6.3: Review/Evaluation
The third part of the Plan-Do-Review cycle involves reflecting on what was planned and implemented. Remember the section in Chapter 3 (The Early Childhood Teaching Profession), regarding the importance of reflection in all aspects of our teaching? Curriculum planning is one of the primary duties teachers engage in, and as such requires a great deal of reflection and review. Some of this will be done informally as you go about your day. Other times it may be helpful to more formally reflect, in order to capture strengths and areas of growth, both in yourself, the children, and the curriculum that you are planning for them. As a form of “assessment”, this feedback proves extremely valuable for teachers and programs. Below are examples of two types of forms teachers might use in their reviews. These will be used in future ECE courses at College of the Canyons.
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Curriculum Implementation Evaluation/Reflection A. Overall impression / comments about your activity (Be specific): B. What went well? C. What did not? D. What type of interactions took place during the implementation of your activity? (child – child, child – adult, …) E. How did individual children respond to your implementation? Did they respond the way you anticipated? (Please be specific and use examples whenever possible) F. If you were to implement this activity again, how would you modify it? Think about: Encouraging more participation from the children Meeting the needs of the wide range of developmental levels in the room Implementing the plan more successfully Additional activities to enhance the same concept(s) / webbing |
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Daily Curriculum Reflection
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Do these review/reflections make sense to you? What might you add or delete from the blank forms? Why?
Some programs will set up areas of the indoor and outdoor classroom with a variety of materials for children to choose from. Others will set up stations for children to participate in. Chapter 7 The Learning Environment will look specifically at setting up environments. Some portions of the day will include individual, small, and large group experiences. All should be carefully planned with intention and meaning for the children that will be engaging in them.


14.1: Demand for Labor
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Marginal Product of Labor (Physical)
The marginal product of labor is the change in output that results from employing an added unit of labor.
learning objectives
- Define the marginal product of labor
In economics, the marginal product of labor (MPL) is the change in output that results from employing an added unit of labor. This is not always equivalent to the output directly produced by that added unit of labor; for example, employing an additional cook at a restaurant may make the other cooks more efficient by allowing more specialization of tasks, creating a marginal product that is greater than that produced directly by the new employee. Conversely, hiring an additional worker onto an already crowded factory floor may make the other employees less productive, leading to a marginal product that is lower than the work done by the additional employee.
When production is discrete, we can define the marginal product of labor as ΔY/ΔL where Y is output. If a factory that is initially producing 100 widgets hires another employee and is then able to produce 106 widgets, the MPL is simply six. When production is continuous, the MPL is the first derivative of the production function in terms of L. Graphically, the MPL is the slope of the production function.
gives another example of marginal product of labor. The second column shows total production with different quantities of labor, while the third column shows the increase (or decrease) as labor is added to the production process.
Marginal Product of Labor : This table shows hypothetical returns and marginal product of labor. Note that in reality this firm would never hire more than seven employees, since a negative marginal product is bad for the firm regardless of the wage rate.
The law of diminishing marginal returns ensures that in most industries, the MPL will eventually be decreasing. The law states that “as units of one input are added (with all other inputs held constant) a point will be reached where the resulting additions to output will begin to decrease; that is marginal product will decline.” The law of diminishing marginal returns applies regardless of whether the production function exhibits increasing, decreasing or constant returns to scale. The key factor is that the variable input is being changed while all other factors of production are being held constant. Under such circumstances diminishing marginal returns are inevitable at some level of production.
Marginal Product of Labor (Revenue)
The marginal revenue product of labor is the change in revenue that results from employing an additional unit of labor.
learning objectives
- Define the marginal product of labor under the marginal revenue productivity theory of wages
The marginal revenue product of labor (MRPL) is the change in revenue that results from employing an additional unit of labor, holding all other inputs constant. The marginal revenue product of a worker is equal to the product of the marginal product of labor (MPL) and the marginal revenue (MR) of output, given by MR×MP: = MRPL. This can be used to determine the optimal number of workers to employ at an exogenously determined market wage rate. Theory states that a profit maximizing firm will hire workers up to the point where the marginal revenue product is equal to the wage rate, because it is not efficient for a firm to pay its workers more than it will earn in revenues from their labor.
For example, if a firm can sell t-shirts for $10 each and the wage rate is $20/hour, the firm will continue to hire workers until the marginal product of an additional hour of work is two t-shirts. If the MPL is three t-shirts the first will hire more workers until the MPL reaches two; if the MPL is one t-shirt then the firm will remove workers until the MPL reaches two.
Let TR=Total Revenue; L=Labor; Q=Quantity. Mathematically:
- \(\mathrm{MRPL= \frac{∆TR}{∆L}}\)
- \(\mathrm{MR = \frac{TR}{∆Q}}\)
- \(\mathrm{MPL =\frac{ ∆Q}{∆L}}\)
- \(\mathrm{MR \times MPL = (\frac{∆TR}{∆Q}) \times (\frac{∆Q}{∆L}) = \frac{∆TR}{∆L}}\)
Note that the change in output is not limited to that directly attributable to the additional worker. Assuming that the firm is operating with diminishing marginal returns then the addition of an extra worker reduces the average productivity of every other worker (and every other worker affects the marginal productivity of the additional worker) – in other words, everybody is getting in each other’s way.
Because the MRPL is equal to the marginal product of labor times the price of output, any variable that affects either MPL or price will affect the MRPL. For example, changes in technology or the quantity of other inputs will change the marginal product of labor, and changes in the product demand or changes in the price of complements or substitutes will affect the price of output. These will all cause shifts in the MRPL.
Deriving the Labor Demand Curve
Firms will demand labor until the marginal revenue product of labor is equal to the wage rate.
learning objectives
- Explain how a company uses marginal revenue product in hiring decisions
Firms demand labor and an input to production. The cost of labor to a firm is called the wage rate. This can be thought of as the firm’s marginal cost. The additional revenue generated by hiring one more unit of labor is the marginal revenue product of labor (MRPL). This can be thought of as the marginal benefit.
The marginal revenue product of labor (MRPL) is the additional amount of revenue a firm can generate by hiring one additional employee. It is found by multiplying the marginal product of labor (MPL) – the amount of additional output one additional worker can generate – by the price of output. If an employee of a customer support call center can take eight calls an hour (the MPL) and each call earns the company $3, then the MRPL is $24.
We can use the MRPL curve to determine the quantity of labor a company will hire. Suppose workers are available at an hourly rate of $10. The amount a factor adds to a firm’s total cost per period is the marginal cost of that factor, so in this case the marginal cost of labor is $10. Firms maximize profit when marginal costs equal marginal revenues, and in the labor market this means that firms will hire more employees until the wage rate (marginal cost of labor) equals the MRPL. At a price of $10, the company will hire workers until the last worker hired gives a marginal revenue product of $10.
Marginal Product of Labor : The MPL falls as the amount of labor employed increases. The optimum demand for labor falls where the real wage rate (w/P) is equal to the MPL.
Thus, the downward-sloping portion of the marginal revenue product curve shows the number of employees a company will hire at each price (wage), so we can interpret this part of the curve as the firm’s demand for labor. As with other demand curves, the market demand curve for labor is the sum of all firm’s individual demand curves.
Shifting the Demand for Labor
There are three main reasons why the demand curve for labor may shift:
- Changes to the marginal productivity of labor: Technology, for instance, may increase the marginal productivity of labor, shifting the demand curve to the right. For example, computer technology has increased the productivity (marginal product) of many types of workers. This has led to an increase in the marginal revenue product of labor for these jobs, shifting firms’ demand for labor to the right. This both increases the number of employed workers and increases the wage rate.
- The prices of other factors of production: The change in the relative price of labor will increase or decrease demand for labor. For example, is capital becomes more expensive relative to labor, the demand for labor will increase as firms seek to substitute labor for capital.
- The price of the firm’s output: Since the price of the output is a component of MRPL, changes will shift the demand curve for labor. If the price that a firm can charge for its output increases, for example, the MRPL will increase. This is reflected in an outward shift of the demand for labor.
Key Points
- The marginal product of labor is not always equivalent to the output directly produced by that added unit of labor.
- When production is discrete, we can define the marginal product of labor (MPL) as ΔY/ΔL.
- When production is continuous, the MPL is the first derivative of the production function in terms of L.
- Graphically, the MPL is the slope of the production function.
- The law of diminishing marginal returns ensures that in most industries, the MPL will eventually be decreasing.
- The marginal revenue product of a worker is equal to the product of the marginal product of labor (MP:) and the marginal revenue (MR) of output.
