8.2.2B: Solutions of Gaseous Solutes in Liquid Solvents
Make sure you thoroughly understand the following essential ideas:
- Explain why, in contrast, gases tend to be only slightly soluble in liquids and solids. And why are some combinations, such as the dissolution of ammonia or hydrogen chloride in water, significant exceptions to this rule?
- State Henry's law and explain
- Why do fish in rivers and streams sometimes become asphyxiated (oxygen-starved) in hot weather?
Solutions of Gases in Liquids
Gases dissolve in liquids, but usually only to a small extent. When a gas dissolves in a liquid, the ability of the gas molecules to move freely throughout the volume of the solvent is greatly restricted. If this latter volume is small, as is often the case, the gas is effectively being compressed. Both of these effects amount to a decrease in the entropy of the gas that is not usually compensated by the entropy increase due to mixing of the two kinds of molecules. Such processes greatly restrict the solubility of gases in liquids.
| liquid solvent, solute → | gas |
|---|---|
| energy to disperse solute | nil |
| energy to introduce into solvent | medium to large |
| increase in entropy | negative |
| miscibility | usually very limited |
Solubility of gases in water
One important consequence of the entropy decrease when a gas dissolves in a liquid is that the solubility of a gas decreases at higher temperatures; this is in contrast to most other situations, where a rise in temperature usually leads to increased solubility. Bringing a liquid to its boiling point will completely remove a gaseous solute. Some typical gas solubilities, expressed in the number of moles of gas at 1 atm pressure that will dissolve in a liter of water at 25° C, are given below:
| solute | formula | solubility, mol L –1 atm –1 |
|---|---|---|
| ammonia | NH 3 | 57 |
| carbon dioxide | CO 2 | 0.0308 |
| methane | CH 4 | 0.00129 |
| nitrogen | N 2 | 0.000661 |
| oxygen | O 2 | 0.00126 |
| sulfur dioxide | SO 2 | 1.25 |
As we indicated above, the only gases that are readily soluble in water are those whose polar character allows them to interact strongly with it.
Ammonia is remarkably soluble in water
Inspection of the above table reveals that ammonia is a champion in this regard. At 0° C, one liter of water will dissolve about 90 g (5.3 mol) of ammonia. The reaction of ammonia with water according to
\[\ce{NH_3 + H_2O → NH_4^{+} + OH^{–}}\]
makes no significant contribution to its solubility; the equilibrium lies heavily on the left side (as evidenced by the strong odor of ammonia solutions). Only about four out of every 1000 NH 3 molecules are in the form of ammonium ions at equilibrium. This is truly impressive when one calculates that this quantity of NH 3 would occupy (5.3 mol) × (22.4 L mol –1 ) = 119 L at STP. Thus one volume of water will dissolve over 100 volumes of this gas. It is even more impressive when you realize that in order to compress 119 L of an ideal gas into a volume of 1 L, a pressure of 119 atm would need to be applied! This, together with the observation that dissolution of ammonia is accompanied by the liberation of a considerable amount of heat, tells us that the high solubility of ammonia is due to the formation of more hydrogen bonds (to H 2 O) than are broken within the water structure in order to accommodate the NH 3 molecule.
If we actually compress 90 g of pure NH 3 gas to 1 L, it will liquefy, and the vapor pressure of the liquid would be about 9 atm. In other words, the escaping tendency of NH 3 molecules from H 2 O is only about 1/9th of what it is from liquid NH 3 . One way of interpreting this is that the strong intermolecular (dipole-dipole) attractions between NH 3 and the solvent H 2 O give rise to a force that has the effect of a negative pressure of 9 atm.
Solubility of gases decreases with Temperature
Recall that entropy is a measure of the ability of thermal energy to spread and be shared and exchanged by molecules in the system. Higher temperature exerts a kind of multiplying effect on a positive entropy change by increasing the amount of thermal energy available for sharing. Have you ever noticed the tiny bubbles that form near the bottom of a container of water when it is placed on a hot stove? These bubbles contain air that was previously dissolved in the water, but reaches its solubility limit as the water is warmed. You can completely rid a liquid of any dissolved gases (including unwanted ones such as Cl 2 or H 2 S) by boiling it in an open container.
This is quite different from the behavior of most (but not all) solutions of solid or liquid solutes in liquid solvents. The reason for this behavior is the very large entropy increase that gases undergo when they are released from the confines of a condensed phase .
Solubility of Oxygen in water
Fresh water at sea level dissolves 14.6 mg of oxygen per liter at 0°C and 8.2 mg/L at 25°C. These saturation levels ensure that fish and other gilled aquatic animals are able to extract sufficient oxygen to meet their respiratory needs. But in actual aquatic environments, the presence of decaying organic matter or nitrogenous runoff can reduce these levels far below saturation. The health and survival of these organisms is severely curtailed when oxygen concentrations fall to around 5 mg/L.
The temperature dependence of the solubility of oxygen in water is an important consideration for the well-being of aquatic life; thermal pollution of natural waters (due to the influx of cooling water from power plants) has been known to reduce the dissolved oxygen concentration to levels low enough to kill fish. The advent of summer temperatures in a river can have the same effect if the oxygen concentration has already been partially depleted by reaction with organic pollutants.
Solubility of gases increases with pressure: Henry's Law
The pressure of a gas is a measure of its "escaping tendency" from a phase. So it stands to reason that raising the pressure of a gas in contact with a solvent will cause a larger fraction of it to "escape" into the solvent phase. The direct-proportionality of gas solubility to pressure was discovered by William Henry (1775-1836) and is known as Henry's Law . It is usually written as
\[P = k_H C \label{7b.2.1}\]
with
- \(P\) is the partial pressure of the gas above the liquid,
- \(C\) is the concentration of gas dissolved in the liquid, and
- \(k_H\) is the Henry's law constant , which can be expressed in various units, and in some instances is defined in different ways, so be very careful to note these units when using published values.
For Table 7b.2.X, k H is given as
\[ k_H = \dfrac{\text{partial pressure of gas in atm}}{\text{concentration in liquid} \; mol \;L^{–1}}\]
| gas | He | N 2 | O 2 | CO 2 | CH 4 | NH 3 |
|---|---|---|---|---|---|---|
| K H | 2703 | 1639 | 769 | 29.4 | 0.00129 | 57 |
Some vendors of bottled waters sell pressurized "oxygenated water" that is (falsely) purported to enhance health and athletic performance by supplying more oxygen to the body.
- How many moles of O 2 will be in equilibrium with one liter of water at 25° C when the partial pressure of O 2 above the water is 2.0 atm?
- How many mL of air (21% O 2 v/v) must you inhale in order to introduce an equivalent quantity of O 2 into the lungs (where it might actually do some good?)
Solution:
- Solving Henry's law for the concentration, we get
\[C = \dfrac{P}{k_H} = \dfrac{2.0\; atm}{769\; L\; atm \;mol^{–1}} = 0.0026\; mol\; L^{–1}\]
- At 25° C, 0.0026 mol of O 2 occupies (22.4 L) × (.0026 mol) × (298/273) = 0.063 L. The equivalent volume of air would be (0.063 L) (.21) = 0.303 L. Given that the average tidal volume of the human lung is around 400 mL, this means that taking one extra breath would take in more O 2 than is present in 1 L of "oxygenated water".
Carbonated beverages: the history of "Fizz-ics"
Artificially carbonated water was first prepared by Joseph Priestley (who later discovered oxygen) in 1767 and was commercialized in 1783 by Joseph Schweppe, a Swiss-German jeweler. Naturally-carbonated spring waters have long been reputed to have curative values, and these became popular tourist destinations in the 19th century. The term "seltzer water" derives from one such spring in Niederselters, Germany. Of course, carbonation produced by fermentation has been known since ancient times. The tingling sensation that carbonated beverages produce in the mouth comes from the carbonic acid produced when bubbles of carbon dioxide come into contact with the mucous membranes of the mouth and tongue:
\[CO_2 + H_2O → H_2CO_3\]


6.1: Robert Schumann
The early Romantic composer Robert Schumann was, like Schubert, a prolific composer of art songs. Also like Schubert, his life was tragically cut short, though unlike the younger composer, Robert enjoyed a good deal more success and recognition. This short introduction is a good summary of the composer’s career. If you’d like to know more, I’d certainly encourage you to read the rest of the article here . The article includes links to additional pieces not on our listening exam. It also tells the story of Robert and Clara Schumann, which is one of the great love stories of music history. Clara Schumann was as formidable a musician as her husband, and as a performer (concert pianist), she was quite famous. She and her husband were both very influential in the life of one of the other composers we will study in this period: Johannes Brahms.
Robert Schumann (8 June 1810–29 July 1856) was a German composer and influential music critic. He is widely regarded as one of the greatest composers of the Romantic era. Schumann left the study of law, intending to pursue a career as a virtuoso pianist. He had been assured by his teacher Friedrich Wieck that he could become the finest pianist in Europe, but a hand injury ended this dream. Schumann then focused his musical energies on composing.
Schumann’s published compositions were written exclusively for the piano until 1840; he later composed works for piano and orchestra; manyLieder (songs for voice and piano); four symphonies; an opera; and other orchestral, choral, and chamber works. Works such as Kinderszenen,Album für die Jugend, Blumenstück, the Sonatas and Albumblätter are among his most famous. His writings about music appeared mostly in the Neue Zeitschrift für Musik (New Journal for Music), a Leipzig-based publication which he jointly founded.
In 1840, against the wishes of her father, Schumann married Friedrich Wieck’s daughter Clara, following a long and acrimonious legal battle, which found in favor of Clara and Robert. Clara also composed music and had a considerable concert career as a pianist, the earnings from which formed a substantial part of her father’s fortune.
