5.7: Wave-Particle Duality
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Learning Objectives
By the end of this section, you will be able to:
- Identify phenomena in which electromagnetic waves behave like a beam of photons and particles behave like waves
- Describe the physics principles behind electron microscopy
- Summarize the evolution of scientific thought that led to the development of quantum mechanics
The energy of radiation detected by a radio-signal receiving antenna comes as the energy of an electromagnetic wave. The same energy of radiation detected by a photocurrent in the photoelectric effect comes as the energy of individual photon particles. Therefore, the question arises about the nature of electromagnetic radiation: Is a photon a wave or is it a particle? Similar questions may be asked about other known forms of energy. For example, an electron that forms part of an electric current in a circuit behaves like a particle moving in unison with other electrons inside the conductor. The same electron behaves as a wave when it passes through a solid crystalline structure and forms a diffraction image. Is an electron a wave or is it a particle? The same question can be extended to all particles of matter—elementary particles, as well as compound molecules—asking about their true physical nature. At our present state of knowledge, such questions about the true nature of things do not have conclusive answers. All we can say is that wave-particle duality exists in nature: Under some experimental conditions, a particle appears to act as a particle, and under different experimental conditions, a particle appears to act a wave. Conversely, under some physical circumstances electromagnetic radiation acts as a wave, and under other physical circumstances, radiation acts as a beam of photons.
This dualistic interpretation is not a new physics concept brought about by specific discoveries in the twentieth century. It was already present in a debate between Isaac Newton and Christiaan Huygens about the nature of light, beginning in the year 1670. According to Newton, a beam of light is a collection of corpuscles of light. According to Huygens, light is a wave. The corpuscular hypothesis failed in 1803, when Thomas Young announced his double-slit interference experiment with light (see Figure \(\PageIndex{1}\)), which firmly established light as a wave. In James Clerk Maxwell’s theory of electromagnetism (completed by the year 1873), light is an electromagnetic wave. Maxwell’s classical view of radiation as an electromagnetic wave is still valid today; however, it is unable to explain blackbody radiation and the photoelectric effect, where light acts as a beam of photons.
A similar dichotomy existed in the interpretation of electricity. From Benjamin Franklin’s observations of electricity in 1751 until J.J. Thomson’s discovery of the electron in 1897, electric current was seen as a flow in a continuous electric medium. Within this theory of electric fluid, the present theory of electric circuits was developed, and electromagnetism and electromagnetic induction were discovered. Thomson’s experiment showed that the unit of negative electric charge (an electron) can travel in a vacuum without any medium to carry the charge around, as in electric circuits. This discovery changed the way in which electricity is understood today and gave the electron its particle status. In Bohr’s early quantum theory of the hydrogen atom, both the electron and the proton are particles of matter. Likewise, in the Compton scattering of X-rays on electrons, the electron is a particle. On the other hand, in electron-scattering experiments on crystalline structures, the electron behaves as a wave.
A skeptic may raise a question that perhaps an electron might always be nothing more than a particle, and that the diffraction images obtained in electron-scattering experiments might be explained within some macroscopic model of a crystal and a macroscopic model of electrons coming at it like a rain of ping-pong balls. As a matter of fact, to investigate this question, we do not need a complex model of a crystal but just a couple of simple slits in a screen that is opaque to electrons. In other words, to gather convincing evidence about the nature of an electron, we need to repeat the Young double-slit experiment with electrons. If the electron is a wave, we should observe the formation of interference patterns typical for waves, such as those described in Figure \(\PageIndex{1}\), even when electrons come through the slits one by one. However, if the electron is a not a wave but a particle, the interference fringes will not be formed.
The very first double-slit experiment with a beam of electrons, performed by Claus Jönsson in Germany in 1961, demonstrated that a beam of electrons indeed forms an interference pattern, which means that electrons collectively behave as a wave. The first double-slit experiments with single electrons passing through the slits one-by-one were performed by Giulio Pozzi in 1974 in Italy and by Akira Tonomura in 1989 in Japan. They show that interference fringes are formed gradually, even when electrons pass through the slits individually. This demonstrates conclusively that electron-diffraction images are formed because of the wave nature of electrons. The results seen in double-slit experiments with electrons are illustrated by the images of the interference pattern in Figure \(\PageIndex{2}\).