- The marginal revenue productivity theory states that a profit maximizing firm will hire workers up to the point where the marginal revenue product is equal to the wage rate.
- The change in output from hiring one more employee is not limited to that directly attributable to the additional worker.
- The marginal revenue product of labor (MRPL) is the additional amount of revenue a firm can generate by hiring one additional employee. It is found by multiplying the marginal product of labor by the price of output.
- Firms will demand labor until the MRPL equals the wage rate.
- The demand curve for labor can be shifted by shifted by changes in the productivity of labor, the relative price of labor, or the price of the output.
- It will also change as a result of a change in technology, a change in the price of the good being produced, or a change in the number of firms hiring the labor.
Key Terms
- returns to scale : A term referring to changes in output resulting from a proportional change in all inputs (where all inputs increase by a constant factor).
- marginal product : The extra output that can be produced by using one more unit of the input.
- diminishing marginal returns : The decrease in the per-unit output of a production process as the amount of a single factor of production is increased.
- marginal revenue product : The change in total revenue earned by a firm that results from employing one more unit of labor.
- factor of production : A resource employed to produce goods and services, such as labor, land, and capital.
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12.5: Interpretive Research Designs
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Case research . As discussed in the previous chapter, case research is an intensive longitudinal study of a phenomenon at one or more research sites for the purpose of deriving detailed, contextualised inferences, and understanding the dynamic process underlying a phenomenon of interest. Case research is a unique research design in that it can be used in an interpretive manner to build theories, or in a positivist manner to test theories. The previous chapter on case research discusses both techniques in depth and provides illustrative exemplars. Furthermore, the case researcher is a neutral observer (direct observation) in the social setting, rather than an active participant (participant observation). As with any other interpretive approach, drawing meaningful inferences from case research depends heavily on the observational skills and integrative abilities of the researcher.
Action research . Action research is a qualitative but positivist research design aimed at theory testing rather than theory building. This is an interactive design that assumes that complex social phenomena are best understood by introducing changes, interventions, or ‘actions’ into those phenomena, and observing the outcomes of such actions on the phenomena of interest. In this method, the researcher is usually a consultant or an organisational member embedded into a social context —such as an organisation—who initiates an action in response to a social problem, and examines how their action influences the phenomenon, while also learning and generating insights about the relationship between the action and the phenomenon. Examples of actions may include organisational change programs—such as the introduction of new organisational processes, procedures, people, or technology or the replacement of old ones—initiated with the goal of improving an organisation’s performance or profitability. The researcher’s choice of actions must be based on theory, which should explain why and how such actions may bring forth the desired social change. The theory is validated by the extent to which the chosen action is successful in remedying the targeted problem. Simultaneous problem-solving and insight generation are the central feature that distinguishes action research from other research methods (which may not involve problem solving), and from consulting (which may not involve insight generation). Hence, action research is an excellent method for bridging research and practice.
There are several variations of the action research method. The most popular of these methods is participatory action research , designed by Susman and Evered (1978). This method follows an action research cycle consisting of five phases: diagnosing, action-planning, action-taking, evaluating, and learning (see Figure 12.1). Diagnosing involves identifying and defining a problem in its social context. Action-planning involves identifying and evaluating alternative solutions to the problem, and deciding on a future course of action based on theoretical rationale. Action-taking is the implementation of the planned course of action. The evaluation stage examines the extent to which the initiated action is successful in resolving the original problem—i.e., whether theorised effects are indeed realised in practice. In the learning phase, the experiences and feedback from action evaluation are used to generate insights about the problem and suggest future modifications or improvements to the action. Based on action evaluation and learning, the action may be modified or adjusted to address the problem better, and the action research cycle is repeated with the modified action sequence. It is suggested that the entire action research cycle be traversed at least twice so that learning from the first cycle can be implemented in the second cycle. The primary mode of data collection is participant observation, although other techniques such as interviews and documentary evidence may be used to corroborate the researcher’s observations.
Ethnography . The ethnographic research method—derived largely from the field of anthropology—emphasises studying a phenomenon within the context of its culture. The researcher must be deeply immersed in the social culture over an extended period of time—usually eight months to two years—and should engage, observe, and record the daily life of the studied culture and its social participants within their natural setting. The primary mode of data collection is participant observation, and data analysis involves a ‘sense-making’ approach. In addition, the researcher must take extensive field notes, and narrate her experience in descriptive detail so that readers may experience the same culture as the researcher. In this method, the researcher has two roles: rely on her unique knowledge and engagement to generate insights (theory), and convince the scientific community of the transsituational nature of the studied phenomenon.
The classic example of ethnographic research is Jane Goodall’s study of primate behaviours. While living with chimpanzees in their natural habitat at Gombe National Park in Tanzania, she observed their behaviours, interacted with them, and shared their lives. During that process, she learnt and chronicled how chimpanzees seek food and shelter, how they socialise with each other, their communication patterns, their mating behaviours, and so forth. A more contemporary example of ethnographic research is Myra Bluebond-Langer’s (1996) [3] study of decision-making in families with children suffering from life-threatening illnesses, and the physical, psychological, environmental, ethical, legal, and cultural issues that influence such decision-making. The researcher followed the experiences of approximately 80 children with incurable illnesses and their families for a period of over two years. Data collection involved participant observation and formal/informal conversations with children, their parents and relatives, and healthcare providers to document their lived experience.
Phenomenology. Phenomenology is a research method that emphasises the study of conscious experiences as a way of understanding the reality around us. It is based on the ideas of early twentieth century German philosopher, Edmund Husserl, who believed that human experience is the source of all knowledge. Phenomenology is concerned with the systematic reflection and analysis of phenomena associated with conscious experiences such as human judgment, perceptions, and actions. Its goal is (appreciating and describing social reality from the diverse subjective perspectives of the participants involved, and understanding the symbolic meanings (‘deep structure’) underlying these subjective experiences. Phenomenological inquiry requires that researchers eliminate any prior assumptions and personal biases, empathise with the participant’s situation, and tune into existential dimensions of that situation so that they can fully understand the deep structures that drive the conscious thinking, feeling, and behaviour of the studied participants.
Some researchers view phenomenology as a philosophy rather than as a research method. In response to this criticism, Giorgi and Giorgi (2003) developed an existential phenomenological research method to guide studies in this area. This method, illustrated in Figure 12.2, can be grouped into data collection and data analysis phases. In the data collection phase, participants embedded in a social phenomenon are interviewed to capture their subjective experiences and perspectives regarding the phenomenon under investigation. Examples of questions that may be asked include ‘Can you describe a typical day?’ or ‘Can you describe that particular incident in more detail?’. These interviews are recorded and transcribed for further analysis. During data analysis , the researcher reads the transcripts to: get a sense of the whole, and establish ‘units of significance’ that can faithfully represent participants’ subjective experiences. Examples of such units of significance are concepts such as ‘felt-space’ and ‘felt-time’, which are then used to document participants’ psychological experiences. For instance, did participants feel safe, free, trapped, or joyous when experiencing a phenomenon (‘felt-space’)? Did they feel that their experience was pressured, slow, or discontinuous (‘felt-time’)? Phenomenological analysis should take into account the participants’ temporal landscape (i.e., their sense of past, present, and future), and the researcher must transpose his/herself in an imaginary sense into the participant’s situation (i.e., temporarily live the participant’s life). The participants’ lived experience is described in the form of a narrative or using emergent themes. The analysis then delves into these themes to identify multiple layers of meaning while retaining the fragility and ambiguity of subjects’ lived experiences.


27.2: Quasars
By the end of this section, you will be able to:
- Describe how quasars were discovered
- Explain how astronomers determined that quasars are at the distances implied by their redshifts
- Justify the statement that the enormous amount of energy produced by quasars is generated in a very small volume of space
The name “ quasar s” started out as short for “quasi-stellar radio sources” (here “quasi-stellar” means “sort of like stars”). The discovery of radio sources that appeared point-like, just like stars, came with the use of surplus World War II radar equipment in the 1950s.  As they improved the images that their new radio telescopes could make, scientists discovered that some radio sources were in the same location as faint blue “stars.” No known type of star in our Galaxy emits such powerful radio radiation. What then were these “quasi-stellar radio sources”?
Redshifts: The Key to Quasars
The answer came when astronomers obtained visible-light spectra of two of those faint “blue stars” that were strong sources of radio waves (Figure 27.2). Spectra of these radio “stars” only deepened the mystery: they had emission lines, but astronomers at first could not identify them with any known substance. By the 1960s, astronomers had a century of experience in identifying elements and compounds in the spectra of stars. Elaborate tables had been published showing the lines that each element would produce under a wide range of conditions. A “star” with unidentifiable lines in the ordinary visible light spectrum had to be something completely new.