Schumann suffered from a lifelong mental disorder, first manifesting itself in 1833 as a severe melancholic depressive episode, which recurred several times alternating with phases of ‘exaltation’ and increasingly also delusional ideas of being poisoned or threatened with metallic items. After a suicide attempt in 1854, Schumann was admitted to a mental asylum, at his own request, in Endenich near Bonn. Diagnosed with “psychotic melancholia”, Schumann died two years later in 1856 without having recovered from his mental illness.
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Robert Schumann. Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/Robert_Schumann . License : CC BY-SA: Attribution-ShareAlike


13.4E: Exercises
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Practice Makes Perfect
Exercise \(\PageIndex{15}\) Graph an Ellipse with Center at the Origin
In the following exercises, graph each ellipse.
- \(\frac{x^{2}}{4}+\frac{y^{2}}{25}=1\)
- \(\frac{x^{2}}{9}+\frac{y^{2}}{25}=1\)
- \(\frac{x^{2}}{25}+\frac{y^{2}}{36}=1\)
- \(\frac{x^{2}}{16}+\frac{y^{2}}{36}=1\)
- \(\frac{x^{2}}{36}+\frac{y^{2}}{16}=1\)
- \(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\)
- \(x^{2}+\frac{y^{2}}{4}=1\)
- \(\frac{x^{2}}{9}+y^{2}=1\)
- \(4 x^{2}+25 y^{2}=100\)
- \(16 x^{2}+9 y^{2}=144\)
- \(16 x^{2}+36 y^{2}=576\)
- \(9 x^{2}+25 y^{2}=225\)
- Answer
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Exercise \(\PageIndex{16}\) Find the Equation of an Ellipse with Center at the Origin
In the following exercises, find the equation of the ellipse shown in the graph.
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- Answer
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1. \(\frac{x^{2}}{9}+\frac{y^{2}}{25}=1\)
3. \(\frac{x^{2}}{9}+\frac{y^{2}}{16}=1\)
Exercise \(\PageIndex{17}\) Graph an Ellipse with Center Not at the Origin
In the following exercises, graph each ellipse.
- \(\frac{(x+1)^{2}}{4}+\frac{(y+6)^{2}}{25}=1\)
- \(\frac{(x-3)^{2}}{25}+\frac{(y+2)^{2}}{9}=1\)
- \(\frac{(x+4)^{2}}{4}+\frac{(y-2)^{2}}{9}=1\)
- \(\frac{(x-4)^{2}}{9}+\frac{(y-1)^{2}}{16}=1\)
- Answer
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Exercise \(\PageIndex{18}\) Graph an Ellipse with Center Not at the Origin
In the following exercises, graph each equation by translation.
- \(\frac{(x-3)^{2}}{4}+\frac{(y-7)^{2}}{25}=1\)
- \(\frac{(x+6)^{2}}{16}+\frac{(y+5)^{2}}{4}=1\)
- \(\frac{(x-5)^{2}}{9}+\frac{(y+4)^{2}}{25}=1\)
- \(\frac{(x+5)^{2}}{36}+\frac{(y-3)^{2}}{16}=1\)
- Answer
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Exercise \(\PageIndex{19}\) Graph an Ellipse with Center Not at the Origin
In the following exercises,
- Write the equation in standard form and
- Graph.
- \(25 x^{2}+9 y^{2}-100 x-54 y-44=0\)
- \(4 x^{2}+25 y^{2}+8 x+100 y+4=0\)
- \(4 x^{2}+25 y^{2}-24 x-64=0\)
- \(9 x^{2}+4 y^{2}+56 y+160=0\)
- Answer
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- \(\frac{(x-2)^{2}}{9}+\frac{(y-3)^{2}}{25}=1\)
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- \(\frac{y^{2}}{4}+\frac{(x-3)^{2}}{25}=1\)
Exercise \(\PageIndex{20}\) Graph an Ellipse with Center Not at the Origin
In the following exercises, graph the equation.
- \(x=-2(y-1)^{2}+2\)
- \(x^{2}+y^{2}=49\)
- \((x+5)^{2}+(y+2)^{2}=4\)
- \(y=-x^{2}+8 x-15\)
- \(\frac{(x+3)^{2}}{16}+\frac{(y+1)^{2}}{4}=1\)
- \((x-2)^{2}+(y-3)^{2}=9\)
- \(\frac{x^{2}}{25}+\frac{y^{2}}{36}=1\)
- \(x=4(y+1)^{2}-4\)
- \(x^{2}+y^{2}=64\)
- \(\frac{x^{2}}{9}+\frac{y^{2}}{25}=1\)
- \(y=6 x^{2}+2 x-1\)
- \(\frac{(x-2)^{2}}{9}+\frac{(y+3)^{2}}{25}=1\)
- Answer
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Exercise \(\PageIndex{21}\) Solve Application with Ellipses
1. A planet moves in an elliptical orbit around its sun. The closest the planet gets to the sun is approximately \(10\) AU and the furthest is approximately \(30\) AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the planet.
2. A planet moves in an elliptical orbit around its sun. The closest the planet gets to the sun is approximately \(10\) AU and the furthest is approximately \(70\) AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the planet.
3. A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately \(15\) AU and the furthest is approximately \(85\) AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.
4. A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately \(15\) AU and the furthest is approximately \(95\) AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.
- Answer
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1. \(\frac{x^{2}}{400}+\frac{y^{2}}{300}=1\)
3. \(\frac{x^{2}}{2500}+\frac{y^{2}}{1275}=1\)
Exercise \(\PageIndex{22}\) Writing Exercises
- In your own words, define an ellipse and write the equation of an ellipse centered at the origin in standard form. Draw a sketch of the ellipse labeling the center, vertices and major and minor axes.
- Explain in your own words how to get the axes from the equation in standard form.
- Compare and contrast the graphs of the equations \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) and \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\).
- Explain in your own words, the difference between a vertex and a focus of the ellipse.
- Answer
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1. Answers may vary
3. Answers may vary
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. What does this checklist tell you about your mastery of this section? What steps will you take to improve?


29.5: The Cosmic Microwave Background
By the end of this section, you will be able to:
- Explain why we can observe the afterglow of the hot, early universe
- Discuss the properties of this afterglow as we see it today, including its average temperature and the size of its temperature fluctuations
- Describe open, flat, and curved universes and explain which type of universe is supported by observations
- Summarize our current knowledge of the basic properties of the universe including its age and contents
The description of the first few minutes of the universe is based on theoretical calculations. It is crucial, however, that a scientific theory should be testable. What predictions does it make? And do observations show those predictions to be accurate? One success of the theory of the first few minutes of the universe is the correct prediction of the amount of helium in the universe.
Another prediction is that a significant milestone in the history of the universe occurred about 380,000 years after the Big Bang. Scientists have directly observed what the universe was like at this early stage, and these observations offer some of the strongest support for the Big Bang theory. To find out what this milestone was, let’s look at what theory tells us about what happened during the first few hundred thousand years after the Big Bang.
The fusion of helium and lithium was completed when the universe was about 4 minutes old. The universe then continued to resemble the interior of a star in some ways for a few hundred thousand years more. It remained hot and opaque, with radiation being scattered from one particle to another. It was still too hot for electrons to “settle down” and become associated with a particular nucleus; such free electrons are especially effective at scattering photons, thus ensuring that no radiation ever got very far in the early universe without having its path changed. In a way, the universe was like an enormous crowd right after a popular concert; if you get separated from a friend, even if he is wearing a flashing button, it is impossible to see through the dense crowd to spot him. Only after the crowd clears is there a path for the light from his button to reach you.
The Universe Becomes Transparent
Not until a few hundred thousand years after the Big Bang, when the temperature had dropped to about 3000 K and the density of atomic nuclei to about 1000 per cubic centimeter, did the electrons and nuclei manage to combine to form stable atoms of hydrogen and helium (Figure 29.14). With no free electrons to scatter photons, the universe became transparent for the first time in cosmic history. From this point on, matter and radiation interacted much less frequently; we say that they decoupled from each other and evolved separately. Suddenly, electromagnetic radiation could really travel, and it has been traveling through the universe ever since.
Discovery of the Cosmic Background Radiation
If the model of the universe described in the previous section is correct, then—as we look far outward in the universe and thus far back in time—the first “afterglow” of the hot, early universe should still be detectable. Observations of it would be very strong evidence that our theoretical calculations about how the universe evolved are correct. As we shall see, we have indeed detected the radiation emitted at this photon decoupling time , when radiation began to stream freely through the universe without interacting with matter (Figure 29.15).
The detection of this afterglow was initially an accident. In the late 1940s, Ralph Alpher and Robert Herman , working with George Gamow , realized that just before the universe became transparent, it must have been radiating like a blackbody at a temperature of about 3000 K—the temperature at which hydrogen atoms could begin to form. If we could have seen that radiation just after neutral atoms formed, it would have resembled radiation from a reddish star. It was as if a giant fireball filled the whole universe.
But that was nearly 14 billion years ago, and, in the meantime, the scale of the universe has increased a thousand fold. This expansion has increased the wavelength of the radiation by a factor of 1000 (see Figure 29.7). According to Wien’s law, which relates wavelength and temperature, the expansion has correspondingly lowered the temperature by a factor of 1000 (see the chapter on Radiation and Spectra). The cosmic background behaves like a blackbody and should therefore have a spectrum that obeys Wien’s Law.
Alpher and Herman predicted that the glow from the fireball should now be at radio wavelengths and should resemble the radiation from a blackbody at a temperature only a few degrees above absolute zero. Since the fireball was everywhere throughout the universe, the radiation left over from it should also be everywhere. If our eyes were sensitive to radio wavelengths, the whole sky would appear to glow very faintly. However, our eyes can’t see at these wavelengths, and at the time Alpher and Herman made their prediction, there were no instruments that could detect the glow. Over the years, their prediction was forgotten.
