enhance corporate decision-making (Byers & Stanberry, 2020).
Communicating with those who differ from us may require us to make an extra effort and even change our viewpoint, but that communication leads to better collaboration and more favorable outcomes overall, according to David Rock, director of the Neuro-Leadership Institute in New York City, who says diverse coworkers “challenge their own and others’ thinking.”
According to the Society for Human Resource Management (SHRM), organizational diversity now includes more than just racial, gender, and religious differences. Organizational diversity also encompasses different thinking styles and personality types, as well as factors such as physical and cognitive abilities and sexual orientation, all of which influence the way people perceive the world. “Finding the right mix of individuals to work on teams, and creating the conditions in which they can excel, are key business goals for today’s leaders, given that collaboration has become a paradigm of the twenty-first century workplace,” according to an SHRM article.
Attracting workers who are not all alike is an important first step in the process of achieving greater diversity. However, managers also must have goals encompassing inclusion, or the engagement of all employees in the corporate culture. “The far bigger challenge is how people interact with each other once they’re on the job,” says Howard J. Ross, founder and chief learning officer at Cook Ross, a consulting firm specializing in diversity. “Diversity is being invited to the party; inclusion is being asked to dance. Diversity is about the ingredients, the mix of people and perspectives. Inclusion is about the container—the place that allows employees to feel they belong, to feel both accepted and different” (Byers & Stanberry, 2020).
The origins of workplace diversity date back to the passage of the Civil Rights Act of 1964 (CRA) or before. Census figures show that women made up less than 29 percent of the civilian workforce when Congress passed Title VII of the CRA prohibiting workplace discrimination. After passage of the law, gender diversity in the workplace expanded significantly. According to the U.S. Bureau of Labor Statistics (BLS), the percentage of women in the labor force increased from 48 percent in 1977 to a peak of 60 percent in 1999. Over the past forty years, the total number of women in the labor force has risen from 41 million in 1977 to 71 million in 2017. The BLS projects the number of women in the U.S. labor force will reach 92 million in 2050, an increase that far outstrips population growth.
The statistical data show a similar trend for African American, Asian American, and Hispanic workers (see Figure 6.1 below). Just before passage of the CRA in 1964, the percentages of minorities in official workforce statistics were relatively small compared with their representation in the total population. In 1966, Asians accounted for just 0.5 percent of private-sector employment, with Hispanics at 2.5 percent and African Americans at 8.2 percent.
However, Hispanic employment numbers have increased significantly since the CRA became law; they are expected to more than double from 15 percent in 2010 to 30 percent of the labor force in 2050. Similarly, Asian Americans are projected to increase their share of employment from 5 to 8 percent between 2010 and 2050 (Byers & Stanberry, 2020).
Figure 6.1 There is a distinct contrast in workforce demographics between 2010 and projected numbers for 2050. (Image by OpenStax is licensed under CC BY 4.0)
Much more progress remains to be made, however, in workplace diversity. For example, many people think of the technology sector as the workplace of open-minded Millennials. Yet Google revealed in recent diversity statistics that progress toward a more inclusive workforce may be steady but is very slow. Men still account for the great majority of employees at the corporation; only about 30 percent are women, and women fill fewer than 20 percent of Google’s technical roles (see Figure 6.2 below). The company has shown a similar lack of gender diversity in leadership roles, where women hold fewer than 25 percent of positions. Despite modest progress, an ocean-sized gap remains to be narrowed. In terms of ethnicity, approximately 56 percent of Google employees are white. About 35 percent are Asian, 3.5 percent are Latino, and 2.4 percent are Black, and of the company’s management and leadership roles, 68 percent are held by whites (Byers & Stanberry, 2020).
Figure 6.2 Google is emblematic of the technology sector, and this graphic shows just how far from equality and diversity the industry remains. (Image by OpenStax is licensed under CC BY 4.0)
Diversity and inclusion are positive steps for business organizations, and despite their sometimes slow pace, the majority of employers are moving in the right direction. Diversity strengthens the company’s internal relationships with employees and improves employee morale, as well as strengthening external relationships with customer groups. Communication, a core value of most successful businesses, becomes more effective with a diverse workforce. Performance improves for multiple reasons, not the least of which is that acknowledging diversity and respecting differences is the ethical thing to do (OpenStax, 2020).
