15.18: Formatting the MLA Works Cited Page
- E valuate the components of a properly formatted MLA Works Cited page
In MLA style, all the sources you cite throughout the text of your paper are listed together in full in the Works Cited section, which comes after the main text of your paper.
Formatting the Works Cited Section
In MLA style, all the sources you cite throughout the text of your paper are listed together in full in the Works Cited section, which comes after the main text of your paper.
- Page numbers: Just as the rest of your paper, the top of the page should retain the right-justified header with your last name and the page number.
- Title: On the first line, the title of the page—“Works Cited”—should appear centered, and not italicized or bolded.
- Spacing: Like the rest of your paper, this page should be double-spaced and have 1-inch margins (don’t skip an extra line between citations).
- Alphabetical order: Starting on the next line after the page title, your references should be listed in alphabetical order by author. Multiple sources by the same author should be listed chronologically by year within the same group. If your reference has no author, use the title to alphabetize, leaving out any articles (for example, alphabetize “The Awakening” under A).
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Hanging indents:
Each reference should be formatted with what is called a hanging indent. This means the first line of each reference should be flush with the left margin (i.e., not indented), but the rest of that reference should be indented 0.5 inches further.
- Any word-processing program will let you format this automatically so you don’t have to do it by hand. (In Microsoft Word, for example, you simply highlight your citations, click on the small arrow right next to the word “Paragraph” on the home tab, and in the popup box choose “hanging indent” under the “Special” section. Click OK, and you’re done. In Google Docs, highlight the area you want to indent then choose Format > Align & Indent > Indentation options > Select “Special,” then “Hanging” > Apply.)
Watch this video to review the process for creating a Works Cited page.
https://lumenlearning.h5p.com/content/1290981892674151068/embed
You can view the transcript for “MLA Works Cited” here (opens in new window) .
Now that you’ve learned the basic formatting guidelines and techniques for citing sources using MLA style, use the following checklist to ensure your own works are formatting and cited correctly.
Contributors and Attributions
- Revision and Adaptation. Provided by : Lumen Learning. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- MLA: The Works Cited Section. Authored by : Catherine McCarthy. Provided by : Boundless. Located at : www.boundless.com/writing/textbooks/boundless-writing-textbook/writing-a-paper-in-mla-style-humanities-255/mla-citations-and-references-303/mla-the-works-cited-section-319-16905/. Project : Boundless Writing. License : CC BY-SA: Attribution-ShareAlike
- Image of How to Cite an Essay. Provided by : Wikihow. Located at : http://www.wikihow.com/Cite-an-Essay . License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike


1.0.s: The Clapeyron Equation
The analysis in the two previous sections can be repeated for any phase change of a pure substance. Let \(\alpha\) and \(\beta\) denote the two phases that are at equilibrium.
\[\alpha \rightleftharpoons \beta\]
Let \(\overline{G}_{\alpha }\), \(\overline{S}_{\alpha }\), and \({\overline{V}}_{\alpha }\) represent the Gibbs free energy, the entropy, and the volume of one mole of pure phase \(\alpha\) at pressure \(P\) and temperature \(T\). Let \(\overline{G}_{\beta }\), \(\overline{S}_{\beta }\), and \(\overline{V}_{\beta }\) represent the corresponding properties of one mole of pure phase \(\beta\). The equations
\[d\overline{G}\left(\alpha \right)=\overline{V}_\alpha dP-\overline{S}_{\alpha }dT\]
and
\[d\overline{G}\left(\beta \right)= \overline{V}_{\beta } dP- \overline{S}_{\beta }dT\]
describe the changes in the Gibbs free energy of a mole of \(\alpha\) and a mole of \(\beta\) when they go from one \(\alpha-\beta\)-equilibrium state at \(P\) and \(T\) to a second \(\alpha-\beta\)-equilibrium state at \(P+dP\) and \(T+dT\). Since these Gibbs free energy changes must be equal, we have
\[ \begin{align*} d\overline{G}\left(\beta \right)-d\overline{G}\left(\alpha \right) &=\left({\overline{V}}_{\beta }-{\overline{V}}_{\alpha }\right)dP-\left({\overline{S}}_{\beta }-\overline{S}_{\alpha }\right)dT \\[4pt] &=\Delta \overline{V}dP-\Delta \overline{S}dT \\[4pt] &=0 \end{align*}\]
and
\[\frac{dP}{dT}=\frac{\Delta \overline{S}}{\Delta \overline{V}}\]
where \(\Delta \overline{S}\) and \(\Delta \overline{V}\) are the entropy and volume changes that occur when one mole of the substance goes from phase \(\alpha\) to phase \(\beta\). Since \(\Delta \overline{S}=\Delta \overline{H}/T\), the condition for equilibrium between phases \(\alpha\) and \(\beta\) becomes
\[\frac{dP}{dT}=\frac{\Delta \overline{H}}{T\ \Delta \overline{V}} \label{Clap1}\]
Equation \ref{Clap1} is known as the Clapeyron equation .


9.3: Interview Survey
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Interviews are a more personalized form of data collection method than questionnaires, and are conducted by trained interviewers using the same research protocol as questionnaire surveys (i.e., a standardized set of questions). However, unlike a questionnaire, the interview script may contain special instructions for the interviewer that is not seen by respondents, and may include space for the interviewer to record personal observations and comments. In addition, unlike mail surveys, the interviewer has the opportunity to clarify any issues raised by the respondent or ask probing or follow-up questions. However, interviews are timeconsuming and resource-intensive. Special interviewing skills are needed on part of the interviewer. The interviewer is also considered to be part of the measurement instrument, and must proactively strive not to artificially bias the observed responses.
The most typical form of interview is personal or face-to-face interview , where the interviewer works directly with the respondent to ask questions and record their responses. Personal interviews may be conducted at the respondent’s home or office location. This approach may even be favored by some respondents, while others may feel uncomfortable in allowing a stranger in their homes. However, skilled interviewers can persuade respondents to cooperate, dramatically improving response rates.
A variation of the personal interview is a group interview, also called focus group . In this technique, a small group of respondents (usually 6-10 respondents) are interviewed together in a common location. The interviewer is essentially a facilitator whose job is to lead the discussion, and ensure that every person has an opportunity to respond. Focus groups allow deeper examination of complex issues than other forms of survey research, because when people hear others talk, it often triggers responses or ideas that they did not think about before. However, focus group discussion may be dominated by a dominant personality, and some individuals may be reluctant to voice their opinions in front of their peers or superiors, especially while dealing with a sensitive issue such as employee underperformance or office politics. Because of their small sample size, focus groups are usually used for exploratory research rather than descriptive or explanatory research.
A third type of interview survey is telephone interviews . In this technique, interviewers contact potential respondents over the phone, typically based on a random selection of people from a telephone directory, to ask a standard set of survey questions. A more recent and technologically advanced approach is computer-assisted telephone interviewing (CATI), increasing being used by academic, government, and commercial survey researchers, where the interviewer is a telephone operator, who is guided through the interview process by a computer program displaying instructions and questions to be asked on a computer screen. The system also selects respondents randomly using a random digit dialing technique, and records responses using voice capture technology. Once respondents are on the phone, higher response rates can be obtained. This technique is not ideal for rural areas where telephone density is low, and also cannot be used for communicating non-audio information such as graphics or product demonstrations.
Role of interviewer
The interviewer has a complex and multi-faceted role in the interview process, which includes the following tasks:
- Prepare for the interview : Since the interviewer is in the forefront of the data collection effort, the quality of data collected depends heavily on how well the interviewer is trained to do the job. The interviewer must be trained in the interview process and the survey method, and also be familiar with the purpose of the study, how responses will be stored and used, and sources of interviewer bias. He/she should also rehearse and time the interview prior to the formal study.
- Locate and enlist the cooperation of respondents : Particularly in personal, in-home surveys, the interviewer must locate specific addresses, and work around respondents’ schedule sometimes at undesirable times such as during weekends. They should also be like a salesperson, selling the idea of participating in the study.
- Motivate respondents : Respondents often feed off the motivation of the interviewer. If the interviewer is disinterested or inattentive, respondents won’t be motivated to provide useful or informative responses either. The interviewer must demonstrate enthusiasm about the study, communicate the importance of the research to respondents, and be attentive to respondents’ needs throughout the interview.
- Clarify any confusion or concerns : Interviewers must be able to think on their feet and address unanticipated concerns or objections raised by respondents to the respondents’ satisfaction. Additionally, they should ask probing questions as necessary even if such questions are not in the script.
- Observe quality of response : The interviewer is in the best position to judge the quality of information collected, and may supplement responses obtained using personal observations of gestures or body language as appropriate.


1.9: Early Fraction Concepts
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- Janet Stramel
- Fort Hays State University
What are Fractions?
According to the Kansas Mathematics Standards (2017), the formal definition of fraction is a number expressible in the form }"> where a is the number of equal parts being referenced and b is the number of equal parts in the whole. But what does this mean?
The word “fraction” comes from the Latin word fractus or broken. A fraction describes how many parts of a certain size there are, for example, one-half, three-fourths, etc. Additionally, the top number (numerator) says how many parts you have, and the bottom number (denominator) says how many equal parts in the whole amount. Most importantly, a fraction is a number.
An understanding of fractions begins in first grade when students partition circles and rectangles into two and four equal parts, and describe the parts with the words halves, fourths, and quarters. In second grade , students partition circles and rectangles in two, three, and four equal parts, and describe those parts with the words halves, thirds, and fourths. Fractions become the major emphasis in third grade as students look at fraction symbols, and explore unit fractions.
“Students need significant time and experiences to develop a deep conceptual understanding” of fractions (Van de Walle, Karp, Bay-Williams, 2019, p. 338). Teachers typically begin fraction instruction with objects, paper folding, or even plastic fraction circles. But often we move too quickly past these models to abstract computation. Some experts even argue that drawings of fractions fail to be concrete enough for some students. The act of cutting and folding and manipulating is critical for all students.
Watch this video from Graham Fletcher, “The Progression of Fractions.”