Example \(\PageIndex{1}\): Double-Slit Experiment with Electrons
In one experimental setup for studying interference patterns of electron waves, two slits are created in a gold-coated silicon membrane. Each slit is 62-nm wide and 4-μm long, and the separation between the slits is 272 nm. The electron beam is created in an electron gun by heating a tungsten element and by accelerating the electrons across a 600-V potential. The beam is subsequently collimated using electromagnetic lenses, and the collimated beam of electrons is sent through the slits. Find the angular position of the first-order bright fringe on the viewing screen.
Strategy
Recall that the angular position θ of the n th order bright fringe that is formed in Young’s two-slit interference pattern (discussed in a previous chapter) is related to the separation, d , between the slits and to the wavelength, λ, of the incident light by the equation dsin θ = nλ, where n = 0, \(\pm 1\), \(\pm 2\),.... The separation is given and is equal to d = 272 nm. For the first-order fringe, we take n = 1. The only thing we now need is the wavelength of the incident electron wave.
Since the electron has been accelerated from rest across a potential difference of ΔV = 600 V, its kinetic energy is K = e ΔV = 600 eV. The rest-mass energy of the electron is \(E_0\) = 511 keV.
We compute its de Broglie wavelength as that of a nonrelativistic electron because its kinetic energy K is much smaller than its rest energy \(E_0\), K ≪ \(E_0\).
Solution
The electron’s wavelength is
\[\lambda = \frac{h}{p} = \frac{h}{\sqrt{2m_eK}} = \frac{h}{\sqrt{2E_0/c^2K}} = \frac{hc}{\sqrt{2E_0K}} = \frac{1.241 \times 10^{-6} \, eV \cdot m}{\sqrt{2(511 \, keV)(600 \, eV)}} = 0.050 \, nm. \nonumber \]
This λ is used to obtain the position of the first bright fringe:
\[\sin \, \theta = \frac{1 \cdot \lambda}{d} = \frac{0.050 \, nm}{272 \, nm} = 0.000184 \Rightarrow θ = 0.010°. \nonumber \]
Significance
Notice that this is also the angular resolution between two consecutive bright fringes up to about n = 1000. For example, between the zero-order fringe and the first-order fringe, between the first-order fringe and the second-order
Exercise \(\PageIndex{1}\)
For the situation described in Example \(\PageIndex{1}\), find the angular position of the fifth-order bright fringe on the viewing screen.
- Answer
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\(0.052^o\)
The wave-particle dual nature of matter particles and of radiation is a declaration of our inability to describe physical reality within one unified classical theory because separately neither a classical particle approach nor a classical wave approach can fully explain the observed phenomena. This limitation of the classical approach was realized by the year 1928, and a foundation for a new statistical theory, called quantum mechanics, was put in place by Bohr, Edwin Schrödinger , Werner Heisenberg , and Paul Dirac . Quantum mechanics takes de Broglie’s idea of matter waves to be the fundamental property of all particles and gives it a statistical interpretation. According to this interpretation, a wave that is associated with a particle carries information about the probable positions of the particle and about its other properties. A single particle is seen as a moving wave packet such as the one shown in Figure \(\PageIndex{3}\). We can intuitively sense from this example that if a particle is a wave packet , we will not be able to measure its exact position in the same sense as we cannot pinpoint a location of a wave packet in a vibrating guitar string. The uncertainty, Δx, in measuring the particle’s position is connected to the uncertainty, Δp, in the simultaneous measuring of its linear momentum by Heisenberg’s uncertainty principle:
\[\Delta x \Delta p \geq \frac{1}{2}\hbar. \label{6.63} \]
Heisenberg’s principle expresses the law of nature that, at the quantum level, our perception is limited. For example, if we know the exact position of a body (which means that Δx = 0 in Equation \ref{6.63}) at the same time we cannot know its momentum, because then the uncertainty in its momentum becomes infinite (because Δp ≥ 0.5 ℏ/Δx in Equation \ref{6.63}). The Heisenberg uncertainty principle sets the limit on the precision of simultaneous measurements of position and momentum of a particle; it shows that the best precision we can obtain is when we have an equals sign (=) in Equation \ref{6.63}, and we cannot do better than that, even with the best instruments of the future. Heisenberg’s principle is a consequence of the wave nature of particles.