In the mid-1960s, in Holmdel, New Jersey, Arno Penzias and Robert Wilson of AT&T’s Bell Laboratories had built a delicate microwave antenna (Figure 29.16) to measure astronomical sources, including supernova remnants like Cassiopeia A (see the chapter on The Death of Stars). They were plagued with some unexpected background noise, just like faint static on a radio, which they could not get rid of. The puzzling thing about this radiation was that it seemed to be coming from all directions at once. This is very unusual in astronomy: after all, most radiation has a specific direction where it is strongest—the direction of the Sun, or a supernova remnant, or the disk of the Milky Way, for example.
Penzias and Wilson at first thought that any radiation appearing to come from all directions must originate from inside their telescope, so they took everything apart to look for the source of the noise. They even found that some pigeons had roosted inside the big horn-shaped antenna and had left (as Penzias delicately put it) “a layer of white, sticky, dielectric substance coating the inside of the antenna.” However, nothing the scientists did could reduce the background radiation to zero, and they reluctantly came to accept that it must be real, and it must be coming from space.
Penzias and Wilson were not cosmologists, but as they began to discuss their puzzling discovery with other scientists, they were quickly put in touch with a group of astronomers and physicists at Princeton University (a short drive away). These astronomers had—as it happened—been redoing the calculations of Alpher and Herman from the 1940s and also realized that the radiation from the decoupling time should be detectable as a faint afterglow of radio waves. The different calculations of what the observed temperature would be for this cosmic microwave background (CMB) 2 were uncertain, but all predicted less than 40 K.
Penzias and Wilson found the distribution of intensity at different radio wavelengths to correspond to a temperature of 3.5 K. This is very cold—closer to absolute zero than most other astronomical measurements—and a testament to how much space (and the waves within it) has stretched. Their measurements have been repeated with better instruments, which give us a reading of 2.73 K. So Penzias and Wilson came very close. Rounding this value, scientists often refer to “the 3-degree microwave background.”
Many other experiments on Earth and in space soon confirmed the discovery by Penzias and Wilson: The radiation was indeed coming from all directions (it was isotropic) and matched the predictions of the Big Bang theory with remarkable precision. Penzias and Wilson had inadvertently observed the glow from the primeval fireball. They received the Nobel Prize for their work in 1978. And just before his death in 1966, Lemaître learned that his “vanished brilliance” had been discovered and confirmed.
You may enjoy watching Three Degrees , a 26-minute video from Bell Labs about Penzias and Wilson’s discovery of the cosmic background radiation (with interesting historical footage).
Properties of the Cosmic Microwave Background
One issue that worried astronomers is that Penzias and Wilson were measuring the background radiation filling space through Earth’s atmosphere. What if that atmosphere is a source of radio waves or somehow affected their measurements? It would be better to measure something this important from space.
The first accurate measurements of the CMB were made with a satellite orbiting Earth. Named the Cosmic Background Explorer ( COBE ), it was launched by NASA in November 1989. The data it received quickly showed that the CMB closely matches that expected from a blackbody with a temperature of 2.73 K (Figure 29.17). This is exactly the result expected if the CMB was indeed redshifted radiation emitted by a hot gas that filled all of space shortly after the universe began.
The first important conclusion from measurements of the CMB, therefore, is that the universe we have today has indeed evolved from a hot, uniform state. This observation also provides direct support for the general idea that we live in an evolving universe, since the universe is cooler today than it was in the beginning.
Small Differences in the CMB
It was known even before the launch of COBE that the CMB is extremely isotropic . In fact, its uniformity in every direction is one of the best confirmations of the cosmological principle— that the universe is homogenous and isotropic.
According to our theories, however, the temperature could not have been perfectly uniform when the CMB was emitted. After all, the CMB is radiation that was scattered from the particles in the universe at the time of decoupling. If the radiation were completely smooth, then all those particles must have been distributed through space absolutely evenly. Yet it is those particles that have become all the galaxies and stars (and astronomy students) that now inhabit the cosmos. Had the particles been completely smoothly distributed, they could not have formed all the large-scale structures now present in the universe—the clusters and superclusters of galaxies discussed in the last few chapters.
The early universe must have had tiny density fluctuations from which such structures could evolve. Regions of higher-than-average density would have attracted additional matter and eventually grown into the galaxies and clusters that we see today. It turns out that these denser regions would appear to us to be colder spots, that is, they would have lower-than-average temperatures.
The reason that temperature and density are related can be explained this way. At the time of decoupling, photons in a slightly denser portion of space had to expend some of their energy to escape the gravitational force exerted by the surrounding gas. In losing energy, the photons became slightly colder than the overall average temperature at the time of decoupling. Vice versa, photons that were located in a slightly less dense portion of space lost less energy upon leaving it than other photons, thus appearing slightly hotter than average. Therefore, if the seeds of present-day galaxies existed at the time that the CMB was emitted, we should see some slight variations in the CMB temperature as we look in different directions in the sky.
Scientists working with the data from the COBE satellite did indeed detect very subtle temperature differences—about 1 part in 100,000—in the CMB. The regions of lower-than-average temperature come in a variety of sizes, but even the smallest of the colder areas detected by COBE is far too large to be the precursor of an individual galaxy, or even a supercluster of galaxies. This is because the COBE instrument had “blurry vision” (poor resolution) and could only measure large patches of the sky. We needed instruments with “sharper vision.”
The most detailed measurements of the CMB have been obtained by two satellites launched more recently than COBE. The results from the first of these satellites, the Wilkinson Microwave Anisotropy Probe ( WMAP ) spacecraft, were published in 2003. In 2015, measurements from the Planck satellite extended the WMAP measurements to even-higher spatial resolution and lower noise (Figure 29.18).
Theoretical calculations show that the sizes of the hot and cold spots in the CMB depend on the geometry of the universe and hence on its total density. (It’s not at all obvious that it should do so, and it takes some pretty fancy calculations—way beyond the level of our text—to make the connection, but having such a dependence is very useful.) The total density we are discussing here includes both the amount of mass in the universe and the mass equivalent of the dark energy. That is, we must add together mass and energy: ordinary matter, dark matter, and the dark energy that is speeding up the expansion.
To see why this works, remember (from the chapter on Black Holes and Curved Spacetime) that with his theory of general relativity , Einstein showed that matter can curve space and that the amount of curvature depends on the amount of matter present. Therefore, the total amount of matter in the universe (including dark matter and the equivalent matter contribution by dark energy), determines the overall geometry of space. Just like the geometry of space around a black hole has a curvature to it, so the entire universe may have a curvature. Let’s take a look at the possibilities (Figure 29.19).
If the density of matter is higher than the critical density, the universe will eventually collapse. In such a closed universe, two initially parallel rays of light will eventually meet. This kind of geometry is referred to as spherical geometry. If the density of matter is less than critical, the universe will expand forever. Two initially parallel rays of light will diverge, and this is referred to as hyperbolic geometry. In a critical-density universe, two parallel light rays never meet, and the expansion comes to a halt only at some time infinitely far in the future. We refer to this as a flat universe , and the kind of Euclidean geometry you learned in high school applies in this type of universe.
If the density of the universe is equal to the critical density, then the hot and cold spots in the CMB should typically be about a degree in size. If the density is greater than critical, then the typical sizes will be larger than one degree. If the universe has a density less than critical, then the structures will appear smaller. In Figure 29.20, you can see the differences easily. WMAP and Planck observations of the CMB confirmed earlier experiments that we do indeed live in a flat, critical-density universe.
Key numbers from an analysis of the Planck data give us the best values currently available for some of the basic properties of the universe:
- Age of universe: billion years (Note: That means we know the age of the universe to within 38 million years. Amazing!)
- Hubble constant: kilometers/second/million parsecs (in the units we’ve been using, 20.65 kilometers/second/million light-years)
- Fraction of universe’s content that is “dark energy”:
- Fraction of the universe’s content that is matter:
Note that this value for the Hubble constant is slightly smaller than the value of 70 kilometers/second/million parsecs that we have adopted in this book. In fact, the value derived from measurements of redshifts is 73 kilometers/second/million parsecs. So precise is modern cosmology these days that scientists are working hard to resolve this discrepancy. The fact that the difference between these two independent measurements is so small is actually a remarkable achievement. Only a few decades ago, astronomers were arguing about whether the Hubble constant was around 50 kilometers/second/million parsecs or 100 kilometers/second/million parsecs.
Analysis of Planck data also shows that ordinary matter (mainly protons and neutrons) makes up 4.9% of the total density. Dark matter plus normal matter add up to 31.5% of the total density. Dark energy contributes the remaining 68.5%. The age of the universe at decoupling—that is, when the CMB was emitted—was 380,000 years.
Perhaps the most surprising result from the high-precision measurements by WMAP and the even higher-precision measurements from Planck is that there were no surprises. The model of cosmology with ordinary matter at about 5%, dark matter at about 25%, and dark energy about 70% has survived since the late 1990s when cosmologists were forced in that direction by the supernovae data. In other words, the very strange universe that we have been describing, with only about 5% of its contents being made up of the kinds of matter we are familiar with here on Earth, really seems to be the universe we live in.
After the CMB was emitted, the universe continued to expand and cool off. By 400 to 500 million years after the Big Bang, the very first stars and galaxies had already formed. Deep in the interiors of stars, matter was reheated, nuclear reactions were ignited, and the more gradual synthesis of the heavier elements that we have discussed throughout this book began.
We conclude this quick tour of our model of the early universe with a reminder. You must not think of the Big Bang as a localized explosion in space , like an exploding superstar. There were no boundaries and there was no single site where the explosion happened. It was an explosion of space (and time and matter and energy) that happened everywhere in the universe. All matter and energy that exist today, including the particles of which you are made, came from the Big Bang.


2.7: Reading- Physical Characteristics of Minerals
What are Minerals?
All rocks except obsidian and coal are made of minerals. (Obsidian is a volcanic rock made of glass and coal is made of organic carbon.) Most rocks contain several minerals in a mixture characteristic of the particular rock type. When identifying a rock you must first identify the individual minerals that make up that rock.
