2.2.14: Henrik Ibsen (1828-1906)
A Doll's House Norwegian Realism Henrik Ibsen is called both "the father of Realism" and "the father of modern theater" in Europe, which is to say that he was the first playwright to use Realism on stage. Ibsen's impact on theater makes him the most influential European playwright since Shakespeare. For Ibsen, art should be both challenging and a force for social change; his plays often expose what he saw as the moral hypocrisy of society. In particular, Ibsen's plays peel back the veneer of respectability of the Norwegian middle class, revealing what happens when people only pretend to be moral. No group or ideology was safe from his criticism, and often there are no characters in a play who are completely without blame. For example, in An Enemy of the People (1882), the outright villains may be the businessmen who are poisoning the local water source, but the locals are equally at fault for refusing to believe the truth for selfish reasons, and the supposed hero of the story makes matters worse with his stubborn temper. In Ghosts (1881), Ibsen broke several taboos in his depiction of how a husband's repeated infidelities lead to passing on syphilis to his unborn son. As guilty as the husband was, everyone from the pastor to the wife bear some responsibility for looking the other way, even after the husband's death. Ibsen's goals for A Doll's House (1879) are every bit as broad as his other works. Nora and Torvald try to live up to their society's ideals for how men and women should behave, but both of them become victims to society's unrealistic expectations. The truth in this case is a lit match that leads to a metaphorical explosion. The fact that Nora and Torvald do not agree on the definition of what is right appears to be a product of which gender holds the power in society, rather than an actual gender issue. A Doll's House does not offer a conventional happy ending, which so shocked audiences that some theaters actually rewrote the ending when staging it. The ending is also complicated by the fact that Nora's rebellion against expectations has no guarantee of success in a society where women could not even borrow money without a man's signature. A common theme in Ibsen's plays, therefore, is that truth does not always set you free; in fact, sometimes the very best intentions are doomed to failure if society refuses to listen or change. It is a problem that Ibsen faced himself, since his efforts to influence change were invariably seen as shocking and controversial. It is a testament to his persistence and talent that audiences now expect the theater to address social issues. Consider while reading:
- What are the possible interpretations for the play's title?
- Does Nora change during the course of the play? Explain.
- Do you agree with what Kristine does? Why or why not?
- How do the time period and the location affect the play? How would the play be different if it were set in another time period or location?
Written by Laura Getty


9.2: Solve Quadratic Equations Using the Square Root Property
By the end of this section, you will be able to:
- Solve quadratic equations of the form \(ax^{2}=k\) using the Square Root Property
- Solve quadratic equations of the form \(a(x–h)^{2}=k\) using the Square Root Property
Before you get started, take this readiness quiz.
-
Simplify: \(\sqrt{128}\).
If you missed this problem, review Example 8.13. -
Simplify: \(\sqrt{\frac{32}{5}}\).
If you missed this problem, review Example 8.50. -
Factor: \(9 x^{2}-12 x+4\).
If you missed this problem, review Example 6.23.
A quadratic equation is an equation of the form \(a x^{2}+b x+c=0\) , where \(a≠0\). Quadratic equations differ from linear equations by including a quadratic term with the variable raised to the second power of the form \(ax^{2}\). We use different methods to solve quadratic equations than linear equations, because just adding, subtracting, multiplying, and dividing terms will not isolate the variable.
We have seen that some quadratic equations can be solved by factoring. In this chapter, we will learn three other methods to use in case a quadratic equation cannot be factored.
Solve Quadratic Equations of the Form \(ax^{2}=k\) using the Square Root Property
We have already solved some quadratic equations by factoring. Let’s review how we used factoring to solve the quadratic equation \(x^{2}=9\).
\(x^{2}=9\)
Put the equation in standard form.
\(x^{2}-9=0\)
Factor the difference of squares.
\((x-3)(x+3)=0\)
Use the Zero Produce Property.
\(x-3=0 \quad x-3=0\)
Solve each equation.
\(x=3 \quad x=-3\)
We can easily use factoring to find the solutions of similar equations, like \(x^{2}=16\) and \(x^{2}=25\), because \(16\) and \(25\) are perfect squares. In each case, we would get two solutions, \(x=4, x=-4\) and \(x=5, x=-5\)
But what happens when we have an equation like \(x^{2}=7\)? Since \(7\) is not a perfect square, we cannot solve the equation by factoring.
Previously we learned that since \(169\) is the square of \(13\), we can also say that \(13\) is a square root of \(169\). Also, \((-13)^{2}=169\), so \(−13\) is also a square root of \(169\). Therefore, both \(13\) and \(−13\) are square roots of \(169\). So, every positive number has two square roots—one positive and one negative. We earlier defined the square root of a number in this way:
If \(n^{2}=m\), then \(n\) is a square root of \(m\).
Since these equations are all of the form \(x^{2}=k\), the square root definition tells us the solutions are the two square roots of \(k\). This leads to the Square Root Property .
Square Root Property
If \(x^{2}=k\), then
\(x=\sqrt{k} \quad\) or \(\quad x=-\sqrt{k} \quad\) or \(\quad x=\pm \sqrt{k}\)
Notice that the Square Root Property gives two solutions to an equation of the form \(x^{2}=k\), the principal square root of \(k\) and its opposite. We could also write the solution as \(x=\pm \sqrt{k}\). We read this as \(x\) equals positive or negative the square root of \(k\).
Now we will solve the equation \(x^{2}=9\) again, this time using the Square Root Property.
\(\begin{aligned} &x^{2} =9 \\ \text { Use the Square Root Property. } \quad& x=\pm \sqrt{9} \\& x =\pm 3 \end{aligned}\)
So \(x=3\) or \(x=-3\)
What happens when the constant is not a perfect square? Let’s use the Square Root Property to solve the equation \(x^{2}=7\).
\(x^{2}=7\)
Use the Square Root Property. \(x=\sqrt{7}, \quad x=-\sqrt{7}\)
We cannot simplify \(\sqrt{7}\), so we leave the answer as a radical.
Solve: \(x^{2}-50=0\).
Solution :
| Step 1 : Isolate the quadratic term and make its coefficient one. | Add \(50\) to both sides to get \(x^{2}\) by itself. | \(\begin{aligned} x^{2}-50 &=0 \\ x^{2} &=50 \end{aligned}\) |
| Step 2 : Use the Square Root Property. | Remember to write the \(\pm\) symbol. | \(x=\pm \sqrt{50}\) |
| Step 3 : Simplify the radical. | Rewrite to show two solutions. | \(\begin{array}{l}{x=\pm \sqrt{25} \cdot \sqrt{2}} \\ {x=\pm 5 \sqrt{2}} \\ {}x=5\sqrt{2}, \:x=-5\sqrt{2}\end{array}\) |
| Step 4 : Check the solutions. | Substitute in \(x=5 \sqrt{2}\) and \(x=-5 \sqrt{2}\) |
\(\begin{array}{r}{x^{2}-50=0} \\ {(\color{red}{5 \sqrt{2}}\color{black}{)}^{2}-50 \stackrel{?}{=} 0} \\ {25 \cdot 2-50 \stackrel{?}{=} 0} \\ {0=0}\end{array}\) \(\begin{array}{r}{x^{2}-50=0} \\ {(\color{red}{-5 \sqrt{2}}\color{black}{)}^{2}-50 \stackrel{?}{=} 0} \\ {25 \cdot 2-50 \stackrel{?}{=} 0} \\ {0=0}\end{array}\) |
Solve: \(x^{2}-48=0\).
- Answer
-
\(x=4 \sqrt{3}, x=-4 \sqrt{3}\)
Solve: \(y^{2}-27=0\).
- Answer
-
\(y=3 \sqrt{3}, y=-3 \sqrt{3}\)
The steps to take to use the Square Root Property to solve a quadratic equation are listed here.
Solve a Quadratic Equation Using the Square Root Property
- Isolate the quadratic term and make its coefficient one.
- Use Square Root Property.
- Simplify the radical.
- Check the solutions.
In order to use the Square Root Property, the coefficient of the variable term must equal one. In the next example, we must divide both sides of the equation by the coefficient \(3\) before using the Square Root Property.
Solve: \(3 z^{2}=108\).
Solution :
| \(3 z^{2}=108\) | |
| The quadratic term is isolated. Divide by \(3\) to make its coefficient \(1\). | \(\frac{3 z^{2}}{3}=\frac{108}{3}\) |
| Simplify. | \(z^{2}=36\) |
| Use the Square Root Property. | \(z=\pm \sqrt{36}\) |
| Simplify the radical. | \(z=\pm 6\) |
| Rewrite to show two solutions. | \(z=6, \quad z=-6\) |
|
Check the solutions: |
Solve: \(2x^{2}=98\).
- Answer
-
\(x=7, x=-7\)
Solve: \(5m^{2}=80\).
- Answer
-
\(m=4, m=-4\)
The Square Root Property states ‘If \(x^{2}=k\),’ What will happen if \(k<0\)? This will be the case in the next example.
Solve: \(x^{2}+72=0\).
Solution :
| \(x^{2}+72=0\) | |
| Isolate the quadratic term. | \(x^{2}=-72\) |
| Use the Square Root Property. | \(x=\pm \sqrt{-72}\) |
| Simplify using complex numbers. | \(x=\pm \sqrt{72} i\) |
| Simplify the radical. | \(x=\pm 6 \sqrt{2} i\) |
| Rewrite to show two solutions | \(x=6 \sqrt{2} i, x=-6 \sqrt{2} i\) |
|
Check the solutions:
\(\begin{array}{r}{x^{2}+72=0}
|
\(\begin{array}{r}{x^{2}+72=0}
|
Solve: \(c^{2}+12=0\).
- Answer
-
\(c=2 \sqrt{3} i, \quad c=-2 \sqrt{3} i\)
Solve: \(q^{2}+24=0\).
- Answer
-
\(c=2 \sqrt{6} i, \quad c=-2 \sqrt{6} i\)
Our method also works when fractions occur in the equation, we solve as any equation with fractions. In the next example, we first isolate the quadratic term, and then make the coefficient equal to one.
Solve: \(\frac{2}{3} u^{2}+5=17\).
Solution :
| \(\frac{2}{3} u^{2}+5=17\) | |
| Isolate the quadratic term. | |
| Multiply by \(\frac{3}{2}\) to make the coefficient \(1\). | |
| Simplify. | |
| Use the Square Root Property. | |
| Simplify the radical. | |
| Simplify. | |
| Rewrite to show two solutions. | |
|
Check:
\(\begin{array}{r}{\frac{2}{3}u^{2}+5=17}
|
\(\begin{array}{r}{\frac{2}{3}u^{2}+5=17}
|
Solve: \(\frac{1}{2} x^{2}+4=24\).
- Answer
-
\(x=2 \sqrt{10}, x=-2 \sqrt{10}\)
Solve: \(\frac{3}{4} y^{2}-3=18\).
- Answer
-
\(y=2 \sqrt{7}, y=-2 \sqrt{7}\)
The solutions to some equations may have fractions inside the radicals. When this happens, we must rationalize the denominator .
Solve: \(2 x^{2}-8=41\).
Solution :
| Isolate the quadratic term. | |
| Divide by \(2\) to make the coefficient \(1\). | |
| Simplify. | |
| Use the Square Root Property. | |
| Rewrite the radical as a fraction of square roots. | |
| Rationalize the denominator. | |
| Simplify. | |
| Rewrite to show two solutions. | |
|
Check: We leave te check for you. |
Solve: \(5 r^{2}-2=34\).
- Answer
-
\(r=\frac{6 \sqrt{5}}{5}, \quad r=-\frac{6 \sqrt{5}}{5}\)
Solve: \(3 t^{2}+6=70\).