In 1963 at Caltech’s Palomar Observatory, Maarten Schmidt (Figure 27.3) was puzzling over the spectrum of one of the radio stars, which was named 3C 273 because it was the 273rd entry in the third Cambridge catalog of radio sources (part (b) of Figure 27.3). There were strong emission lines in the spectrum, and Schmidt recognized that they had the same spacing between them as the Balmer lines of hydrogen (see Radiation and Spectra). But the lines in 3C 273 were shifted far to the red of the wavelengths at which the Balmer lines are normally located. Indeed, these lines were at such long wavelengths that if the redshifts were attributed to the Doppler effect, 3C 273 was receding from us at a speed of 45,000 kilometers per second, or about 15% the speed of light! Since stars don’t show Doppler shifts this large, no one had thought of considering high redshifts to be the cause of the strange spectra.
The puzzling emission lines in other star-like radio sources were then reexamined to see if they, too, might be well-known lines with large redshifts. This proved to be the case, but the other objects were found to be receding from us at even greater speeds. Their astounding speeds showed that the radio “stars” could not possibly be stars in our own Galaxy. Any true star moving at more than a few hundred kilometers per second would be able to overcome the gravitational pull of the Galaxy and completely escape from it. (As we shall see later in this chapter, astronomers eventually discovered that there was also more to these “stars” than just a point of light.)
It turns out that these high-velocity objects only look like stars because they are compact and very far away. Later, astronomers discovered objects with large redshifts that appear star-like but have no radio emission. Observations also showed that quasars were bright in the infrared and X-ray bands too, and not all these X-ray or infrared-bright quasars could be seen in either the radio or the visible-light bands of the spectrum. Today, all these objects are referred to as quasi-stellar objects ( QSOs ), or, as they are more popularly known, quasars . (The name was also soon appropriated by a manufacturer of home electronics.)
Read an interview with Maarten Schmidt on the fiftieth anniversary of his insight about the spectrum of quasars and their redshifts.
Over a million quasars have now been discovered, and spectra are available for over a hundred thousand. All these spectra show redshifts, none show blueshifts, and their redshifts can be very large. Yet in a photo they look just like stars (Figure 27.4).
In the record-holding quasars, the first Lyman series line of hydrogen, with a laboratory wavelength of 121.5 nanometers in the ultraviolet portion of the spectrum, is shifted all the way through the visible region to the infrared. At such high redshifts, the simple formula for converting a Doppler shift to speed (Radiation and Spectra) must be modified to take into account the effects of the theory of relativity. If we apply the relativistic form of the Doppler shift formula, we find that these redshifts correspond to velocities of about 96% of the speed of light.
The formula for the Doppler shift, which astronomers denote by the letters z , is
where λ is the wavelength emitted by a source of radiation that is not moving, Δλ is the difference between that wavelength and the wavelength we measure, v is the speed with which the source moves away, and c (as usual) is the speed of light.
A line in the spectrum of a galaxy is at a wavelength of 393 nanometers (nm, or 10 –9 m) when the source is at rest. Let’s say the line is measured to be longer than this value (redshifted) by 7.86 nm. Then its redshift , so its speed away from us is 2% of the speed of light
This formula is fine for galaxies that are relatively nearby and are moving away from us slowly in the expansion of the universe. But the quasars and distant galaxies we discuss in this chapter are moving away at speeds close to the speed of light. In that case, converting a Doppler shift (redshift) to a distance must include the effects of the special theory of relativity, which explains how measurements of space and time change when we see things moving at high speeds. The details of how this is done are way beyond the level of this text, but we can share with you the relativistic formula for the Doppler shift:
Let’s do an example. Suppose a distant quasar has a redshift of 3. At what fraction of the speed of light is the quasar moving away?
Solution
We calculate the following:
The quasar is thus receding from us at about 88% the speed of light.
Several lines of hydrogen absorption in the visible spectrum have rest wavelengths of 410 nm, 434 nm, 486 nm, and 656 nm. In a spectrum of a distant galaxy, these same lines are observed to have wavelengths of 492 nm, 521 nm, 583 nm, and 787 nm respectively. What is the redshift of this galaxy? What is the recession speed of this galaxy?
- Answer
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Because this is the same galaxy, we could pick any one of the four wavelengths and calculate how much it has shifted. If we use a rest wavelength of 410 nm and compare it to the shifted wavelength of 492 nm, we see that
In the classical view, this galaxy is receding at 20% of the speed of light; however, at 20% of the speed of light, relativistic effects are starting to become important. So, using the relativistic Doppler equation, we compute the true recession rate as
Therefore, the actual recession speed is only 18% of the speed of light. While this may not initially seem like a big difference from the classical measurement, there is already an 11% deviation between the classical and the relativistic solutions; and at greater recession speeds, the divergence between the classical and relativistic speeds increases rapidly!
Quasars Obey the Hubble Law
The first question astronomers asked was whether quasars obeyed the Hubble law and were really at the large distances implied by their redshifts. If they did not obey the rule that large redshift means large distance, then they could be much closer, and their luminosity could be a lot less. One straightforward way to show that quasars had to obey the Hubble law was to demonstrate that they were actually part of galaxies, and that their redshift was the same as the galaxy that hosted them. Since ordinary galaxies do obey the Hubble law, anything within them would be subject to the same rules.
Observations with the Hubble Space Telescope provided the strongest evidence showing that quasars are located at the centers of galaxies. Hints that this is true had been obtained with ground-based telescopes, but space observations were required to make a convincing case. The reason is that quasars can outshine their entire galaxies by factors of 10 to 100 or even more. When this light passes through Earth’s atmosphere, it is blurred by turbulence and drowns out the faint light from the surrounding galaxy—much as the bright headlights from an oncoming car at night make it difficult to see anything close by.
The Hubble Space Telescope, however, is not affected by atmospheric turbulence and can detect the faint glow from some of the galaxies that host quasars (Figure 27.5). Quasars have been found in the cores of both spiral and elliptical galaxies, and each quasar has the same redshift as its host galaxy. A wide range of studies with the Hubble Space Telescope now clearly demonstrate that quasars are indeed far away. If so, they must be producing a truly impressive amount of energy to be detectable as points of light that are much brighter than their galaxy. Interestingly, many quasar host galaxies are found to be involved in a collision with a second galaxy, providing, as we shall see, an important clue to the source of their prodigious energy output.
The Size of the Energy Source
Given their large distances, quasars have to be extremely luminous to be visible to us at all—far brighter than any normal galaxy. In visible light alone, most are far more energetic than the brightest elliptical galaxies. But, as we saw, quasars also emit energy at X-ray and ultraviolet wavelengths, and some are radio sources as well. When all their radiation is added together, some QSOs have total luminosities as large as a hundred trillion Suns (10 14 L Sun ), which is 10 to 100 times the brightness of luminous elliptical galaxies.
Finding a mechanism to produce the large amount of energy emitted by a quasar would be difficult under any circumstances. But there is an additional problem. When astronomers began monitoring quasars carefully, they found that some vary in luminosity on time scales of months, weeks, or even, in some cases, days. This variation is irregular and can change the brightness of a quasar by a few tens of percent in both its visible light and radio output.
Think about what such a change in luminosity means. A quasar at its dimmest is still more brilliant than any normal galaxy. Now imagine that the brightness increases by 30% in a few weeks. Whatever mechanism is responsible must be able to release new energy at rates that stagger our imaginations. The most dramatic changes in quasar brightness are equivalent to the energy released by 100,000 billion Suns. To produce this much energy we would have to convert the total mass of about ten Earths into energy every minute.
Moreover, because the fluctuations occur in such short times, the part of a quasar that is varying must be smaller than the distance light travels in the time it takes the variation to occur—typically a few months. To see why this must be so, let’s consider a cluster of stars 10 light-years in diameter at a very large distance from Earth (see Figure 27.6, in which Earth is off to the right). Suppose every star in this cluster somehow brightens simultaneously and remains bright. When the light from this event arrives at Earth, we would first see the brighter light from stars on the near side; 5 years later we would see increased light from stars at the center. Ten years would pass before we detected more light from stars on the far side.
Even though all stars in the cluster brightened at the same time, the fact that the cluster is 10 light-years wide means that 10 years must elapse before the increased light from every part of the cluster reaches us. From Earth we would see the cluster get brighter and brighter, as light from more and more stars began to reach us. Not until 10 years after the brightening began would we see the cluster reach maximum brightness. In other words, if an extended object suddenly flares up, it will seem to brighten over a period of time equal to the time it takes light to travel across the object from its far side.