Generational Differences in the Workforce
Today we have five different generations in the workforce, and each generation differs in terms of values, communication style, and life experiences. Each group brings valuable contributions to the workplace.
Each generation is a subculture with a sense of reality based on formative world and national events, technological innovations and socio-cultural values. To understand how that experience impacts communication, we need to consider how the different generations view technology and communications media. The following examples are based on an analysis of generational differences (Lumen Learning, 2022):
|
Traditionalists |
Baby Boomers |
Generation X |
Millennials |
Generation Z |
|
|---|---|---|---|---|---|
|
Technology is |
Hoover Dam |
The microwave |
Internet |
Hand-held devices |
Virtual |
|
Communicate via |
Rotary phones |
Four-tone phones |
Cell phones |
Internet & Text |
Social Media |
Every generation develops expertise with communication formats and media that reflect their situational reality. For example, Traditionalists tend to have a more formal communication style, with a strict adherence to written grammatical rules and a strong cultural structure. Baby Boomers tend to prefer a more informal and collaborative approach. Gen X communications tend to be more blunt and direct: just the facts. Millennial and Gen Z communication is technology-dependent. As an Ad Council article notes, these generations are driving a truncation of the English language, shortening words (e.g., “totally” becomes “totes”) and abbreviating phrases into one or two-syllable words or acronyms which may or may not be spoken aloud (e.g., “FOMO” for “fear of missing out” and “TIL” for “today I learned”) (Lumen Learning, 2022).
Bridging the Generation Gap
Each generation brings not only a frame of reference but also a set of competencies—and expectations—based on how they view the world and their place in it. The challenge for both businesses and individuals is that we now have five generations in the workforce. Differences in generational communication style and media are, effectively, language barriers. To the extent that individuals can’t translate, the communication gaps are a hindrance to effective collaboration and, by extension, achievement of critical goals and objectives. The communication disconnect can also affect employee morale and productivity.
The opportunity in this situation is to leverage specific generational strengths and decrease points of friction. The best case scenario is to create a culture and opportunities that encourage cross-generational sharing and mentoring. As Nora Zelevansky wrote in a piece for Coca-Cola: “In order to master intergenerational communication, it is necessary to understand some broad generalizations about the generations and then move beyond those to connect as individuals” (Zelevansky, 2014).
In a related trend, the model of talent management is changing. As discussed in a Sodexo report on 2017 Workplace Trends, global companies are moving to a model of shared learning, where workers of all ages contribute to each other’s growth and development. Indeed, the researchers identified “intergenerational agility” as a critical aspect of the employee and employer value proposition (Sodexo, 2017). Business benefits of intergenerational learning include increased efficiency, productivity and competitive positioning. Two statistics that suggest the culture and communication gaps can be bridged are listed below (The Hartford, 2014).
- 96 percent of Millennials believe that Boomers bring substantial experience and knowledge to the workplace
- 90 percent of Baby Boomers believe that Millennials bring new skills and ideas to the workplace.
The diversity of the intergenerational workplace is not just a development or trend; instead, that diversity is a creative opportunity for businesses and organizations (The Hartford, 2014).
Professor Mariano Sánchez of the University of Granada in Spain sees the opportunity in cultivating ”generational intelligence;” specifically, “organizing activities that raise generational awareness, connect generations and help them work better together—exchanging knowledge, ideas, skills and more to enhance the broad skill sets everyone needs in today’s jobs” (Sodexo, 2017).