- Answer
-
\(t=\frac{8 \sqrt{3}}{3}, \quad t=-\frac{8 \sqrt{3}}{3}\)
Solve Quadratic Equation of the Form \(a(x-h)^{2}=k\) Using the Square Root Property
We can use the Square Root Property to solve an equation of the form \(a(x-h)^{2}=k\) as well. Notice that the quadratic term, \(x\), in the original form \(ax^{2}=k\) is replaced with \((x-h)\).
The first step, like before, is to isolate the term that has the variable squared. In this case, a binomial is being squared. Once the binomial is isolated, by dividing each side by the coefficient of \(a\), then the Square Root Property can be used on \((x-h)^{2}\).
Solve: \(4(y-7)^{2}=48\).
Solution :
| \(4(y-7)^{2}=48\) | |
| Divide both sides by the coefficient \(4\). | \((y-7)^{2}=12\) |
| Use the Square Root Property on the binomial. | \(y-7=\pm \sqrt{12}\) |
| Simplify the radical. | \(y-7=\pm 2 \sqrt{3}\) |
| Solve for \(y\). | \(y=7 \pm 2 \sqrt{3}\) |
| Rewrite to show two solutions. |
\(y=7+2 \sqrt{3}\)
\(y=7-2 \sqrt{3}\) |
|
Check: |
Solve: \(3(a-3)^{2}=54\).
- Answer
-
\(a=3+3 \sqrt{2}, \quad a=3-3 \sqrt{2}\)
Solve: \(2(b+2)^{2}=80\).
- Answer
-
\(b=-2+2 \sqrt{10}, \quad b=-2-2 \sqrt{10}\)
Remember when we take the square root of a fraction, we can take the square root of the numerator and denominator separately.
Solve: \(\left(x-\frac{1}{3}\right)^{2}=\frac{5}{9}\).
Solution :
\(\left(x-\frac{1}{3}\right)^{2}=\frac{5}{9}\)
Use the Square Root Property.
\(x-\frac{1}{3}=\pm \sqrt{\frac{5}{9}}\)
Rewrite the radical as a fraction of square roots.
\(x-\frac{1}{3}=\pm \frac{\sqrt{5}}{\sqrt{9}}\)
Simplify the radical.
\(x-\frac{1}{3}=\pm \frac{\sqrt{5}}{3}\)
Solve for \(x\).
\(x=\frac{1}{3} \pm \frac{\sqrt{5}}{3}\)
Rewrite to show two solutions.
\(x=\frac{1}{3}+\frac{\sqrt{5}}{3}, x=\frac{1}{3}-\frac{\sqrt{5}}{3}\)
Check:
We leave the check for you.
Solve: \(\left(x-\frac{1}{2}\right)^{2}=\frac{5}{4}\).
- Answer
-
\(x=\frac{1}{2}+\frac{\sqrt{5}}{2}, x=\frac{1}{2}-\frac{\sqrt{5}}{2}\)
Solve: \(\left(y+\frac{3}{4}\right)^{2}=\frac{7}{16}\).
- Answer
-
\(y=-\frac{3}{4}+\frac{\sqrt{7}}{4}, y=-\frac{3}{4}-\frac{\sqrt{7}}{4}\)
We will start the solution to the next example by isolating the binomial term.
Solve: \(2(x-2)^{2}+3=57\).
Solution :
\(2(x-2)^{2}+3=57\)
Subtract \(3\) from both sides to isolate the binomial term.
\(2(x-2)^{2}=54\)
Divide both sides by \(2\).
\((x-2)^{2}=27\)
Use the Square Root Property.
\(x-2=\pm \sqrt{27}\)
Simplify the radical.
\(x-2=\pm 3 \sqrt{3}\)
Solve for \(x\).
\(x=2 \pm 3 \sqrt{3}\)
Rewrite to show two solutions.
\(x=2+3 \sqrt{3}, x=2-3 \sqrt{3}\)
Check:
We leave the check for you.
Solve: \(5(a-5)^{2}+4=104\).
- Answer
-
\(a=5+2 \sqrt{5}, a=5-2 \sqrt{5}\)
Solve: \(3(b+3)^{2}-8=88\).
- Answer
-
\(b=-3+4 \sqrt{2}, \quad b=-3-4 \sqrt{2}\)
Sometimes the solutions are complex numbers.
Solve: \((2 x-3)^{2}=-12\).
Solution :
\((2 x-3)^{2}=-12\)
Use the Square Root Property.
\(2 x-3=\pm \sqrt{-12}\)
Simplify the radical.
\(2 x-3=\pm 2 \sqrt{3} i\)
Add \(3\) to both sides.
\(2 x=3 \pm 2 \sqrt{3} i\)
Divide both sides by \(2\).
\(x=\frac{3 \pm 2 \sqrt{3 i}}{2}\)
Rewrite in standard form.
\(x=\frac{3}{2} \pm \frac{2 \sqrt{3} i}{2}\)
Simplify.
\(x=\frac{3}{2} \pm \sqrt{3} i\)
Rewrite to show two solutions.
\(x=\frac{3}{2}+\sqrt{3} i, x=\frac{3}{2}-\sqrt{3} i\)
Check:
We leave the check for you.
Solve: \((3 r+4)^{2}=-8\).
- Answer
-
\(r=-\frac{4}{3}+\frac{2 \sqrt{2} i}{3}, r=-\frac{4}{3}-\frac{2 \sqrt{2} i}{3}\)
Solve: \((2 t-8)^{2}=-10\).
- Answer
-
\(t=4+\frac{\sqrt{10} i}{2}, t=4-\frac{\sqrt{10 i}}{2}\)
The left sides of the equations in the next two examples do not seem to be of the form \(a(x-h)^{2}\). But they are perfect square trinomials, so we will factor to put them in the form we need.
Solve: \(4 n^{2}+4 n+1=16\).
Solution :
We notice the left side of the equation is a perfect square trinomial. We will factor it first.
| \(4 n^{2}+4 n+1=16\) | |
| Factor the perfect square trinomial. | \((2 n+1)^{2}=16\) |
| Use the Square Root Property. | \(2 n+1=\pm \sqrt{16}\) |
| Simplify the radical. | \(2 n+1=\pm 4\) |
| Solve for \(n\). | \(2 n=-1 \pm 4\) |
| Divide each side by \(2\). | \(\begin{aligned} \frac{2 n}{2} &=\frac{-1 \pm 4}{2} \\ n &=\frac{-1 \pm 4}{2} \end{aligned}\) |
| Rewrite to show two solutions. | \(n=\frac{-1+4}{2}, n=\frac{-1-4}{2}\) |
| Simplify each equation. | \(n=\frac{3}{2}, \quad n=-\frac{5}{2}\) |
|
Check: |
Solve: \(9 m^{2}-12 m+4=25\).
- Answer
-
\(m=\frac{7}{3}, \quad m=-1\)
Solve: \(16 n^{2}+40 n+25=4\).
- Answer
-
\(n=-\frac{3}{4}, \quad n=-\frac{7}{4}\)
Access this online resource for additional instruction and practice with using the Square Root Property to solve quadratic equations.
Key Concepts
-
Square Root Property
- If \(x^{2}=k\), then \(x=\sqrt{k}\) or \(x=-\sqrt{k}\)or \(x=\pm \sqrt{k}\)
- Isolate the quadratic term and make its coefficient one.
- Use Square Root Property.
- Simplify the radical.
- Check the solutions.


5.11: Learning Activities
Learning Activities
(Answers to “Learning Activities” can be found in the “Answer Key” at the end of the book. Answers to interactive activity elements will be provided within the element as immediate feedback.)
Assessing a patient’s risk for falls and planning interventions to prevent falls are common safety strategies completed by nurses. This section uses a patient scenario to demonstrate how to use the nursing process to assess a patient and then create a nursing care plan to prevent falls. Begin by reading the Handoff Report received from the nurse on the previous shift.
Handoff Report
Mr. Moore is a 72-year-old widower recovering in the hospital after sustaining injuries he received from a fall he sustained at home. See Figure 5.15 for an image of Mr. Moore. [1] He fractured his right hip and underwent surgical repair two days ago. He is receiving IV fluids and morphine for pain control. He has a history of hypertension and cardiovascular disease. He wears glasses and hearing aids. Per recommendations from the physical therapist, he is able to transfer with one assist with a walker, but is weak on his right side. He has an order to ambulate at least 100 feet four times daily with a wheeled walker. He is 6 feet tall and weighs 165 pounds. Prior to the fall, he lived at home alone independently, and he is looking forward to returning home.
Assessment
The nurse collects the following assessment findings:
- Vital Signs: Blood pressure 90/60, heart rate 56, respiratory rate 18, temperature 37 degrees Celsius, pulse oximetry reading 92%, current pain level 0
- Alert and oriented x 3 to person, place, and time
- Lungs clear
- Cardiovascular Assessment: Heart rate is regular, capillary refill less than 3 seconds in fingers and toes, pedal pulses 2+
- Right lower extremity strength is 1+ (weak)
- Ambulates with walker with assistance; gait is unsteady
Critical Thinking Questions
1. Describe the fall risk factors for Mr. Moore.
2. Use the Morse Fall Risk Assessment tool to assess Mr. Moore’s risk for falling.
Diagnosis
The NANDA-I nursing diagnosis is established: Risk for Falls as evidenced by lower extremity weakness and difficulty with gait .
Outcome Identification
Overall Goal: Mr. Moore will remain free from falls during his hospitalization stay.
SMART Expected Outcomes:
- Mr. Moore will not experience a fall during hospitalization.
- Mr. Moore will correctly use his assistive device (walker) every time he ambulates during hospitalization.
Planning Interventions
The following interventions are planned based on Mr. Moore’s fall risk factors.
- Remove clutter from the floor.
- Provide adequate lighting with a night-light at the bedside.
- Use half side rails to prevent falls from the bed.
- Monitor gait, balance, and fatigue with ambulation and encourage resting as needed.
- Place personal items within easy reach of the patient at the bedside.
- Provide an elevated toilet seat.
- Encourage the use of prescribed glasses and hearing aids when walking.
- Obtain orthostatic blood pressures daily and notify the provider as indicated.
- Ensure the patient wears shoes that fit properly, are fastened securely, and have no-skid soles.
- Suggest home adaptations to improve safety after discharge, such as adjusting the toilet seat height, installing grab bars in the bathroom, and using a rubber mat in the shower.
Critical Thinking Question
3. What additional interventions could be implemented for Mr. Moore to reduce his risk of falls that target his specific risk factors?
Implementation of Interventions
The following day, upon entering the room, you find Mr. Moore has climbed out of bed and is on his way to the bathroom. He states, “I need to go to the bathroom for a bowel movement and didn’t have time to ring the call light and wait.” You assist him with his walker, but he seems unsteady on his feet as he walks toward the bathroom. You’re not sure if he will make it to the toilet without falling. He says, “We need to hurry or I’m not going to make it.”
Critical Thinking Question:
4. What is the best response?
Evaluation
The nurse evaluates Mr. Moore’s progress based on the established expected outcomes:
- Mr. Moore will not experience a fall during hospitalization: Outcome Met.
- Mr. Moore will use his assistive device (walker) correctly during hospitalization: Outcome Partially Met.
Mr. Moore forgets to call for assistance and uses a walker when he needs to go to the bathroom. A “stop” sign has been placed within patient view to remind him to use the call light before getting up. In addition to hourly rounding, toileting will be performed at scheduled intervals every two hours. An icon has been posted on the doorframe to alert staff that the patient is at high risk for falls. In addition to the bed being kept low and locked, a mat will be placed next to the bed at night. If Mr. Moore continues to forget to call for assistance, a bed alarm will be placed to alert staff of movement so that quick assistance can be offered.