Fraction Constructs
Fractions include many meanings such as part-whole, measurement, division, operator, and ratio. Understanding fractions includes the need to understand all of these different meanings. As you and your students begin to make sense of fractions, you must also understand all the possible concepts that fractions represent. The following constructs are based on the research from Van de Walle, Karp, and Bay-Williams (2019):
Fractions as Part-whole Comparisons. Part-whole comparisons is a good place to start when building a conceptual understanding of fractions. Part-whole comparisons can be shown by shading a region, part of a group, and measurement.
- Shading a region
For example, a circle can be divided into four equal parts. If three of those parts are shaded pink, then the fraction shaded is .
- Part of a group
For example, I have five pets; three cats and two dogs. The fractional part of cats to all of my pets is .
- Measurement
\[\frac{1}{8}\]
Fractions as Division. Just like with whole numbers, division means making equal-sized groups. Students understand the connection to equal- and fair-sharing which builds from their experiences with sharing and partitioning. For example, “Eight friends are sharing four apples. How much of an apple will each person receive?” Each person will get or of an apple.
A fraction is the quotient of division of two integers. For example, if you want to divide three things equally among 5 people, students know to this can be solved with division, although the quotient of cannot be represented with a whole number, so new numbers need to be introduced; the fraction .
Fractions as Operators . This construct builds on the concepts of seeing a fraction as a multiple of a unit fraction. To find of something, you could divide by 4 and then multiply by 3, or multiply by 3 and then divide by 4. The result would be a smaller quantity than the original. For example, x 12 = 9.
Fractions as Ratios. Ratio is another meaning of fractions. For example, can be seen as the probability of an event being five in eight. Ratios can be expressed in two different ways: part-part, such as the ratio of boys to girls in class, and part-whole, such as the ratio of boys to the whole class.
The Language of Fractions
The language of fractions is critical to students developing a conceptual understanding of fractions. But this does not mean that teachers teach only the definitions of fractions, but instead teachers should first focus on making sense of what fractions are.
Students first begin to make sense of fractions in first grade when they compose and decompose plane figures, such as putting together two triangles to make a quadrilateral. First grade students must have multiple opportunities and many experiences with the words halves, fourths, and quarters. According to Van de Walle, Karp, and Bay-Williams (2019), the best time to introduce these words to students is during problem solving, not before. For example, when students are cutting a pan of brownies into four equal shares, the teacher can say, “We call these fourths. The whole is cut into four equal-size parts called fourths.”
In second grade, students add to their fraction vocabulary the word “third,” but students are not introduced to the traditional fractional notation until third grade when they develop an understanding of unit fractions. Additionally, the fraction expectations in third grade are limited to fractions with denominators of 2, 3, 4, 6, and 8. It is in third grade that students develop an understanding of fractions as numbers, use the fraction symbols, and use area and linear models. Third grade lays the foundation for an understanding of fractions and is critical for students to be successful in future mathematics.
The numerator is the “counting number,” or the number of parts we have.
The denominator is how many parts are in the whole.
When talking about these terms, be sure to ask students the following questions:
- “What does the numerator in the fraction tell us?”
- “What does the denominator in the fraction tell us?”
- “What would a fraction equal to one look like?”
- “How do you know if a fraction is less than or greater than one?”
Area – Fractions are determined on how much of a part or area relates to the whole area.
Length – Fractions can be represented on a number line, or as a distance between 0 and another point on the number line.
Set -Fractions can be represented based on how many items are in part of the whole set.
Fractions are Numbers
The most important thing young children need to understand is that fractions are numbers; they represent quantities that have values (Van de Walle, Karp, & Bay-Williams, 2019). Partitioning and iterating are two important actions that emphasize the numerical nature of fractions.
Partitioning is when students section or segment a shape into equal-sized parts. When partitioning, start with a unit fraction such as . One-third ( ) is the size of piece you would get by taking a whole and dividing it into 3 equal parts.
When students draw and see diagrams such as the one above, it emphasizes that fractions are a fraction of something . You would teach this by saying, “We call these pieces thirds because we cut the whole into three parts . All of these parts are the same size – thirds.”
Students are introduced to fractions in first grade as they are presented with the names for fractional parts – halves and fourths; in second grade, students partition circles and rectangles into two, three, and four equal parts, and describe those parts using the words halves, thirds, and fourths; fractional symbols are not introduced until third grade.
Iterating is when students perform the action of aligning, copying, or combining equal units to verify a fraction. According to Siebert (2007), “To conceive of a fraction from an iteration perspective, first start with a unit fraction, such as . How can you tell if something is ? An amount is if five copies of it equals 1. The image here is of taking and iterating is 5 more times to make a whole.”
Equivalent Fractions and Fraction Comparisons
Equivalence is a core concept in mathematics. Equivalent fractions have the same value, even though they may look different. For example 12 and 24 are equivalent because they are both “half.”
Teachers must devote enough time so that their students have a deep understanding of this concept. Equivalent fractions and comparing fractions is first addressed in third grade, and then applied in fourth grade. The focus of both the third and fourth grade is to use visual fraction models so that students can explore the idea of equivalent fractions instead of using algorithms.
An important concept when comparing fractions is both the size of the parts and the number of parts. For example, is lesson than because when the whole is cut into 4 pieces, the pieces are smaller than when the whole is cut into 2 pieces. Students need to see this.
Students are not expected to generate a rule for equivalent fractions until fourth grade, although third grade students should notice the connections between the models and the fractions.
Fraction Equivalence
When teaching fraction equivalence, do not move to the traditional algorithm too quickly. Allow your students to make sense of the relationship between the two fractions first. After students have a conceptual understanding using visual representations and models, then you can ask if they see patterns in the way the fractional parts are counted (Stramel, 2021).


15.22D: Bovine Spongiform Encephalopathy
- Recognize the route of transmission and causes of bovine spongiform encephalopathy (BSE)
Bovine spongiform encephalopathy (BSE), commonly known as mad cow disease, is a fatal neurodegenerative disease in cattle that causes a spongy degeneration in the brain and spinal cord. BSE has a long incubation period, about 30 months to eight years, usually affecting adult cattle at a peak age of four to five years, all breeds being equally susceptible.
In the United Kingdom, the country worst affected, more than 180,000 cattle have been infected and 4.4 million slaughtered during the eradication program. The disease may be most easily transmitted to human beings by eating food contaminated with the brain, spinal cord or digestive tract of infected carcasses. However, it should also be noted that the infectious agent, although most highly concentrated in nervous tissue, can be found in virtually all tissues throughout the body, including blood. In humans, it is known as new variant Creutzfeldt–Jakob disease (vCJD or nvCJD), and by October 2009, had killed 166 people in the United Kingdom and 44 elsewhere. Between 460,000 and 482,000 BSE-infected animals had entered the human food chain before controls on high-risk offal were introduced in 1989.
The infectious agent in BSE is believed to be a specific type of misfolded protein called a prion. Prions will not disappear even if the beef containing them is cooked. Prion proteins carry the disease between individuals and cause deterioration of the brain.
BSE is a type of transmissible spongiform encephalopathy (TSE). TSEs can arise in animals that carry an allele which causes previously normal protein molecules to contort by themselves from an alpha helix arrangement to a beta sheet, which is the disease-causing shape for the particular protein. Transmission can occur when healthy animals come in contact with tainted tissues from others with the disease. In the brain, these proteins cause native cellular prion protein to deform into the infectious state, which then goes on to deform further prion protein in an exponential cascade. This results in protein aggregates, which then form dense plaque fibers, leading to the microscopic appearance of “holes” in the brain, degeneration of physical and mental abilities, and ultimately death.
Different hypotheses exist for the origin of prion proteins in cattle. Two leading hypotheses suggest it may have jumped species from the disease scrapie in sheep, or that it evolved from a spontaneous form of “mad cow disease” that has been seen occasionally in cattle for many centuries. In the fifth century BC, Hippocrates described a similar illness in cattle and sheep, which he believed also occurred in man. Publius Flavius Vegetius Renatus recorded cases of a disease with similar characteristics in the fourth and fifth centuries. Recent research suggests mad cow disease is caused by a genetic mutation within a gene called the prion protein gene. The research shows, for the first time, that a 10-year-old cow from Alabama with an atypical form of bovine spongiform encephalopathy had the same type of prion protein gene mutation as found in human patients with the genetic form of Creutzfeldt–Jakob disease, also called genetic CJD, for short. Besides having a genetic origin, other human forms of prion diseases can be sporadic, as in sporadic CJD, as well as foodborne. They are contracted when people eat products contaminated with mad cow disease. This form of Creutzfeldt-Jakob disease is called variant CJD.
Key Points
- BSE is probably caused by a misfolded protein known as a prion; a prion can cause other correctly folded proteins to misfold thus propagating the disease.
- Cooking of prions does not destroy them.
- Evidence exists that when the prions in cows that cause BSE are consumed by humans this causes prion transmission to humans and the neurodegenerative Creutzfeldt-Jakob disease.
Key Terms
- prion : A self-propagating misfolded conformer of a protein that is responsible for a number of diseases that affect the brain and other neural tissue.
- beta sheet : A secondary structure in proteins consisting of multiple strands connected laterally.
- alpha helix : A secondary structure found in many proteins, where the amino acids are arranged in a coil, or helix, with almost no free space on the inside and all side chains being pointed towards the outside.


3.6: Sinusoidal Steady State and the Series RLC Circuit
Phasors may be used to analyze the behavior of electrical and mechanical systems that have reached a kind of equilibrium called sinusoidal steady state. In the sinusoidal steady state, every voltage and current (or force and velocity) in a system is sinusoidal with angular frequency \(ω\). However, the amplitudes and phases of these sinusoidal voltages and currents are all different. For example, the voltage across a resistor might lead the voltage across a capacitor by 90 ∘ (\(\frac π 2\) radians) and lag the voltage across an inductor by 90 ∘ (\(\frac π 2\) radians).