According to Jason Dorsey, Millennial and Gen Z researcher and co-founder of The Center for Generational Kinetics, “the key is getting each person to recognize that everyone has different communication skills that can be harnessed to best support the organization” (Zelevansky, 2014). Incorporating multiple communication media in meetings and to facilitate ongoing discussion/collaboration allows members of different generations to share expertise and demonstrate the value of a particular medium. Selecting technology that supports multiple ways of communicating and collaborating can also leverage collective strengths and create fertile ground. For example, using a videoconferencing platform such as Zoom allows for participants to connect visually and participate virtually, with audio, screen sharing and recording capabilities.
Generational Differences
After reading the section on generational differences in the workforce, reflect on your experiences at school, work and in your community and answer the following questions:
With which generation do you identify?
How closely do you resemble some of the descriptors used to describe this generation? Explain and give examples.
Why is understanding generational differences in the workplace important? Explain and give examples.
Please note that being aware of broad generalizations about generations is important to help understand different work styles and preferences, but moving beyond generalizations is important to connect as individuals and colleagues.


12.2: Security Return- The Holding Pattern Return (Raw Calculation)
I n order to figure the return o n an asset (a percent) , it is useful to first outline the various components of the return on the asset – and then to calculate the return. The two principal components of return are income and profit, i.e., sales or “exit” price less investment cost . Income is common to many assets including bonds (interest) , stocks (dividends) , and real estate (rent) . The exit price may be equal to, greater or less than the asset’s original investment cost , hence “profit” is included in return .
Example :
- Stock/bond/real estate cost or investment (C) = $1,000. This is an “outflow.”
- Income (I) received over the course of the holding period = $100.
- Sales (exit) price (P) = $1,050.
- The income and sales price are “inflows.”
Holding Period Return (HPR) :
HPR = inflows ÷ outflows -1 = [(I + P) ¸ C] – 1
HPR = [(100 + 1,050) ¸ 1,000] – 1 = + 15%
Alternatively, we could have calculated the HPR by focusing only on the profit ( “ Π ” ), in which case, we would have left out the “-1” expression at the end. The result is the same.
HPR = [(I + Π ) ¸ C]
HPR = [(100 + 50) ¸ 1,000] = + 15%
Note :
While the HPR has severe conceptual limits, it provides key information upon which a better model may be constructed. In this section, we will present the calculation for dollar price and return of a bond.
When you make a sale to your fellow…do not victimize one another.
-Leviticus 25:1
And when you transact a purchase to your friend, or acquire from the hand of your friend, you shall not defraud one another.
-Leviticus 25:14


4.4: Approaches to Theorizing
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How do researchers build theories? Steinfeld and Fulk (1990)2 recommend four such approaches. The first approach is to build theories inductively based on observed patterns of events or behaviors. Such approach is often called “grounded theory building”, because the theory is grounded in empirical observations. This technique is heavily dependent on the observational and interpretive abilities of the researcher, and the resulting theory may be subjective and non-confirmable. Furthermore, observing certain patterns of events will not necessarily make a theory, unless the researcher is able to provide consistent explanations for the observed patterns. We will discuss the grounded theory approach in a later chapter on qualitative research.
The second approach to theory building is to conduct a bottom-up conceptual analysis to identify different sets of predictors relevant to the phenomenon of interest using a predefined framework. One such framework may be a simple input-process-output framework, where the researcher may look for different categories of inputs, such as individual, organizational, and/or technological factors potentially related to the phenomenon of interest (the output), and describe the underlying processes that link these factors to the target phenomenon. This is also an inductive approach that relies heavily on the inductive abilities of the researcher, and interpretation may be biased by researcher’s prior knowledge of the phenomenon being studied.
The third approach to theorizing is to extend or modify existing theories to explain a new context, such as by extending theories of individual learning to explain organizational learning. While making such an extension, certain concepts, propositions, and/or boundary conditions of the old theory may be retained and others modified to fit the new context. This deductive approach leverages the rich inventory of social science theories developed by prior theoreticians, and is an efficient way of building new theories by building on existing ones.