Query \(\PageIndex{1}\)
Query \(\PageIndex{2}\)
Query \(\PageIndex{3}\)
Query \(\PageIndex{4}\)
Functional Ability Case Study by Susan Jepsen for Lansing Community College is licensed under CC BY 4.0
- “ old-man-1208210_960_720.jpg ” by Free-Photos at Pixabay is licensed under CC0 ↵


2.7: The Kinetic Theory of Gases Introduction (Exercises)
Conceptual Questions
2.1 Molecular Model of an Ideal Gas
1. Two \(\displaystyle H_2\) molecules can react with one (O_2\) molecule to produce two \(\displaystyle H_2O\) molecules. How many moles of hydrogen molecules are needed to react with one mole of oxygen molecules?
2. Under what circumstances would you expect a gas to behave significantly differently than predicted by the ideal gas law?
3. A constant-volume gas thermometer contains a fixed amount of gas. What property of the gas is measured to indicate its temperature?
4. Inflate a balloon at room temperature. Leave the inflated balloon in the refrigerator overnight. What happens to the balloon, and why?
5. In the last chapter, free convection was explained as the result of buoyant forces on hot fluids. Explain the upward motion of air in flames based on the ideal gas law.
2.2 Pressure, Temperature, and RMS Speed
6. How is momentum related to the pressure exerted by a gas? Explain on the molecular level, considering the behavior of molecules.
7. If one kind of molecule has double the radius of another and eight times the mass, how do their mean free paths under the same conditions compare? How do their mean free times compare?
8. What is the average velocity of the air molecules in the room where you are right now?
9. Why do the atmospheres of Jupiter, Saturn, Uranus, and Neptune, which are much more massive and farther from the Sun than Earth is, contain large amounts of hydrogen and helium?
10. Statistical mechanics says that in a gas maintained at a constant temperature through thermal contact with a bigger system (a “reservoir”) at that temperature, the fluctuations in internal energy are typically a fraction \(\displaystyle 1/\sqrt{N}\) of the internal energy. As a fraction of the total internal energy of a mole of gas, how big are the fluctuations in the internal energy? Are we justified in ignoring them?
11. Which is more dangerous, a closet where tanks of nitrogen are stored, or one where tanks of carbon dioxide are stored?
2.3 Heat Capacity and Equipartition of Energy
12. Experimentally it appears that many polyatomic molecules’ vibrational degrees of freedom can contribute to some extent to their energy at room temperature. Would you expect that fact to increase or decrease their heat capacity from the value R? Explain.
13. One might think that the internal energy of diatomic gases is given by \(\displaystyle E_{int}=5RT/2\).. Do diatomic gases near room temperature have more or less internal energy than that? Hint: Their internal energy includes the total energy added in raising the temperature from the boiling point (very low) to room temperature.
14. You mix 5 moles of \(\displaystyle H_2\) at 300 K with 5 moles of He at 360 K in a perfectly insulated calorimeter. Is the final temperature higher or lower than 330 K?
2.4 Distribution of Molecular Speeds
15. One cylinder contains helium gas and another contains krypton gas at the same temperature. Mark each of these statements true, false, or impossible to determine from the given information.
(a) The rms speeds of atoms in the two gases are the same.
(b) The average kinetic energies of atoms in the two gases are the same.
(c) The internal energies of 1 mole of gas in each cylinder are the same.
(d) The pressures in the two cylinders are the same.
16. Repeat the previous question if one gas is still helium but the other is changed to fluorine, \(\displaystyle F_2\).
17. An ideal gas is at a temperature of 300 K. To double the average speed of its molecules, what does the temperature need to be changed to?
Problems
2.1 Molecular Model of an Ideal Gas
18. The gauge pressure in your car tires is \(\displaystyle 2.50×10^5N/m^2\) a temperature of 35.0°C when you drive it onto a ship in Los Angeles to be sent to Alaska. What is their gauge pressure on a night in Alaska when their temperature has dropped to −40.0°C ? Assume the tires have not gained or lost any air.
19. Suppose a gas-filled incandescent light bulb is manufactured so that the gas inside the bulb is at atmospheric pressure when the bulb has a temperature of 20.0°C .
(a) Find the gauge pressure inside such a bulb when it is hot, assuming its average temperature is 60.0°C (an approximation) and neglecting any change in volume due to thermal expansion or gas leaks.
(b) The actual final pressure for the light bulb will be less than calculated in part (a) because the glass bulb will expand. Is this effect significant?
20. People buying food in sealed bags at high elevations often notice that the bags are puffed up because the air inside has expanded. A bag of pretzels was packed at a pressure of 1.00 atm and a temperature of 22.0°C .When opened at a summer picnic in Santa Fe, New Mexico, at a temperature of 32.0°C , the volume of the air in the bag is 1.38 times its original volume. What is the pressure of the air?
21. How many moles are there in
(a) 0.0500 g of \(\displaystyle N_2\) gas ( M=28.0g/mol )?
(b) 10.0 g of \(\displaystyle CO_2\) gas ( M=44.0g/mol )?
(c) How many molecules are present in each case?
22. A cubic container of volume 2.00 L holds 0.500 mol of nitrogen gas at a temperature of 25.0°C . What is the net force due to the nitrogen on one wall of the container? Compare that force to the sample’s weight.
23. Calculate the number of moles in the 2.00-L volume of air in the lungs of the average person. Note that the air is at 37.0°C (body temperature) and that the total volume in the lungs is several times the amount inhaled in a typical breath as given in Example 2.2.
24. An airplane passenger has \(\displaystyle 100cm^3\) of air in his stomach just before the plane takes off from a sea-level airport. What volume will the air have at cruising altitude if cabin pressure drops to \(\displaystyle 7.50×10^4N/m^2\)?
25. A company advertises that it delivers helium at a gauge pressure of \(\displaystyle 1.72×10^7Pa\) in a cylinder of volume 43.8 L. How many balloons can be inflated to a volume of 4.00 L with that amount of helium? Assume the pressure inside the balloons is 1.01×105Pa1.01×105Pa and the temperature in the cylinder and the balloons is 25.0°C25.0°C.
26. According to http://hyperphysics.phy-astr.gsu.edu.../venusenv.html , the atmosphere of Venus is approximately \(\displaystyle 96.5%CO_2\) and \(\displaystyle 3.5%N_2\) by volume. On the surface, where the temperature is about 750 K and the pressure is about 90 atm, what is the density of the atmosphere?
27. An expensive vacuum system can achieve a pressure as low as \(\displaystyle 1.00×10^{−7}N/m^2\) at 20.0°C . How many molecules are there in a cubic centimeter at this pressure and temperature?
28. The number density N/V of gas molecules at a certain location in the space above our planet is about \(\displaystyle 1.00×10^{11}m^{−3}\), and the pressure is \(\displaystyle 2.75×10^{−10N}/m^2\) in this space. What is the temperature there?
29. A bicycle tire contains 2.00 L of gas at an absolute pressure of \(\displaystyle 7.00×10^5N/m^2\) and a temperature of 18.0°C . What will its pressure be if you let out an amount of air that has a volume of \(\displaystyle 100cm^3\) at atmospheric pressure? Assume tire temperature and volume remain constant.
30.  When the bottle is cooled, the pressure difference between inside and outside forces the egg into the bottle. Suppose the bottle has a volume of 0.500 L and the temperature inside it is raised to 80.0°C while the pressure remains constant at 1.00 atm because the bottle is open.
(a) How many moles of air are inside?
(b) Now the egg is put in place, sealing the bottle. What is the gauge pressure inside after the air cools back to the ambient temperature of 25°C but before the egg is forced into the bottle?
31. A high-pressure gas cylinder contains 50.0 L of toxic gas at a pressure of \(\displaystyle 1.40×10^7N/m^2\) and a temperature of 25.0°C . The cylinder is cooled to dry ice temperature ( −78.5°C ) to reduce the leak rate and pressure so that it can be safely repaired.
(a) What is the final pressure in the tank, assuming a negligible amount of gas leaks while being cooled and that there is no phase change?
(b) What is the final pressure if one-tenth of the gas escapes? (c) To what temperature must the tank be cooled to reduce the pressure to 1.00 atm (assuming the gas does not change phase and that there is no leakage during cooling)? (d) Does cooling the tank as in part
(c) appear to be a practical solution?
32. Find the number of moles in 2.00 L of gas at 35.0°C and under \(\displaystyle 7.41×10^7N/m^2\) of pressure.
33. Calculate the depth to which Avogadro’s number of table tennis balls would cover Earth. Each ball has a diameter of 3.75 cm. Assume the space between balls adds an extra 25.0% to their volume and assume they are not crushed by their own weight.
34. a) What is the gauge pressure in a 25.0°C car tire containing 3.60 mol of gas in a 30.0-L volume?
(b) What will its gauge pressure be if you add 1.00 L of gas originally at atmospheric pressure and 25.0°C ? Assume the temperature remains at 25.0°C and the volume remains constant.
2.2 Pressure, Temperature, and RMS Speed
In the problems in this section, assume all gases are ideal.
35. A person hits a tennis ball with a mass of 0.058 kg against a wall. The average component of the ball’s velocity perpendicular to the wall is 11 m/s, and the ball hits the wall every 2.1 s on average, rebounding with the opposite perpendicular velocity component.
(a) What is the average force exerted on the wall?
(b) If the part of the wall the person hits has an area of \(\displaystyle 3.0m^2\), what is the average pressure on that area?
36. A person is in a closed room (a racquetball court) with \(\displaystyle V=453m^3\) hitting a ball ( m=42.0g )(m=42.0g) around at random without any pauses. The average kinetic energy of the ball is 2.30 J.
(a) What is the average value of \(\displaystyle v^2_x\)? Does it matter which direction you take to be x?
(b) Applying the methods of this chapter, find the average pressure on the walls?
(c) Aside from the presence of only one “molecule” in this problem, what is the main assumption in Pressure, Temperature, and RMS Speed that does not apply here?
37. Five bicyclists are riding at the following speeds: 5.4 m/s, 5.7 m/s, 5.8 m/s, 6.0 m/s, and 6.5 m/s. (a) What is their average speed? (b) What is their rms speed?
38. Some incandescent light bulbs are filled with argon gas. What is \(\displaystyle v_{rms}\) for argon atoms near the filament, assuming their temperature is 2500 K?
39. Typical molecular speeds (\(\displaystyle v_{rms}\)) are large, even at low temperatures. What is \(\displaystyle v_{rms}\) for helium atoms at 5.00 K, less than one degree above helium’s liquefaction temperature?
40. What is the average kinetic energy in joules of hydrogen atoms on the 5500°C surface of the Sun?
(b) What is the average kinetic energy of helium atoms in a region of the solar corona where the temperature is \(\displaystyle 6.00×10^5K\) ?
41. What is the ratio of the average translational kinetic energy of a nitrogen molecule at a temperature of 300 K to the gravitational potential energy of a nitrogen-molecule−Earth system at the ceiling of a 3-m-tall room with respect to the same system with the molecule at the floor?
42. What is the total translational kinetic energy of the air molecules in a room of volume \(\displaystyle 23m^3\) if the pressure is \(\displaystyle 9.5×10^4Pa\) (the room is at fairly high elevation) and the temperature is 21°C ? Is any item of data unnecessary for the solution?
43. The product of the pressure and volume of a sample of hydrogen gas at 0.00°C is 80.0 J.
(a) How many moles of hydrogen are present?
(b) What is the average translational kinetic energy of the hydrogen molecules?
(c) What is the value of the product of pressure and volume at 200°C?
44. What is the gauge pressure inside a tank of \(\displaystyle 4.86×10^4mol\) of compressed nitrogen with a volume of \(\displaystyle 6.56m^3\) if the rms speed is 514 m/s?
45. If the rms speed of oxygen molecules inside a refrigerator of volume \(\displaystyle 22.0ft.^3\) is 465 m/s, what is the partial pressure of the oxygen? There are 5.71 moles of oxygen in the refrigerator, and the molar mass of oxygen is 32.0 g/mol.