We can apply this idea to brightness changes in quasars to estimate their diameters. Because quasars typically vary (get brighter and dimmer) over periods of a few months, the region where the energy is generated can be no larger than a few light-months across. If it were larger, it would take longer than a few months for the light from the far side to reach us.
How large is a region of a few light-months? Pluto, usually the outermost (dwarf) planet in our solar system, is about 5.5 light-hours from us, while the nearest star is 4 light-years away. Clearly a region a few light months across is tiny relative to the size of the entire Galaxy. And some quasars vary even more rapidly, which means their energy is generated in an even smaller region. Whatever mechanism powers the quasars must be able to generate more energy than that produced by an entire galaxy in a volume of space that, in some cases, is not much larger than our solar system.
Earlier Evidence
Even before the discovery of quasars, there had been hints that something very strange was going on in the centers of at least some galaxies. Back in 1918, American astronomer Heber Curtis used the large Lick Observatory telescope to photograph the galaxy Messier 87 in the constellation Virgo. On that photograph, he saw what we now call a jet coming from the center, or nucleus, of the galaxy (Figure 27.7). This jet literally and figuratively pointed to some strange activity going on in that galaxy nucleus. But he had no idea what it was. No one else knew what to do with this space oddity either.
The random factoid that such a central jet existed lay around for a quarter century, until Carl Seyfert , a young astronomer at Mount Wilson Observatory, also in California, found half a dozen galaxies with extremely bright nuclei that were almost stellar, rather than fuzzy in appearance like most galaxy nuclei. Using spectroscopy, he found that these nuclei contain gas moving at up to two percent the speed of light. That may not sound like much, but it is 6 million miles per hour, and more than 10 times faster than the typical motions of stars in galaxies.
After decades of study, astronomers identified many other strange objects beyond our Milky Way Galaxy; they populate a whole “zoo” of what are now called active galaxies or active galactic nuclei (AGN) . Astronomers first called them by many different names, depending on what sorts of observations discovered each category, but now we know that we are always looking at the same basic mechanism. What all these galaxies have in common is some activity in their nuclei that produces an enormous amount of energy in a very small volume of space. In the next section, we describe a model that explains all these galaxies with strong central activity—both the AGNs and the QSOs.
To see a jet for yourself, check out a time-lapse video of the jet ejected from NGC 3862.


3.3: Other Graphical Representations of Data
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There are many other types of graphs. Some of the more common ones are the frequency polygon, the dot plot, the stem plot, scatter plot, and a time-series plot. There are also many different graphs that have emerged lately for qualitative data. Many are found in publications and websites. The following is a description of the stem plot, the scatter plot, and the time-series plot.
Stem Plots
Stem plots are a quick and easy way to look at small samples of numerical data. You can look for any patterns or any strange data values. It is easy to compare two samples using stem plots.
The first step is to divide each number into 2 parts, the stem (such as the leftmost digit) and the leaf (such as the rightmost digit). There are no set rules, you just have to look at the data and see what makes sense.
Example \(\PageIndex{1}\) stem plot for grade distribution
The following are the percentage grades of 25 students from a statistics course. Draw a stem plot of the data.
| 62 | 87 | 81 | 69 | 87 | 62 | 45 | 95 | 76 | 76 |
| 62 | 71 | 65 | 67 | 72 | 80 | 40 | 77 | 87 | 58 |
| 84 | 73 | 93 | 64 | 89 |
Solution
Divide each number so that the tens digit is the stem and the ones digit is the leaf. 62 becomes 6|2.
Make a vertical chart with the stems on the left of a vertical bar. Be sure to fill in any missing stems. In other words, the stems should have equal spacing (for example, count by ones or count by tens). The Graph 2.3.1 shows the stems for this example.
Now go through the list of data and add the leaves. Put each leaf next to its corresponding stem. Don’t worry about order yet just get all the leaves down.
When the data value 62 is placed on the plot it looks like the plot in Graph 2.3.2 .
When the data value 87 is placed on the plot it looks like the plot in Graph 2.3.3 .
Filling in the rest of the leaves to obtain the plot in Graph 2.3.4 .
Now you have to add labels and make the graph look pretty. You need to add a label and sort the leaves into increasing order. You also need to tell people what the stems and leaves mean by inserting a legend. Be careful to line the leaves up in columns . You need to be able to compare the lengths of the rows when you interpret the graph. The final stem plot for the test grade data is in Graph 2.3.5.
Now you can interpret the stem-and-leaf display. The data is bimodal and somewhat symmetric. There are no gaps in the data. The center of the distribution is around 70.
You can create a stem and leaf plot on R. the command is:
stem(variable) – creates a stem and leaf plot, if you do not get a stem plot that shows all of the stems then use scale = a number. Adjust the number until you see all of the stems. So you would have stem(variable, scale = a number)
For Example \(\PageIndex{1}\), the command would be
grades<-c(62, 87, 81, 69, 87, 62, 45, 95, 76, 76, 62, 71, 65, 67, 72, 80, 40, 77, 87, 58, 84, 73, 93, 64, 89)
stem(grades, scale = 2)
Output:
The decimal point is 1 digit(s) to the right of the |
Now just put a title on the stem plot.
Scatter Plot
Sometimes you have two different variables and you want to see if they are related in any way. A scatter plot helps you to see what the relationship would look like. A scatter plot is just a plotting of the ordered pairs.
Example \(\PageIndex{2}\) scatter plot
Is there any relationship between elevation and high temperature on a given day? The following data are the high temperatures at various cities on a single day and the elevation of the city.
|
Elevation
(in feet) |
7000 | 4000 | 6000 | 3000 | 7000 | 4500 | 5000 |
|---|---|---|---|---|---|---|---|
| Temperature (°F) | 50 | 60 | 48 | 70 | 55 | 55 | 60 |
Solution
Preliminary: State the random variables
Let x = altitude
y = high temperature
Now plot the x values on the horizontal axis, and the y values on the vertical axis. Then set up a scale that fits the data on each axes. Once that is done, then just plot the x and y values as an ordered pair. In R, the command is:
independent variable<-c(type in data with commas in between values)
dependent variable<-c(type in data with commas in between values)
plot(independent variable, dependent variable, main="type in a title you want", xlab="type in a label for the horizontal axis", ylab="type in a label for the vertical axis", ylim=c(0, number above maximum y value)
For this example, that would be:
elevation<-c(7000, 4000, 6000, 3000, 7000, 4500, 5000)
temperature<-c(50, 60, 48, 70, 55, 55, 60)
plot(elevation, temperature, main="Temperature versus Elevation", xlab="Elevation (in feet)", ylab="Temperature (in degrees F)", ylim=c(0, 80))
Looking at the graph, it appears that there is a linear relationship between temperature and elevation. It also appears to be a negative relationship, thus as elevation increases, the temperature decreases.
Time-Series
A time-series plot is a graph showing the data measurements in chronological order, the data being quantitative data. For example, a time-series plot is used to show profits over the last 5 years. To create a time-series plot, the time always goes on the horizontal axis, and the other variable goes on the vertical axis. Then plot the ordered pairs and connect the dots. The purpose of a time-series graph is to look for trends over time. Caution, you must realize that the trend may not continue. Just because you see an increase, doesn’t mean the increase will continue forever. As an example, prior to 2007, many people noticed that housing prices were increasing. The belief at the time was that housing prices would continue to increase. However, the housing bubble burst in 2007, and many houses lost value, and haven’t recovered.
Example \(\PageIndex{3}\) Time-series plot
The following table tracks the weight of a dieter, where the time in months is measuring how long since the person started the diet
| Time (months) | 0 | 1 | 2 | 3 | 4 | 5 |
| Weight (pounds) | 200 | 195 | 192 | 193 | 190 | 187 |
Make a time-series plot of this data
Solution
In R, the command would be:
variable1<-c(type in data with commas in between values, this should be the time variable)
variable2<-c(type in data with commas in between values)
plot(variable1, variable2, ylim=c(0,number over max), main="type in a title you want", xlab="type in a label for the horizontal axis", ylab="type in a label for the vertical axis")
lines(variable1, variable2) – connects the dots
For this example:
time<-c(0, 1, 2, 3, 4, 5)
weight<-c(200, 195, 192, 193, 190, 187)
plot(time, weight, ylim=c(0,250), main="Weight over Time", xlab="Time (Months) ", ylab="Weight (pounds)")
ines(time, weight)
Notice, that over the 5 months, the weight appears to be decreasing. Though it doesn’t look like there is a large decrease.