In order to make our application of phasors to electrical systems concrete, we consider the series RLC circuit illustrated in Figure \(\PageIndex{1}\). The arrow labeled \(i(t)\) denotes a current that flows in response to the voltage applied, and the + and - on the voltage source indicate that the polarity of the applied voltage is positive on the top and negative on the bottom. Our convention is that current flows from positive to negative, in this case clockwise in the circuit.
We will assume that the voltage source is an audio oscillator that produces the voltage
\[V(t)=A\cos(ωt+φ) \nonumber \]
We represent this voltage as the complex signal
\[V(t)↔Ae^{jφ}e^{jωt} \nonumber \]
and give it the phasor representation
\[V(t)↔V;V=Ae^{jφ} \nonumber \]
We then describe the voltage source by the phasor V and remember that we can always compute the actual voltage by multiplying by \(e^{jωt}\) and taking the real part:
\[V(t)=\mathrm{Re}\{Ve^{jωt}\} \nonumber \]
Show that
\[\mathrm{Re}[Ve^{jωt}]=A\cos(ωt+φ) \nonumber \]
when \(V=Ae^{jφ}\).
Circuit Laws
In your circuits classes you will study the Kirchhoff laws that govern the low frequency behavior of circuits built from resistors (R), inductors (L), and capacitors (C). In your study you will learn that the voltage dropped across a resistor is related to the current that flows through it by the equation
\[V_R(t)=Ri(t) \nonumber \]
You will learn that the voltage dropped across an inductor is proportional to the derivative of the current that flows through it, and the voltage dropped across a capacitor is proportional to the integral of the current that flows through it:
\[V_L(t)=L\frac {di} {dt} (t) \nonumber \]
\[V_C(t)=\frac 1 C ∫i(t)dt \nonumber \]
Phasors and Complex Impedance
Now suppose that the current in the preceding equations is sinusoidal, of the form
\[i(t)=B\cos(ωt+θ) \nonumber \]
We may rewrite \(i(t)\) as
\[i(t)=\mathrm{Re}\{Ie^{jωt}\} \nonumber \]
where \(I\) is the phasor representation of \(i(t)\).
Find the phasor \(I\) in terms of \(B\) and \(θ\) in the above Equation.
The voltage dropped across the resistor is
\[\begin{align} V_R(t) &=Ri(t) \\[4pt] &=R\mathrm{Re}\{Ie^{jωt}\} \\[4pt] &=\mathrm{Re}\{RIe^{jωt}\} \end{align} \nonumber \]
Thus the phasor representation for \(V_R(t)\) is
\[V_R(t)↔V_R;V_R=RI \nonumber \]
We call R the impedance of the resistor because R is the scale constant that relates the “phasor voltage V R ' to the “phasor current I .”
The voltage dropped across the inductor is
\[V_L(t)=L\frac {di}{dt}(t)=L\frac d {dt}\mathrm{Re}\{Ie^{jωt}\} \nonumber \]
The derivative may be moved through the Re[] operator (see Exercise) to produce the result
\[\begin{align} V_L(t)&=L\mathrm{Re}\{jωIe^{jωt}\} \\[4pt] &=\mathrm{Re}\{jωLIe^{jωt}\} \end{align} \nonumber \]
Thus the phasor representation of \(V_L(t)\)
\[V_L(t)↔V_L;V_L=jωLI \nonumber \]
We call \(jωL\) the impedance of the inductor because \(jωL\) is the complex scale constant that relates “phasor voltage \(V_L\)' to “phasor current \(I\).
Prove that the operators \(\frac d {dt}\) and Re[] commute:
\(\frac d {dt}\mathrm{Re}\{e^{jωt}\}=\mathrm{Re}\{\frac d {dt} e^{jωt}\}\)
The voltage dropped across the capacitor is
\[V_C(t)=\frac 1 C ∫i(t)dt = \frac 1 C ∫ \mathrm{Re}\{Ie^{jωt}\}dt \nonumber \]
The integral may be moved through the Re[] operator to produce the result
\[ \begin{align} V_C(t) &=\frac 1 C \mathrm{Re}\{\frac I {jω} e^{jωt}\} \\[4pt] &=\mathrm{Re}\{\frac I {jωC} e^{jωt}\} \end{align} \nonumber \]
Thus the phasor representation of V C (t) is
\[V_C(t)↔V_C;V_C=\frac I {jωC} \nonumber \]
We call \(\frac 1 {jωC}\) the impedance of the capacitor because \(\frac 1 {jωC}\) is the complex scale constant that relates “phasor voltage V C " to “phasor current I.”
Kirchhoff's Voltage Law
Kirchhoff's voltage law says that the voltage dropped in the series combination of R, L, and C illustrated in Figure equals the voltage generated by the source (this is one of two fundamental conservation laws in circuit theory, the other being a conservation law for current):
\[V(t)=V_R(t)+V_L(t)+V_C(t) \nonumber \]
If we replace all of these voltages by their complex representations, we have
\[\mathrm{Re}\{Ve^{jωt}\}=\mathrm{Re}\{(V_R+V_L+V_C)e^{jωt}\} \nonumber \]
An obvious solution is
\[V=VR+VL+VC \nonumber \]
\[=(R+jωL+\frac 1 {jωC})I \nonumber \]
where I is the phasor representation for the current that flows in the circuit. This solution is illustrated in the Figure, where the phasor voltages \(RI\), \(jωLI\), and \(\frac 1 {jωC} I\) are forced to add up to the phasor voltage \(V\).
Redraw the above Figure for \(R=ωL=\frac 1 {ωC} = 1\)
Impedance
We call the complex number \(R+jωL+\frac 1 {jωC}\) the complex impedance for the series RLC network because it is the complex number that relates the phasor voltage V to the phasor current I:
\[V=ZI \nonumber \]
\[Z=R+jωL+\frac 1 {jωC} \nonumber \]
The complex number \(Z\) depends on the numerical values of resistance \((R)\), inductance \((L)\), and capacitance \((C)\), but it also depends on the angular frequency \((ω)\) used for the sinusoidal source. This impedance may be manipulated as follows to put it into an illuminating form:
\[Z=R+j(ωL−\frac 1 {ωC}) \nonumber \]
\[= R+j \sqrt{\frac L C}(ω \sqrt{LC}−\frac 1 {/sqrt{ωLC}}) \nonumber \]
The parameter \(ω_0=\frac 1 {\sqrt{LC}}\) is a parameter that you will learn to call an "undamped natural frequency" in your more advanced circuits courses. With it, we may write the impedance as
\[Z=R+jω_0L(\frac ω {ω_0}−\frac {ω_0} ω) \nonumber \]
The frequency \(\frac ω {ω_0}\) is a normalized frequency that we denote by \(ν\). Then the impedence, as a function of normalized frequency, is
\[Z(ν)=R+jω_0L(ν−\frac 1 ν) \nonumber \]
When the normalized frequency equals one \((ν=1)\), then the impedance is entirely real and \(Z=R\). The circuit looks like it is a single resistor.
\[∣Z(ν)∣∣=R[1+(/frac {ω_0L} R)^2(ν−\frac 1 ν)^2]^{1/2} \nonumber \]
\[\mathrm{arg}Z(ν)=\tan^{−1}\frac {ω_0L} R (ν−\frac 1 ν) \nonumber \]
The impedance obeys the following symmetries around \(ν=1\):
\[Z(ν)=Z^∗(\frac 1 ν) \nonumber \]
\[∣Z(ν)∣=∣Z(\frac 1 ν)∣ \nonumber \]
\[\mathrm{arg}Z(ν)=−\mathrm{arg}Z(\frac 1 ν) \nonumber \]
In the next paragraph we show how this impedance function influences the current that flows in the circuit.
Resonance
The phasor representation for the current that flows the current that flows in the series RLC circuit is
\[\begin{align} I &=VZ(ν) \\[4pt] &=1|Z(ν)|e−jargZ(ν)V \end{align} \nonumber \]
The function \(H(ν)=\frac 1 {Z(ν)}\) displays a "resonance phenomenon." that is, \(|H(ν)|\) peaks at \(ν=1\) and decreases to zero and ν=0 and ν=∞:
\[|H(ν)|= \begin{cases} 0, ν=0 \\ \frac 1 R, ν=1 \\ 0, ν=∞ \end{cases} \nonumber \]
When \(|H(ν)|=0\), no current flows.
The function \(|H(ν)|\) is plotted against the normalized frequency \(ν=\frac ω {ω_0}\) in the Figure. The resonance peak occurs at ν=1, where \(|H(ν)|=\frac 1 R\) meaning that the circuit looks purely resistive. Resonance phenomena underlie the frequency selectivity of all electrical and mechanical networks.
(MATLAB) Write a MATLAB program to compute and plot \(|H(ν)|\) and \(argH(ν)\) versus ν for ν ranging from 0.1 to 10 in steps of 0.1. Carry out your computations for \(ω_0LR=10,1,0.1\), and 0.01, and overplot your results.