46. The escape velocity of any object from Earth is 11.1 km/s. At what temperature would oxygen molecules (molar mass is equal to 32.0 g/mol) have root-mean-square velocity \(\displaystyle v_{rms}\) equal to Earth’s escape velocity of 11.1 km/s?
47. The escape velocity from the Moon is much smaller than that from the Earth, only 2.38 km/s. At what temperature would hydrogen molecules (molar mass is equal to 2.016 g/mol) have a root-mean-square velocity vrmsvrms equal to the Moon’s escape velocity?
48. Nuclear fusion, the energy source of the Sun, hydrogen bombs, and fusion reactors, occurs much more readily when the average kinetic energy of the atoms is high—that is, at high temperatures. Suppose you want the atoms in your fusion experiment to have average kinetic energies of \(\displaystyle 6.40×10^{−14}J\). What temperature is needed?
49. Suppose that the typical speed (\(\displaystyle v_{rms}\)) of carbon dioxide molecules (molar mass is 44.0 g/mol) in a flame is found to be 1350 m/s. What temperature does this indicate?
50. (a) Hydrogen molecules (molar mass is equal to 2.016 g/mol) have vrmsvrms equal to 193 m/s. What is the temperature? (b) Much of the gas near the Sun is atomic hydrogen (H rather than \(\displaystyle H_2\)). Its temperature would have to be \(\displaystyle 1.5×10^7K\) for the rms speed vrmsvrms to equal the escape velocity from the Sun. What is that velocity?
51. There are two important isotopes of uranium, \(\displaystyle ^{235}U\) and \(\displaystyle ^{238}U\); these isotopes are nearly identical chemically but have different atomic masses. Only \(\displaystyle ^{235}U\) is very useful in nuclear reactors. Separating the isotopes is called uranium enrichment (and is often in the news as of this writing, because of concerns that some countries are enriching uranium with the goal of making nuclear weapons.) One of the techniques for enrichment, gas diffusion, is based on the different molecular speeds of uranium hexafluoride gas, \(\displaystyle UF_6\).
(a) The molar masses of \(\displaystyle ^{235}U\) and \(\displaystyle ^{238}UF_6\) are 349.0 g/mol and 352.0 g/mol, respectively. What is the ratio of their typical speeds vrmsvrms?
(b) At what temperature would their typical speeds differ by 1.00 m/s?
(c) Do your answers in this problem imply that this technique may be difficult?
52. The partial pressure of carbon dioxide in the lungs is about 470 Pa when the total pressure in the lungs is 1.0 atm. What percentage of the air molecules in the lungs is carbon dioxide? Compare your result to the percentage of carbon dioxide in the atmosphere, about 0.033%.
53. Dry air consists of approximately 78%nitrogen,21%oxygen,and 1% argon by mole, with trace amounts of other gases. A tank of compressed dry air has a volume of 1.76 cubic feet at a gauge pressure of 2200 pounds per square inch and a temperature of 293 K. How much oxygen does it contain in moles?
54. (a) Using data from the previous problem, find the mass of nitrogen, oxygen, and argon in 1 mol of dry air. The molar mass of \(\displaystyle N_2\) is 28.0 g/mol, that of \(\displaystyle O_2\) is 32.0 g/mol, and that of argon is 39.9 g/mol.
(b) Dry air is mixed with pentane (\(\displaystyle C_5H_{12}\), molar mass 72.2 g/mol), an important constituent of gasoline, in an air-fuel ratio of 15:1 by mass (roughly typical for car engines). Find the partial pressure of pentane in this mixture at an overall pressure of 1.00 atm.
55. (a) Given that air is 21% oxygen, find the minimum atmospheric pressure that gives a relatively safe partial pressure of oxygen of 0.16 atm.
(b) What is the minimum pressure that gives a partial pressure of oxygen above the quickly fatal level of 0.06 atm?
(c) The air pressure at the summit of Mount Everest (8848 m) is 0.334 atm. Why have a few people climbed it without oxygen, while some who have tried, even though they had trained at high elevation, had to turn back?
56. (a) If the partial pressure of water vapor is 8.05 torr, what is the dew point? (760torr=1atm=101,325Pa)
(b) On a warm day when the air temperature is 35°C and the dew point is 25°C , what are the partial pressure of the water in the air and the relative humidity?
2.3 Heat Capacity and Equipartition of Energy
57. To give a helium atom nonzero angular momentum requires about 21.2 eV of energy (that is, 21.2 eV is the difference between the energies of the lowest-energy or ground state and the lowest-energy state with angular momentum). The electron-volt or eV is defined as \(\displaystyle 1.60×10^{−19}J.\) Find the temperature T where this amount of energy equals \(\displaystyle k_BT/2\). Does this explain why we can ignore the rotational energy of helium for most purposes? (The results for other monatomic gases, and for diatomic gases rotating around the axis connecting the two atoms, have comparable orders of magnitude.)
58. (a) How much heat must be added to raise the temperature of 1.5 mol of air from 25.0°C to 33.0°C at constant volume? Assume air is completely diatomic.
(b) Repeat the problem for the same number of moles of xenon, Xe.
59. A sealed, rigid container of 0.560 mol of an unknown ideal gas at a temperature of 30.0°C is cooled to −40.0°C . In the process, 980 J of heat are removed from the gas. Is the gas monatomic, diatomic, or polyatomic?
60. A sample of neon gas (Ne, molar mass M=20.2g/mol ) at a temperature of 13.0°C is put into a steel container of mass 47.2 g that’s at a temperature of −40.0°C . The final temperature is −28.0°C . (No heat is exchanged with the surroundings, and you can neglect any change in the volume of the container.) What is the mass of the sample of neon?
61. A steel container of mass 135 g contains 24.0 g of ammonia, \(\displaystyle NH_3\), which has a molar mass of 17.0 g/mol. The container and gas are in equilibrium at 12.0°C . How much heat has to be removed to reach a temperature of −20.0°C ? Ignore the change in volume of the steel.
62. A sealed room has a volume of \(\displaystyle 24m^3\). It’s filled with air, which may be assumed to be diatomic, at a temperature of 24°C and a pressure of 9.83×104Pa.9.83×104Pa. A 1.00-kg block of ice at its melting point is placed in the room. Assume the walls of the room transfer no heat. What is the equilibrium temperature?
63. Heliox, a mixture of helium and oxygen, is sometimes given to hospital patients who have trouble breathing, because the low mass of helium makes it easier to breathe than air. Suppose helium at 25°C is mixed with oxygen at 35°C to make a mixture that is 70% helium by mole. What is the final temperature? Ignore any heat flow to or from the surroundings, and assume the final volume is the sum of the initial volumes.
64. Professional divers sometimes use heliox, consisting of 79% helium and 21% oxygen by mole. Suppose a perfectly rigid scuba tank with a volume of 11 L contains heliox at an absolute pressure of \(\displaystyle 2.1×10^7Pa\) at a temperature of 31°C
(a) How many moles of helium and how many moles of oxygen are in the tank?
(b) The diver goes down to a point where the sea temperature is 27°C while using a negligible amount of the mixture. As the gas in the tank reaches this new temperature, how much heat is removed from it?
65. In car racing, one advantage of mixing liquid nitrous oxide (\(\displaystyle N_2O\)) with air is that the boiling of the “nitrous” absorbs latent heat of vaporization and thus cools the air and ultimately the fuel-air mixture, allowing more fuel-air mixture to go into each cylinder. As a very rough look at this process, suppose 1.0 mol of nitrous oxide gas at its boiling point, −88°C , is mixed with 4.0 mol of air (assumed diatomic) at 30°C30°C. What is the final temperature of the mixture? Use the measured heat capacity of \(\displaystyle N_2O\) at 25°C , which is 30.4J/mol°C . (The primary advantage of nitrous oxide is that it consists of 1/3 oxygen, which is more than air contains, so it supplies more oxygen to burn the fuel. Another advantage is that its decomposition into nitrogen and oxygen releases energy in the cylinder.)
2.4 Distribution of Molecular Speeds
66. In a sample of hydrogen sulfide ( M=34.1g/mol ) at a temperature of \(\displaystyle 3.00×10^2K\), estimate the ratio of the number of molecules that have speeds very close to \(\displaystyle v_{rms}\) to the number that have speeds very close to \(\displaystyle 2v_{rms}\).
67. Using the approximation \(\displaystyle ∫^{v_1+Δv}_{v_1}f(v)dv≈f(v1)Δv\) for small Δv , estimate the fraction of nitrogen molecules at a temperature of \(\displaystyle 3.00×10^2K\) that have speeds between 290 m/s and 291 m/s.
68. Using the method of the preceding problem, estimate the fraction of nitric oxide (NO) molecules at a temperature of 250 K that have energies between \(\displaystyle 3.45×10^{−21}J\) and \(\displaystyle 3.50×10^{−21}J\).
69. By counting squares in the following figure, estimate the fraction of argon atoms at T= 300K that have speeds between 600 m/s and 800 m/s. The curve is correctly normalized. The value of a square is its length as measured on the x-axis times its height as measured on the y-axis, with the units given on those axes.
70. Using a numerical integration method such as Simpson’s rule, find the fraction of molecules in a sample of oxygen gas at a temperature of 250 K that have speeds between 100 m/s and 150 m/s. The molar mass of oxygen (\(\displaystyle O_2\)) is 32.0 g/mol. A precision to two significant digits is enough.
71. Find (a) the most probable speed,
(b) the average speed, and
(c) the rms speed for nitrogen molecules at 295 K.
72. Repeat the preceding problem for nitrogen molecules at 2950 K.
73. At what temperature is the average speed of carbon dioxide molecules ( M=44.0g/mol ) 510 m/s?
74. The most probable speed for molecules of a gas at 296 K is 263 m/s. What is the molar mass of the gas? (You might like to figure out what the gas is likely to be.)
75. a) At what temperature do oxygen molecules have the same average speed as helium atoms ( M=4.00g/mol ) have at 300 K?
b) What is the answer to the same question about most probable speeds?
c) What is the answer to the same question about rms speeds?
Additional Problems
76. In the deep space between galaxies, the density of molecules (which are mostly single atoms) can be as low as \(\displaystyle 10^6atoms/m^3\), and the temperature is a frigid 2.7 K. What is the pressure?
(b) What volume (in \(\displaystyle m^3\)) is occupied by 1 mol of gas?
(c) If this volume is a cube, what is the length of its sides in kilometers?
77. (a) Find the density in SI units of air at a pressure of 1.00 atm and a temperature of 20°C 20°C, assuming that air is \(\displaystyle 78%N_2,21%O_2,\) and \(\displaystyle 1%Ar\)
(b) Find the density of the atmosphere on Venus, assuming that it’s \(\displaystyle 96%CO_2\) and \(\displaystyle 4%N_2\), with a temperature of 737 K and a pressure of 92.0 atm.
78. The air inside a hot-air balloon has a temperature of 370 K and a pressure of 101.3 kPa, the same as that of the air outside. Using the composition of air as \(\displaystyle 78%N_2,21%O_2,\) and \(\displaystyle 1%Ar\), find the density of the air inside the balloon.
79. When an air bubble rises from the bottom to the top of a freshwater lake, its volume increases by 80% . If the temperatures at the bottom and the top of the lake are 4.0 and 10 °C , respectively, how deep is the lake?
80. (a) Use the ideal gas equation to estimate the temperature at which 1.00 kg of steam (molar mass M=18.0g/ mol ) at a pressure of \(\displaystyle 1.50×10^6Pa\) occupies a volume of \(\displaystyle 0.220m^3\).
(b) The van der Waals constants for water are \(\displaystyle a=0.5537Pa⋅m^6/mol^2\) and \(\displaystyle b=3.049×10^{−5}m^3/mol\). Use the Van der Waals equation of state to estimate the temperature under the same conditions.