Minerals are naturally occurring, inorganic solids with a definite chemical composition and a crystal lattice structure. Although thousands of minerals in the earth have been identified, just ten minerals make up most of the volume of the earth’s crust—plagioclase, quartz, orthoclase, amphibole, pyroxene, olivine, calcite, biotite, garnet, and clay.
Together, the chemical formula (the types and proportions of the chemical elements) and the crystal lattice (the geometry of how the atoms are arranged and bonded together) determine the physical properties of minerals.
The chemical formula and crystal lattice of a mineral can only be determined in a laboratory, but by examining a mineral and determining several of its physical properties, you can identify the mineral. First, you need to become familiar with the physical properties of minerals and how to recognize them.
Minerals can be identified by their physical characteristics. The physical properties of minerals are related to their chemical composition and bonding. Some characteristics, such as a mineral’s hardness, are more useful for mineral identification. Color is readily observable and certainly obvious, but it is usually less reliable than other physical properties.
How are Minerals Identified?
Mineralogists are scientists who study minerals. One of the things mineralogists must do is identify and categorize minerals. While a mineralogist might use a high-powered microscope to identify some minerals, most are recognizable using physical properties.
Check out the mineral in figure 1. What is the mineral’s color? What is its shape? Are the individual crystals shiny or dull? Are there lines (striations) running across the minerals?
Color, Streak, and Luster
Diamonds are popular gemstones because the way they reflect light makes them very sparkly. Turquoise is prized for its striking greenish-blue color. Notice that specific terms are being used to describe the appearance of minerals.
Color
Color is often useful, but should not be relied upon. Different minerals may be the same color. Real gold, as seen in figure 2, is very similar in color to the pyrite in figure 1.
Additionally, Some minerals come in many different colors. Quartz, for example, may be clear, white, gray, brown, yellow, pink, red, or orange. So color can help, but do not rely on color as the determining property. Figure 3 shows one sample of quartz that is colorless and another quartz that is purple. A tiny amount of iron makes the quartz purple. Many minerals are colored by chemical impurities.
Luster
Luster describes the reflection of light off a mineral’s surface. Mineralogists have special terms to describe luster. One simple way to classify luster is based on whether the mineral is metallic or non-metallic. Minerals that are opaque and shiny, such as pyrite, have a metallic luster. Minerals such as quartz have a non-metallic luster.
Luster is how the surface of a mineral reflects light. It is not the same thing as color, so it crucial to distinguish luster from color. For example, a mineral described as “shiny yellow” is being described in terms of luster (“shiny”) and color (“yellow”), which are two different physical properties. Standard names for luster include metallic, glassy, pearly, silky, greasy, and dull. It is often useful to first determine if a mineral has a metallic luster. A metallic luster means shiny like polished metal. For example cleaned polished pieces of chrome, steel, titanium, copper, and brass all exhibit metallic luster as do many other minerals. Of the nonmetallic lusters, glassy is the most common and means the surface of the mineral reflects light like glass. Pearly luster is important in identifying the feldspars, which are the most common type of mineral. Pearly luster refers to a subtle irridescence or color play in the reflected light, same way pearls reflect light. Silky means relecting light with a silk-like sheen. Greasy luster looks similar to the luster of solidified bacon grease. Minerals with dull luster reflect very little light. Identifying luster takes a little practice. Remember to distinguish luster from color.
Different types of non-metallic luster are described in table 1.
| Table 1. Six types of non-metallic luster. | |
|---|---|
| Luster | Appearance |
| Adamantine | Sparkly |
| Earthy | Dull, clay-like |
| Pearly | Pearl-like |
| Resinous | Like resins, such as tree sap |
| Silky | Soft-looking with long fibers |
| Vitreous | Glassy |
Can you match the minerals in figure 4 with the correct luster from table 1?
Streak
Streak is the color of a mineral’s powder. Streak is a more reliable property than color because streak does not vary. Minerals that are the same color may have a different colored streak. Many minerals, such as the quartz in the figure 3, do not have streak.
To check streak, scrape the mineral across an unglazed porcelain plate (Figure 5). Yellow-gold pyrite has a blackish streak, another indicator that pyrite is not gold, which has a golden yellow streak.
Specific Gravity
Density describes how much matter is in a certain amount of space: density = mass/volume.
Mass is a measure of the amount of matter in an object. The amount of space an object takes up is described by its volume. The density of an object depends on its mass and its volume. For example, the water in a drinking glass has the same density as the water in the same volume of a swimming pool.
The specific gravity of a substance compares its density to that of water. Substances that are more dense have higher specific gravity.
Hardness
Hardness is the strength with which a mineral resists its surface being scraped or punctured. In working with hand samples without specialized tools, mineral hardness is specified by the Mohs hardness scale. The Mohs hardness scale is based 10 reference minerals, from talc the softest (Mohs hardness of 1), to diamond the hardest (Mohs hardness of 10). It is a relative, or nonlinear, scale. A hardness of 2.5 simply means that the mineral is harder than gypsum (Mohs hardness of 2) and softer than calcite (Mohs hardness of 3). To compare the hardness of two minerals see which mineral scratches the surface of the other.
| Table 2. Mohs Hardness Scale | ||
|---|---|---|
| Hardness | Index Minerals | Common Objects |
| 1 | talc | |
| 2 | gypsum | 2.5-fingernail |
| 3 | calcite | 3.5-pure, untarnished copper |
| 4 | fluorite | |
| 5 | feldspar | 5 to 5.5-stainless steel |
| 5.5 to 6-glass | ||
| 6 | apatite | 6 to 6.5-hard steel file |
| 7 | quartz | |
| 8 | topaz | |
| 9 | corundum | |
| 10 | diamond |
With a Mohs scale, anyone can test an unknown mineral for its hardness. Imagine you have an unknown mineral. You find that it can scratch fluorite or even feldspar, but apatite scratches it. You know then that the mineral’s hardness is between 5 and 6. Note that no other mineral can scratch diamond.
Cleavage and Fracture
Breaking a mineral breaks its chemical bonds. Since some bonds are weaker than other bonds, each type of mineral is likely to break where the bonds between the atoms are weaker. For that reason, minerals break apart in characteristic ways.
Cleavage
Cleavage is the tendency of a mineral to break along certain planes to make smooth surfaces. Halite breaks between layers of sodium and chlorine to form cubes with smooth surfaces (figure 6).
A mineral that naturally breaks into perfectly flat surfaces is exhibiting cleavage. Not all minerals have cleavage. A cleavage represents a direction of weakness in the crystal lattice. Cleavage surfaces can be distinguished by how they consistently reflect light, as if polished, smooth, and even. The cleavage properties of a mineral are described in terms of the number of cleavages and, if more than one cleavage, the angles between the cleavages. The number of cleavages is the number or directions in which the mineral cleaves. A mineral may exhibit 100 cleavage surfaces parallel to each other. Those represent a single cleavage because the surfaces are all oriented in the same diretion. The possible number of cleavages a mineral may have are 1,2,3,4, or 6. If more than 1 cleavage is present, and a device for measuring angles is not available, simply state whether the cleavages intersect at 90° or not 90°.
To see mineral cleavage, hold the mineral up beneath a strong light and move it around, move it around some more, to see how the different sides reflect light. A cleavage direction will show up as a smooth, shiny, evenly bright sheen of light reflected by one set of parallel surfaces on the mineral.
Mica has cleavage in one direction and forms sheets (figure 7).
Minerals can cleave into polygons. Fluorite forms octahedrons (figure 8).
One reason gemstones are beautiful is that the cleavage planes make an attractive crystal shape with smooth faces.
Fracture
Fracture is a break in a mineral that is not along a cleavage plane. Fracture is not always the same in the same mineral because fracture is not determined by the structure of the mineral.
Minerals may have characteristic fractures (figure 9). Metals usually fracture into jagged edges. If a mineral splinters like wood, it may be fibrous. Some minerals, such as quartz, form smooth curved surfaces when they fracture.
All minerals have fracture. Fracture is breakage, which occurs in directions that are not cleavage directions. Some minerals, such as quartz, have no cleavage whatsoever. When a mineral with no cleavage is broken apart by a hammer, it fractures in all directions. Quartz is said to exhibit conchoidal fracture. Conchoidal fracture is the way a thick piece of glass breaks with concentric, curving ridges on the broken surfaces. However, some quartz crystals have so many flaws that instead of exhibiting conchoidal fracture they simply exhibit irregular fracture. Irregular fracture is a standard term for fractures that do not exhibit any of the qualities of the other fracture types. In introductory geology, the key fracture types to remember are irregular, which most minerals exhibit, and conchoidal, seen in quartz.
Crystal Shape
All minerals are crystalline, but only some have the opportunity to exhibit the shapes of their crystals, their crystal forms. Many minerals in an introductory geology lab do not exhibit their crystal form. If a mineral has space while it grows, it may form natural crystals, with a crystal shape reflecting the geometry of the mineral’s internal crystal lattice. The shape of a crystal follows the symmetry of its crystal lattice. Quartz, for instance, forms six-sided crystals, showing the hexagonal symmetry of its crystal lattice. There are two complicating factors to remember here: (1) minerals do not always form nice crystals when they grow, and (2) a crystal face is different from a cleavage surface. A crystal face forms during the growth of the mineral. A cleavage surface is formed when the mineral is broken.
Other Identifying Characteristics
There are some properties that only help to distinguish a small number of minerals, or even just a single mineral. An example of such a special property is the effervescent reaction of calcite to a weak solution of hydrochloric acid (5% HCl). Calcite fizzes or effervesces as the HCl solution dissolves it and creates CO 2 gas. Calcite is easy to identify even without testing the reaction to HCl, by its hardness, luster and cleavage.
Another special property is magnetism. This can be tested by seeing if a small magnet responds to the mineral. The most common mineral that is strongly magnetic is the mineral magnetite. A special property that shows up in some sample of plagioclase feldspar is its tendency to exhibit striations on cleavage surfaces. Striations are perfectly straight, fine, parallel lines. Magnification may be required to see striations on plagioclase cleavage surfaces. Other special properties may be encountered on a mineral to mineral basis.