Circle Criterion and Power Factor
Our study of the impedance \(Z(ν)\) and the function \(H(ν)=\frac 1 {Z(ν)}\) brings insight into the resonance of an RLC circuit and illustrates the frequency selectivity of the circuit. But there is more that we can do to illuminate the behavior of the circuit.
\[V=RI+j(ωL−\frac 1 {ωC})I \nonumber \]
This equation shows how voltage is divided between resistor voltage RI and inductor-capacitor voltage \(j(ωL−1ωC)I\).
\[V=RI+jω_0L(\frac ω {ω_0}−\frac {ω_0} ω)I \nonumber \]
or
\[V=RI+\frac {jω_0L} R (ν−/frac 1 ν)RI \nonumber \]
In order to simplify our notation, we can write this equation as
\[V=V_R+jk(ν)V_R \nonumber \]
where \(V_R\) is the phasor voltage \(RI\) and \(k(ν)\) is the real variable
\[k(ν)=\frac {ω_0L} R (ν−\frac 1 ν) \nonumber \]
The equation brings very important geometrical insights. First, even though the phasor voltage \(V_R\) in the RLC circuit is complex, the terms \(V_R\) and \(jk(ν)V_R\) are out of phase by \(\frac π 2\) radians. This means that, for every allowable value of \(V_R\), the corresponding \(jk(ν)V_R\) must add in a right triangle to produce the source voltage V. This is illustrated in Figure. As the frequency ν changes, then \(k(ν)\) changes, producing other values of \(V_R\) and \(jk(ν)V_R\) that sum to V. Several such solutions for \(V_R\) and \(jk(ν)V_R\) are illustrated in Figure 3.15(b). From the figure we gain the clear impression that the phasor voltage \(V_{>R}\) lies on a circle of radius \(\frac V 2\) centered at \(\frac V 2\) Let's try this solution,
\[V_R=\frac V 2 +\frac V 2 e^{jψ}=\frac V 2 (1+e^{jψ}) \nonumber \]
and explore its consequences. When this solution is substituted into Equation, the result is
\[V=\frac V 2 (1+e^{jψ})+jk(ν)V_2(1+e^{jψ}) \nonumber \]
or
\[2=(1+e^{jψ})[1+jk(ν)] \nonumber \]
If we multiply the left-hand side by its complex conjugate and the right-hand side by its complex conjugate, we obtain the identity
\[4=2(1+\cosψ)[1+k^2(ν)] \nonumber \]
This equation tells us how the angle ψ depends on \(k(ν)\) and, conversely, how \(k(ν)\) depends on ψ:
\[\cosψ = \frac {1−k^2(ν)} {1+k^2(ν)} \nonumber \]
\[k^2(ν)=\frac {1−\cosψ} {1+\cosψ} \nonumber \]
The number \(\cosψ\) lies between −1 and +1, so a circular solution does indeed work.
Check \(−1≤\cosψ≤1\) for \(−∞<k<∞\) and \(−∞<k<∞\) for \(−π≤ψ≤π\). Sketch \(k\) versus \(ψ\) and \(ψ\) versus \(k\).
The equation \(V_R=\frac V 2 (1+e{jψ})\) is illustrated in Figure. The angle that \(V_R\) makes with \(V\) is determined from the equation
\[2φ+π−ψ=π⇒φ=\frac ψ 2 \nonumber \]
In the study of power systems, cosφ is a "power factor" that determines how much power is delivered to the resistor. We may denote the power factor as
\[η=\cosφ=\cos\frac ψ 2 \nonumber \]
But \(\cosψ\) may be written as
\[η=cosφ=cos\frac ψ 2 \nonumber \]
But \(\cosψ\) may be written as
\[\begin{align} \cosψ=\cos(φ+φ) &= \cos^2φ−−\sin^2φ \\ &=\cos^2φ−(1−\cos^2φ) \\ &=2\cos^2φ−1 \\ &=2η^2−1 \end{align} \nonumber \]
Therefore the square of the power factor η is
\[ \eta ^2 = \frac{\cos \psi + 1}{2} = \frac{1}{1+k^2(\nu)} \nonumber \]
The power factor is a maximum of 1 for \(k(ν)=0\), corresponding to \(ν=1(ω=ω_0)\). It is a minimum of 0 for \(k(ν)=±∞\), corresponding to \(ν=0,∞(ω=0,∞)\).
With \(k\) defined as \(k(ν)=\frac {ω_0L} R (ν−\frac 1 ν)\), plot \(k^2(ν)\), \(\cosψ\), and \(η^2\) versus \(ν\).
Find the value of \(ν\) that makes the power factor \(η=0.707\).


4.2: Defining Values and Setting Goals
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Questions to consider:
- What beliefs help shape your decision-making and goals?
- How do you set manageable goals that will help you stay on track?
 Your life is a product of choices.”
— Dr. Kathleen Hall, CEO of the Stress Institute and Mindful Living Network 1
A recent high school graduate, Mateo was considering his options for the future. He knew he wanted to go to college, but he wasn’t quite sure what he would study. At a family picnic to celebrate his graduation, he talked about his indecision with his two uncles. One uncle, his Uncle Nico, told him that his best bet was to find out what types of jobs would be hiring in a couple years at high enough salary for Mateo to afford to live however he desired. His other uncle, who rarely agreed with Uncle Nico, nodded and said, “Hey, that’s one way to look at it, but don’t you want to enjoy what you do every day regardless of how much money you make? You should do whatever interests you. After all, don’t they say that if you love what you do, you’ll never work a day in your life?”
Mateo appreciated the advice of his uncles and realized that they might both be right. He wanted to do something that interested him, but he also wanted to be employable and to make money. Clarifying his interests and recognizing his values would be key to helping Mateo decide his path.
Values
Values are the basic beliefs that guide our thinking and actions. Whether we are consciously aware of them or not, values influence both our attitudes and our actions. They help us determine what is important and what makes us happy. It is important to think about and reflect on your values, especially as you make decisions.
ACTIVITY
Determining Your Values
To begin to identify some of your personal values, consider the examples listed below. As a first step, select the five that you find most important, that bring you the greatest happiness, or that make you feel the most proud. Then, rank those five values in order of importance. Feel encouraged to write in other options that are relevant to you.
| Achievement | Efficiency | Hard Work | Positivity |
| Adventure | Empathy | Health | Security |
| Ambition | Equality | Honesty | Selflessness |
| Balance | Excellence | Honor | Service |
| Belonging | Exploration | Humility | Simplicity |
| Calm | Fairness | Independence | Spontaneity |
| Challenge | Faith | Intelligence | Stability |
| Commitment | Family | Joy | Strength |
| Community | Fitness | Justice | Success |
| Competition | Flexibility | Love | Trustworthiness |
| Contribution | Freedom | Loyalty | Understanding |
| Control | Friends | Making a Difference | Uniqueness |
| Creativity | Fun | Merit | |
| Curiosity | Generosity | Openness | |
| Dependability | Growth | Originality | |
| Diversity | Happiness | Perfection |
Another way to recognize the important influence of values is to consider if you have ever made a decision that you later regretted. Did you reflect on your values prior to making that choice? Sometimes others ask us to do things that are inconsistent with our values. Knowing what you value and making plans accordingly is an important effort to help you stay on track toward your goals.
ANALYSIS QUESTION
Recall a decision that you have recently made (for example, a smaller decision about how to spend your Saturday, or maybe a larger decision about where to apply for part-time work). Did the values you identified through this exercise influence that decision? If so, how?
Goals and Planning
Have you ever put together a jigsaw puzzle? Many people start by looking for the edge and corner pieces to assemble the border. Some will then group pieces with similar colors, while others just try to fit in new pieces as they pick them up. Regardless of strategy, a jigsaw puzzle is most easily solved when people have a picture to reference. When you know what the picture should look like, you can gauge your progress and avoid making mistakes. If you were to put a puzzle together facedown (cardboard side up, rather than picture side up), you could still connect the pieces, but it would take you much longer to understand how it should fit together. Your attempts, beyond the border, would be mostly by trial and error. Pursuing anything without goals and a plan is like putting together an upside-down puzzle. You can still finish, or get to where you’re meant to be, but it will take you much longer to determine your steps along the way.
In Chapter 3, you learned about the SMART goal method for setting actionable goals, or goals that are planned and stated with enough clarity for the goal-setter to take realistic action toward meeting those goals. SMART goals help you focus on your priorities and manage your time while also providing a means of organizing your thinking and actions into manageable steps. Long- and short-term goals help to connect the action steps.
Long-Term Goals
Long-term goals are future goals that often take years to complete. An example of a long-term goal might be to complete a bachelor of arts degree within four years. Another example might be purchasing a home or running a marathon. While this chapter focuses on academic and career planning, long-term goals are not exclusive to these areas of your life. You might set long-term goals related to fitness, wellness, spirituality, and relationships, among many others. When you set a long-term goal in any aspect of your life, you are demonstrating a commitment to dedicate time and effort toward making progress in that area. Because of this commitment, it is important that your long-term goals are aligned with your values.
Short-Term Goals
Setting short-term goals helps you consider the necessary steps you’ll need to take, but it also helps to chunk a larger effort into smaller, more manageable tasks. Even when your long-term goals are SMART, it’s easier to stay focused and you’ll become less overwhelmed in the process of completing short-term goals.
You might assume that short-term and long-term goals are different goals that vary in the length of time they take to complete. Given this assumption, you might give the example of a long-term goal of learning how to create an app and a short-term goal of remembering to pay your cell phone bill this weekend. These are valid goals, but they don’t exactly demonstrate the intention of short- and long-term goals for the purposes of effective planning.
Instead of just being bound by the difference of time, short-term goals are the action steps that take less time to complete than a long-term goal, but that help you work toward your long-term goals. To determine your best degree option, it might make sense to do some research to determine what kind of career you’re most interested in pursuing. Or, if you recall that short-term goal of paying your cell phone bill this weekend, perhaps this short-term goal is related to a longer-term goal of learning how to better manage your budgeting and finances.
Setting Long- and Short-Term Goals
Sunil’s story provides an example of effective goal setting. While meeting with an academic advisor at his college to discuss his change of major, Sunil was tasked with setting long- and short-term goals aligned with that major. He selected a degree plan in business administration, sharing with his advisor his intention to work in business and hopefully human relations in particular. His advisor discussed with him how he could set short-term goals that would help his progress on that plan. Sunil wondered if he should be as specific as setting short-term goals week by week or for the successful completion of every homework assignment or exam. His advisor shared that he could certainly break his goals down into that level of specificity if it helped him to stay focused, but recommended that he start by outlining how many credits or courses he would hope to complete. Sunil drafted his goals and planned to meet again with his advisor in another week to discuss.
Sunil worried that his list of short-term goals looked more like a checklist of tasks than anything. His advisor reassured him, sharing that short-term goals can absolutely look like a checklist of tasks because their purpose is to break the long-term goal down into manageable chunks that are easier to focus on and complete. His advisor then recommended that Sunil add to his plan an additional note at the end of every other semester to “check in” with his advisor to make certain that he was on track.