(c) The actual temperature is 779 K. Which estimate is better?
81. One process for decaffeinating coffee uses carbon dioxide ( M=44.0g/mol ) at a molar density of about \(\displaystyle 14,600mol/m^3\) and a temperature of about 60°C .
(a) Is \(\displaystyle CO_2\) a solid, liquid, gas, or supercritical fluid under those conditions?
(b) The van der Waals constants for carbon dioxide are \(\displaystyle a=0.3658Pa⋅m^6/mol^2\) and \(\displaystyle b=4.286×10^{−5}m^3/mol\). Using the van der Waals equation, estimate the pressure of \(\displaystyle CO_2\) at that temperature and density.
82. On a winter day when the air temperature is 0°C , the relative humidity is 50% . Outside air comes inside and is heated to a room temperature of 20°C . What is the relative humidity of the air inside the room. (Does this problem show why inside air is so dry in winter?)
83. On a warm day when the air temperature is 30°C , a metal can is slowly cooled by adding bits of ice to liquid water in it. Condensation first appears when the can reaches 15°C . What is the relative humidity of the air?
84. (a) People often think of humid air as “heavy.” Compare the densities of air with 0% relative humidity and 100% 100% relative humidity when both are at 1 atm and 30°C . Assume that the dry air is an ideal gas composed of molecules with a molar mass of 29.0 g/mol and the moist air is the same gas mixed with water vapor.
(b) As discussed in the chapter on the applications of Newton’s laws, the air resistance felt by projectiles such as baseballs and golf balls is approximately \(\displaystyle F_D=CρAv^2/2\), where ρ is the mass density of the air, A is the cross-sectional area of the projectile, and C is the projectile’s drag coefficient. For a fixed air pressure, describe qualitatively how the range of a projectile changes with the relative humidity.
(c) When a thunderstorm is coming, usually the humidity is high and the air pressure is low. Do those conditions give an advantage or disadvantage to home-run hitters?
85. The mean free path for helium at a certain temperature and pressure is \(\displaystyle 2.10×10^{−7}m\). The radius of a helium atom can be taken as \(\displaystyle 1.10×10^{−11}m\). What is the measure of the density of helium under those conditions
(a) in molecules per cubic meter and
(b) in moles per cubic meter?
86. The mean free path for methane at a temperature of 269 K and a pressure of \(\displaystyle 1.11×10^5Pa\) is \(\displaystyle 4.81×10^{−8}m\). Find the effective radius r of the methane molecule.
87. In the chapter on fluid mechanics, Bernoulli’s equation for the flow of incompressible fluids was explained in terms of changes affecting a small volume dV of fluid. Such volumes are a fundamental idea in the study of the flow of compressible fluids such as gases as well. For the equations of hydrodynamics to apply, the mean free path must be much less than the linear size of such a volume, \(\displaystyle a≈dV^{1/3}\). For air in the stratosphere at a temperature of 220 K and a pressure of 5.8 kPa, how big should a be for it to be 100 times the mean free path? Take the effective radius of air molecules to be \(\displaystyle 1.88×10^{−11}m\), which is roughly correct for \(\displaystyle N_2\).
88. Find the total number of collisions between molecules in 1.00 s in 1.00 L of nitrogen gas at standard temperature and pressure ( 0°C , 1.00 atm). Use \(\displaystyle 1.88×10^{−10}m\) as the effective radius of a nitrogen molecule. (The number of collisions per second is the reciprocal of the collision time.) Keep in mind that each collision involves two molecules, so if one molecule collides once in a certain period of time, the collision of the molecule it hit cannot be counted.
89. (a) Estimate the specific heat capacity of sodium from the Law of Dulong and Petit. The molar mass of sodium is 23.0 g/mol.
(b) What is the percent error of your estimate from the known value, 1230J/kg⋅°C ?
90. A sealed, perfectly insulated container contains 0.630 mol of air at 20.0°C and an iron stirring bar of mass 40.0 g. The stirring bar is magnetically driven to a kinetic energy of 50.0 J and allowed to slow down by air resistance. What is the equilibrium temperature?
91. Find the ratio \(\displaystyle f(v_p)/f(v_{rms})\) for hydrogen gas ( M=2.02g/mol ) at a temperature of 77.0 K.
92. Unreasonable results. (a) Find the temperature of 0.360 kg of water, modeled as an ideal gas, at a pressure of \(\displaystyle 1.01×10^5Pa\) if it has a volume of \(\displaystyle 0.615m^3\).
(b) What is unreasonable about this answer? How could you get a better answer?
93. Unreasonable results. (a) Find the average speed of hydrogen sulfide, \(\displaystyle H_2S\), molecules at a temperature of 250 K. Its molar mass is 31.4 g/mol
(b) The result isn’t very unreasonable, but why is it less reliable than those for, say, neon or nitrogen?
Challenge Problems
94. An airtight dispenser for drinking water is 25cm×10cm in horizontal dimensions and 20 cm tall. It has a tap of negligible volume that opens at the level of the bottom of the dispenser. Initially, it contains water to a level 3.0 cm from the top and air at the ambient pressure, 1.00 atm, from there to the top. When the tap is opened, water will flow out until the gauge pressure at the bottom of the dispenser, and thus at the opening of the tap, is 0. What volume of water flows out? Assume the temperature is constant, the dispenser is perfectly rigid, and the water has a constant density of \(\displaystyle 1000kg/m^3\).
95. Eight bumper cars, each with a mass of 322 kg, are running in a room 21.0 m long and 13.0 m wide. They have no drivers, so they just bounce around on their own. The rms speed of the cars is 2.50 m/s. Repeating the arguments of Pressure, Temperature, and RMS Speed, find the average force per unit length (analogous to pressure) that the cars exert on the walls.
96. Verify that \(\displaystyle v_p=\sqrt{\frac{2k_BT}{m}}\).
97. Verify the normalization equation \(\displaystyle ∫^∞_0f(v)dv=1\). In doing the integral, first make the substitution \(\displaystyle u=\sqrt{\frac{m}{2k_BT}}v=\frac{v}{v_p}\).This “scaling” transformation gives you all features of the answer except for the integral, which is a dimensionless numerical factor. You’ll need the formula \(\displaystyle ∫^∞_0x^2e^{−x^2}dx=\frac{\sqrt{π}}{4}\) find the numerical factor and verify the normalization.
98. Verify that \(\displaystyle \bar{v}=\sqrt{\frac{8}{π}\frac{k_BT}{m}}.\). Make the same scaling transformation as in the preceding problem.
99. Verify that \(\displaystyle v_{rms}=\sqrt{\bar{v^2}=\sqrt{\frac{3k_BT}{m}}\).


7.10: Chapter 7 Problems
An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I.
7.1 Derive the following relations from the definitions of \(\alpha, \kappa_{T}\), and \(\rho\) :
\[
\alpha=-\frac{1}{\rho}\left(\frac{\partial \rho}{\partial T}\right)_{p} \quad \kappa_{T}=\frac{1}{\rho}\left(\frac{\partial \rho}{\partial p}\right)_{T}
\]
7.2 Use equations in this chapter to derive the following expressions for an ideal gas:
\[
\alpha=1 / T \quad \kappa_{T}=1 / p
\]
7.3 For a gas with the simple equation of state
\[
V_{\mathrm{m}}=\frac{R T}{p}+B
\]
(Eq. 2.2.8), where \(B\) is the second virial coefficient (a function of \(T\) ), find expressions for \(\alpha\), \(\kappa_{T}\), and \(\left(\partial U_{\mathrm{m}} / \partial V\right)_{T}\) in terms of \(\mathrm{d} B / \mathrm{d} T\) and other state functions.
7.4 Show that when the virial equation \(p V_{\mathrm{m}}=R T\left(1+B_{p} p+C_{p} p^{2}+\cdots\right)\) (Eq. 2.2.3) adequately represents the equation of state of a real gas, the Joule-Thomson coefficient is given by
\[
\mu_{\mathrm{JT}}=\frac{R T^{2}\left[\mathrm{~d} B_{p} / \mathrm{d} T+\left(\mathrm{d} C_{p} / \mathrm{d} T\right) p+\cdots\right]}{C_{p, \mathrm{~m}}}
\]
Note that the limiting value at low pressure, \(R T^{2}\left(\mathrm{~d} B_{p} / \mathrm{d} T\right) / C_{p, \mathrm{~m}}\), is not necessarily equal to zero even though the equation of state approaches that of an ideal gas in this limit.
7.5 The quantity \((\partial T / \partial V)_{U}\) is called the Joule coefficient. James Joule attempted to evaluate this quantity by measuring the temperature change accompanying the expansion of air into a vacuum - the "Joule experiment." Write an expression for the total differential of \(U\) with \(T\) and \(V\) as independent variables, and by a procedure similar to that used in Sec. 7.5.2 show that the Joule coefficient is equal to
\[
\frac{p-\alpha T / \kappa_{T}}{C_{V}}
\]
7.6 \(p-V-T\) data for several organic liquids were measured by Gibson and Loeffler. \({ }^{11}\) The following formulas describe the results for aniline.
Molar volume as a function of temperature at \(p=1\) bar \((298-358 \mathrm{~K})\) :
\[
V_{\mathrm{m}}=a+b T+c T^{2}+d T^{3}
\]
where the parameters have the values
\[
\begin{array}{ll}
a=69.287 \mathrm{~cm}^{3} \mathrm{~mol}^{-1} & c=-1.0443 \times 10^{-4} \mathrm{~cm}^{3} \mathrm{~K}^{-2} \mathrm{~mol}^{-1} \\
b=0.08852 \mathrm{~cm}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} & d=1.940 \times 10^{-7} \mathrm{~cm}^{3} \mathrm{~K}^{-3} \mathrm{~mol}^{-1}
\end{array}
\]
Molar volume as a function of pressure at \(T=298.15 \mathrm{~K}\) (1-1000 bar):
\[
V_{\mathrm{m}}=e-f \ln (g+p / \text { bar })
\]
where the parameter values are
\[
e=156.812 \mathrm{~cm}^{3} \mathrm{~mol}^{-1} \quad f=8.5834 \mathrm{~cm}^{3} \mathrm{~mol}^{-1} \quad g=2006.6
\]
(a) Use these formulas to evaluate \(\alpha, \kappa_{T},(\partial p / \partial T)_{V}\), and \((\partial U / \partial V)_{T}\) (the internal pressure) for aniline at \(T=298.15 \mathrm{~K}\) and \(p=1.000\) bar.
(b) Estimate the pressure increase if the temperature of a fixed amount of aniline is increased by \(0.10 \mathrm{~K}\) at constant volume.
7.7 (a) From the total differential of \(H\) with \(T\) and \(p\) as independent variables, derive the relation \(\left(\partial C_{p, \mathrm{~m}} / \partial p\right)_{T}=-T\left(\partial^{2} V_{\mathrm{m}} / \partial T^{2}\right)_{p}\)
(b) Evaluate \(\left(\partial C_{p, \mathrm{~m}} / \partial p\right)_{T}\) for liquid aniline at \(300.0 \mathrm{~K}\) and 1 bar using data in Prob. \(7.6 .\)
7.8 (a) From the total differential of \(V\) with \(T\) and \(p\) as independent variables, derive the relation \((\partial \alpha / \partial p)_{T}=-\left(\partial \kappa_{T} / \partial T\right)_{p}\).
(b) Use this relation to estimate the value of \(\alpha\) for benzene at \(25^{\circ} \mathrm{C}\) and 500 bar, given that the value of \(\alpha\) is \(1.2 \times 10^{-3} \mathrm{~K}^{-1}\) at \(25^{\circ} \mathrm{C}\) and 1 bar. (Use information from Fig. \(7.2\) on page 168.)