We routinely use many electronic devices that exploit wave-particle duality without even realizing the sophistication of the physics underlying their operation. One example of a technology based on the particle properties of photons and electrons is a charge-coupled device, which is used for light detection in any instrumentation where high-quality digital data are required, such as in digital cameras or in medical sensors. An example in which the wave properties of electrons is exploited is an electron microscope.
In 1931, physicist Ernst Ruska - building on the idea that magnetic fields can direct an electron beam just as lenses can direct a beam of light in an optical microscope—developed the first prototype of the electron microscope. This development originated the field of electron microscopy . In the transmission electron microscope (TEM) , shown in Figure \(\PageIndex{4}\), electrons are produced by a hot tungsten element and accelerated by a potential difference in an electron gun, which gives them up to 400 keV in kinetic energy. After leaving the electron gun, the electron beam is focused by electromagnetic lenses (a system of condensing lenses) and transmitted through a specimen sample to be viewed. The image of the sample is reconstructed from the transmitted electron beam. The magnified image may be viewed either directly on a fluorescent screen or indirectly by sending it, for example, to a digital camera or a computer monitor. The entire setup consisting of the electron gun, the lenses, the specimen, and the fluorescent screen are enclosed in a vacuum chamber to prevent the energy loss from the beam. Resolution of the TEM is limited only by spherical aberration (discussed in a previous chapter). Modern high-resolution models of a TEM can have resolving power greater than 0.5 Å and magnifications higher than 50 million times. For comparison, the best resolving power obtained with light microscopy is currently about 97 nm. A limitation of the TEM is that the samples must be about 100-nm thick and biological samples require a special preparation involving chemical “fixing” to stabilize them for ultrathin slicing.
Such limitations do not appear in the scanning electron microscope (SEM) , which was invented by Manfred von Ardenne in 1937. In an SEM, a typical energy of the electron beam is up to 40 keV and the beam is not transmitted through a sample but is scattered off its surface. Surface topography of the sample is reconstructed by analyzing back-scattered electrons, transmitted electrons, and the emitted radiation produced by electrons interacting with atoms in the sample. The resolving power of an SEM is better than 1 nm, and the magnification can be more than 250 times better than that obtained with a light microscope. The samples scanned by an SEM can be as large as several centimeters but they must be specially prepared, depending on electrical properties of the sample.
High magnifications of the TEM and SEM allow us to see individual molecules. High resolving powers of the TEM and SEM allow us to see fine details, such as those shown in the SEM micrograph of pollen at the beginning of this chapter ( Figure 6.1.1 ).
Example \(\PageIndex{2}\): Resolving Power of an Electron Microscope
If a 1.0-pm electron beam of a TEM passes through a 2.0-μm circular opening, what is the angle between the two just-resolvable point sources for this microscope?
Solution
We can directly use a formula for the resolving power, \(Δθ\), of a microscope when the wavelength of the incident radiation is \(λ=1.0\, pm\) and the diameter of the aperture is \(D = 2.0\, μm\):
\[\Delta \theta = 1.22 \frac{\lambda}{D} = 1.22 \frac{1.0 \, pm}{2.0 \, \mu m} = 6.10 \times 10^{-7} rad = 3.50 \times 10.5^o. \nonumber \]
Significance
Note that if we used a conventional microscope with a 400-nm light, the resolving power would be only 14°, which means that all of the fine details in the image would be blurred.
Exercise \(\PageIndex{2}\)
Suppose that the diameter of the aperture in Example \(\PageIndex{2}\) is halved. How does it affect the resolving power?
- Answer
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doubles it


5.3: The Existence of Primitive Roots
In this section, we demonstrate which integers have primitive roots. We start by showing that every power of an odd prime has a primitive root and to do this we start by showing that every square of an odd prime has a primitive root.
If \(p\) is an odd prime with primitive root \(r\), then one can have either \(r\) or \(r+p\) as a primitive root modulo \(p^2\).