Be careful when making a graph. If you don’t start the vertical axis at 0, then the change can look much more dramatic than it really is. As an example, Graph 2.3.8 shows the Graph 2.3.7 with a different scaling on the vertical axis. Notice the decrease in weight looks much larger than it really is.
Homework
Exercise \(\PageIndex{1}\)
-
Students in a statistics class took their first test. The data in Example \(\PageIndex{4}\) are the scores they earned. Create a stem plot.
80 79 89 74 73 67 79 93 70 70 76 88 83 73 81 79 80 85 79 80 79 58 93 94 74 Table \(\PageIndex{4}\): Data of Test 1 Grades -
Students in a statistics class took their first test. The data in Example \(\PageIndex{5}\) are the scores they earned. Create a stem plot. Compare to the graph in question 1.
67 67 76 47 85 70 87 76 80 72 84 98 84 64 65 82 81 81 88 74 87 83 Table \(\PageIndex{5}\): Data of Test 1 Grades -
When an anthropologist finds skeletal remains, they need to figure out the height of the person. The height of a person (in cm) and the length of one of their metacarpal bone (in cm) were collected and are in Example \(\PageIndex{6}\) ("Prediction of height," 2013). Create a scatter plot and state if there is a relationship between the height of a person and the length of their metacarpal.
Length of Metacarpal Height of Person 45 171 51 178 39 157 41 163 48 172 49 183 46 173 43 175 47 173 Table \(\PageIndex{6}\): Data of Metacarpal versus Height -
Example \(\PageIndex{7}\) contains the value of the house and the amount of rental income in a year that the house brings in ("Capital and rental," 2013). Create a scatter plot and state if there is a relationship between the value of the house and the annual rental income.
Value Rental Value Rental Value Rental Value Rental 81000 6656 77000 4576 75000 7280 67500 6864 95000 7904 94000 8736 90000 6240 85000 7072<PHONE_NUMBER>4<PHONE_NUMBER><PHONE_NUMBER><PHONE_NUMBER><PHONE_NUMBER><PHONE_NUMBER><PHONE_NUMBER><PHONE_NUMBER><PHONE_NUMBER><PHONE_NUMBER><PHONE_NUMBER><PHONE_NUMBER><PHONE_NUMBER>2<PHONE_NUMBER><PHONE_NUMBER><PHONE_NUMBER><PHONE_NUMBER>6<PHONE_NUMBER>0<PHONE_NUMBER><PHONE_NUMBER>8<PHONE_NUMBER>2<PHONE_NUMBER>8<PHONE_NUMBER>2<PHONE_NUMBER><PHONE_NUMBER><PHONE_NUMBER>0<PHONE_NUMBER>0<PHONE_NUMBER><PHONE_NUMBER>2<PHONE_NUMBER>4<PHONE_NUMBER>8<PHONE_NUMBER>0<PHONE_NUMBER>8<PHONE_NUMBER>6<PHONE_NUMBER>2<PHONE_NUMBER>2<PHONE_NUMBER>0<PHONE_NUMBER>0<PHONE_NUMBER>2<PHONE_NUMBER>0 Table \(\PageIndex{7}\): Data of House Value versus Rental -
The World Bank collects information on the life expectancy of a person in each country ("Life expectancy at," 2013) and the fertility rate per woman in the country ("Fertility rate," 2013). The data for 24 randomly selected countries for the year 2011 are in Example \(\PageIndex{8}\). Create a scatter plot of the data and state if there appears to be a relationship between life expectancy and the number of births per woman.
Life Expectancy Fertility Rate Life Expectancy Fertility rate 77.2 1.7 72.3 3.9 55.4 5.8 76.0 1.5 69.9 2.2 66.0 4.2 76.4 2.1 5.9 5.2 75.0 1.8 54.4 6.8 78.2 2.0 62.9 4.7 73.0 2.6 78.3 2.1 70.8 2.8 72.1 2.9 82.6 1.4 80.7 1.4 68.9 2.6 74.2 2.5 81.0 1.5 73.3 1.5 54.2 6.9 67.1 2.4 Table \(\PageIndex{8}\): Data of Life Expectancy versus Fertility Rate -
The World Bank collected data on the percentage of gross domestic product (GDP) that a country spends on health expenditures ("Health expenditure," 2013) and the percentage of woman receiving prenatal care ("Pregnant woman receiving," 2013). The data for the countries where this information is available for the year 2011 is in Example \(\PageIndex{9}\). Create a scatter plot of the data and state if there appears to be a relationship between percentage spent on health expenditure and the percentage of woman receiving prenatal care.
Prenatal Care (%) Health Expenditure (% of GDP) 47.9 9.6 54.6 3.7 93.7 5.2 84.7 5.2 100.0 10.0 42.5 4.7 96.4 4.8 77.1 6.0 58.3 5.4 95.4 4.8 78.0 4.1 93.3 6.0 93.3 9.5 93.7 6.8 89.8 6.1 Table \(\PageIndex{9}\): Data of Prenatal Care versus Health Expenditure -
The Australian Institute of Criminology gathered data on the number of deaths (per 100,000 people) due to firearms during the period 1983 to 1997 ("Deaths from firearms," 2013). The data is in Example \(\PageIndex{10}\). Create a time-series plot of the data and state any findings you can from the graph.
Year 1983 1984 1985 1986 1987 1988 1989 1990 Rate 4.31 4.42 4.52 4.35 4.39 4.21 3.40 3.61 Year 1991 1992 1993 1994 1995 1996 1997 Rate 3.67 3.61 2.98 2.95 2.72 2.95 2.3 Table \(\PageIndex{10}\): Data of Year versus Number of Deaths due to Firearms -
The economic crisis of 2008 affected many countries, though some more than others. Some people in Australia have claimed that Australia wasn’t hurt that badly from the crisis. The bank assets (in billions of Australia dollars (AUD)) of the Reserve Bank of Australia (RBA) for the time period of March 2007 through March 2013 are contained in Example \(\PageIndex{11}\) ("B1 assets of," 2013). Create a time-series plot and interpret any findings.
Date Assets in Billions of AUD Mar-2006 96.9 Jun-2006 107.4 Sep-2006 107.2 Dec-2006 116.2 Mar-2007 123.7 Jun-2007 134.0 Sep-2007 123.0 Dec-2007 93.2 Mar-2008 93.7 Jun-2008 105.6 Sep-2008 101.5 Dec-2008 158.8 Mar-2009 118.7 Jun-2009 111.9 Sep-2009 87.0 Dec-2009 86.1 Mar-2010 83.4 Jun-2010 85.7 Sep-2010 74.8 Dec-2010 76.0 Mar-2011 75.7 Jun-2011 75.9 Sep-2011 75.2 Dec-2011 87.9 Mar-2012 91.0 Jun-2012 90.1 Sep-2012 83.9 Dec-2012 95.8 Mar-2013 90.5 Table \(\PageIndex{11}\): Data of Date versus RBA Assets -
The consumer price index (CPI) is a measure used by the U.S. government to describe the cost of living. Example \(\PageIndex{12}\) gives the cost of living for the U.S. from the years 1947 through 2011, with the year 1977 being used as the year that all others are compared (DeNavas-Walt, Proctor & Smith, 2012). Create a time-series plot and interpret.
Year CPI-U-RS1 index (December 1977=100) Year CPI-U-RS1 index (December 1977=100) 1947 37.5 1980 127.1 1948 40.5 1981 139.2 1949 40.0 1982 147.6 1950 40.5 1983 153.9 1951 43.7 1984 160.2 1952 44.5 1985 165.7 1953 44.8 1986 168.7 1954 45.2 1987 174.4 1955 45.0 1988 180.8 1956 45.7 1989 188.6 1957 47.2 1990 198.0 1958 48.5 1991 205.1 1959 48.9 1992 210.3 1960 49.7 1993 215.5 1961 50.2 1994 220.1 1962 50.7 1995 225.4 1963 51.4 1996 231.4 1964 52.1 1997 236.4 1965 52.9 1998 239.7 1966 54.4 1999 244.7 1967 56.1 2000 252.9 1968 58.3 2001 260.0 1969 60.9 2002 264.2 1970 63.9 2003 270.1 1971 66.7 2004 277.4 1972 68.7 2005 286.7 1973 73.0 2006 296.1 1974 80.3 2007 304.5 1975 86.9 2008 316.2 1976 91.9 2009 315.0 1977 97.7 2010 320.2 1978 104.4 2011 330.3 1979 114.4 Table \(\PageIndex{12}\): Data of Time versus CPI -
The median incomes for all households in the U.S. for the years 1967 to 2011 are given in Example \(\PageIndex{13}\) (DeNavas-Walt, Proctor & Smith, 2012). Create a time-series plot and interpret.