Some minerals have other unique properties, some of which are listed in table 3. Can you name a unique property that would allow you to instantly identify a mineral that’s been described quite a bit in this chapter? (Hint: It is most likely found on your dinner table.)
| Table 3. Some minerals have unusual properties that can be used for identification. | ||
|---|---|---|
| Property | Description | Example of Mineral |
| Fluorescence | Mineral glows under ultraviolet light | Fluorite |
| Magnetism | Mineral is attracted to a magnet | Magnetite |
| Radioactivity | Mineral gives off radiation that can be measured with Geiger counter | Uraninite |
| Reactivity | Bubbles form when mineral is exposed to a weak acid | Calcite |
| Smell | Some minerals have a distinctive smell | Sulfur (smells like rotten eggs) |
| Taste | Some minerals taste salty | Halite |
Contributors and Attributions
- Revision and Adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
- 3.3: Mineral Identification. Provided by : CK-12. Located at : http://www.ck12.org/book/CK-12-Earth-Science-For-High-School/section/3.3/ . License : CC BY-NC: Attribution-NonCommercial
- Basics - Minerals. Authored by : Ralph L. Dawes and Cheryl D. Dawes. Provided by : Wenatchee Valley College. Located at : http://commons.wvc.edu/rdawes/G101OCL/Basics/minerals.html . Project : Geology 101 - Introduction to Physical Geology. License : CC BY: Attribution


24.16: Chapter 16 (Administration of Medications via Other Routes)
Answer Key to Chapter 16 Learning Activities
- Patients suffering from nausea or experiencing difficulty swallowing may not tolerate medications administered via the enteral route. For a patient experiencing pain, appropriate alternate methods of medication administration may be transdermal medication administration. Transdermal medications are beneficial in that they provide a consistent level of drug into the bloodstream during distribution.
- B – when placing a patch, the nurse should press firmly to the skin to ensure adequate adherence, C – gloves are required for patch application and removal, E – date and location of patch application should be promptly documented in the medication administration record (MAR).
Answers to interactive elements are given within the interactive element.


8.4: Health Care Reimbursement Models
As discussed in the previous section, hospitals and health care providers are paid for services provided to individuals by government insurance programs (such as Medicare and Medicaid), private insurance companies, or people using their out-of-pocket funds. Traditionally, health care institutions were paid based on a “fee-for-service” model. For example, if a patient was admitted to a hospital with pneumonia, the hospital billed that individual’s insurance program for the cost of care.
However, as part of a recent national strategy to reduce health care costs, insurance providers have transitioned to “Pay for Performance” reimbursement models that are based on overall agency performance and patient outcomes.
Pay for Performance
Pay for Performance , also known as value-based payment, refers to reimbursement models that attach financial incentives to the performance of health care agencies and providers. Pay for Performance models tie higher reimbursement payments to positive patient outcomes, best practices, and patient satisfaction, thus aligning payment with value and quality. [1] Nurses support higher reimbursement levels to their employers based on their documentation related to nursing care plans and achievement of expected patient outcomes.
There are two Pay for Performance models. The first model rewards hospitals and providers with higher reimbursement payments based on how well they perform on process, quality, and efficiency measures. The second model penalizes hospitals and providers for subpar performance by reducing reimbursement amounts. [2] For example, Medicare no longer reimburses hospitals to treat patients who acquire certain preventable conditions during their hospital stay, such as pressure injuries or urinary tract infections associated with use of catheters. [3]
The Centers for Medicare and Medicaid Services (CMS), spurred by the Affordable Care Act, has led the way in value-based payment with a variety of payment models. CMS is the largest health care funder in the United States with almost 40% of overall health care spending for Medicare and Medicaid. CMS developed three Pay for Performance models that impact hospitals’ reimbursement by Medicare. These models are called the Hospital Value-Based Purchasing Program, the Hospital Readmissions Reduction Program, and the Hospital-Acquired Condition Reduction Program. Private insurers are also committed to performance-based payment models. In 2017 Forbes reported that almost 50% of insurers’ reimbursements were in the form of value-based care models. [4]
Hospital Value-Based Purchasing Program
The Hospital Value-Based Purchasing Program (VBP) was designed to improve health care quality and patient experience by using financial incentives that encourage hospitals to follow established best clinical practices and improve patient satisfaction scores via patient satisfaction surveys. Reimbursement is based on hospital performance on measures divided into four quality domains: safety, clinical care, efficiency and cost reduction, and patient and caregiver-centered experience. [5]
Read more about patient satisfaction surveys .
Hospital Readmissions Reduction Program
The Hospital Readmissions Reduction Program (HRRP) penalizes hospitals with higher rates of patient readmissions compared to other hospitals. HRRP was established by the Affordable Care Act and applies to patients with specific conditions, such as heart attacks, heart failure, pneumonia, chronic obstructive pulmonary disease (COPD), hip or knee replacements, and coronary bypass surgery. Hospitals with poor performance receive a 3% reduction of their Medicare payments. However, it was discovered that hospitals with higher proportions of low-income patients were penalized the most, so Congress passed legislation in 2019 that divided hospitals into groups for comparison based on the socioeconomic status of their patient populations. [6]
Hospital-Acquired Condition Reduction Program
The Hospital-Acquired Condition Reduction Program (HACRP) was established by the Affordable Care Act. This Pay for Performance model reduces payments to hospitals based on poor performance regarding patient safety and hospital-acquired conditions, such as surgical site infections, hip fractures resulting from falls, and pressure injuries. This model has saved Medicare approximately $350 million per year. [7]
The HACRP model measures the incidence of hospital-acquired conditions, including central line-associated bloodstream infections (CLABSI), catheter-associated urinary tract infections (CAUTI), surgical site infections (SSI), Methicillin-Resistant Staphylococcus Aureus (MRSA), and Clostridium Difficile (C. diff). [8] As a result, nurses have seen changes in daily practices based on evidence-based practices related to these conditions. For example, stringent documentation is now required for clients with Foley catheters that indicates continued need and associated infection control measures.
Other CMS Pay for Performance Models
CMS has created other value-based payment programs for agencies other than hospitals, including the End-Stage Renal Disease (ESRD) Quality Initiative Program, the Skilled Nursing Facility Value-Based Program (SNFVBP), the Home Health Value-Based Program (HHVBP), and the Value Modifier (VM) Program. The VM program is aimed at Medicare Part B providers who receive high, average, or low ratings based on quality and cost measurements as compared to peer agencies.
Impacts of Value-Based Payment
Pay for Performance (i.e., value-based payment) stresses quality over quantity of care and allows health care payers to use reimbursement to encourage best clinical practices and promote positive health outcomes. It focuses on transparency by using metrics that are publicly reported, thus incentivizing organizations to protect and strengthen their reputations. In this manner, Pay for Performance models encourage accountability and consumer-informed choice. [9] See Figure 8.8 [10] for an illustration of Pay for Performance.
Pay for Performance models have reduced health care costs and decreased the incidence of poor patient outcomes. For example, 30-day hospital readmission rates have been falling since 2012, indicating HRRP and HACRP are having an impact. [11]
However, there are also disadvantages to value-based payment. As previously discussed, initial research indicated hospitals with higher proportions of low-income patients were being penalized the most, resulting in additional legislation to compare hospital performance in groups based on their clients’ socioeconomic status. Nursing leaders continue to emphasize strategies that further address social determinants of health and promote health equity. [12] Read more about equity and social determinants of health in the following subsection.
Nursing Considerations
Nurses have a direct impact on activities related to quality care and reimbursement rates received by their employer. There are several categories of actions nurses can take to improve quality patient care, reduce costs, and improve reimbursement. By incorporating these actions into their daily care, nurses can help ensure the funding they need to provide quality patient care is received by their employer and resources are allocated appropriately to their patients.
The following categories of actions to improve quality of care are based on the Institute of Medicine (IOM) report To Err Is Human: Building a Safer Health Care System and Crossing the Quality Chasm [13] :
- Effectiveness and Efficiency: Nurses support their institution’s effectiveness and efficiency with individualized nursing care planning, good documentation, and care coordination. With accurate and timely documentation and care coordination, there is reduced care duplication and waste. Coordinating care also helps to reduce the risk of hospital readmissions.
- Timeliness: Nurses positively impact timeliness by prioritizing and delegating care. This helps reduce patient wait times and delays in care.
Read more about these concepts in the “ Delegation and Supervision ” and “ Prioritization ” chapters in this book.
- Safety: Nurses pay attention to their patients’ changing conditions and effectively communicate these changes with appropriate health care team members. They take any concerns about client care up the chain of command until their concerns are resolved.
- Patient-Centered Care: Nurses support this quality measure by ensuring nursing care plans are individualized for each patient. Effective care plans can improve patient compliance, resulting in improved patient outcomes.
- Evidence-Based Practice: Nurses provide care based on evidence-based practice. Evidence-Based Practice (EBP) is defined by the American Nurses Association as, “A lifelong problem-solving approach that integrates the best evidence from well-designed research studies and evidence-based theories; clinical expertise and evidence from assessment of the health care consumer’s history and condition, as well as health care resources; and patient, family, group, community, and population preferences and values.” [14] EBP is a component of Scholarly Inquiry , one of the ANA’s Standards of Professional Practice. Nurses’ implementation of EBP ensures proper resources are allocated to the appropriate clients. EBP promotes safe, efficient, and effective health care. [15] , [16]
Read more information about EBP in the “ Quality and Evidence-Based Practice ” chapter of this book.