Planning for Adjustments
You will recall from the SMART goals goal-setting model that goals should be both measurable and attainable. Far too often, however, we set goals with the best of intentions but then fail to keep track of our progress or adjust our short-term goals if they’re not helping us to progress as quickly as we’d like. When setting goals, the most successful planners also consider when they will evaluate their progress. At that time, perhaps after each short-term goal should have been met, they may reflect on the following:
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Am I meeting my short-term goals as planned?
- If so, celebrate!
- If not, you may want to additionally consider:
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Are my short-term goals still planned across time in a way where they will meet my long-term goals?
- If so, continue on your path.
- If not, reconsider the steps you need to take to meet your long-term goal. If you’ve gotten off track or if you’ve learned that other steps must be taken, set new short-term goals with timelines appropriate to each step. You may also want to seek some additional advice from others who have successfully met long-term goals that are similar to your own.
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Are my long-term goals still relevant, or have my values changed since I set my goals?
- If your goals are still relevant to your interests and values, then continue on your path, seeking advice and support as needed to stay on track.
- If your goals are no longer relevant or aligned with your values, give careful consideration to setting new goals.
While departing from your original goals may seem like a failure, taking the time to reflect on goals before you set them aside to develop new ones is a success. Pivoting from a goal to new, better-fitting goal involves increased self-awareness and increased knowledge about the processes surrounding your specific goal (such as the details of a college transfer, for example). With careful reflection and information seeking, your change in plans may even demonstrate learning and increased maturity!
APPLICATION
Take a moment to practice setting long- and short-term goals. Your short-term goal should help you progress toward your long-term goal. Include a plan for when and how you will know if you’re on track or if you need to adjust your goals to match new priorities.
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My Long-Term Goal: |
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My Short-Term Goal:
|
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My Plan for Checking My Progress: |
Keep in mind that values and goals may change over time as you meet new people, your life circumstances change, and you gain more wisdom or self-awareness. In addition to setting goals and tracking your progress, you should also periodically reflect on your goals to ensure their consistency with your values.
ANALYSIS QUESTION
Now that you’ve set some goals, what is your plan to track your progress on those goals? Can you identify a time you will set aside to intentionally reflect on your progress and whether you need to set any new short-term goals or perhaps adjust your larger plans?
Footnotes
- 1 Hall, Kathleen. Alter Your Life: Overbooked, Overworked, Overwhelmed? Oak Haven Press. Georgia. 2005.


1.4: The Wave Behavior of Matter
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Learning Objectives
- To understand the wave–particle duality of matter.
Einstein’s photons of light were individual packets of energy having many of the characteristics of particles. Recall that the collision of an electron (a particle) with a sufficiently energetic photon can eject a photoelectron from the surface of a metal. Any excess energy is transferred to the electron and is converted to the kinetic energy of the ejected electron. Einstein’s hypothesis that energy is concentrated in localized bundles, however, was in sharp contrast to the classical notion that energy is spread out uniformly in a wave. We now describe Einstein’s theory of the relationship between energy and mass, a theory that others built on to develop our current model of the atom.
The Wave Character of Matter
Einstein initially assumed that photons had zero mass, which made them a peculiar sort of particle indeed. In 1905, however, he published his special theory of relativity, which related energy and mass according to the famous equation:
\[ E=h u=h\dfrac{c}{\lambda }=mc^{2} \label{6.4.1} \]
According to this theory, a photon of wavelength \(λ\) and frequency \( u\) has a nonzero mass, which is given as follows:
\[ m=\dfrac{E}{c^{2}}=\dfrac{h u }{c^{2}}=\dfrac{h}{\lambda c} \label{6.4.2} \]
That is, light, which had always been regarded as a wave, also has properties typical of particles, a condition known as wave–particle duality (a principle that matter and energy have properties typical of both waves and particles). Depending on conditions, light could be viewed as either a wave or a particle.
In 1922, the American physicist Arthur Compton (1892–1962) reported the results of experiments involving the collision of x-rays and electrons that supported the particle nature of light. At about the same time, a young French physics student, Louis de Broglie (1892–1972), began to wonder whether the converse was true: Could particles exhibit the properties of waves? In his PhD dissertation submitted to the Sorbonne in 1924, de Broglie proposed that a particle such as an electron could be described by a wave whose wavelength is given by
\[\lambda =\dfrac{h}{mv} \label{6.4.3} \]
where
- \(h\) is Planck’s constant,
- \(m\) is the mass of the particle, and
- \(v\) is the velocity of the particle.
This revolutionary idea was quickly confirmed by American physicists Clinton Davisson (1881–1958) and Lester Germer (1896–1971), who showed that beams of electrons, regarded as particles, were diffracted by a sodium chloride crystal in the same manner as x-rays, which were regarded as waves. It was proven experimentally that electrons do exhibit the properties of waves. For his work, de Broglie received the Nobel Prize in Physics in 1929.
If particles exhibit the properties of waves, why had no one observed them before? The answer lies in the numerator of de Broglie’s equation, which is an extremely small number. As you will calculate in Example \(\PageIndex{1}\), Planck’s constant (6.63 × 10 −34 J•s) is so small that the wavelength of a particle with a large mass is too short (less than the diameter of an atomic nucleus) to be noticeable.
The de Broglie Equation: The de Broglie Equation, YouTube(opens in new window) [youtu.be]
Example \(\PageIndex{1}\): Wavelength of a Baseball in Motion
Calculate the wavelength of a baseball, which has a mass of 149 g and a speed of 100 mi/h.
Given: mass and speed of object
Asked for: wavelength
Strategy:
- Convert the speed of the baseball to the appropriate SI units: meters per second.
- Substitute values into Equation \(\ref{6.4.3}\) and solve for the wavelength.
Solution:
The wavelength of a particle is given by \(λ = h/mv\). We know that m = 0.149 kg, so all we need to find is the speed of the baseball:
\( v=\left ( \dfrac{100\; \cancel{mi}}{\cancel{h}} \right )\left ( \dfrac{1\; \cancel{h}}{60\; \cancel{min}} \right )\left ( \dfrac{1.609\; \cancel{km}}{\cancel{mi}} \right )\left ( \dfrac{1000\; m}{\cancel{km}} \right ) \)
B Recall that the joule is a derived unit, whose units are (kg•m 2 )/s 2 . Thus the wavelength of the baseball is
\[ \lambda =\dfrac{6.626\times 10^{-34}\; J\cdot s}{\left ( 0.149\; kg \right )\left ( 44.69\; m\cdot s \right )}= \dfrac{6.626\times 10^{-34}\; \cancel{kg}\cdot m{^\cancel{2}\cdot \cancel{s}{\cancel{^{-2}}\cdot \cancel{s}}}}{\left ( 0.149\; \cancel{kg} \right )\left ( 44.69\; \cancel{m}\cdot \cancel{s^{-1}} \right )}=9.95\times 10^{-35}\; m \nonumber \]
(You should verify that the units cancel to give the wavelength in meters.) Given that the diameter of the nucleus of an atom is approximately 10 −14 m, the wavelength of the baseball is almost unimaginably small.
Exercise \(\PageIndex{1}\): Wavelength of a Neutron in Motion
Calculate the wavelength of a neutron that is moving at 3.00 × 10 3 m/s.
- Answer
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1.32 Å, or 132 pm
As you calculated in Example \(\PageIndex{1}\), objects such as a baseball or a neutron have such short wavelengths that they are best regarded primarily as particles. In contrast, objects with very small masses (such as photons) have large wavelengths and can be viewed primarily as waves. Objects with intermediate masses, however, such as electrons, exhibit the properties of both particles and waves. Although we still usually think of electrons as particles, the wave nature of electrons is employed in an electron microscope , which has revealed most of what we know about the microscopic structure of living organisms and materials. Because the wavelength of an electron beam is much shorter than the wavelength of a beam of visible light, this instrument can resolve smaller details than a light microscope can (Figure \(\PageIndex{1}\)).
An Important Wave Property: Phase And Interference
A wave is a disturbance that travels in space. The magnitude of the wave at any point in space and time varies sinusoidally. While the absolute value of the magnitude of one wave at any point is not very important, the relative displacement of two waves, called the phase difference, is vitally important because it determines whether the waves reinforce or interfere with each other. Figure \(\PageIndex{2A}\) shows an arbitrary phase difference between two wave and Figure \(\PageIndex{2B}\) shows what happens when the two waves are 180 degrees out of phase. The green line is their sum. Figure \(\PageIndex{2C}\) shows what happens when the two lines are in phase, exactly superimposed on each other. Again, the green line is the sum of the intensities. A pattern of constructive and destructive interference is obtained when two (or more) diffracting waves interact with each other. This principle of diffraction and interference was used to prove the wave properties of electrons and is the basis for how electron microscopes work.
Photograph of an interference pattern produced by circular water waves in a ripple tank.
For a mathematical analysis of phase aspects in sinusoids, check the math Libretexts library.
Standing Waves
De Broglie also investigated why only certain orbits were allowed in Bohr’s model of the hydrogen atom. He hypothesized that the electron behaves like a standing wave (a wave that does not travel in space). An example of a standing wave is the motion of a string of a violin or guitar. When the string is plucked, it vibrates at certain fixed frequencies because it is fastened at both ends (Figure \(\PageIndex{3}\)). If the length of the string is \(L\), then the lowest-energy vibration (the fundamental) has wavelength
\[ \begin{align} \dfrac{\lambda }{2} & =L \nonumber \\ \lambda &= 2L \nonumber \end{align} \label{6.4.4} \]
Higher-energy vibrations are called overtones (the vibration of a standing wave that is higher in energy than the fundamental vibration) and are produced when the string is plucked more strongly; they have wavelengths given by
\[ \lambda=\dfrac{2L}{n} \label{6.4.5} \]
where n has any integral value. When plucked, all other frequencies die out immediately. Only the resonant frequencies survive and are heard. Thus, we can think of the resonant frequencies of the string as being quantized. Notice in Figure \(\PageIndex{3}\) that all overtones have one or more nodes, points where the string does not move. The amplitude of the wave at a node is zero.