Year Median Income Year Median Income 1967 42,056 1990 49,950 1968 43,868 1991 48,516 1969 45,499 1992 48,117 1970 45,146 1993 47,884 1971 44,707 1994 48,418 1972 46,622 1995 49,935 1973 47,563 1996 50,661 1974 46,057 1997 51,704 1975 44,851 1998 53,582 1976 45,595 1999 54,932 1977 45,884 2000 54,841 1978 47,659 2001 53,646 1979 47,527 2002 53,019 1980 46,024 2003 52,973 1981 45,260 2004 52,788 1982 45,139 2005 53,371 1983 44,823 2006 53,768 1984 46,215 2007 54,489 1985 47,079 2008 52,546 1986 48,746 2009 52,195 1987 49,358 2010 50,831 1988 49,737 2011 50,054 1989 50,624 Table \(\PageIndex{13}\): Data of Time versus Median Income -
State everything that makes
Graph 2.3.9
a misleading or poor graph.
Graph 2.3.9 : Example of a Poor Graph -
State everything that makes
Graph 2.3.10
a misleading or poor graph (Benen, 2011).
Graph 2.3.10 : Example of a Poor Graph -
State everything that makes
Graph 2.3.11
a misleading or poor graph ("United States unemployment," 2013).
Graph 2.3.11 : Example of a Poor Graph -
State everything that makes
Graph 2.3.12
a misleading or poor graph.
Graph 2.3.12 : Example of a Poor Graph
- Answer
-
See solutions
Data Sources:
B1 assets of financial institutions. (2013, June 27). Retrieved from www.rba.gov.au/statistics/tables/xls/b01hist.xls
Benen, S. (2011, September 02). [Web log message]. Retrieved from http://www.washingtonmonthly.com/pol...edit031960.php
Capital and rental values of Auckland properties . (2013, September 26). Retrieved from http://www.statsci.org/data/oz/rentcap.html
Contraceptive use . (2013, October 9). Retrieved from http://www.prb.org/DataFinder/Topic/...gs.aspx?ind=35
Deaths from firearms . (2013, September 26). Retrieved from http://www.statsci.org/data/oz/firearms.html
DeNavas-Walt, C., Proctor, B., & Smith, J. U.S. Department of Commerce, U.S. Census Bureau. (2012). Income, poverty, and health insurance coverage in the United States: 2011 (P60-243). Retrieved from website: www.census.gov/prod/2012pubs/p60-243.pdf
Density of people in Africa . (2013, October 9). Retrieved from http://www.prb.org/DataFinder/Topic/...249,250,251,25 2,253,254,34227,255,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,27 2,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,294, 295,296,297,298,299,300,301,302,304,305,306,307,308
Department of Health and Human Services, ASPE. (2013). Health insurance marketplace premiums for 2014. Retrieved from website: aspe.hhs.gov/health/reports/2...b_premiumsland scape.pdf
Electricity usage . (2013, October 9). Retrieved from http://www.prb.org/DataFinder/Topic/...s.aspx?ind=162
Fertility rate. (2013, October 14). Retrieved from http://data.worldbank.org/indicator/SP.DYN.TFRT.IN
Fuel oil usage. (2013, October 9). Retrieved from http://www.prb.org/DataFinder/Topic/...s.aspx?ind=164
Gas usage. (2013, October 9). Retrieved from http://www.prb.org/DataFinder/Topic/...s.aspx?ind=165
Health expenditure. (2013, October 14). Retrieved from http://data.worldbank.org/indicator/SH.XPD.TOTL.ZS Hinatov, M. U.S. Consumer Product Safety Commission, Directorate of Epidemiology. (2012). Incidents, deaths, and in-depth investigations associated with non-fire carbon monoxide from engine-driven generators and other engine-driven tools, 1999-2011. Retrieved from website: www.cpsc.gov/PageFiles/129857/cogenerators.pdf
Life expectancy at birth. (2013, October 14). Retrieved from http://data.worldbank.org/indicator/SP.DYN.LE00.IN
Median income of males. (2013, October 9). Retrieved from http://www.prb.org/DataFinder/Topic/...s.aspx?ind=137
Median income of males. (2013, October 9). Retrieved from http://www.prb.org/DataFinder/Topic/...s.aspx?ind=136
Prediction of height from metacarpal bone length . (2013, September 26). Retrieved from http://www.statsci.org/data/general/stature.html
Pregnant woman receiving prenatal care. (2013, October 14). Retrieved from http://data.worldbank.org/indicator/SH.STA.ANVC.ZS
United States unemployment. (2013, October 14). Retrieved from http://www.tradingeconomics.com/unit...mployment-rate
Weissmann, J. (2013, March 20). A truly devastating graph on state higher education spending. The Atlantic. Retrieved from http://www.theatlantic.com/business/...ending/274199/


8.7: Ionization of Water and The pH Scale
↵
- Define and use the ion product constant for water to calculate concentrations of H 3 O + and OH − in aqueous solutions.
- Define the pH scale and use it to describe acids and bases.
Ionization of Water
As we have already seen, H 2 O can act as an acid or a base. Within any given sample of water, some H 2 O molecules are acting as acids, and other H 2 O molecules are acting as bases. The chemical equation is as follows:
Similar to a weak acid, the autoionization of water is an equilibrium process, and is more properly written as follows:
H 2 O(ℓ)+H 2 O(ℓ) ⇌ H 3 O + (aq)+OH − (aq)
The equilibrium constant for the autoionization of water is referred to as the ion-product for water and is given the symbol K w.
Kw = [H 3 O + ][OH − ]
The ion-product constant for water ( K w ) is the mathematical product of the concentration of hydronium ions and hydroxide ions. Note that H 2 O is not included in the ion-product expression because it is a pure liquid. The value of K w is very small, in accordance with a reaction that favors the reactants. At 25 o C, the experimentally determined value of K w in pure water is 1.0×10 −14 .
Kw = [H 3 O + ][OH − ] = 1.0×10 −14
In a sample of pure water, the concentrations of hydronium and hydroxide ions are equal to one another. Pure water or any other aqueous solution in which this ratio holds is said to be neutral .
[H 3 O + ] = [OH − ] = 1.0×10 −7
The product of these two concentrations is 1.0×10 −14
[H 3 O + ]×[OH − ]=(1.0×10 −7 )(1.0×10 −7 )=1.0×10 −14
- For acids, the concentration of H 3 O + is greater than 1.0×10 −7 M
- For bases, the concentration of OH − is greater than 1.0×10 −7 M.
No matter whether the aqueous solution is an acid, a base, or neutral the Kw remains constant.
- For acidic solutions, [H 3 O + ] is greater than [OH − ].
- For basic solutions, [OH − ] is greater than [H 3 O + ].
- For neutral solutions, [H 3 O + ] = [OH − ] = 1.0×10 −7 M
NOTE: It is rare to truly have pure water. Water exposed to air will usually be slightly acidic because dissolved carbon dioxide gas, or carbonic acid, decreases the pH slightly below 7. Alternatively, dissolved minerals, like calcium carbonate (limestone), can make water slightly basic.
Knowing the amount of acid and base in solutions is extremely important for a wide variety of applications ranging from brewing beer or wine to studying the effects of river and creek water acidification to health and medicine. Scientists are good at calculating and measuring the concentration of hydronium in a solution, however, there is a more convenient way to make comparisons between solutions – the pH scale .
The pH Scale
One qualitative measure of the strength of an acid or a base solution is the pH scale, which is based on the concentration of the hydronium in aqueous solution.
\[pH = -\log[H_3O^+] \nonumber \]
Figure \(\PageIndex{1}\) illus trates this relationship, along with some examples of various solutions. Because hydronium ion concentrations are generally less than one (for example \(1.3 \times 10^{-3}\,M\), th e log of the number will be a negative number. To make pH even easier to work with, pH is defined as the negative log of \([H_3O^+]\) , which will give a positive value for pH. Note that [H 3 O + ] may be addressed as [H + ].
A neutral (neither acidic nor basic) solution has a pH of 7. A pH below 7 means that a solution is acidic, with lower values of pH corresponding to increasingly acidic solutions. A pH greater than 7 indicates a basic solution, with higher values of pH corresponding to increasingly basic solutions. Thus, given the pH of several solutions, you can state which ones are acidic, which ones are basic, and which are more acidic or basic than others. These are summarized in table below.