- Equity: Health care institutions care for all members of their community regardless of client demographics and their associated social determinants of health (SDOH) . SDOH are conditions in the places where people live, learn, work, and play that affect a wide range of health risks and outcomes. Health disparities in communities with poor SDOH have been consistently documented in reports by the Agency for Healthcare Research and Quality (AHRQ). [17]
Nurses address negative determinants of health by advocating for interventions that reduce health disparities and promote the delivery of equitable health care resources. The term health disparities describes the differences in health outcomes that result from SDOH. Advocating for resources that enhance quality of life can significantly influence a community’s health outcomes. Examples of resources that promote health include safe and affordable housing, access to education, public safety, availability of healthy foods, local emergency/health services, and environments free of life-threatening toxins.
A related term is health care disparity that refers to differences in access to health care and insurance coverage. Health disparities and health care disparities can lead to decreased quality of life, increased personal costs, and lower life expectancy. More broadly, these disparities also translate to greater societal costs, such as the financial burden of uncontrolled chronic illnesses. An example of nurses addressing health care disparities are nurse practitioners providing health care according to their scope of practice to underserved populations in rural communities.
The ANA promotes nurse advocacy in workplaces and local communities. There are many ways nurses can promote health and wellness within their communities through a variety of advocacy programs at the federal, state, and community level. [18] Read more about advocacy and reducing health disparities in the following boxes.
Read more about ANA Policy and Advocacy .
Read more information in the “ Advocacy ” chapter of this book.
Read more about addressing health disparities in the “ Diverse Patients ” chapter in Open RN Nursing Fundamentals .
- Pay for Performance. (2018). NEJM Catalyst . https://catalyst.nejm.org/doi/full/10.1056/CAT.18.0245 ↵
- Pay for Performance. (2018). NEJM Catalyst . https://catalyst.nejm.org/doi/full/10.1056/CAT.18.0245 ↵
- James, J. (2012, October 11). Pay-for-performance. Health Affairs. https://www.healthaffairs.org/do/10.1377/hpb20121011.90233/full/ ↵
- Pay for Performance. (2018). NEJM Catalyst . https://catalyst.nejm.org/doi/full/10.1056/CAT.18.0245 ↵
- Pay for Performance. (2018). NEJM Catalyst . https://catalyst.nejm.org/doi/full/10.1056/CAT.18.0245 ↵
- Pay for Performance. (2018). NEJM Catalyst . https://catalyst.nejm.org/doi/full/10.1056/CAT.18.0245 ↵
- Pay for Performance. (2018). NEJM Catalyst . https://catalyst.nejm.org/doi/full/10.1056/CAT.18.0245 ↵
- Pay for Performance. (2018). NEJM Catalyst . https://catalyst.nejm.org/doi/full/10.1056/CAT.18.0245 ↵
- Pay for Performance. (2018). NEJM Catalyst . https://catalyst.nejm.org/doi/full/10.1056/CAT.18.0245 ↵
- “Pillars of Pay Performance.png” by Meredith Pomietlo for Chippewa Valley Technical College is licensed under CC BY 4.0 ↵
- Pay for Performance. (2018). NEJM Catalyst . https://catalyst.nejm.org/doi/full/10.1056/CAT.18.0245 ↵
- Pay for Performance. (2018). NEJM Catalyst . https://catalyst.nejm.org/doi/full/10.1056/CAT.18.0245 ↵
- Avalere Health LLC. (2015). Optimal nurse staffing to improve quality of care and patient outcomes: Executive summary [White paper]. https://cdn2.hubspot.net/hubfs/4850206/ANA/NurseStaffingWhitePaper_Final.pdf?__hstc=53609399.b25284991e95c5d4f1c55a49e826489f.1612299494138.1628353446072.1628356381954.16&__hssc=53609399.3.1628356381954&__hsfp=1865500357 ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- Stevens, K. (2013, May 31). The impact of evidence-based practice in nursing and the next big ideas. OJIN: The Online Journal of Issues in Nursing, 18 (2). https://ojin.nursingworld.org/MainMenuCategories/ANAMarketplace/ANAPeriodicals/OJIN/TableofContents/Vol-18-2013/No2-May-2013/Impact-of-Evidence-Based-Practice.html ↵
- Centers for Disease Control and Prevention. (2020, May 6). Social determinants of health: Know what affects health . https://www.cdc.gov/socialdeterminants/index.htm ↵
- Agency for Healthcare Research and Quality. (2021, June). 2019 national healthcare quality and disparities report. https://www.ahrq.gov/research/findings/nhqrdr/nhqdr19/index.html ↵


1.1.3: Healthy People
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We want people to be healthy. A healthy population is a nations greatest resource, it means a productive population, a prosperous population, and a healthy economy. Your individual commitment to striving for personal wellness may contribute to our national goals for a healthy population!
Every decade since 1980, the U.S. Department of Health and Human Services (HHS) has published healthy goals for our U.S. population called the Healthy People initiative. Healthy People 2030 is the 5th iteration of the Healthy People initiative. The Healthy People initiative is designed to guide national health promotion and disease prevention efforts to improve the health of the nation.
The 2030 broad goals for this decade include the following:
- Attain healthy, thriving lives and well-being, free of preventable disease, disability, injury and premature death.
- Eliminate health disparities, achieve health equity, and attain health literacy to improve the health and well-being of all.
- Create social, physical, and economic environments that promote attaining full potential for health and well-being for all.
- Promote healthy development, healthy behaviors and well-being across all life stages.
- Engage leadership, key constituents, and the public across multiple sectors to take action and design policies that improve the health and well-being of all.
The five broad goals are further divided into over 400 objectives: 355 core objectives, 115 developmental objectives and 40 newly added research objectives. The science-based objectives include targets to monitor progress and motivate action. The Healthy People 2030 objectives were carefully chosen based on national data. The goal is to work together as a nation to achieve the achieve the objectives and goals to hopefully improve health and well-being nationwide.
Challenge: Healthy People Goals and You
How might you benefit from the work being done to achieve the Healthy People goals?
Go to the Healthy People website and browse the Healthy People Objectives
Your challenge is to find an objective that either impacts you or a loved one, or is something that you have been curious about.
Click on the objective and begin by reading the overview to gain a better understanding of what they are hoping to achieve and then navigate to the menu item titled Healthy People In Action to see how communities across the nation are working to achieve the goal. You can then review the Evidence Based Resources to see if you can locate helpful evidence-based resources to achieve the objective.
Health Literacy
The internet brought us the ability to have a wealth of information at our fingertips.
With so much information available to us, it is imperative that we understand how to critically evaluate information we see or hear. The need to be critical consumers of information is reflected in the Healthy People broad goal to “eliminate health disparities, achieve health equity, and attain health literacy to improve the health and well-being of all.” We need to increase our health literacy.
The CDC defines personal health literacy as the degree to which individuals have the ability to find, understand, and use information and services to inform health-related decisions and actions for themselves and others [7] . We must be critical consumers of information which starts with evaluating the source of the information.
Evaluating and Finding Health Information
One strategy for assessing a website is to use the CRAAP [8] method which stands for Currency, Relevance, Authority, Accuracy, and Purpose.
As you read information critically, ask yourself the following questions:
Currency: The timeliness of the information.
- When was the information published or posted?
- Has the information been revised or updated?
- Does your topic of interest require current information, or will older sources work as well?
- Are the links functional?
Relevance: The importance of the information for your needs.
- Does the information relate to your topic or answer your question?
- Who is the intended audience?
- Is the information at an appropriate level (i.e. not too elementary or advanced for your needs)?
- Have you looked at a variety of sources before determining this is one you will use?
- Would you be comfortable citing this source in a research paper?
Authority: The source of the information.
- Who is the author/publisher/source/sponsor?
- What are the author’s credentials or organizational affiliations?
- Is the author qualified to write on the topic?
- Is there contact information, such as a publisher or email address?
- Does the URL reveal anything about the author or source? examples: .com .edu .gov .org .net
Accuracy: The reliability, truthfulness and correctness of the content.
- Where does the information come from?
- Is the information supported by evidence?
- Has the information been reviewed or refereed?
- Can you verify any of the information in another source or from personal knowledge?
- Does the language or tone seem unbiased and free of emotion?
- Are there spelling, grammar or typographical errors?
Purpose: The reason the information exists.
- What is the purpose of the information? Is it to inform, teach, sell, entertain or persuade?
- Do the authors/sponsors make their intentions or purpose clear?
- Is the information fact, opinion or propaganda?
- Does the point of view appear objective and impartial?
- Are there political, ideological, cultural, religious, institutional or personal biases?
Examples of Trusted Source of Health Information
The following is a short list of websites that are most likely to pass the CRAAP test for health information:
- The Center for Disease Control and Prevention
- World Health Organization
- PubMed Central (PMC) is a free full-text archive of biomedical and life sciences journal literature at the U.S. National Institutes of Health’s National Library of Medicine (NIH/NLM).
- MyHealthFinder is a prevention and wellness resource that includes evidence-based health information in English and Spanish that’s actionable and easy to use.
- U.S. Department of Health and Human Services


23.5: The Han Dynasty
Philosophy and Art of the Han Dynasty
Spanning more than four centuries, the Han Dynasty period is considered a golden age in Chinese history. This period was strongly influenced by Confucianism.
Discuss the influence of Han philosophy and art on this golden age in Chinese history
Key Points
- The Han Dynasty was an imperial dynasty of China, preceded by the Qin Dynasty (221–207 BCE) and succeeded by the Three Kingdoms (220–280 CE).
- The Dynasty is separated into two periods: the Western Han (206 BCE – 9 CE) and the Eastern Han (25–220 CE).
- The early Western Han court simultaneously accepted the philosophical teachings of Legalism , Huang-Lao Daoism, and Confucianism in making state decisions and shaping government policy.
- However, the Han court under Emperor Wu gave Confucianism exclusive patronage . In 136 BCE, he abolished all academic chairs not associated with the Confucian Five Classics and encouraged officials to receive a Confucian-based education at the Imperial University that established in 124 BCE.
- Han Dynasty poetry was dominated by the fu genre , intermediary pieces between poetry and prose in which a place, object, feeling, or other subject is described in detail and from as many angles as possible.