Quantized vibrations and overtones containing nodes are not restricted to one-dimensional systems, such as strings. A two-dimensional surface, such as a drumhead, also has quantized vibrations. Similarly, when the ends of a string are joined to form a circle, the only allowed vibrations are those with wavelength
\[2πr = nλ \label{6.4.6} \]
where \(r\) is the radius of the circle. De Broglie argued that Bohr’s allowed orbits could be understood if the electron behaved like a standing circular wave (Figure \(\PageIndex{4}\)). The standing wave could exist only if the circumference of the circle was an integral multiple of the wavelength such that the propagated waves were all in phase, thereby increasing the net amplitudes and causing constructive interference . Otherwise, the propagated waves would be out of phase, resulting in a net decrease in amplitude and causing destructive interference. The nonresonant waves interfere with themselves! De Broglie’s idea explained Bohr’s allowed orbits and energy levels nicely: in the lowest energy level, corresponding to \(n = 1\) in Equation \(\ref{6.4.6}\), one complete wavelength would close the circle. Higher energy levels would have successively higher values of n with a corresponding number of nodes.
Like all analogies, although the standing wave model helps us understand much about why Bohr's theory worked, it also, if pushed too far, can mislead. As you will see, some of de Broglie’s ideas are retained in the modern theory of the electronic structure of the atom: the wave behavior of the electron and the presence of nodes that increase in number as the energy level increases. Unfortunately, his (and Bohr's) explanation also contains one major feature that we now know to be incorrect: in the currently accepted model, the electron in a given orbit is not always at the same distance from the nucleus.
The Heisenberg Uncertainty Principle
Because a wave is a disturbance that travels in space, it has no fixed position. One might therefore expect that it would also be hard to specify the exact position of a particle that exhibits wavelike behavior. A characteristic of light is that is can be bent or spread out by passing through a narrow slit. You can literally see this by half closing your eyes and looking through your eye lashes. This reduces the brightness of what you are seeing and somewhat fuzzes out the image, but the light bends around your lashes to provide a complete image rather than a bunch of bars across the image. This is called diffraction.
This behavior of waves is captured in Maxwell's equations (1870 or so) for electromagnetic waves and was and is well understood. An "uncertainty principle" for light is, if you will, merely a conclusion about the nature of electromagnetic waves and nothing new. De Broglie's idea of wave-particle duality means that particles such as electrons which exhibit wavelike characteristics will also undergo diffraction from slits whose size is on the order of the electron wavelength.
This situation was described mathematically by the German physicist Werner Heisenberg (1901–1976; Nobel Prize in Physics, 1932), who related the position of a particle to its momentum. Referring to the electron, Heisenberg stated that “at every moment the electron has only an inaccurate position and an inaccurate velocity, and between these two inaccuracies there is this uncertainty relation.” Mathematically, the Heisenberg uncertainty principle states that the uncertainty in the position of a particle (Δ x ) multiplied by the uncertainty in its momentum [Δ( mv )] is greater than or equal to Planck’s constant divided by 4π:
\[ \left ( \Delta x \right )\left ( \Delta \left [ mv \right ] \right )\ge \dfrac{h}{4\pi } \label{6.4.7} \]
Because Planck’s constant is a very small number, the Heisenberg uncertainty principle is important only for particles such as electrons that have very low masses. These are the same particles predicted by de Broglie’s equation to have measurable wavelengths.
If the precise position \(x\) of a particle is known absolutely (Δ x = 0), then the uncertainty in its momentum must be infinite:
\[ \left ( \Delta \left [ mv \right ] \right )= \dfrac{h}{4\pi \left ( \Delta x \right ) }=\dfrac{h}{4\pi \left ( 0 \right ) }=\infty \label{6.4.8} \]
Because the mass of the electron at rest (\(m\)) is both constant and accurately known, the uncertainty in \(Δ(mv)\) must be due to the \(Δv\) term, which would have to be infinitely large for \(Δ(mv)\) to equal infinity. That is, according to Equation \(\ref{6.4.8}\), the more accurately we know the exact position of the electron (as \(Δx → 0\)), the less accurately we know the speed and the kinetic energy of the electron (1/2 mv 2 ) because \(Δ(mv) → ∞\). Conversely, the more accurately we know the precise momentum (and the energy) of the electron [as \(Δ(mv) → 0\)], then \(Δx → ∞\) and we have no idea where the electron is.
Bohr’s model of the hydrogen atom violated the Heisenberg uncertainty principle by trying to specify simultaneously both the position (an orbit of a particular radius) and the energy (a quantity related to the momentum) of the electron. Moreover, given its mass and wavelike nature, the electron in the hydrogen atom could not possibly orbit the nucleus in a well-defined circular path as predicted by Bohr. You will see, however, that the most probable radius of the electron in the hydrogen atom is exactly the one predicted by Bohr’s model.
Example \(\PageIndex{1}\): Quantum Nature of Baseballs
Calculate the minimum uncertainty in the position of the pitched baseball from Example \(\ref{6.4.1}\) that has a mass of exactly 149 g and a speed of 100 ± 1 mi/h.
Given: mass and speed of object
Asked for: minimum uncertainty in its position
Strategy:
- Rearrange the inequality that describes the Heisenberg uncertainty principle (Equation \(\ref{6.4.7}\)) to solve for the minimum uncertainty in the position of an object (Δ x ).
- Find Δ v by converting the velocity of the baseball to the appropriate SI units: meters per second.
- Substitute the appropriate values into the expression for the inequality and solve for Δ x .
Solution:
A The Heisenberg uncertainty principle (Equation \ref{6.4.7}) tells us that \[(Δx)(Δ(mv)) = h/4π \nonumber \]. Rearranging the inequality gives
\( \Delta x \ge \left( {\dfrac{h}{4\pi }} \right)\left( {\dfrac{1}{\Delta (mv)}} \right)\)
B We know that h = 6.626 × 10 −34 J•s and m = 0.149 kg. Because there is no uncertainty in the mass of the baseball, Δ( mv ) = m Δ v and Δ v = ±1 mi/h. We have
\[ \Delta u =\left ( \dfrac{1\; \cancel{mi}}{\cancel{h}} \right )\left ( \dfrac{1\; \cancel{h}}{60\; \cancel{min}} \right )\left ( \dfrac{1\; \cancel{min}}{60\; s} \right )\left ( \dfrac{1.609\; \cancel{km}}{\cancel{mi}} \right )\left ( \dfrac{1000\; m}{\cancel{km}} \right )=0.4469\; m/s \nonumber \]
C Therefore,
\[ \Delta x \ge \left ( \dfrac{6.626\times 10^{-34}\; J\cdot s}{4\left ( 3.1416 \right )} \right ) \left ( \dfrac{1}{\left ( 0.149\; kg \right )\left ( 0.4469\; m\cdot s^{-1} \right )} \right ) \nonumber \]
Inserting the definition of a joule (1 J = 1 kg•m 2 /s 2 ) gives
\[ \Delta x \ge \left ( \dfrac{6.626\times 10^{-34}\; \cancel{kg} \cdot m^{\cancel{2}} \cdot s}{4\left ( 3.1416 \right )\left ( \cancel{s^{2}} \right )} \right ) \left ( \dfrac{1\; \cancel{s}}{\left ( 0.149\; \cancel{kg} \right )\left ( 0.4469\; \cancel{m} \right )} \right ) \nonumber \]
\[ \Delta x \ge 7.92 \pm \times 10^{-34}\; m \nonumber \]
This is equal to \(3.12 \times 10^{−32}\) inches. We can safely say that if a batter misjudges the speed of a fastball by 1 mi/h (about 1%), he will not be able to blame Heisenberg’s uncertainty principle for striking out.
Exercise \(\PageIndex{2}\)
Calculate the minimum uncertainty in the position of an electron traveling at one-third the speed of light, if the uncertainty in its speed is ±0.1%. Assume its mass to be equal to its mass at rest.
- Answer
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6 × 10 −10 m, or 0.6 nm (about the diameter of a benzene molecule)
Summary
An electron possesses both particle and wave properties. The modern model for the electronic structure of the atom is based on recognizing that an electron possesses particle and wave properties, the so-called wave–particle duality . Louis de Broglie showed that the wavelength of a particle is equal to Planck’s constant divided by the mass times the velocity of the particle.
\[\lambda =\dfrac{h}{mv} \nonumber \]
The electron in Bohr’s circular orbits could thus be described as a standing wave , one that does not move through space. Standing waves are familiar from music: the lowest-energy standing wave is the fundamental vibration, and higher-energy vibrations are overtones and have successively more nodes , points where the amplitude of the wave is always zero. Werner Heisenberg’s uncertainty principle states that it is impossible to precisely describe both the location and the speed of particles that exhibit wavelike behavior.
\[ \left ( \Delta x \right )\left ( \Delta \left [ mv \right ] \right )\geqslant \dfrac{h}{4\pi } \nonumber \]


6.7: Experimental Test of the Maxwell-Boltzmann Probability Density
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There are numerous applications of the Maxwell-Boltzmann equation. These include predictions of collision frequencies, mean-free paths, effusion and diffusion rates, the thermal conductivity of gases, and gas viscosities. These applications are important, but none of them is a direct test of the validity of the Maxwell-Boltzmann equation.
The validity of the equation has been demonstrated directly in experiments in which a gas of metal atoms is produced in an oven at a very high temperature. As sketched in Figure 4, the gas is allowed to escape into a vacuum chamber through a very small hole in the side of the oven. The escaping atoms impinge on one or more metal plates. Narrow slits cut in these plates stop any metal atoms whose flight paths do not pass though the slits. This produces a beambeam:of metal atoms of metal atoms whose velocity distribution is the same as that of the metal-atom gas inside the oven. The rate at which metal atoms arrive at a detector is measured. Various methods are used to translate the atom-arrival rate into a measurement of their speed.