Table \(\PageIndex{1}\): Acidic, Basic and Neutral pH Values
| Classification | Relative Ion Concentrations | pH at 25 °C |
|---|---|---|
| acidic | [H 3 O + ] > [OH − ] | pH < 7 |
| neutral | [H 3 O + ] = [OH − ] | pH = 7 |
| basic | [H 3 O + ] < [OH − ] | pH > 7 |
Find the pH, given the [H 3 O + ] of the following:
- 1 ×10 -3 M
- 2.5 ×10 -11 M
- 4.7 ×10 -9 M
Solution
pH = - log [H 3 O + ]
Substitute the known quantity into the equation and solve. Use a scientific calculator for b and c.
- pH = - log [ 1 × 10 −3 ] = 3. 0 (1 decimal place since 1 has 1 significant figure)
- pH = - log [ 2.5 ×10 -11 ] = 10. 60 (2 decimal places since 2.5 has 2 significant figures)
- pH = - log [ 4.7 ×10 -9 ] = 8. 33 (2 decimal places since 4.7 has 2 significant figures)
Note on significant figures:
Because the number(s) before the decimal point in the pH value relate to the power on 10, the number of digits after the decimal point (underlined) is what determines the number of significant figures in the final answer.
Exercise \(\PageIndex{1}\)
Find the pH, given [H 3 O + ] of the following:
- 5.8 ×10 -4 M
- 1.0×10 -7 M
- Answer
-
a. 3.24
b. 7.00
Figure \(\PageIndex{2}\) lists the pH of several common solutions. The most aci dic among the listed solutions is 1 M HCl with the lowest pH value (0.0): battery acid is the next most acidic solution with a pH value of 0.3. The most basic is 1M NaOH solution with the highest pH value of 14.0. Notice that some biological fluids (gastric juices and blood) are not neutral. You may also notice that many food products are slightly acidic. They are acidic because they contain solutions of weak acids. If the acid components of these foods were strong acids, the food would likely be inedible.
Label each solution as acidic, basic, or neutral based only on the stated \(pH\).
- milk of magnesia, pH = 10.5
- pure water, pH = 7
- wine, pH = 3.0
Solution
- With a pH greater than 7, milk of magnesia is basic. (Milk of magnesia is largely Mg(OH) 2 .)
- Pure water, with a pH of 7, is neutral.
- With a pH of less than 7, wine is acidic.
Exercise \(\PageIndex{2}\)
Identify each substance as acidic, basic, or neutral based only on the stated \(pH\).
- human blood with \(pH\) = 7.4
- household ammonia with \(pH\) = 11.0
- cherries with \(pH\) = 3.6
- Answer
-
a. slightly basic
b. basic
c. acidic
Measuring pH
Tools have been developed that make the measurement of pH simple and convenient (Figures 8.7.3). For example, pH paper consists of strips of paper impregnated with one or more acid–base indicators , which are intensely colored organic molecules whose colors change dramatically depending on the pH of the solution. Placing a drop of a solution on a strip of pH paper and comparing its color with standards give the solution’s approximate pH. A more accurate tool, the pH meter, uses a glass electrode, a device whose voltage depends on the H + ion concentration (Figure \(\PageIndex{3}\)).
Normal rainwater has a pH between 5 and 6 due to the presence of dissolved CO 2 which forms carbonic acid:
\[\ce{H2O (l) + CO2(g) ⇌ H2CO3(aq)} \label{14} \]
H 2 CO 3 (aq) + H 2 O (l) ⇌ H 3 O + (aq) + HCO 3 - (aq)
Acid rain is rainwater that has a pH of less than 5, due to a variety of nonmetal oxides, including CO 2 , SO 2 , SO 3 , NO, and NO 2 being dissolved in the water and reacting with it to form not only carbonic acid, but sulfuric acid and nitric acid. The formation and subsequent ionization of sulfuric acid are shown here:
\[\ce{H2O (l) + SO3(g) ⇌ H2SO4(aq)} \label{16} \]
H 2 SO 4 (aq) + H 2 O (l) → H 3 O + (aq)+ HSO 4 - (aq)
Carbon dioxide is naturally present in the atmosphere because we and most other organisms produce it as a waste product of metabolism. Carbon dioxide is also formed when fires release carbon stored in vegetation or when we burn wood or fossil fuels. Sulfur trioxide in the atmosphere is naturally produced by volcanic activity, but it also stems from burning fossil fuels, which have traces of sulfur, and from the process of “roasting” ores of metal sulfides in metal-refining processes. Oxides of nitrogen are formed in internal combustion engines where the high temperatures make it possible for the nitrogen and oxygen in air to chemically combine.
Acid rain is a particular problem in industrial areas where the products of combustion and smelting are released into the air without being stripped of sulfur and nitrogen oxides. In North America and Europe until the 1980s, it was responsible for the destruction of forests and freshwater lakes, when the acidity of the rain actually killed trees, damaged soil, and made lakes uninhabitable for all but the most acid-tolerant species. Acid rain also corrodes statuary and building facades that are made of marble and limestone (Figure \(\PageIndex{4}\)). Regulations limiting the amount of sulfur and nitrogen oxides that can be released into the atmosphere by industry and automobiles have reduced the severity of acid damage to both natural and manmade environments in North America and Europe. It is now a growing problem in industrial areas of China and India.
Figure \(\PageIndex{4}\): (a) Acid rain makes trees more susceptible to drought and insect infestation, and depletes nutrients in the soil. (b) It also is corrodes statues that are carved from marble or limestone. (credit a: modification of work by Chris M Morris; credit b: modification of work by “Eden, Janine and Jim”/Flickr)


17.E: Analyzing Starlight (Exercises)
For Further Exploration
Articles
Berman, B. “Magnitude Cum Laude.” Astronomy (December 1998): 92. How we measure the apparent brightnesses of stars is discussed.
Dvorak, J. “The Women Who Created Modern Astronomy [including Annie Cannon].” Sky & Telescope (August 2013): 28.
Hearnshaw, J. “Origins of the Stellar Magnitude Scale.” Sky & Telescope (November 1992): 494. A good history of how we have come to have this cumbersome system is discussed.
Hirshfeld, A. “The Absolute Magnitude of Stars.” Sky & Telescope (September 1994): 35.
Kaler, J. “Stars in the Cellar: Classes Lost and Found.” Sky & Telescope (September 2000): 39. An introduction is provided for spectral types and the new classes L and T.
Kaler, J. “Origins of the Spectral Sequence.” Sky & Telescope (February 1986): 129.
Skrutskie, M. “2MASS: Unveiling the Infrared Universe.” Sky & Telescope (July 2001): 34. This article focuses on an all-sky survey at 2 microns.
Sneden, C. “Reading the Colors of the Stars.” Astronomy (April 1989): 36. This article includes a discussion of what we learn from spectroscopy.
Steffey, P. “The Truth about Star Colors.” Sky & Telescope (September 1992): 266. The color index and how the eye and film “see” colors are discussed.
Tomkins, J. “Once and Future Celestial Kings.” Sky & Telescope (April 1989): 59. Calculating the motion of stars and determining which stars were, are, and will be brightest in the sky are discussed.
Websites
Discovery of Brown Dwarfs: http://w.astro.berkeley.edu/~basri/b...SciAm-book.pdf .
Listing of Nearby Brown Dwarfs: http://www.solstation.com/stars/pc10bd.htm .
Spectral Types of Stars: http://www.skyandtelescope.com/astro...ypes-of-stars/ .
Stellar Velocities https://www.e-education.psu.edu/astr...ent/l4_p7.html .
Unheard Voices! The Contributions of Women to Astronomy: A Resource Guide: http://multiverse.ssl.berkeley.edu/women and http://www.astrosociety.org/educatio...esource-guide/ .
Videos
When You Are Just Too Small to be a Star: https://www.youtube.com/watch?v=zXCDsb4n4KU . 2013 Public Talk on Brown Dwarfs and Planets by Dr. Gibor Basri of the University of California–Berkeley (1:32:52).
Collaborative Group Activities
- The Voyagers in Astronomy feature on Annie Cannon: Classifier of the Stars in Section 17.3 discusses some of the difficulties women who wanted to do astronomy faced in the first half of the twentieth century. What does your group think about the situation for women today? Do men and women have an equal chance to become scientists? Discuss with your group whether, in your experience, boys and girls were equally encouraged to do science and math where you went to school.
- In the section on magnitudes in The Brightness of Stars, we discussed how this old system of classifying how bright different stars appear to the eye first developed. Your authors complained about the fact that this old system still has to be taught to every generation of new students. Can your group think of any other traditional systems of doing things in science and measurement where tradition rules even though common sense says a better system could certainly be found. Explain. (Hint: Try Daylight Savings Time, or metric versus English units.)