Key Terms
- Confucianism : An ethical and philosophical system developed from the teachings of the Chinese philosopher by the same name (551–479 BCE). It emphasizes the ethics of filial piety, harmonious relationships, ritual, and righteousness.
- Legalism : A utilitarian political and realist reform-oriented philosophy meant to strengthen government and reinforce adherence to the law, stressing a strict system of punishments and rewards to maintain law and order.
Overview: The Han Dynasty
The Han Dynasty was an imperial dynasty of China, preceded by the Qin Dynasty (221–207 BCE) and succeeded by the Three Kingdoms (220–280 CE). It was founded by the rebel leader Liu Bang, known posthumously as Emperor Gaozu of Han. It was briefly interrupted by the Xin Dynasty (9–23 CE) of the former regent Wang Mang, an interim that separates the Han into two periods: the Western Han (206 BCE – 9 CE) and the Eastern Han (25–220 CE). Spanning more than four centuries, the Han Dynasty is considered a golden age in Chinese history. To this day, China’s majority ethnic group refers to itself as the “Han people,” and Chinese characters are referred to as “Han characters”.
Han Philosophy
The early Western Han court simultaneously accepted the philosophical teachings of Legalism, Huang-Lao Daoism, and Confucianism in making state decisions and shaping government policy. However, the Han court under Emperor Wu gave Confucianism exclusive patronage.
Confucianism
Confucianism is an ethical and philosophical system developed from the teachings of the Chinese philosopher Confucious (551–479 BCE). Confucianism originated as an “ethical-sociopolitical teaching” during the Spring and Autumn Periods, but during the Han Dynasty it developed metaphysical and cosmological elements. At the core of Confucian ethics were the virtues of filial piety, harmonious relationships, ritual , and righteousness.
Unlike the original ideology espoused by Confucius , Han Confucianism in Emperor Wu’s reign was the creation of Dong Zhongshu (179–104 BCE). Dong was a scholar and minor official who aggregated the ethical Confucian ideas of ritual, filial piety, and harmonious relationships with the five phases (the natural cycles that governed Heaven, Earth, and Man) and yin-yang cosmologies. Dong’s synthesis justified the imperial system of government within the natural order of the universe.
Education
In 136 BCE, Emperor Wu abolished all academic chairs (boshi 博) not dealing with the Confucian Five Classics and encouraged nominees for office to receive a Confucian-based education at the Imperial University that he established in 124 BCE. The Imperial University grew in importance as the student body grew to over 30,000 by the 2 nd century CE. A Confucian-based education was also made available at commander-level schools, and private schools opened in small towns where teachers earned respectable incomes from tuition payments.
Notable Texts
Some important texts were created and studied by scholars during this time. Philosophical works written by Yang Xiong (53 BCE – 18 CE), Huan Tan (43 BCE – 28 CE), Wang Chong (27–100 CE), and Wang Fu (78–163 CE) questioned whether human nature was innately good or evil and posed challenges to Dong’s universal order. The Records of the Grand Historian by Sima Tan (d. 110 BCE) and his son Sima Qian (145–86 BCE) established the standard model for all of imperial China’s Standard Histories, such as the Book of Han written by Ban Biao (3–54 CE), his son Ban Gu (32–92 CE), and his daughter Ban Zhao (45–116 CE). There were dictionaries such as the Shuowen Jiezi by Xu Shen (c. 58 – c. 147 CE) and the Fangyan by Yang Xiong. Biographies on important figures were written by various gentrymen.
Han dynasty poetry was dominated by the fu genre, which achieved its greatest prominence during the reign of Emperor Wu. Fu are intermediary pieces between poetry and prose in which a place, object, feeling, or other subject is described and rhapsodized in exhaustive detail and from as many angles as possible. Classical fu composers attempted to use an extensive vocabulary and included many rare and archaic terms in their compositions . Fu poems employ alternating rhyme and prose, varying line length, close alliteration, onomatopoeia, loose parallelism, and extensive cataloging of their topics.
Burial Goods of the Han Dynasty
The Han Dynasty is considered a golden age in Chinese history in which a great deal of art was produced, including burial goods.
Describe the materials individuals were buried with during the Han Dyansty
Key Points
- One of the most well-known styles of art during the Han Dynasty was burial art, which evolved between the Western and Eastern Han periods.
- During the Western Han period, burial goods were usually wares and pieces of art that had been used by the tomb occupant during life.
- During the Eastern Han period, stylistic goods, wares, and artwork found in tombs were usually made exclusively for burial.
- Common items used for burial during the Eastern Han period included miniature models of ceramic towers, querns, water wells, pigsties, pestling shops, and farm fields with pottery pigs, dogs, sheep, chickens, and ducks.
- Although many items placed in tombs were commonly used wares and utensils, it was considered taboo to bring objects specified for burial into living quarters or the imperial palace.
- The Han Dynasty was known for jade burial suits, ceremonial suits made of pieces of jade in which members of the royalty were buried.
- One of the earliest known depictions of a landscape in Chinese art comes from a pair of hollow-tile door panels from a Western Han Dynasty tomb near Zhengzhou, dated 60 BCE.
Key Terms
- jade : An ornamental rock used for hardstone carving since prehistoric periods.
Overview: The Han Dynasty
The period of the Han Dynasty spanned more than four centuries and is considered a golden age in Chinese history during which a great deal of art was produced. The dynasty has been divided into the Western Han (206 BCE – 9 CE) and the Eastern Han (25–220 CE) periods. One of the most well-known legacies of the Han Dynasty is burial art, which evolved between the Western and Eastern Han periods.
Art of the Han Dynasty
Burial Goods
During the Western Han period, burial goods consisted of wares and pieces of art used by the tomb occupant when he or she was alive. During the Eastern Han period, however, stylistic goods, wares, and artwork found in tombs were usually made exclusively for burial and not previously used by the deceased.
Common items for burial during the Eastern Han period included miniature ceramic towers—usually watchtowers and urban residential towers—which provide historians with clues about lost wooden architecture. In addition to towers, there were also miniature models of querns (hand mills for grinding grain), water wells, pigsties, pestling shops, and farm fields with pottery pigs, dogs, sheep, chickens, and ducks. Although many items placed in tombs were commonly used wares and utensils, it was taboo to bring objects specified for burial into living quarters or the imperial palace. This ban was lifted once the objects were properly announced at funeral ceremonies , where they became known as mingqi (明/冥, “fearsome artifacts,” “objects for the dead,” or “brilliant artifacts”).
Other Tomb Art
The Han Dynasty was known for jade burial suits, ceremonial suits made of pieces of jade in which members of royalty were buried. One of the earliest known landscape depictions in Chinese art comes from a pair of hollow-tile door panels from a Western Han Dynasty tomb near Zhengzhou, dated 60 BCE. A scene of continuous depth recession is conveyed by the zigzag of lines representing roads and garden walls, giving the impression that one is looking down from the top of a hill. This scene was made by the repeated impression of stamps on the clay while it was still soft, before firing . The oldest known classically painted Chinese landscape is a work by Zhan Ziqian of the later Sui Dynasty (581–618 CE).
Architecture of the Han Dynasty
Remains of Han Dynasty architecture include ruins of brick and rammed earth walls, rammed earth platforms, and funerary stone pillar gates.
Describe the building materials, layout, and architectural characteristics of Han palace halls, towers, tombs, and other abodes
Key Points
- Surviving architecture from the Han Dynasty includes ruins of brick and rammed earth walls (including above- ground city walls and underground tomb walls), rammed earth platforms for terraced altars and halls, funerary stone or brick pillar -gates, and scattered ceramic roof tiles. Timber was the chief building material in Han architecture, used for grand palace halls, multi-story towers, multi-story residential halls, and humble abodes.
- Walls of frontier towns and forts in Inner Mongolia were typically constructed with stamped clay bricks instead of rammed earth. Thatched or tiled roofs were supported by wooden pillars, since the addition of brick, rammed earth, or mud walls did not actually support the roof. Stone and plaster were also used for domestic architecture.
- Valuable clues about Han architecture can be found in burial artwork of ceramic models, paintings, and carved or stamped bricks discovered in tombs and other sites.
Key Terms
- pillar : A large post, often used as supporting architecture.
- crenellations : The battlements of a castle or other building.
- dougong : A unique structural element of interlocking wooden brackets, one of the most important elements in traditional Chinese architecture.
- rammed earth : A construction material made by compressing dirt.
Surviving architecture from the Han Dynasty includes ruins of brick and rammed earth walls (including above-ground city walls and underground tomb walls), rammed earth platforms for terraced altars and halls, funerary stone or brick pillar-gates, and scattered ceramic roof tiles that once adorned timber halls. Sections of the Han-era rammed earth Great Wall still exist in Gansu province, along with the frontier ruins of thirty beacon towers and two fortified castles with crenellations .
Building Materials
Timber was the chief building material in Han Dynasty architecture, used for grand palace halls, multi-story towers, multi-story residential halls, and humble abodes. However, due to the rapid decay of wood over time and its susceptibility to fire, the oldest wooden buildings found in China (which include several temple halls of Mount Wutai) date no earlier than the Tang Dynasty (618–907 CE).
Walls of frontier towns and forts in Inner Mongolia were typically constructed with stamped clay bricks instead of rammed earth. Thatched or tiled roofs were supported by wooden pillars, since the addition of brick, rammed earth, or mud walls did not support the roof. Stone and plaster were used for domestic architecture. Tiled eaves projecting outward were built to distance falling rainwater from the walls; they were supported by dougong brackets that were sometimes elaborately decorated. Molded designs usually decorated the ends of roof tiles, as seen in artistic models of buildings and in surviving tile pieces.