One device uses a solid cylindrical drum, which rotates on its cylindrical axis. As sketched in Figure 5, a spiral groove is cut into the cylindrical face of this drum. This groove is cut with a constant pitch. When the drum rotates at a constant rate, an atom traveling at a constant
velocity parallel to the cylindrical axis can traverse the length of the drum while remaining within the groove. That is, for a given rotation rate, there is one critical velocity at which an atom can travel in a straight line while remaining in the middle of the groove all the way from one end of the drum to the other. If the atom moves significantly faster or slower than this critical velocity, it collides with—and sticks to—one side or the other of the groove.
Since the groove has a finite width, atoms whose velocities lie in a narrow range about the critical velocity can traverse the groove without hitting one of the sides.
Let us assume that the groove is cut so that the spiral travels half way around the cylinder. That is, if we project the spiral onto one of the circular faces of the drum, the projection traverses an angle of \({180}^o\) on the face. In order to remain in the middle of this groove all the way from one end of the drum to the other, the atom must travel the length of the cylindrical drum in exactly the same time that it takes the drum to make a half-rotation. Let the critical velocity be \(v_{critical}\). Then the time required for the atom to traverse the length, d , of the drum is \({d}/{v_{critial}}\). If the drum rotates at u cycles/sec, the time required for the drum to make one-half rotation is \({1}/{2u}\). Thus, the atom will remain in the middle of the groove all the way through the drum if
\[v_{critial}=2ud \nonumber \]
By varying the rotation rate, we can vary the critical velocity.
Because the groove has a finite width, atoms whose velocities are in a range \(v_{min}
\[t_{max}=\left(\pi r+w\right)/\left(2\pi ru\right) \nonumber \] and \[v_{min}=d/t_{max}=2\pi rud/\left(\pi r+w\right) \nonumber \]
A fast atom that enters the groove at the last possible moment—when the trailing edge of the grove just leaves the beam of atoms—can still traverse the groove if it does so in the time, \(t_{min}\) that it takes the trailing edge of the groove to travel a distance \(\pi r-w\). So,
\[t_{min}=\left(\pi r-w\right)/\left(2\pi ru\right) \nonumber \] and \[v_{max}=d/t_{min}=2\pi rud/\left(\pi r-w\right) \nonumber \]
At a given rotation rate, the drum will pass atoms whose speeds are in the range
\[\Delta v=v_{max}-v_{min}=2ud\left(\frac{\pi r}{\pi r-w}-\frac{\pi r}{\pi r+w}\right)=2ud\left(\frac{2\pi rw}{\left(\pi r\right)^2-w^2}\right)\approx v_{critical}\left(\frac{2w}{\pi r}\right) \nonumber \]
So that \[\frac{\Delta v}{v_{critial}}\approx \frac{2w}{\pi r} \nonumber \]
The fraction of the incident atoms that successfully traverse the groove is equal to the fraction that have velocities in the interval \(\Delta v\) centered on the critical velocity, \(v_{critical}=2ud\).


The main tissue types of the ground tissue system are parenchyma, collenchyma and sclerenchyma. Parenchyma have thin walls of cellulose, whereas collenchyma have cell walls with thickened areas of additional cellulose. Sclerenchyma cells have lignified cell walls. They can be further categorised into narrow long cells (fibers) and cells of various other shapes (sclereids).
Parenchyma
When parenchyma cells are modified to create tissues with air spaces for buoyancy or aeration of tissues, then the tissue is described as aerenchyma rather than parenchyma.
Collenchyma
Sclerenchyma
Sclerenchyma cells have lignified cell walls. They can be of two broad types: sclereids and fiber cells.
<IP_ADDRESS> Sclereids
<IP_ADDRESS> Fibers
Fiber cells are sclerenchyma cells that are long and thin.


3.8: Cultural Assessment
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- Ernstmeyer & Christman (Eds.)
- Chippewa Valley Technical College via OpenRN
After establishing a culturally sensitive environment, nurses should incorporate a cultural assessment when caring for all patients. There are many assessment guides used for patient interviews that are adaptable to a variety of health care settings and are designed to facilitate understanding and communication. The Four Cs of Culture model [1] is an example of a quick cultural assessment tool that asks questions about what the patient C onsiders to be a problem, the C ause of the problem, how they are C oping with the problem, and how C oncerned they are about the problem. See the following box for examples of sample answers to the four Cs assessment.
Four Cs of Culture [2]
1. What do you think is wrong? What is worrying you? (In other words, discover what the patient C onsiders to be the problem and what they C all it.)
- A patient with a diagnosis of pneumonia believes his body is “unbalanced.”
2. What do you think C aused this problem? How did this happen?
- The patient believes this illness is a punishment for a misdeed.
- The patient avoids eating certain foods to treat the illness while also using home remedies such as herbal tea.
3. How serious is this problem for you? How C oncerned are you?
- A patient views the illness as being “God’s will” and states, “It’s in God’s hands.”
A more comprehensive cultural assessment tool, inspired by R. E. Spector’s Heritage Assessment interview, [3] is described in the following box.
Sample Cultural Assessment Interview (Adapted from Spector’s Heritage Assessment Tool) [4]
- Where were you born? Where were your parents born?
- What pronoun do you use (he, she, they)?
- In what language are you most comfortable speaking and reading?
- Did you grow up in a city or a town or a rural setting?
- When you were growing up, who lived with you and your family?
- Are your friends from the same cultural background as you?
- What is your religious preference?
- Do you have any dietary preferences related to your religious or cultural beliefs?
- In your culture, how do you celebrate the birth of a baby? A wedding?
- When a woman is pregnant, are there any special customs she needs to follow? Any special foods?
- When someone in your family is ill, who cares for them? What foods are prepared? Is there anything the ill person should avoid or refrain from doing?
- What home remedies might be used if someone is ill?
- As a family member is approaching death, what actions do you find comforting?
- After a loved one dies, what rituals are performed?
- What do you think a nurse should know about your culture if a family member is hospitalized?
- Who makes the decisions in your family?
- How are elders viewed in your culture?
- Are there any special beliefs regarding organ donation or blood transfusions that are held in your culture?
- Is your culture known for any special customs (e.g., rites of passage, foods, holidays, etc.)?
- Galanti, G. A. (2014). Caring for patients from different cultures (5th ed.). University of Pennsylvania Press. ↵
- Galanti, G. A. (2014). Caring for patients from different cultures (5th ed.). University of Pennsylvania Press. ↵
- Spector, R. E. (2017). Cultural diversity in health and illness (9th ed.). Pearson Education. ↵
- Spector, R. E. (2017). Cultural diversity in health and illness (9th ed.). Pearson Education. ↵


23.3: Evolution of Massive Stars- An Explosive Finish
By the end of this section, you will be able to:
- Describe the interior of a massive star before a supernova
- Explain the steps of a core collapse and explosion
- List the hazards associated with nearby supernovae
Thanks to mass loss, then, stars with starting masses of at least 8 M Sun (and perhaps even more) probably end their lives as white dwarfs. But we know stars can have masses as large as 150 (or more) M Sun . They have a different kind of death in store for them. As we will see, these stars die with a bang.
Nuclear Fusion of Heavy Elements
After the helium in its core is exhausted (see The Evolution of More Massive Stars), the evolution of a massive star takes a significantly different course from that of lower-mass stars. In a massive star, the weight of the outer layers is sufficient to force the carbon core to contract until it becomes hot enough to fuse carbon into oxygen, neon, and magnesium. This cycle of contraction, heating, and the ignition of another nuclear fuel repeats several more times. After each of the possible nuclear fuels is exhausted, the core contracts again until it reaches a new temperature high enough to fuse still-heavier nuclei. The products of carbon fusion can be further converted into silicon, sulfur, calcium, and argon. And these elements, when heated to a still-higher temperature, can combine to produce iron. Massive stars go through these stages very, very quickly. In really massive stars, some fusion stages toward the very end can take only months or even days! This is a far cry from the millions of years they spend in the main-sequence stage.
At this stage of its evolution, a massive star resembles an onion with an iron core. As we get farther from the center, we find shells of decreasing temperature in which nuclear reactions involve nuclei of progressively lower mass—silicon and sulfur, oxygen, neon, carbon, helium, and finally, hydrogen (Figure 23.6).
But there is a limit to how long this process of building up elements by fusion can go on. The fusion of silicon into iron turns out to be the last step in the sequence of nonexplosive element production. Up to this point, each fusion reaction has produced energy because the nucleus of each fusion product has been a bit more stable than the nuclei that formed it. As discussed in The Sun: A Nuclear Powerhouse, light nuclei give up some of their binding energy in the process of fusing into more tightly bound, heavier nuclei. It is this released energy that maintains the outward pressure in the core so that the star does not collapse. But of all the nuclei known, iron is the most tightly bound and thus the most stable.
You might think of the situation like this: all smaller nuclei want to “grow up” to be like iron, and they are willing to pay ( produce energy) to move toward that goal. But iron is a mature nucleus with good self-esteem, perfectly content being iron; it requires payment (must absorb energy) to change its stable nuclear structure. This is the exact opposite of what has happened in each nuclear reaction so far: instead of providing energy to balance the inward pull of gravity, any nuclear reactions involving iron would remove some energy from the core of the star.
Unable to generate energy, the star now faces catastrophe.
Collapse into a Ball of Neutrons
When nuclear reactions stop, the core of a massive star is supported by degenerate electrons, just as a white dwarf is. For stars that begin their evolution with masses of at least 10 M Sun , this core is likely made mainly of iron. (For stars with initial masses in the range 8 to 10 M Sun , the core is likely made of oxygen, neon, and magnesium, because the star never gets hot enough to form elements as heavy as iron. The exact composition of the cores of stars in this mass range is very difficult to determine because of the complex physical characteristics in the cores, particularly at the very high densities and temperatures involved.) We will focus on the more massive iron cores in our discussion.
While no energy is being generated within the white dwarf core of the star, fusion still occurs in the shells that surround the core. As the shells finish their fusion reactions and stop producing energy, the ashes of the last reaction fall onto the white dwarf core, increasing its mass. As Figure 23.2 shows, a higher mass means a smaller core. The core can contract because even a degenerate gas is still mostly empty space. Electrons and atomic nuclei are, after all, extremely small. The electrons and nuclei in a stellar core may be crowded compared to the air in your room, but there is still lots of space between them.