- Suppose you could observe a star that has only one spectral line. Could you tell what element that spectral line comes from? Make a list of reasons with your group about why you answered yes or no.
- A wealthy alumnus of your college decides to give $50 million to the astronomy department to build a world-class observatory for learning more about the characteristics of stars. Have your group discuss what kind of equipment they would put in the observatory. Where should this observatory be located? Justify your answers. (You may want to refer back to the Astronomical Instruments chapter and to revisit this question as you learn more about the stars and equipment for observing them in future chapters.)
- For some astronomers, introducing a new spectral type for the stars (like the types L, T, and Y discussed in the text) is similar to introducing a new area code for telephone calls. No one likes to disrupt the old system, but sometimes it is simply necessary. Have your group make a list of steps an astronomer would have to go through to persuade colleagues that a new spectral class is needed.
Review Questions
- What two factors determine how bright a star appears to be in the sky?
- Explain why color is a measure of a star’s temperature.
- What is the main reason that the spectra of all stars are not identical? Explain.
- What elements are stars mostly made of? How do we know this?
- What did Annie Cannon contribute to the understanding of stellar spectra?
- Name five characteristics of a star that can be determined by measuring its spectrum. Explain how you would use a spectrum to determine these characteristics.
- How do objects of spectral types L, T, and Y differ from those of the other spectral types?
- Do stars that look brighter in the sky have larger or smaller magnitudes than fainter stars?
- The star Antares has an apparent magnitude of 1.0, whereas the star Procyon has an apparent magnitude of 0.4. Which star appears brighter in the sky?
- Based on their colors, which of the following stars is hottest? Which is coolest? Archenar (blue), Betelgeuse (red), Capella (yellow).
- Order the seven basic spectral types from hottest to coldest.
- What is the defining difference between a brown dwarf and a true star?
Thought Questions
- If the star Sirius emits 23 times more energy than the Sun, why does the Sun appear brighter in the sky?
- How would two stars of equal luminosity—one blue and the other red—appear in an image taken through a filter that passes mainly blue light? How would their appearance change in an image taken through a filter that transmits mainly red light?
- Table \(17.3.1\) in Section 17.3 lists the temperature ranges that correspond to the different spectral types. What part of the star do these temperatures refer to? Why?
- Suppose you are given the task of measuring the colors of the brightest stars, listed in Appendix J, through three filters: the first transmits blue light, the second transmits yellow light, and the third transmits red light. If you observe the star Vega, it will appear equally bright through each of the three filters. Which stars will appear brighter through the blue filter than through the red filter? Which stars will appear brighter through the red filter? Which star is likely to have colors most nearly like those of Vega?
- Star X has lines of ionized helium in its spectrum, and star Y has bands of titanium oxide. Which is hotter? Why? The spectrum of star Z shows lines of ionized helium and also molecular bands of titanium oxide. What is strange about this spectrum? Can you suggest an explanation?
- The spectrum of the Sun has hundreds of strong lines of nonionized iron but only a few, very weak lines of helium. A star of spectral type B has very strong lines of helium but very weak iron lines. Do these differences mean that the Sun contains more iron and less helium than the B star? Explain.
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What are the approximate spectral classes of stars with the following characteristics?
- Balmer lines of hydrogen are very strong; some lines of ionized metals are present.
- The strongest lines are those of ionized helium.
- Lines of ionized calcium are the strongest in the spectrum; hydrogen lines show only moderate strength; lines of neutral and metals are present.
- The strongest lines are those of neutral metals and bands of titanium oxide.
- Look at the chemical elements in Appendix K. Can you identify any relationship between the abundance of an element and its atomic weight? Are there any obvious exceptions to this relationship?
- Appendix I lists some of the nearest stars. Are most of these stars hotter or cooler than the Sun? Do any of them emit more energy than the Sun? If so, which ones?
- Appendix J lists the stars that appear brightest in our sky. Are most of these hotter or cooler than the Sun? Can you suggest a reason for the difference between this answer and the answer to the previous question? (Hint: Look at the luminosities.) Is there any tendency for a correlation between temperature and luminosity? Are there exceptions to the correlation?
- What star appears the brightest in the sky (other than the Sun)? The second brightest? What color is Betelgeuse? Use Appendix J to find the answers.
- Suppose hominids one million years ago had left behind maps of the night sky. Would these maps represent accurately the sky that we see today? Why or why not?
- Why can only a lower limit to the rate of stellar rotation be determined from line broadening rather than the actual rotation rate? (Refer to Figure \(17.4.6\) in Section 17.4.)
- Why do you think astronomers have suggested three different spectral types (L, T, and Y) for the brown dwarfs instead of M? Why was one not enough?
- Sam, a college student, just bought a new car. Sam’s friend Adam, a graduate student in astronomy, asks Sam for a ride. In the car, Adam remarks that the colors on the temperature control are wrong. Why did he say that?
- Would a red star have a smaller or larger magnitude in a red filter than in a blue filter?
- Two stars have proper motions of one arcsecond per year. Star A is 20 light-years from Earth, and Star B is 10 light-years away from Earth. Which one has the faster velocity in space?
- Suppose there are three stars in space, each moving at 100 km/s. Star A is moving across (i.e., perpendicular to) our line of sight, Star B is moving directly away from Earth, and Star C is moving away from Earth, but at a 30° angle to the line of sight. From which star will you observe the greatest Doppler shift? From which star will you observe the smallest Doppler shift?
- What would you say to a friend who made this statement, “The visible-light spectrum of the Sun shows weak hydrogen lines and strong calcium lines. The Sun must therefore contain more calcium than hydrogen.”?
Figuring for Yourself
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In Appendix J, how much more luminous is the most luminous of the stars than the least luminous?
For Exercise 2 through Exercise 7 below, use the equations relating magnitude and apparent brightness given in the section on the magnitude scale in The Brightness of Stars and Example \(17.1.1\) in that section.
- Verify that if two stars have a difference of five magnitudes, this corresponds to a factor of 100 in the ratio \(\left( \frac{b_2}{b_1} \right)\); that 2.5 magnitudes corresponds to a factor of 10; and that 0.75 magnitudes corresponds to a factor of 2.
- As seen from Earth, the Sun has an apparent magnitude of about −26.7. What is the apparent magnitude of the Sun as seen from Saturn, about 10 AU away? (Remember that one AU is the distance from Earth to the Sun and that the brightness decreases as the inverse square of the distance.) Would the Sun still be the brightest star in the sky?
- An astronomer is investigating a faint star that has recently been discovered in very sensitive surveys of the sky. The star has a magnitude of 16. How much less bright is it than Antares, a star with magnitude roughly equal to 1?
- The center of a faint but active galaxy has magnitude 26. How much less bright does it look than the very faintest star that our eyes can see, roughly magnitude 6?
- You have enough information from this chapter to estimate the distance to Alpha Centauri, the second nearest star, which has an apparent magnitude of 0. Since it is a G2 star, like the Sun, assume it has the same luminosity as the Sun and the difference in magnitudes is a result only of the difference in distance. Estimate how far away Alpha Centauri is. Describe the necessary steps in words and then do the calculation. (As we will learn in the Celestial Distances chapter, this method—namely, assuming that stars with identical spectral types emit the same amount of energy—is actually used to estimate distances to stars.) If you assume the distance to the Sun is in AU, your answer will come out in AU.
- Do the previous problem again, this time using the information that the Sun is 150,000,000 km away. You will get a very large number of km as your answer. To get a better feeling for how the distances compare, try calculating the time it takes light at a speed of 299,338 km/s to travel from the Sun to Earth and from Alpha Centauri to Earth. For Alpha Centauri, figure out how long the trip will take in years as well as in seconds.
- Star A and Star B have different apparent brightnesses but identical luminosities. If Star A is 20 light-years away from Earth and Star B is 40 light-years away from Earth, which star appears brighter and by what factor?
- Star A and Star B have different apparent brightnesses but identical luminosities. Star A is 10 light-years away from Earth and appears 36 times brighter than Star B. How far away is Star B?
- The star Sirius A has an apparent magnitude of −1.5. Sirius A has a dim companion, Sirius B, which is 10,000 times less bright than Sirius A. What is the apparent magnitude of Sirius B? Can Sirius B be seen with the naked eye?
- Our Sun, a type G star, has a surface temperature of 5800 K. We know, therefore, that it is cooler than a type O star and hotter than a type M star. Given what you learned about the temperature ranges of these types of stars, how many times hotter than our Sun is the hottest type O star? How many times cooler than our Sun is the coolest type M star?