The fourth approach is to apply existing theories in entirely new contexts by drawing upon the structural similarities between the two contexts. This approach relies on reasoning by analogy, and is probably the most creative way of theorizing using a deductive approach. For instance, Markus (1987)3 used analogic similarities between a nuclear explosion and uncontrolled growth of networks or network-based businesses to propose a critical mass theory of network growth. Just as a nuclear explosion requires a critical mass of radioactive material to sustain a nuclear explosion, Markus suggested that a network requires a critical mass of users to sustain its growth, and without such critical mass, users may leave the network, causing an eventual demise of the network.


16.1: Faculty Resources Overview
We’ve seen overwhelming demand for high quality, openly-licensed course materials, including supplemental resources to enrich teaching and learning and to make life easier for instructors. To support this need, we’ve developed and curated faculty resources to use with this course.
Free and Open Supplemental Materials
On the following pages, you will find supplemental resources that are freely available to use with the interactive learning materials for this course. Since these resources are openly licensed, you may use them as is or adapt them to your needs.
Additional Faculty Resources
Additional supplemental resources, learning tools, and support services are available to faculty who adopt Waymaker, OHM, or Candela courses with paid support from Lumen Learning. For many courses, these include some combination of summative assessments, answer keys, solutions manuals, or other materials shared only with authorized instructors in order to protect academic integrity.
Click here to learn more about additional instructor tools and resources available to faculty who adopt Lumen-supported courseware. Information about pricing and payment options is available on this page . Lumen’s low-cost support fees replace the cost of expensive textbooks and may be paid by students or by the institution directly.
Continuously Improving Learning Materials
Are you interested in collaborating with us to make these course materials better? We use learning data to identify where content improvements are needed, and then we invite faculty and subject matter experts to work with us developing continuous improvements aimed at increasing learning.
Learn more from this blog post , or sign up here to join our continuous improvement mailing list and stay up to date about upcoming OER hackathons and other continuous improvement activities.
Contributors and Attributions
- Request Access. Provided by : Lumen Learning. License : CC BY: Attribution
- Lock Icon. Authored by : Hopkins. Provided by : Noun Project. Located at : https://thenounproject.com/term/lock/942574/ . License : CC BY: Attribution


3.3B: Osmosis
Osmosis is the movement of water across a membrane from an area of low solute concentration to an area of high solute concentration.
- Describe the process of osmosis and explain how concentration gradient affects osmosis
Key Points
- Osmosis occurs according to the concentration gradient of water across the membrane, which is inversely proportional to the concentration of solutes.
- Osmosis occurs until the concentration gradient of water goes to zero or until the hydrostatic pressure of the water balances the osmotic pressure.
- Osmosis occurs when there is a concentration gradient of a solute within a solution, but the membrane does not allow diffusion of the solute.
Key Terms
- solute : Any substance that is dissolved in a liquid solvent to create a solution
- osmosis : The net movement of solvent molecules from a region of high solvent potential to a region of lower solvent potential through a partially permeable membrane
- semipermeable membrane : A type of biological membrane that will allow certain molecules or ions to pass through it by diffusion and occasionally by specialized facilitated diffusion
Osmosis and Semipermeable Membranes
Osmosis is the movement of water through a semipermeable membrane according to the concentration gradient of water across the membrane, which is inversely proportional to the concentration of solutes. Semipermeable membranes, also termed selectively permeable membranes or partially permeable membranes, allow certain molecules or ions to pass through by diffusion.
While diffusion transports materials across membranes and within cells, osmosis transports only water across a membrane. The semipermeable membrane limits the diffusion of solutes in the water. Not surprisingly, the aquaporin proteins that facilitate water movement play a large role in osmosis, most prominently in red blood cells and the membranes of kidney tubules.
Mechanism of Osmosis
Osmosis is a special case of diffusion. Water, like other substances, moves from an area of high concentration to one of low concentration. An obvious question is what makes water move at all? Imagine a beaker with a semipermeable membrane separating the two sides or halves. On both sides of the membrane the water level is the same, but there are different concentrations of a dissolved substance, or solute, that cannot cross the membrane (otherwise the concentrations on each side would be balanced by the solute crossing the membrane). If the volume of the solution on both sides of the membrane is the same but the concentrations of solute are different, then there are different amounts of water, the solvent, on either side of the membrane. If there is more solute in one area, then there is less water; if there is less solute in one area, then there must be more water.