7.9 Certain equations of state supposed to be applicable to nonpolar liquids and gases are of the form \(p=T f\left(V_{\mathrm{m}}\right)-a / V_{\mathrm{m}}^{2}\), where \(f\left(V_{\mathrm{m}}\right)\) is a function of the molar volume only and \(a\) is a constant.
(a) Show that the van der Waals equation of state \(\left(p+a / V_{\mathrm{m}}^{2}\right)\left(V_{\mathrm{m}}-b\right)=R T\) (where \(a\) and \(b\) are constants) is of this form.
(b) Show that any fluid with an equation of state of this form has an internal pressure equal to \(a / V_{\mathrm{m}}^{2}\).
7.10 Suppose that the molar heat capacity at constant pressure of a substance has a temperature dependence given by \(C_{p, \mathrm{~m}}=a+b T+c T^{2}\), where \(a, b\), and \(c\) are constants. Consider the heating of an amount \(n\) of the substance from \(T_{1}\) to \(T_{2}\) at constant pressure. Find expressions for \(\Delta H\) and \(\Delta S\) for this process in terms of \(a, b, c, n, T_{1}\), and \(T_{2}\).
7.11 At \(p=1 \mathrm{~atm}\), the molar heat capacity at constant pressure of aluminum is given by
\[
C_{p, \mathrm{~m}}=a+b T
\]
where the constants have the values
\[
a=20.67 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} \quad b=0.01238 \mathrm{JK}^{-2} \mathrm{~mol}^{-1}
\]
Calculate the quantity of electrical work needed to heat \(2.000 \mathrm{~mol}\) of aluminum from \(300.00 \mathrm{~K}\) to \(400.00 \mathrm{~K}\) at \(1 \mathrm{~atm}\) in an adiabatic enclosure.
7.12 The temperature dependence of the standard molar heat capacity of gaseous carbon dioxide in the temperature range \(298 \mathrm{~K}-2000 \mathrm{~K}\) is given by
\[
C_{p, \mathrm{~m}}^{\circ}=a+b T+\frac{c}{T^{2}}
\]
where the constants have the values
\[
a=44.2 \mathrm{JK}^{-1} \mathrm{~mol}^{-1} \quad b=8.8 \times 10^{-3} \mathrm{~J} \mathrm{~K}^{-2} \mathrm{~mol}^{-1} \quad c=-8.6 \times 10^{5} \mathrm{~J} \mathrm{~K} \mathrm{~mol}^{-1}
\]
Calculate the enthalpy and entropy changes when one mole of \(\mathrm{CO}_{2}\) is heated at 1 bar from \(300.00 \mathrm{~K}\) to \(800.00 \mathrm{~K}\). You can assume that at this pressure \(C_{p, \mathrm{~m}}\) is practically equal to \(C_{p, \mathrm{~m}}^{\circ}\).
7.13 This problem concerns gaseous carbon dioxide. At \(400 \mathrm{~K}\), the relation between \(p\) and \(V_{\mathrm{m}}\) at pressures up to at least 100 bar is given to good accuracy by a virial equation of state truncated
at the second virial coefficient, \(B\). In the temperature range \(300 \mathrm{~K}-800 \mathrm{~K}\) the dependence of \(B\) on temperature is given by
\[
B=a^{\prime}+b^{\prime} T+c^{\prime} T^{2}+d^{\prime} T^{3}
\]
where the constants have the values
\[
\begin{aligned}
&a^{\prime}=-521 \mathrm{~cm}^{3} \mathrm{~mol}^{-1} \\
&b^{\prime}=2.08 \mathrm{~cm}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} \\
&c^{\prime}=-2.89 \times 10^{-3} \mathrm{~cm}^{3} \mathrm{~K}^{-2} \mathrm{~mol}^{-1} \\
&d^{\prime}=1.397 \times 10^{-6} \mathrm{~cm}^{3} \mathrm{~K}^{-3} \mathrm{~mol}^{-1}
\end{aligned}
\]
(a) From information in Prob. 7.12, calculate the standard molar heat capacity at constant pressure, \(C_{p, \mathrm{~m}}^{\circ}\), at \(T=400.0 \mathrm{~K}\).
(b) Estimate the value of \(C_{p, \mathrm{~m}}\) under the conditions \(T=400.0 \mathrm{~K}\) and \(p=100.0\) bar.
7.14 A chemist, needing to determine the specific heat capacity of a certain liquid but not having an electrically heated calorimeter at her disposal, used the following simple procedure known as drop calorimetry. She placed \(500.0 \mathrm{~g}\) of the liquid in a thermally insulated container equipped with a lid and a thermometer. After recording the initial temperature of the liquid, \(24.80^{\circ} \mathrm{C}\), she removed a \(60.17\)-g block of aluminum metal from a boiling water bath at \(100.00^{\circ} \mathrm{C}\) and quickly immersed it in the liquid in the container. After the contents of the container had become thermally equilibrated, she recorded a final temperature of \(27.92^{\circ} \mathrm{C}\). She calculated the specific heat capacity \(C_{p} / m\) of the liquid from these data, making use of the molar mass of aluminum \(\left(M=26.9815 \mathrm{~g} \mathrm{~mol}^{-1}\right)\) and the formula for the molar heat capacity of aluminum given in Prob. \(7.11 .\)
(a) From these data, find the specific heat capacity of the liquid under the assumption that its value does not vary with temperature. Hint: Treat the temperature equilibration process as adiabatic and isobaric \((\Delta H=0)\), and equate \(\Delta H\) to the sum of the enthalpy changes in the two phases.
(b) Show that the value obtained in part (a) is actually an average value of \(C_{p} / m\) over the temperature range between the initial and final temperatures of the liquid given by
\[
\frac{\int_{T_{1}}^{T_{2}}\left(C_{p} / m\right) \mathrm{d} T}{T_{2}-T_{1}}
\]
7.15 Suppose a gas has the virial equation of state \(p V_{\mathrm{m}}=R T\left(1+B_{p} p+C_{p} p^{2}\right)\), where \(B_{p}\) and \(C_{p}\) depend only on \(T\), and higher powers of \(p\) can be ignored.
(a) Derive an expression for the fugacity coefficient, \(\phi\), of this gas as a function of \(p\).
(b) For \(\mathrm{CO}_{2}(\mathrm{~g})\) at \(0.00^{\circ} \mathrm{C}\), the virial coefficients have the values \(B_{p}=-6.67 \times 10^{-3}\) bar \(^{-1}\) and \(C_{p}=-3.4 \times 10^{-5} \mathrm{bar}^{-2}\). Evaluate the fugacity \(f\) at \(0.00^{\circ} \mathrm{C}\) and \(p=20.0\) bar.
7.16 Table \(7.6\) on the next page lists values of the molar volume of gaseous \(\mathrm{H}_{2} \mathrm{O}\) at \(400.00^{\circ} \mathrm{C}\) and 12 pressures.
(a) Evaluate the fugacity coefficient and fugacity of \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) at \(400.00^{\circ} \mathrm{C}\) and 200 bar.
(b) Show that the second virial coefficient \(B\) in the virial equation of state, \(p V_{\mathrm{m}}=R T(1+\) \(\left.B / V_{\mathrm{m}}+C / V_{\mathrm{m}}^{2}+\cdots\right)\), is given by
\[
B=R T \lim _{p \rightarrow 0}\left(\frac{V_{\mathrm{m}}}{R T}-\frac{1}{p}\right)
\]
where the limit is taken at constant \(T\). Then evaluate \(B\) for \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) at \(400.00^{\circ} \mathrm{C}\).
Table 7.6 Molar volume of \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) at \(400.00^{\circ} \mathrm{C}^{a}\)
\[\begin{tabular}{cccc}
\hline\(p / 10^{5} \mathrm{~Pa}\) & \(V_{\mathrm{m}} / 10^{-3} \mathrm{~m}^{3} \mathrm{~mol}^{-1}\) & \(p / 10^{5} \mathrm{~Pa}\) & \(V_{\mathrm{m}} / 10^{-3} \mathrm{~m}^{3} \mathrm{~mol}^{-1}\) \\
\hline 1 & \(55.896\) & 100 & \(0.47575\) \\
10 & \(5.5231\) & 120 & \(0.37976\) \\
20 & \(2.7237\) & 140 & \(0.31020\) \\
40 & \(1.3224\) & 160 & \(0.25699\) \\
60 & \(0.85374\) & 180 & \(0.21447\) \\
80 & \(0.61817\) & 200 & \(0.17918\) \\
\hline
\end{tabular}\]
\({ }^{a}\) based on data in Ref. [75]


10.8: Reading- New Products in the Portfolio
Factors Influencing the Pace of Product Development
New-product introductions are an important component of the product portfolio. This has always been the case, but there have been a number of changes since 2000 that have increased the pace of new-product development and, with it, the importance of new products in the portfolio.
The Internet has fundamentally changed the way that we think about new products. We could devote an entire course to exploring and substantiating this statement, but for our purposes here, we’ll concentrate on the following notable developments:
- The Internet increases our ability to find new products. In the past, if a local store carried version 3.0 of any product, a buyer could consider the attributes of version 3.0 and make a purchase decision. Today, the buyer may enter the store, but she’s more likely to know the improvements incorporated in version 4.0 and the new features expected in version 5.0.
- The Internet supports real-time comparisons of competitive products, including their features and users’ experience with the products.
- The Internet enables new products and services that have changed expectations for service delivery.
- The Internet enables customers to recommend new products and experiences to others.
In addition to these Internet-related developments, there are many new tools, technologies, and methodologies that are speeding the pace of new product development. For example, software development frameworks make it possible to launch, test, and refine a new Web-based service in a fraction of the time that it required in the past. A new retail offering that, in the past, would have required a team of programmers to bring to market can be launched on Etsy today in less than an hour.
New products have never been faster and easier to launch.
New Products Bring Risk
Still, new products are risky. In the strategic opportunity matrix section we reviewed a number of strategies that include new and existing products. Why is it important to consider whether the product and/or market is new in this strategic context? Current products in current markets are known, and new products and new markets are not known.
In this section we have discussed a couple of examples of new products in the context of company strategies. Apple launched five major iPod models in six years, and then followed them with a total of twenty different versions of those models. Perhaps most interesting, Apple launched three different iPod models—the Classic, the Mini, and the Shuffle—before it saw significant sales growth.
The Johnson & Johnson medical practices unit launched more than fifty new products between 2012 and 2014 but has seen only a 0.5 percent increase in sales during that period. With the 2014 sale of a diagnostics product that was showing declining sales numbers, the SBU is better poised to show growth in its 2015 sales numbers.
Both of these examples show that new products do not guarantee new sales, and they certainly don’t guarantee immediate success.
Role of New Products in the Portfolio
New products bring greater risk to the product portfolio but also greater potential reward. The goal of portfolio management is to balance risk in order to create strong performance today and in the future. Though new products can drain resources in the short run, they are positioned to generate new sales in the long run—and to take off when other products are entering stages of maturity and decline.
The early investment in multiple new iPod models that were not growing quickly created a future base for the growth of the iPod. That, in turn, generated returns and positioned the Apple brand for the development and release of the iPhone. Johnson & Johnson’s medical products division has aggressively invested in new products that can spur growth and divested from products with declining sales. Still, in the first three quarters of 2015, overall growth for the division is down. It is not yet clear whether the new products will generate enough growth to replace the growth of their predecessors. [1]
- files.shareholder.com/downloads/JNJ/1279939564x0x854183/3152FFAB-82A7-4F38-AA75-D25F107767DB/Sales_of_Key_Products_Franchises_3Q2015.pdf ↵
Contributors and Attributions
- Reading: New Products in the Portfolio. Provided by : Lumen Learning. License : CC BY: Attribution


7.2.2: Building Polygons (Part 2)
Lesson
Let's build more triangles.