Styles of Architecture
Tombs and Houses
Valuable clues about Han architecture can be found in an artwork of ceramic models, paintings, and carved or stamped bricks discovered in tombs and other sites. Han tombs were laid out like underground houses, comparable to the scenes of courtyard houses found on tomb bricks and in three-dimensional models. Han homes had a courtyard area (some had multiple courtyards), with slightly elevated halls connected by stairways. Multi-story buildings included the main colonnaded residence halls built around the courtyards as well as watchtowers. The halls were built with intersecting crossbeams and rafters usually carved with decorations; stairways and walls were plastered over to produce a smooth surface and then painted.
Tower Architecture
There are Han-era literary references to tall towers in the capital cities. They often served as watchtowers, astronomical observatories, and religious establishments meant to attract the favor of immortals . It is unknown whether miniature ceramic models of residential towers and watchtowers found in Han Dynasty tombs are faithful representations of such timber towers; nevertheless, they reveal vital clues about lost timber architecture.
Only a handful of ceramic models of multi-story towers exist from the pre-Han and Western Han eras, though hundreds of existing models were made during the Eastern Han period. Model towers could be fired as one piece in the kiln or assembled from several different ceramic pieces. Each model is unique, yet they share common features such as a walled courtyard at the bottom, a balcony with balustrades and windows for every floor, and roof tiles capping and concealing the ceiling rafters. There were also door knockers, human figures peering out of the windows or standing on the balconies, and model pets such as dogs in the courtyard. Perhaps the most direct evidence to suggest that miniature ceramic towers represent of real-life Han timber towers are the tile patterns. Artistic patterns found on the circular tiles that cap the eave-ends on the miniature models are exact matches of patterns found on roof tiles excavated at sites such as the royal palaces in Chang’an and Luoyang.
Other Types of Buildings
Other ceramic models from the Han burial sites reveal a variety of building types. These include multi-story storehouses such as granaries, courtyard houses with multi-story halls, kiosks, walled gate towers, mills, factories and workshops, animal pens, outhouses, and water wells. Even models of single-story farmhouses show great detail, including tiled roofs and courtyards. Models of granaries and storehouses had tiled rooftops, dougong brackets, windows, and stilt supports raising them above ground level. Han models of water wells sometimes feature tiny tiled roofs supported by beams that house the rope pulley used for lifting a bucket.
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5.8: The Geodesic Equation
In this section, which can be skipped at a first reading, we show how the Christoffel symbols can be used to find differential equations that describe geodesics.
Characterization of the Geodesic
A geodesic can be defined as a world-line that preserves tangency under parallel transport, Figure \(\PageIndex{1}\). This is essentially a mathematical way of expressing the notion that we have previously expressed more informally in terms of “staying on course” or moving “inertially.”
A curve can be specified by giving functions \(x^{\mu} (\lambda)\) for its coordinates, where \(\lambda\) is a real parameter. A vector lying tangent to the curve can then be calculated using partial derivatives, \(T^{\mu} = \frac{\partial x^{\mu}}{\partial \lambda}\). There are three ways in which a vector function of \(\lambda\) could change: (1) it could change for the trivial reason that the metric is changing, so that its components changed when expressed in the new metric; (2) it could change its components perpendicular to the curve; or (3) it could change its component parallel to the curve. Possibility 1 should not really be considered a change at all, and the definition of the covariant derivative is specifically designed to be insensitive to this kind of thing. 2 cannot apply to T \(\mu\) , which is tangent by construction. It would therefore be convenient if T \(\mu\) happened to be always the same length. If so, then 3 would not happen either, and we could reexpress the definition of a geodesic by saying that the covariant derivative of T \(\mu\) was zero. For this reason, we will assume for the remainder of this section that the parametrization of the curve has this property. In a Newtonian context, we could imagine the x \(\mu\) to be purely spatial coordinates, and \(\lambda\) to be a universal time coordinate. We would then interpret T \(\mu\) as the velocity, and the restriction would be to a parametrization describing motion with constant speed. In relativity, the restriction is that \(\lambda\) must be an affine parameter. For example, it could be the proper time of a particle, if the curve in question is timelike.
Covariant Derivative with Respect to a Parameter
The notation of section 5.6 is not quite adapted to our present purposes, since it allows us to express a covariant derivative with respect to one of the coordinates, but not with respect to a parameter such as \(\lambda\). We would like to notate the covariant derivative of T \(\mu\) with respect to \(\lambda\) as \(\nabla_{\lambda} T^{\mu}\), even though \(\lambda\) isn’t a coordinate. To connect the two types of derivatives, we can use a total derivative. To make the idea clear, here is how we calculate a total derivative for a scalar function f(x, y), without tensor notation:
\[\frac{df}{d \lambda} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial \lambda} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial \lambda}\]
This is just the generalization of the chain rule to a function of two variables. For example, if \(\lambda\) represents time and f temperature, then this would tell us the rate of change of the temperature as a thermometer was carried through space. Applying this to the present problem, we express the total covariant derivative as
\[\begin{split} \nabla_{\lambda} T^{\mu} &= (\nabla_{\kappa} T^{\mu}) \frac{dx^{\kappa}}{d \lambda} \\ &= (\partial_{\kappa} T^{\mu} + \Gamma^{\mu}_{\kappa \nu} T^{\nu}) \frac{d x^{\kappa}}{d \lambda} \ldotp \end{split}\]
The Geodesic Equation
Recognizing \(\frac{\partial_{\kappa} T^{\mu} dx^{\kappa}}{d \lambda}\) as a total non-covariant derivative, we find
\[\nabla_{\lambda} T^{\mu} = \frac{dT^{\mu}}{d \lambda} + \Gamma^{\mu}_{\kappa \nu} T^{\nu} \frac{dx^{\kappa}}{d \lambda} \ldotp\]
This is known as the geodesic equation. There is a factor of two that is a common gotcha when applying this equation. The symmetry of the Christoffel symbols \(\Gamma^{\mu}_{\kappa \nu} = \Gamma^{\mu}_{\nu \kappa}\) implies that when \(\kappa\) and \(\nu\) are distinct, the same term will appear twice in the summation.
If this differential equation is satisfied for one affine parameter \(\lambda\), then it is also satisfied for any other affine parameter \(\lambda' = a \lambda + b\), where a and b are constants (problem 5). Recall that affine parameters are only defined along geodesics, not along arbitrary curves.
We can’t start by defining an affine parameter and then use it to find geodesics using this equation, because we can’t define an affine parameter without first specifying a geodesic. Likewise, we can’t do the geodesic first and then the affine parameter, because if we already had a geodesic in hand, we wouldn’t need the differential equation in order to find a geodesic. The solution to this chicken-and-egg conundrum is to write down the differential equations and try to find a solution, without trying to specify either the affine parameter or the geodesic in advance. We will seldom have occasion to resort to this technique, an exception being example 19 .
Uniqueness
The geodesic equation is useful in establishing one of the necessary theoretical foundations of relativity, which is the uniqueness of geodesics for a given set of initial conditions. This is related to axiom O1 of ordered geometry, that two points determine a line, and is necessary physically for the reasons discussed in section 1.5 ; briefly, if the geodesic were not uniquely determined, then particles would have no way of deciding how to move. The form of the geodesic equation guarantees uniqueness. To see this, consider the following algorithm for determining a numerical approximation to a geodesic:
- Initialize \(\lambda\), the x \(\mu\) and their derivatives \(\frac{dx^{\mu}}{d \lambda}\). Also, set a small step-size \(\Delta \lambda\) by which to increment \(\lambda\) at each step below.
- For each i, calculate \(\frac{d^{2} x^{\mu}}{d \lambda^{2}}\) using the geodesic equation.
- Add \((\frac{d^{2} x^{\mu}}{d \lambda^{2}}) \Delta \lambda\) to the currently stored value of \(\frac{dx^{\mu}}{d \lambda}\).
- Add \((\frac{dx^{\mu}}{d \lambda}) \Delta \lambda\) to x \(\mu\) .
- Add \(\Delta \lambda\) to \(\lambda\).
- Repeat steps 2-5 until the geodesic has been extended to the desired affine distance.
Since the result of the calculation depends only on the inputs at step 1, we find that the geodesic is uniquely determined.
To see that this is really a valid way of proving uniqueness, it may be helpful to consider how the proof could have failed. Omitting some of the details of the tensors and the multidimensionality of the space, the form of the geodesic equation is essentially \(\ddot{x}\ + f \dot{x}^{2} = 0\), where dots indicate derivatives with respect to \(\lambda\). Suppose that it had instead had the form \(\ddot{x}^{2} + f \dot{x} = 0\). Then at step 2 we would have had to pick either a positive or a negative square root for \(\ddot{x}\). Although continuity would usually suffice to maintain a consistent sign from one iteration to the next, that would not work if we ever came to a point where \(\ddot{x}\) vanished momentarily. An equation of this form therefore would not have a unique solution for a given set of initial conditions.
The practical use of this algorithm to compute geodesics numerically is demonstrated in section 5.9 .


30.0: Prelude to Plant Form and Physiology
Plants are as essential to human existence as land, water, and air. Without plants, our day-to-day lives would be impossible because without oxygen from photosynthesis, aerobic life cannot be sustained. From providing food and shelter to serving as a source of medicines, oils, perfumes, and industrial products, plants provide humans with numerous valuable resources.
When you think of plants, most of the organisms that come to mind are vascular plants. These plants have tissues that conduct food and water, and they have seeds. Seed plants are divided into gymnosperms and angiosperms. Gymnosperms include the needle-leaved conifers—spruce, fir, and pine—as well as less familiar plants, such as ginkgos and cycads. Their seeds are not enclosed by a fleshy fruit. Angiosperms, also called flowering plants, constitute the majority of seed plants. They include broadleaved trees (such as maple, oak, and elm), vegetables (such as potatoes, lettuce, and carrots), grasses, and plants known for the beauty of their flowers (roses, irises, and daffodils, for example).
While individual plant species are unique, all share a common structure: a plant body consisting of stems, roots, and leaves. They all transport water, minerals, and sugars produced through photosynthesis through the plant body in a similar manner. All plant species also respond to environmental factors, such as light, gravity, competition, temperature, and predation.