Notice that since \(r\) is a primitive root modulo \(p\), then \[ord_pr=\phi(p)=p-1.\] Let \(m=ord_{p^2}r\), then \[r^m\equiv 1(mod \ p^2).\] Thus \[r^m\equiv 1(mod \ p).\] By Theorem 54, we have \[p-1\mid m.\] By Exercise 7 of section 6.1, we also have that \[m\mid \phi(p^2).\] Also, \(\phi(p^2)=p(p-1)\) and thus \(m\) either divides \(p\) or \(p-1\). And since \(p-1\mid m\) then we have \[m=p-1 \ \ \mbox{or} \ \ m=p(p-1).\] If \(m=p(p-1)\) and \(ord_{p^2}r=\phi(p^2)\) then \(r\) is a primitive root modulo \(p^2\). Otherwise, we have \(m=p-1\) and thus \[r^{p-1}\equiv 1(mod \ p^2).\] Let \(s=r+p\). Then \(s\) is also a primitive root modulo \(p\). Hence, \(ord_{p^2}s\) equals either \(p-1\) or \(p(p-1)\). We will show that \(ord_{p^2}s\neq p-1\) so that \(ord_{p^2}s=p(p-1)\). Note that \[\begin{aligned} s^{p-1}=(r+p)^{p-1}&=&r^{p-1}+(p-1)r^{p-2}p+...+p^{p-1}\\&=& r^{p-1}+(p-1)p.r^{p-2}(mod \ p^2).\end{aligned}\] Hence \[p^2\mid s^{p-1}-(1-pr^{p-2}.\] Note also that if \[p^2 \mid (s^{p-1}-1),\] then \[p^2\mid pr^{p-2}.\] Thus we have \[p\mid r^{p-2}\] which is impossible because \(p\nmid r\). Because \(ord_{p^2}s\neq p-1\), we can conclude that \[ord_{p^2}s=p(p-1)=\phi(p^2).\] Thus, \(s=r+p\) is a primitive root of \(p^2\).
Notice that 7 has 3 as a primitive root. Either \(ord_{49}3=6\) or \(ord_{49}3=42\). But since \(3^6\not\equiv 1(mod \ 49)\). Hence \(ord_{49}3=42\). Hence 3 is a primitive root of 49.
We now show that any power of an odd prime has a primitive root.
Let \(p\) be an odd prime. Then any power of \(p\) is a primitive root. Moreover, if \(r\) is a primitive root modulo \(p^2\), then \(r\) is a primitive root modulo \(p^m\) for all positive integers \(m\).
By Theorem 62, we know that any prime \(p\) has a primitive root \(r\) which is also a primitive root modulo \(p^2\), thus \[\label{1} p^2\nmid (r^{p-1}-1).\] We will prove by induction that \[\label{2} p^m\nmid (r^{p^{m-2}(p-1)}-1)\] for all integers \(m\geq 2\). Once we prove the above congruence, we show that \(r\) is also a primitive root modulo \(p^m\). Let \(n=ord_{p^m}r\). By Theorem 54, we know that \(n\mid \phi(p^m)\). Also, we know that \(\phi(p^m)=p^m(p-1)\). Hence \(n\mid p^m(p-1)\). On the other hand, because \[p^m\mid (r^n- 1),\] we also know that \[p\mid (r^n-1).\] Since \(\phi(p)=p-1\), we see that by Theorem 54, we have \(n=l(p-1)\). also \(n\mid p^{m-1}(p-1)\), we have that \(n=p^s(p-1)\), where \(0 \leq s\leq m-1\). If \(n=p^s(p-1)\) with \(s\leq m-2\), then \[p^k\mid r^{p^{m-2}(p-1)}-1,\] which is a contradiction. Hence \[ord_{p^m}r=\phi(p^m).\]
We prove now ( [2] ) by induction. Assume that our assertion is true for all \(m\geq 2\). Then \[p^m\nmid (r^{p^{m-2}(p-1)}-1).\] Because \((r,p)=1\), we see that \((r,p^{m-1})=1\). We also know from Euler’s theorem that \[p^{m-1}\mid (r^{p^{m-2}(p-1)}-1).\] Thus there exists an integer \(k\) such that \[r^{p^{m-2}(p-1)}=1+kp^{m-1}.\] where \(p\nmid k\) because \(r^{p^{m-2}(p-1)}\not\equiv 1(mod \ p^m)\). Thus we have now \[\begin{aligned} r^{p^{m-1}(p-1)}&=&(1+kp^{m-1})^p\\ &\equiv&1+kp^m(mod \ p^{m+1})\end{aligned}\] Because \(p\nmid k\), we have \[p^{m+1}\nmid (r^{p^{m-1}(p-1)}-1).\]
Since 3 is a primitive root of 7, then 3 is a primitive root for \(7^k\) for all positive integers \(k\).