Exercise \(\PageIndex{1}\): Where is Lin?
At a park, the slide is 5 meters east of the swings. Lin is standing 3 meters away from the slide.
- Draw a diagram of the situation including a place where Lin could be.
- How far away from the swings is Lin in your diagram?
- Where are some other places Lin could be?
Exercise \(\PageIndex{2}\): How Long is the Third Side?
Use the applet to answer the questions.
- Build as many different triangles as you can that have one side length of 5 inches and one of 4 inches. Record the side lengths of each triangle you build.
- Are there any other lengths that could be used for the third side of the triangle but aren't values of the sliders?
- Are there any lengths that are values of the sliders but could not be used as the third side of the triangle?
Are you ready for more?
Assuming you had access to strips of any length, and you used the 9-inch and 5-inch strips as the first two sides, complete the sentences:
- The third side can't be _____ inches or longer.
- The third side can't be _____ inches or shorter.
Exercise \(\PageIndex{3}\): Swinging the Sides Around
We'll explore a method for drawing a triangle that has three specific side lengths. Use the applet to answer the questions.
-
Follow these instructions to mark the possible endpoints of one side:
-
For now, ignore segment \(AC\), the 3-inch side length on the left side
Figure \(\PageIndex{1}\) - Let segment \(BD\) be the 3-unit side length on the right side. Right-click on point \(D\), check Trace On. Rotate the point, drawing all the places where a 3-inch side could end.
-
For now, ignore segment \(AC\), the 3-inch side length on the left side
- What shape have you drawn while moving \(BD\) around? Why? Which tool in your geometry toolkit can do something similar?
- Use your drawing to create two unique triangles, each with a base of length 4 inches and a side of length 3 inches. Use a different color to draw each triangle.
- Repeat the previous instructions, letting segment \(AC\) be the 3-unit side length.
- Using a third color, draw a point where the two traces intersect. Using this third color, draw a triangle with side lengths of 4 inches, 3 inches, and 3 inches.
Summary
If we want to build a polygon with two given side lengths that share a vertex, we can think of them as being connected by a hinge that can be opened or closed:
All of the possible positions of the endpoint of the moving side form a circle:
You may have noticed that sometimes it is not possible to build a polygon given a set of lengths. For example, if we have one really, really long segment and a bunch of short segments, we may not be able to connect them all up. Here's what happens if you try to make a triangle with side lengths 21, 4, and 2:
The short sides don't seem like they can meet up because they are too far away from each other.
If we draw circles of radius 4 and 2 on the endpoints of the side of length 21 to represent positions for the shorter sides, we can see that there are no places for the short sides that would allow them to meet up and form a triangle.
In general, the longest side length must be less than the sum of the other two side lengths. If not, we can’t make a triangle!
If we can make a triangle with three given side lengths, it turns out that the measures of the corresponding angles will always be the same. For example, if two triangles have side lengths 3, 4, and 5, they will have the same corresponding angle measures.
Practice
Exercise \(\PageIndex{4}\)
In the diagram, the length of segment \(AB\) is 10 units and the radius of the circle centered at \(A\) is 4 units. Use this to create two unique triangles, each with a side of length 10 and a side of length 4. Label the sides that have length 10 and 4.
Exercise \(\PageIndex{5}\)
Select all the sets of three side lengths that will make a triangle.
- \(3, 4, 8\)
- \(7, 6, 12\)
- \(5, 11, 13\)
- \(4, 6, 12\)
- \(4, 6, 10\)
Exercise \(\PageIndex{6}\)
Based on signal strength, a person knows their lost phone is exactly 47 feet from the nearest cell tower. The person is currently standing 23 feet from the same cell tower. What is the closest the phone could be to the person? What is the furthest their phone could be from them?
Exercise \(\PageIndex{7}\)
Each row contains the degree measures of two complementary angles. Complete the table.
| measure of an angle | measure of its complement |
|---|---|
| \(80^{\circ}\) | |
| \(25^{\circ}\) | |
| \(54^{\circ}\) | |
| \(x\) |
(From Unit 7.1.2)
Exercise \(\PageIndex{8}\)
Here are two patterns made using identical rhombuses. Without using a protractor, determine the value of \(a\) and \(b\). Explain or show your reasoning.
(From Unit 7.1.1)
Exercise \(\PageIndex{9}\)
Mai’s family is traveling in a car at a constant speed of 65 miles per hour.
- At that speed, how long will it take them to travel 200 miles?
- How far do they travel in 25 minutes?
(From Unit 4.1.3)


18.3: T Lymphocytes
- Describe the process of T-cell maturation and thymic selection
- Explain the genetic events that lead to diversity of T-cell receptors
- Compare and contrast the various classes and subtypes of T cells in terms of activation and function
- Explain the mechanism by which superantigens effect unregulated T-cell activation
As explained in Overview of Specific Adaptive Immunity , the antibodies involved in humoral immunity often bind pathogens and toxins before they can attach to and invade host cells. Thus, humoral immunity is primarily concerned with fighting pathogens in extracellular spaces. However, pathogens that have already gained entry to host cells are largely protected from the humoral antibody-mediated defenses. Cellular immunity, on the other hand, targets and eliminates intracellular pathogens through the actions of T lymphocytes, or T cells (Figure \(\PageIndex{1}\)). T cells also play a more central role in orchestrating the overall adaptive immune response (humoral as well as cellular) along with the cellular defenses of innate immunity.
T Cell Production and Maturation
T cells, like all other white blood cells involved in innate and adaptive immunity, are formed from multipotent hematopoietic stem cells (HSCs) in the bone marrow (see Figure 17.3.1 ). However, unlike the white blood cells of innate immunity, eventual T cells differentiate first into lymphoid stem cells that then become small, immature lymphocytes, sometimes called lymphoblasts. The first steps of differentiation occur in the red marrow of bones (Figure \(\PageIndex{2}\)), after which immature T lymphocytes enter the bloodstream and travel to the thymus for the final steps of maturation (Figure \(\PageIndex{3}\)). Once in the thymus, the immature T lymphocytes are referred to as thymocytes.
The maturation of thymocytes within the thymus can be divided into tree critical steps of positive and negative selection, collectively referred to as thymic selection. The first step of thymic selection occurs in the cortex of the thymus and involves the development of a functional T-cell receptor (TCR) that is required for activation by APCs. Thymocytes with defective TCRs are removed by negative selection through the induction of apoptosis (programmed controlled cell death). The second step of thymic selection also occurs in the cortex and involves the positive selection of thymocytes that will interact appropriately with MHC molecules. Thymocytes that can interact appropriately with MHC molecules receive a positive stimulation that moves them further through the process of maturation, whereas thymocytes that do not interact appropriately are not stimulated and are eliminated by apoptosis. The third and final step of thymic selection occurs in both the cortex and medulla and involves negative selection to remove self-reacting thymocytes, those that react to self-antigens, by apoptosis. This final step is sometimes referred to as central tolerance because it prevents self-reacting T cells from reaching the bloodstream and potentially causing autoimmune disease, which occurs when the immune system attacks healthy “self” cells.
Despite central tolerance, some self-reactive T cells generally escape the thymus and enter the peripheral bloodstream. Therefore, a second line of defense called peripheral tolerance is needed to protect against autoimmune disease. Peripheral tolerance involves mechanisms of anergy and inhibition of self-reactive T cells by regulatory T cells. Anergy refers to a state of nonresponsiveness to antigen stimulation. In the case of self-reactive T cells that escape the thymus, lack of an essential co-stimulatory signal required for activation causes anergy and prevents autoimmune activation. Regulatory T cells participate in peripheral tolerance by inhibiting the activation and function of self-reactive T cells and by secreting anti-inflammatory cytokines.
It is not completely understood what events specifically direct maturation of thymocytes into regulatory T cells. Current theories suggest the critical events may occur during the third step of thymic selection, when most self-reactive T cells are eliminated. Regulatory T cells may receive a unique signal that is below the threshold required to target them for negative selection and apoptosis. Consequently, these cells continue to mature and then exit the thymus, armed to inhibit the activation of self-reactive T cells.
It has been estimated that the three steps of thymic selection eliminate 98% of thymocytes. The remaining 2% that exit the thymus migrate through the bloodstream and lymphatic system to sites of secondary lymphoid organs/tissues, such as the lymph nodes, spleen, and tonsils (Figure \(\PageIndex{3}\)), where they await activation through the presentation of specific antigens by APCs. Until they are activated, they are known as mature naïve T cells.
- What anatomical sites are involved in T cell production and maturation?
- What are the three steps involved in thymic selection?
- Why are central tolerance and peripheral tolerance important? What do they prevent?
Classes of T Cells
T cells can be categorized into three distinct classes: helper T cells, regulatory T cells, and cytotoxic T cells. These classes are differentiated based on their expression of certain surface molecules, their mode of activation, and their functional roles in adaptive immunity (Table \(\PageIndex{1}\)).
All T cells produce cluster of differentiation (CD) molecules, cell surface glycoproteins that can be used to identify and distinguish between the various types of white blood cells. Although T cells can produce a variety of CD molecules, CD4and CD8 are the two most important used for differentiation of the classes. Helper T cells and regulatory T cells are characterized by the expression of CD4 on their surface, whereas cytotoxic T cells are characterized by the expression of CD8.
Classes of T cells can also be distinguished by the specific MHC molecules and APCs with which they interact for activation. Helper T cells and regulatory T cells can only be activated by APCs presenting antigens associated with MHC II. In contrast, cytotoxic T cells recognize antigens presented in association with MHC I, either by APCs or by nucleated cells infected with an intracellular pathogen.
The different classes of T cells also play different functional roles in the immune system. Helper T cells serve as the central orchestrators that help activate and direct functions of humoral and cellular immunity. In addition, helper T cells enhance the pathogen-killing functions of macrophages and NK cells of innate immunity. In contrast, the primary role of regulatory T cells is to prevent undesirable and potentially damaging immune responses. Their role in peripheral tolerance, for example, protects against autoimmune disorders, as discussed earlier. Finally, cytotoxic T cells are the primary effector cells for cellular immunity. They recognize and target cells that have been infected by intracellular pathogens, destroying infected cells along with the pathogens inside.
| Class | Surface CD Molecules | Activation | Functions |
|---|---|---|---|
| Helper T cells | CD4 | APCs presenting antigens associated with MHC II | Orchestrate humoral and cellular immunity |
| Involved in the activation of macrophages and NK cells | |||
| Regulatory T cells | CD4 | APCs presenting antigens associated with MHC II | Involved in peripheral tolerance and prevention of autoimmune responses |
| Cytotoxic T cells | CD8 | APCs or infected nucleated cells presenting antigens associated with MHC I | Destroy cells infected with intracellular pathogens |
- What are the unique functions of the three classes of T cells?
- Which T cells can be activated by antigens presented by cells other than APCs?
T-Cell Receptors
For both helper T cells and cytotoxic T cells, activation is a complex process that requires the interactions of multiple molecules and exposure to cytokines. The T-cell receptor (TCR) is involved in the first step of pathogen epitope recognition during the activation process.
The TCR comes from the same receptor family as the antibodies IgD and IgM, the antigen receptors on the B cell membrane surface, and thus shares common structural elements. Similar to antibodies, the TCR has a variable regionand a constant region, and the variable region provides the antigen-binding site (Figure \(\PageIndex{4}\)). However, the structure of TCR is smaller and less complex than the immunoglobulin molecules ( Figure 18.1.4 ). Whereas immunoglobulins have four peptide chains and Y-shaped structures, the TCR consists of just two peptide chains (α and β chains), both of which span the cytoplasmic membrane of the T cell.