To illustrate this, imagine two full glasses of water. One has a single teaspoon of sugar in it, whereas the second one contains one-quarter cup of sugar. If the total volume of the solutions in both cups is the same, which cup contains more water? Because the large amount of sugar in the second cup takes up much more space than the teaspoon of sugar in the first cup, the first cup has more water in it.
Osmosis : In osmosis, water always moves from an area of higher water concentration to one of lower concentration. In the diagram shown, the solute cannot pass through the selectively permeable membrane, but the water can.
Returning to the beaker example, recall that it has a mixture of solutes on either side of the membrane. A principle of diffusion is that the molecules move around and will spread evenly throughout the medium if they can. However, only the material capable of passing through the membrane will diffuse through it. In this example, the solute cannot diffuse through the membrane, but the water can. Water has a concentration gradient in this system. Thus, water will diffuse down its concentration gradient, crossing the membrane to the side where it is less concentrated. This diffusion of water through the membrane—osmosis—will continue until the concentration gradient of water goes to zero or until the hydrostatic pressure of the water balances the osmotic pressure. In the beaker example, this means that the level of fluid in the side with a higher solute concentration will go up.


33.3: Thank u, Emerson
The lasting effects of early American literature can be seen throughout contemporary pieces in various medias. Ralph Waldo Emerson’s Self-Reliance has set an example of the importance of self and finding one’s identity for generations to come. One way the themes from Emerson’s piece have reoccurred is through Ariana Grande’s 2018 single, “ Thank u, next .” Grande’s song reshapes Emerson’s themes to be about relying on yourself for love, rather than for survival in society. While the context and setting are different than Self-Reliance , this song builds off the essential themes of Emerson’s text. Grande also reshapes Emerson’s piece by making it a song instead of a printed text. The themes from early American literature are transformed and reintroduced in a new technological format with new context.
One of the most prevalent themes and claims that Emerson makes is the idea of trusting yourself and believing in yourself truly. Emerson introduces this idea as a way of survival in society, explaining that it is divine and the correct way of living. Emerson urges to “trust thyself: every heart vibrates to that iron string. Accept the place the divine providence has found you” (Emerson). To trust oneself is a given right and the most important thing to do. This piece was written during the period when Americans were seeking a national identity. Grande’s song reflects this idea of forming an identity and trusting oneself. She sings, “I ain’t worried ’bout nothin’/ Plus, I met someone else/ We havin’ better discussions/ I know they say I move on too fast/ But this one gon’ last/ ‘Cause her name is Ari” (Grande). In this part of the song she is explaining how she’s moved on from her exes to herself. Now, by reflecting and relying on herself for love, in a sense, she is happy and “havin’ better discussions.” Grande reinvents Emerson’s idea of self-reliance by applying it to love and relationships.
Another theme present in both pieces of work is the idea of how divine and excellent self-help is. Emerson further develops the idea of self-reliance and explains that it not only includes self-trust, but also self-help. Emerson’s text declares “the secret of fortune is joy in our hands. Welcome evermore to gods and men is the self-helping man” (Emerson). He states that self-helping yourself is essential to success and god-like. Grande follows Emerson’s ideas by helping herself, revealing in third person that, “she [Grande] taught me love (Love)/ She taught me patience (Patience)/ And she handles pain (Pain)” (Grande). Grande reflects on herself and explains how her own experiences have taught her how to love, how to be patience, and how to handle emotional pain. Grande reinterprets themes seen in Emerson’s text and introduces them to us in the form of song.
Aside from self, the theme of history is used as a way to establish identity. Emerson explains that “he cannot be happy and strong until he too lives with nature in the present, above time” (Emerson). Emerson urges this idea of parting with the past and living in the present. Similarly, Grande writes, “So, look what I got/ Look what you taught me/ And for that, I say/ Thank you, next” (Grande). Grande is saying “thank u” to the past and looks at what she has now, in the present.