In the following theorem, we prove that no power of 2, other than 2 or 4, has a primitive root and that is because when \(m\) is an odd integer, \(ord_2^km\neq \phi(2^k)\) and this is because \(2^k\mid (a^{\phi(2^k)/2}-1)\).
If \(m\) is an odd integer, and if \(k\geq 3\) is an integer, then \[m^{2^{k-2}}\equiv 1(mod \ 2^k).\]
We prove the result by induction. If \(m\) is an odd integer, then \(m=2n+1\) for some integer \(n\). Hence, \[m^2=4n^2+4n+1=4n(n+1)+1.\] It follows that \(8\mid (m^2-1)\).
Assume now that \[2^k\mid (m^{2^{k-2}}-1).\] Then there is an integer \(q\) such that \[m^{2^{k-2}}=1+q.2^{k}.\] Thus squaring both sides, we get \[m^{2^{k-1}}=1+q.2^{k+1}+q^22^{2k}.\] Thus \[2^{k+1}\mid (m^{2^{k-1}}-1).\]
Note now that 2 and 4 have primitive roots 1 and 3 respectively.
We now list the set of integers that do not have primitive roots.
If \(m\) is not \(p^a\) or \(2p^a\), then \(m\) does not have a primitive root.
Let \(m=p_1^{s_1}p_2^{s_2}...p_i^{s_i}\). If \(m\) has a primitive root \(r\) then \(r\) and \(m\) are relatively prime and \(ord_mr=\phi(m)\). We also have, we have \((r,p^s)=1\) where \(p^s\) is of the primes in the factorization of \(m\). By Euler’s theorem, we have \[p^s\mid (r^{\phi(p^s)}-1).\] Now let \[L=[\phi(p_1^{s_1}), \phi(p_2^{s_2}),...,\phi(p_i^{s_i})].\] We know that \[r^L\equiv 1(mod \ p_k^{s_k})\] for all \(1\leq k\leq m\). Thus using the Chinese Remainder Theorem, we get \[m\mid (r^L-1),\] which leads to \(ord_mr=\phi(m)\leq L\). Now because \[\phi(m)=\phi(p_1^{s_1})\phi(p_2^{s_2})...\phi(p_n^{s_n})\leq [\phi(p_1^{s_1}),\phi(p_2^{s_2}),...,\phi(p_n^{s_n})].\] Now the inequality above holds only if \[\phi(p_1^{s_1}),\phi(p_2^{s_2}),...,\phi(p_n^{s_n})\] are relatively prime. Notice now that by Theorem 41, \[\phi(p_1^{s_1}),\phi(p_2^{s_2}),...,\phi(p_n^{s_n})\] are not relatively prime unless \(m=p^s\) or \(m=2p^s\) where \(p\) is an odd prime and \(t\) is any positive integer.
We now show that all integers of the form \(m=2p^s\) have primitive roots.
Consider a prime \(p\neq 2\) and let \(s\) is a positive integer, then \(2p^s\) has a primitive root. In fact, if \(r\) is an odd primitive root modulo \(p^s\), then it is also a primitive root modulo \(2p^s\) but if \(r\) is even, \(r+p^s\) is a primitive root modulo \(2p^s\).