The electrons at first resist being crowded closer together, and so the core shrinks only a small amount. Ultimately, however, the iron core reaches a mass so large that even degenerate electrons can no longer support it. When the density reaches 4 × 10 11 g/cm 3 (400 billion times the density of water), some electrons are actually squeezed into the atomic nuclei, where they combine with protons to form neutrons and neutrinos. This transformation is not something that is familiar from everyday life, but becomes very important as such a massive star core collapses.
Some of the electrons are now gone, so the core can no longer resist the crushing mass of the star’s overlying layers. The core begins to shrink rapidly. More and more electrons are now pushed into the atomic nuclei, which ultimately become so saturated with neutrons that they cannot hold onto them.
At this point, the neutrons are squeezed out of the nuclei and can exert a new force. As is true for electrons, it turns out that the neutrons strongly resist being in the same place and moving in the same way. The force that can be exerted by such degenerate neutrons is much greater than that produced by degenerate electrons, so unless the core is too massive, they can ultimately stop the collapse.
This means the collapsing core can reach a stable state as a crushed ball made mainly of neutrons, which astronomers call a neutron star . We don’t have an exact number (a “Chandrasekhar limit”) for the maximum mass of a neutron star, but calculations tell us that the upper mass limit of a body made of neutrons might only be about 3 M Sun . So if the mass of the core were greater than this, then even neutron degeneracy would not be able to stop the core from collapsing further. The dying star must end up as something even more extremely compressed, which until recently was believed to be only one possible type of object—the state of ultimate compaction known as a black hole (which is the subject of our next chapter). This is because no force was believed to exist that could stop a collapse beyond the neutron star stage.
Collapse and Explosion
When the collapse of a high-mass star’s core is stopped by degenerate neutrons, the core is saved from further destruction, but it turns out that the rest of the star is literally blown apart. Here’s how it happens.
The collapse that takes place when electrons are absorbed into the nuclei is very rapid. In less than a second, a core with a mass of about 1 M Sun , which originally was approximately the size of Earth, collapses to a diameter of less than 20 kilometers. The speed with which material falls inward reaches one-fourth the speed of light. The collapse halts only when the density of the core exceeds the density of an atomic nucleus (which is the densest form of matter we know). A typical neutron star is so compressed that to duplicate its density, we would have to squeeze all the people in the world into a single sugar cube! This would give us one sugar cube’s worth (one cubic centimeter’s worth) of a neutron star.
The neutron degenerate core strongly resists further compression, abruptly halting the collapse. The shock of the sudden jolt initiates a shock wave that starts to propagate outward. However, this shock alone is not enough to create a star explosion. The energy produced by the outflowing matter is quickly absorbed by atomic nuclei in the dense, overlying layers of gas, where it breaks up the nuclei into individual neutrons and protons.
Our understanding of nuclear processes indicates (as we mentioned above) that each time an electron and a proton in the star’s core merge to make a neutron, the merger releases a neutrino . These ghostly subatomic particles, introduced in The Sun: A Nuclear Powerhouse, carry away some of the nuclear energy. It is their presence that launches the final disastrous explosion of the star. The total energy contained in the neutrino s is huge. In the initial second of the star’s explosion, the power carried by the neutrinos (10 46 watts) is greater than the power put out by all the stars in over a billion galaxies.
While neutrinos ordinarily do not interact very much with ordinary matter (we earlier accused them of being downright antisocial), matter near the center of a collapsing star is so dense that the neutrinos do interact with it to some degree. They deposit some of this energy in the layers of the star just outside the core. This huge, sudden input of energy reverses the infall of these layers and drives them explosively outward. Most of the mass of the star (apart from that which went into the neutron star in the core) is then ejected outward into space. As we saw earlier, such an explosion requires a star of at least 8 M Sun , and the neutron star can have a mass of at most 3 M Sun. Consequently, at least five times the mass of our Sun is ejected into space in each such explosive event!
The resulting explosion is called a supernova (Figure 23.7). When these explosions happen close by, they can be among the most spectacular celestial events, as we will discuss in the next section. (Actually, there are at least two different types of supernova explosions: the kind we have been describing, which is the collapse of a massive star, is called, for historical reasons, a type II supernova . We will describe how the types differ later in this chapter).
Table 23.1 summarizes the discussion so far about what happens to stars and substellar objects of different initial masses at the ends of their lives. Like so much of our scientific understanding, this list represents a progress report: it is the best we can do with our present models and observations. The mass limits corresponding to various outcomes may change somewhat as models are improved. There is much we do not yet understand about the details of what happens when stars die.
| Initial Mass (Mass of Sun = 1) 1 | Final State at the End of Its Life |
|---|---|
| < 0.01 | Planet |
| 0.01 to 0.08 | Brown dwarf |
| 0.08 to 0.25 | White dwarf made mostly of helium |
| 0.25 to 8 | White dwarf made mostly of carbon and oxygen |
| 8 to 10 | White dwarf made of oxygen, neon, and magnesium |
| 10 to 40 | Supernova explosion that leaves a neutron star |
| > 40 | Supernova explosion that leaves a black hole |
The Supernova Giveth and the Supernova Taketh Away
After the supernova explosion, the life of a massive star comes to an end. But the death of each massive star is an important event in the history of its galaxy. The elements built up by fusion during the star’s life are now “recycled” into space by the explosion, making them available to enrich the gas and dust that form new stars and planets. Because these heavy elements ejected by supernovae are critical for the formation of planets and the origin of life, it’s fair to say that without mass loss from supernovae and planetary nebulae, neither the authors nor the readers of this book would exist. 2
But the supernova explosion has one more creative contribution to make, one we alluded to in Stars from Adolescence to Old Age when we asked where the atoms in your jewelry came from. The supernova explosion produces a flood of energetic neutrons that barrel through the expanding material. These neutrons can be absorbed by iron and other nuclei where they can turn into protons. Thus, they can build up elements that are more massive than iron, possibly including such terrestrial favorites as gold, silver and uranium. Supernovae (and, as we will shortly see, the explosive mergers of neutron stars) are the only candidates we have for places where such heavier atoms can be made. Next time you wear some gold jewelry (or give some to your sweetheart), bear in mind that those gold atoms were forged long ago in these kinds of celestial explosions!
When supernovae explode, these elements (as well as the ones the star made during more stable times) are ejected into the existing gas between the stars and mixed with it. Thus, supernovae play a crucial role in enriching their galaxy with heavier elements, allowing, among other things, the chemical elements that make up earthlike planets and the building blocks of life to become more common as time goes on (Figure 23.8).
Supernovae are also thought to be the source of many of the high-energy cosmic ray particles discussed in Cosmic Rays. Trapped by the magnetic field of the Galaxy, the particles from exploded stars continue to circulate around the vast spiral of the Milky Way. Scientists speculate that high-speed cosmic rays hitting the genetic material of Earth organisms over billions of years may have contributed to the steady mutations —subtle changes in the genetic code—that drive the evolution of life on our planet. In all the ways we have mentioned, supernovae have played a part in the development of new generations of stars, planets, and life.
But supernovae also have a dark side.  Such life forms may find themselves snuffed out when the harsh radiation and high-energy particles from the neighboring star’s explosion reach their world. If, as some astronomers speculate, life can develop on many planets around long-lived (lower-mass) stars, then the suitability of that life’s own star and planet may not be all that matters for its long-term evolution and survival. Life may well have formed around a number of pleasantly stable stars only to be wiped out because a massive nearby star suddenly went supernova. Just as children born in a war zone may find themselves the unjust victims of their violent neighborhood, life too close to a star that goes supernova may fall prey to having been born in the wrong place at the wrong time.
What is a safe distance to be from a supernova explosion? A lot depends on the violence of the particular explosion, what type of supernova it is (see The Evolution of Binary Star Systems), and what level of destruction we are willing to accept. Calculations suggest that a supernova less than 50 light-years away from us would certainly end all life on Earth, and that even one 100 light-years away would have drastic consequences for the radiation levels here. One minor extinction of sea creatures about 2 million years ago on Earth may actually have been caused by a supernova at a distance of about 120 light-years.
The good news is that there are at present no massive stars that promise to become supernovae within 50 light-years of the Sun. (This is in part because the kinds of massive stars that become supernovae are overall quite rare.) The massive star closest to us, Spica (in the constellation of Virgo), is about 260 light-years away, probably a safe distance, even if it were to explode as a supernova in the near future.
In this section, you were introduced to some very dense objects. How would those objects’ gravity affect you? Recall that the force of gravity, F , between two bodies is calculated as
where G is the gravitational constant, are the masses of the two bodies, and R is their separation. Also, from Newton’s second law,
where a is the acceleration of a body with mass M .
So let’s consider the situation of a mass—say, you—standing on a body, such as Earth or a white dwarf (where we assume you will be wearing a heat-proof space suit). You are M 1 and the body you are standing on is M 2 . The distance between you and the center of gravity of the body on which you stand is its radius, R . The force exerted on you is
Solving for a , the acceleration of gravity on that world, we get
Note that we have replaced the general symbol for acceleration, a , with the symbol scientists use for the acceleration of gravity, g .
Say that a particular white dwarf has the mass of the Sun but the radius of Earth . What is the acceleration of gravity at the surface of the white dwarf?
Solution
The acceleration of gravity at the surface of the white dwarf is
Compare this to g on the surface of Earth, which is 9.8 m/s 2 .
What is the acceleration of gravity at the surface if the white dwarf has the twice the mass of the Sun and is only half the radius of Earth?
- Answer
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Footnotes
- 1 Stars in the mass ranges 0.25–8 and 8–10 may later produce a type of supernova different from the one we have discussed so far. These are discussed in The Evolution of Binary Star Systems.
- 2 To tell the full story, there is another way to make and recycle these heavier elements which we will discuss in The Milky Way Galaxy. That is the merger of two neutron stars, which also causes a huge explosion.