The second half of her chorus is “next,” meaning she is also focusing on the future and accepting her past. Emerson mentions future when he explains that “Act singly, and what you have already done singly will justify you now. Greatness appeals to the future” (Emerson). Although almost contradictory, readers can see how the idea of building an identity is highly reflective of one’s past. Both artist and writer encourage the acceptance of past and focus on the present and future. While Emerson has the aspect of nature in his writings, Grande looks at love. Again, readers and listeners can see how Grande’s piece of work reshapes Emerson’s beliefs and presents it in a new way.
The original format for Emerson’s Self-Reliance was printed and shared through bookstores and stands, whereas Grande’s form of American literature is electronic and in the form of a song. Grande’s song has electronic beats and claps with a repetitive chorus of “thank u, next.” These aspects create a catchy rhythm and song that is able to make its way into everyone’s mind. In fact, Grande sings the phrase, “thank u, next,” 27 times, making sure it’ll get stuck in the listener’s head. By repeating this single phrase so many times, Grande does a similar thing that Emerson did in his writing. In Self-Reliance , Emerson constantly repeats the idea of being true to yourself and that you are all that you need. Grande’s repeated phrase focuses more on reflecting, being true to yourself in love, and looking onto the future. Although they both have a similar tactic, hearing the phrase sung aloud, along with a beat, is more memorable than reading it from a long document. Grande’s song is, therefore, more effective at creating a lasting mark on listeners without them having to analysis lengthy sentences. She can deliver to the audience the importance of self, as Emerson did, but in an easy-to-understand and retainable way. Grande’s piece of work has pulled from Emerson’s early American text and reshaped it into a catchy song about relationships.
Although songs have been around far longer than early America, Grande’s text is special because its online. Being an electronic song enables it to travel anywhere there is a computer. While Emerson’s Self-Reliance reached a lot of readers, it didn’t do it in a day. Grande’s song was so influential, it broke Spotify’s global single-day streaming record for female artists (forbes.com). As America’s people and technologies changed to become seekers of instant-gratification, literature changed as well to become more instantaneous. One technological platform that helped promote Grande’s song was Instagram. As with the case of most things online, even if an Instagram user didn’t listen to the song themselves, they easily may have heard about it within the same day that it was released. “Thank u, Next” was able to bring Emerson’s ideas to a wider and instant audience.
Ariana Grande’s single reshapes ideas and themes from Emerson’s Self- Reliance and presents them in a new way that keeps up with societal changes. Early American literature lives on through pieces like Grande’s song about self-reliant love. Apart from applying Emerson’s ideas to relationships and love, the song uses technology that wasn’t available at the time of early America. American literature, then, is presented through new technologies that represent the current times. In a classroom setting, technology can instantly provide literature and has thus shaped the study of American literature to be more instant. Students are able to access texts easily and quickly, enabling them to analyze and learn no matter where they are, as long as they have internet. Technology is able to influence the way we consume and study American literature to make it more accessible and shareable. Just as Grande says thank you to the past, one can say “next” to what’s becoming of American literature with future technology.
Works Cited
Grande, Ariana. “Thank U, Next.” Republic Records , 2018, https://www.youtube.com/watch?v=5OeR5XBEahU
Rolli, Bryan. “Ariana Grande’s ‘Thank U, Next’ Breaks Single-Day Spotify Record For A Female Artist – Twice.” Forbes , 7 Nov. 2018, https://www.forbes.com/sites/bryanrolli/2018/11/07/ariana-grande-thank-u-next-spotify-record/#3013cf016b92 .