If \(r\) is a primitive root modulo \(p^s\), then \[p^s\mid (r^{\phi(p^s)}-1)\] and no positive exponent smaller than \(\phi(p^s)\) has this property. Note also that \[\phi(2p^s)=\phi(p^s),\] so that \[p^s\mid (r^{\phi(2p^s)}-1).\]
If \(r\) is odd, then \[2\mid (r^{\phi(2p^s)}-1).\] Thus by Theorem 56, we get \[2p^s\mid (r^{\phi(2p^s)}-1).\] It is important to note that no smaller power of \(r\) is congruent to 1 modulo \(2p^s\). This power as well would also be congruent to 1 modulo \(p^s\) contradicting that \(r\) is a primitive root of \(p^s\). It follows that \(r\) is a primitive root modulo \(2p^s\).
While, if \(r\) is even, then \(r+p^s\) is odd. Hence \[2\mid ((r+p^s)^{\phi(2p^s)}-1).\]
Because \(p^s\mid (r+p^s-r)\), we see that \[p^s\mid ((r+p^s)^{\phi(2p^s)}-1).\] As a result, we see that \(2p^s\mid ((r+p^s)^{\phi(2p^s)}-1)\) and since for no smaller power of \(r+p^s\) is congruent to 1 modulo \(2p^s\), we see that \(r+p^s\) is a primitive root modulo \(2p^s\).
As a result, by Theorem 63, Theorem 65 and Theorem 66, we see that
The positive integer \(m\) has a primitive root if and only if \(n=2,4, p^s\) or \(2p^s\)
for prime \(p\neq 2\) and \(s\) is a positive integer.
Exercises
- Which of the following integers 4, 12, 28, 36, 125 have a primitive root.
- Find a primitive root of 4, 25, 18.
- Find all primitive roots modulo 22.
- Show that there are the same number of primitive roots modulo \(2p ^s\) as there are modulo \(p^s\), where \(p\) is an odd prime and \(s\) is a positive integer.
- Find all primitive roots modulo 25.
- Show that the integer \(n\) has a primitive root if and only if the only solutions of the congruence \(x^2\equiv 1(mod n)\) are \(x\equiv \pm1 (mod \ n)\).
Contributors and Attributions
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Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution ( CC BY ) license.


6.3: Glycerolipids
- Define glycerolipids and triglycerides and understand their structures.
- Understand the difference in composition and properties of fats and oils and their primary role as energy storage molecules.
- Understand the chemical properties of fats and oils, including hydrolysis, saponification, and partial or complete hydrogenation.
What is a glycerolipid?
Glycerolipids have two components: propane-1,2,3-triol, also called glycerol, and one, two, or three fatty acids. Recall that a condensation reaction between an alcohol and a carboxylic acid forms an ester by eliminating a water molecule. The following example shows the reaction of glycerol with three molecules of stearic acid, creating a triester.
A mono-, di-, or tri-ester of glycerol is called glycerolipid.
The name of a glycerolipid begins with glyceryl and is followed by carboxylate. For example, the triester of glycerol shown above is glyceryl tristearate, which is commonly known as tristearin.
Glycerolipids that are triesters of glycerol with three fatty acids are called triacylglycerol or triglycerides . For example, glyceryl tristearate shown above is a triglyceride.
Triglycerides may be esters of three molecules of the same fatty acids, e.g., tristearin. Still, often the triglycerides found in nature are esters of two or three different fatty acids. Fatty acids usually found in fats and vegetable oils include lauric, myristic, palmitic, stearic, oleic, linoleic, and linolenic acids. An example of a mixed triglyceride of stearic, oleic, and palmitic acid is shown below.
Fats and oils
Fats and oils are triglycerides used as energy storage molecules in animals and plants.
Energy storage is essential for hibernating animals that live in icy environments. They have plenty of food available during summer but no food and below-freezing temperatures in winter.  For example, the polar bear shown on the right can accumulate up to 14 kg of fat per week in summer when the food is plenty and uses it as an energy source when the animal hibernates for 4 to 7 months in winter.
Although glycogen is a quick energy source, fats and oils are more energy dense for two reasons:
- first, fats and oils are in a more reduced form than glycogen and release more energy per mass upon oxidation, and
- second, glycogen is hydrophilic and absorbs water, increasing its mass.