TCRs are epitope-specific, and it has been estimated that 25 million T cells with unique epitope-binding TCRs are required to protect an individual against a wide range of microbial pathogens. Because the human genome only contains about 25,000 genes, we know that each specific TCR cannot be encoded by its own set of genes. This raises the question of how such a vast population of T cells with millions of specific TCRs can be achieved. The answer is a process called genetic rearrangement, which occurs in the thymus during the first step of thymic selection.
The genes that code for the variable regions of the TCR are divided into distinct gene segments called variable (V), diversity (D), and joining (J) segments. The genes segments associated with the α chain of the TCR consist 70 or more different V α segments and 61 different J α segments. The gene segments associated with the β chain of the TCR consist of 52 different V β segments, two different D β segments, and 13 different J β segments. During the development of the functional TCR in the thymus, genetic rearrangement in a T cell brings together one V α segment and one J α segment to code for the variable region of the α chain. Similarly, genetic rearrangement brings one of the V β segments together with one of the D β segments and one of thetJ β segments to code for the variable region of the β chain. All the possible combinations of rearrangements between different segments of V, D, and J provide the genetic diversity required to produce millions of TCRs with unique epitope-specific variable regions.
- What are the similarities and differences between TCRs and immunoglobulins?
- What process is used to provide millions of unique TCR binding sites?
Activation and Differentiation of Helper T Cells
Helper T cells can only be activated by APCs presenting processed foreign epitopes in association with MHC II. The first step in the activation process is TCR recognition of the specific foreign epitope presented within the MHC II antigen-binding cleft. The second step involves the interaction of CD4 on the helper T cell with a region of the MHC II molecule separate from the antigen-binding cleft. This second interaction anchors the MHC II-TCR complex and ensures that the helper T cell is recognizing both the foreign (“nonself”) epitope and “self” antigen of the APC; both recognitions are required for activation of the cell. In the third step, the APC and T cell secrete cytokines that activate the helper T cell. The activated helper T cell then proliferates, dividing by mitosis to produce clonal naïve helper T cells that differentiate into subtypes with different functions (Figure \(\PageIndex{5}\)).
Activated helper T cells can differentiate into one of four distinct subtypes, summarized in Table \(\PageIndex{2}\). The differentiation process is directed by APC-secreted cytokines. Depending on which APC-secreted cytokines interact with an activated helper T cell, the cell may differentiate into a T helper 1 (T H 1) cell, a T helper 2 (T H 2) cell, or a memory helper T cell. The two types of helper T cells are relatively short-lived effector cells, meaning that they perform various functions of the immediate immune response. In contrast, memory helper T cells are relatively long lived; they are programmed to “remember” a specific antigen or epitope in order to mount a rapid, strong, secondary response to subsequent exposures.
T H 1 cells secrete their own cytokines that are involved in stimulating and orchestrating other cells involved in adaptive and innate immunity. For example, they stimulate cytotoxic T cells, enhancing their killing of infected cells and promoting differentiation into memory cytotoxic T cells. T H 1 cells also stimulate macrophages and neutrophils to become more effective in their killing of intracellular bacteria. They can also stimulate NK cells to become more effective at killing target cells.
T H 2 cells play an important role in orchestrating the humoral immune response through their secretion of cytokines that activate B cells and direct B cell differentiation and antibody production. Various cytokines produced by T H 2 cells orchestrate antibody class switching, which allows B cells to switch between the production of IgM, IgG, IgA, and IgE as needed to carry out specific antibody functions and to provide pathogen-specific humoral immune responses.
A third subtype of helper T cells called T H 17 cells was discovered through observations that immunity to some infections is not associated with T H 1 or T H 2 cells. T H 17 cells and the cytokines they produce appear to be specifically responsible for the body’s defense against chronic mucocutaneous infections. Patients who lack sufficient T H 17 cells in the mucosa (e.g., HIV patients) may be more susceptible to bacteremia and gastrointestinal infections. 1
| Subtype | Functions |
|---|---|
| T H 1 cells | Stimulate cytotoxic T cells and produce memory cytotoxic T cells |
| Stimulate macrophages and neutrophils (PMNs) for more effective intracellular killing of pathogens | |
| Stimulate NK cells to kill more effectively | |
| T H 2 cells | Stimulate B cell activation and differentiation into plasma cells and memory B cells |
| Direct antibody class switching in B cells | |
| T H 17 cells | Stimulate immunity to specific infections such as chronic mucocutaneous infections |
| Memory helper T cells | “Remember” a specific pathogen and mount a strong, rapid secondary response upon re-exposure |
Activation and Differentiation of Cytotoxic T Cells
Cytotoxic T cells (also referred to as cytotoxic T lymphocytes, or CTLs) are activated by APCs in a three-step process similar to that of helper T cells. The key difference is that the activation of cytotoxic T cells involves recognition of an antigen presented with MHC I (as opposed to MHC II) and interaction of CD8 (as opposed to CD4) with the receptor complex. After the successful co-recognition of foreign epitope and self-antigen, the production of cytokines by the APC and the cytotoxic T cell activate clonal proliferation and differentiation. Activated cytotoxic T cells can differentiate into effector cytotoxic T cells that target pathogens for destruction or memory cells that are ready to respond to subsequent exposures.
As noted, proliferation and differentiation of cytotoxic T cells is also stimulated by cytokines secreted from T H 1 cells activated by the same foreign epitope. The co-stimulation that comes from these T H 1 cells is provided by secreted cytokines. Although it is possible for activation of cytotoxic T cells to occur without stimulation from T H 1 cells, the activation is not as effective or long-lasting.
Once activated, cytotoxic T cells serve as the effector cells of cellular immunity, recognizing and kill cells infected with intracellular pathogens through a mechanism very similar to that of NK cells. However, whereas NK cells recognize nonspecific signals of cell stress or abnormality, cytotoxic T cells recognize infected cells through antigen presentation of pathogen-specific epitopes associated with MHC I. Once an infected cell is recognized, the TCR of the cytotoxic T cell binds to the epitope and releases perforin and granzymes that destroy the infected cell (Figure \(\PageIndex{6}\)). Perforin is a protein that creates pores in the target cell, and granzymes are proteases that enter the pores and induce apoptosis. This mechanism of programmed cell death is a controlled and efficient means of destroying and removing infected cells without releasing the pathogens inside to infect neighboring cells, as might occur if the infected cells were simply lysed.
- Compare and contrast the activation of helper T cells and cytotoxic T cells.
- What are the different functions of helper T cell subtypes?
- What is the mechanism of CTL-mediated destruction of infected cells?
Superantigens and Unregulated Activation of T Cells
When T cell activation is controlled and regulated, the result is a protective response that is effective in combating infections. However, if T cell activation is unregulated and excessive, the result can be a life-threatening. Certain bacterial and viral pathogens produce toxins known as superantigens (see Virulence Factors of Bacterial and Viral Pathogens ) that can trigger such an unregulated response. Known bacterial superantigens include toxic shock syndrome toxin (TSST), staphylococcal enterotoxins, streptococcal pyrogenic toxins, streptococcal superantigen, and the streptococcal mitogenic exotoxin. Viruses known to produce superantigens include Epstein-Barr virus (human herpesvirus 4), cytomegalovirus (human herpesvirus 5), and others.
The mechanism of T cell activation by superantigens involves their simultaneous binding to MHC II molecules of APCs and the variable region of the TCR β chain. This binding occurs outside of the antigen-binding cleft of MHC II, so the superantigen will bridge together and activate MHC II and TCR without specific foreign epitope recognition (Figure \(\PageIndex{7}\)). The result is an excessive, uncontrolled release of cytokines, often called a cytokine storm, which stimulates an excessive inflammatory response. This can lead to a dangerous decrease in blood pressure, shock, multi-organ failure, and potentially, death.
- What are examples of superantigens?
- How does a superantigen activate a helper T cell?
- What effect does a superantigen have on a T cell?
Melissa, an otherwise healthy 22-year-old woman, is brought to the emergency room by her concerned boyfriend. She complains of a sudden onset of high fever, vomiting, diarrhea, and muscle aches. In her initial interview, she tells the attending physician that she is on hormonal birth control and also is two days into the menstruation portion of her cycle. She is on no other medications and is not abusing any drugs or alcohol. She is not a smoker. She is not diabetic and does not currently have an infection of any kind to her knowledge.
While waiting in the emergency room, Melissa’s blood pressure begins to drop dramatically and her mental state deteriorates to general confusion. The physician believes she is likely suffering from toxic shock syndrome (TSS). TSS is caused by the toxin TSST-1, a superantigen associated with Staphylococcus aureus , and improper tampon use is a common cause of infections leading to TSS. The superantigen inappropriately stimulates widespread T cell activation and excessive cytokine release, resulting in a massive and systemic inflammatory response that can be fatal.
Vaginal or cervical swabs may be taken to confirm the presence of the microbe, but these tests are not critical to perform based on Melissa’s symptoms and medical history. The physician prescribes rehydration, supportive therapy, and antibiotics to stem the bacterial infection. She also prescribes drugs to increase Melissa’s blood pressure. Melissa spends three days in the hospital undergoing treatment; in addition, her kidney function is monitored because of the high risk of kidney failure associated with TSS. After 72 hours, Melissa is well enough to be discharged to continue her recovery at home.
In what way would antibiotic therapy help to combat a superantigen?
Olivia’s swollen lymph nodes, abdomen, and spleen suggest a strong immune response to a systemic infection in progress. In addition, little Olivia is reluctant to turn her head and appears to be experiencing severe neck pain.  The cerebrospinal fluid (CSF) obtained appears cloudy and is further evaluated by Gram stain assessment and culturing for potential bacterial pathogens. The complete blood count indicates elevated numbers of white blood cells in Olivia’s bloodstream. The white blood cell increases are recorded at 28.5 K/µL (normal range: 6.0–17.5 K/µL). The neutrophil percentage was recorded as 60% (normal range: 23–45%). Glucose levels in the CSF were registered at 30 mg/100 mL (normal range: 50–80 mg/100 mL). The WBC count in the CSF was 1,163/mm3 (normal range: 5–20/mm 3 ).
- Based on these results, do you have a preliminary diagnosis?
- What is a recommended treatment based on this preliminary diagnosis?
Key Concepts and Summary
- Immature T lymphocytes are produced in the red bone marrow and travel to the thymus for maturation.
- Thymic selection is a three-step process of negative and positive selection that determines which T cells will mature and exit the thymus into the peripheral bloodstream.
- Central tolerance involves negative selection of self-reactive T cells in the thymus, and peripheral tolerance involves anergy and regulatory T cells that prevent self-reactive immune responses and autoimmunity.
- The TCR is similar in structure to immunoglobulins, but less complex. Millions of unique epitope-binding TCRs are encoded through a process of genetic rearrangement of V, D, and J gene segments.
- T cells can be divided into three classes— helper T cells, cytotoxic T cells, and regulatory T cells— based on their expression of CD4 or CD8, the MHC molecules with which they interact for activation, and their respective functions.
- Activated helper T cells differentiate into T H 1, T H 2, T H 17 , or memory T cell subtypes . Differentiation is directed by the specific cytokines to which they are exposed. T H 1, T H 2, and T H 17 perform different functions related to stimulation of adaptive and innate immune defenses. Memory T cells are long-lived cells that can respond quickly to secondary exposures.
- Once activated, cytotoxic T cells target and kill cells infected with intracellular pathogens. Killing requires recognition of specific pathogen epitopes presented on the cell surface using MHC I molecules. Killing is mediated by perforin and granzymes that induce apoptosis.
- Superantigens are bacterial or viral proteins that cause a nonspecific activation of helper T cells, leading to an excessive release of cytokines ( cytokine storm ) and a systemic, potentially fatal inflammatory response.
Footnotes
- 1 Blaschitz C., Raffatellu M. “Th17 cytokines and the gut mucosal barrier.” J Clin Immunol. 2010 Mar; 30(2):196-203. doi: 10.1007/s10875-010-9368-7.