Claire-Helps. Ariana Grande Gif. Tumbler.com , https://www.google.com/url?sa=i&source=images&cd=&cad=rja&uact=8&ved=2ahUKEwjfnOzAiIzfAhXjRt8KHaFEBrUQjhx6BAgBEAM&url=http%3A%2F%2Fclaire-helps.tumblr.com%2Fpost%2F88565117624%2Fariana-grande-gif-hunt&psig=AOvVaw2DHvnF2GjkzyjXOq9j5cve&ust=1544215311921555


2.7: The Camera
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Learning Objectives
By the end of this section, you will be able to:
- Describe the optics of a camera
- Characterize the image created by a camera
Cameras are very common in our everyday life. Between 1825 and 1827, French inventor Nicéphore Niépce successfully photographed an image created by a primitive camera. Since then, enormous progress has been achieved in the design of cameras and camera-based detectors.
Initially, photographs were recorded by using the light-sensitive reaction of silver-based compounds such as silver chloride or silver bromide. Silver-based photographic paper was in common use until the advent of digital photography in the 1980s, which is intimately connected to charge-coupled device (CCD) detectors. In a nutshell, a CCD is a semiconductor chip that records images as a matrix of tiny pixels, each pixel located in a “bin” in the surface. Each pixel is capable of detecting the intensity of light impinging on it. Color is brought into play by putting red-, blue-, and green-colored filters over the pixels, resulting in colored digital images (Figure \(\PageIndex{1}\)). At its best resolution, one CCD pixel corresponds to one pixel of the image. To reduce the resolution and decrease the size of the file, we can “bin” several CCD pixels into one, resulting in a smaller but “pixelated” image.
Clearly, electronics is a big part of a digital camera; however, the underlying physics is basic optics. As a matter of fact, the optics of a camera are pretty much the same as those of a single lens with an object distance that is significantly larger than the lens’s focal distance (Figure \(\PageIndex{2}\)).
For instance, let us consider the camera in a smartphone. An average smartphone camera is equipped with a stationary wide-angle lens with a focal length of about 4–5 mm. (This focal length is about equal to the thickness of the phone.) The image created by the lens is focused on the CCD detector mounted at the opposite side of the phone. In a cell phone, the lens and the CCD cannot move relative to each other. So how do we make sure that both the images of a distant and a close object are in focus?
Recall that a human eye can accommodate for distant and close images by changing its focal distance. A cell phone camera cannot do that because the distance from the lens to the detector is fixed. Here is where the small focal distance becomes important. Let us assume we have a camera with a 5-mm focal distance. What is the image distance for a selfie? The object distance for a selfie (the length of the hand holding the phone) is about 50 cm. Using the thin-lens equation, we can write
\[\frac{1}{5mm}=\frac{1}{500mm}+\frac{1}{d_i} \nonumber \]
We then obtain the image distance:
\[\frac{1}{d_i}=\frac{1}{5mm}−\frac{1}{500mm} \nonumber \]
Note that the object distance is 100 times larger than the focal distance. We can clearly see that the 1/(500 mm) term is significantly smaller than 1/(5 mm), which means that the image distance is pretty much equal to the lens’s focal length. An actual calculation gives us the image distance d i =5.05mm. This value is extremely close to the lens’s focal distance.
Now let us consider the case of a distant object. Let us say that we would like to take a picture of a person standing about 5 m from us. Using the thin-lens equation again, we obtain the image distance of 5.005 mm. The farther the object is from the lens, the closer the image distance is to the focal distance. At the limiting case of an infinitely distant object, we obtain the image distance exactly equal to the focal distance of the lens.
As you can see, the difference between the image distance for a selfie and the image distance for a distant object is just about 0.05 mm or 50 microns. Even a short object distance such as the length of your hand is two orders of magnitude larger than the lens’s focal length, resulting in minute variations of the image distance. (The 50-micron difference is smaller than the thickness of an average sheet of paper.) Such a small difference can be easily accommodated by the same detector, positioned at the focal distance of the lens. Image analysis software can help improve image quality.
Conventional point-and-shoot cameras often use a movable lens to change the lens-to-image distance. Complex lenses of the more expensive mirror reflex cameras allow for superb quality photographic images. The optics of these camera lenses is beyond the scope of this textbook.