Physical properties of fats and oils
Pure fats and oils are colorless, odorless, and tasteless. The color, taste, and odor associated with butter -an animal fat, and olive oil -a vegetable oil, is due to a small number of other substances mixed with the triglycerides.
Fats come from animal sources, such as meat, whole milk, lard, butter, and cheese. Vegetable oils come from plant sources, e.g., soybeans, peanuts, sunflowers, etc.
Fats are usually solid at room temperature because they have a higher proportion of long-chain unsaturated fatty acids with higher melting points, e.g., lard and butter shown in Figure \(\PageIndex{1}\). Vegetable oils are usually liquid at room temperature because they contain more unsaturated or short-chain saturated fatty acids.
For example, canola oil, safflower oil, flaxseed oil, sunflower oil, corn oil, olive oil, soyabean oil, peanut oil, and cottonseed oil are vegetable oils that are liquid at room temperature and have a higher proportion of unsaturated fatty acids as shown in Figure \(\PageIndex{2}\). Palm and coconut oils are vegetable oils with a higher balance of saturated fatty acids. Still, they are liquid at room temperature because they have short-chain saturated fatty acids. For example, coconut mainly comprises lauric acid (12:0), a short-chain fatty acid.
Chemical properties of fats and oils
Three important reactions of triglycerides are hydrolysis by water in the presence of acid or lipase enzyme, saponification, i.e., base-promoted hydrolysis, and hydrogenation of double bonds in the fatty acids.
Hydrolysis
Easters are hydrolyzed (split) by water in the presence of an acid catalyst into alcohol and carboxylic acid. The same reaction happens with triglycerides, i.e., fats and vegetable oils—for example, tripalmitin hydrolyzes into glycerol and three palmitic acid molecules, as shown below.
Lipase enzymes do the same reaction during the digestion of triglycerides, e.g., trilaurin is hydrolyzed by lipase to glycerol and three molecules of lauric acid, as shown in the reaction below.
The digestion of lipids starts in the mouth with lingual lipases secreted by glands in the tongue and continues in the stomach with lingual and gastric lipases. Glycerol in the hydrolysis product is soluble in water, but fatty acids are not. Fatty acids are emulsified in the stomach and later mixed with bile and pancreatic juice to continue the process of digestion.
Saponification
Saponification is base-promoted hydrolysis of triglycerides that produces glycerol and salts of the fatty acids, a soap, as shown below for the case of saponification of tripalmitin.
Different varieties of soaps are shown in the figure on the right. Sodium salts of fatty acids make hard soaps, while potassium salts make soft soaps. Three of the most common soaps are sodium stearate, oleate, and linoleate. Salts of saturated fatty acids make rigid, and those of polyunsaturated fatty acids make soft soap. Perfumes are added for scent and dyes for color, sand is added in scouring soap, and the air is blown into the soap to make it float on water.
Hydrogenation
Alkenes add \(\ce{H2}\) and covert to alkane in the presence of \(\ce{Ni}\), \(\ce{Pt}\), or \(\ce{Pd}\) catalyst. The same reaction happens with the \(\ce{C=C}\) in unsaturated fatty acids found in triglycerides. The conversion of unsaturated fatty acids to saturated fatty acids increases the melting point of triglycerides. Therefore, vegetable oils with more unsaturated fatty acids become semisolid or solid after partial or complete hydrogenation, as illustrated by the reaction in Figure \(\PageIndex{3}\). Hydrogenation of vegetable oils is a commercial process to convert vegetable oils to semisolid products like margarine and shortening.
The driving force for the industrial process of hydrogenation of vegetable oils is that margarine or shortenings are used as cheaper alternatives to butter with a longer shelf-life. Complete hydrogenation of vegetable oils converts them into hard margarine. Often partial hydrogenation is performed as the partially hydrogenated product is semisolid and mimics butter better than hard shortening. Dyes and flavors are mixed with it to mimic butter's color, taste, and odor.
Like all other processed food products, hydrogenated vegetable oil products are associated with health problems. Partial hydrogenation converts some of the cis -\(\ce{C=C}\) bonds to trans -\(\ce{C=C}\) bonds, as illustrated in Figure \(\PageIndex{3}\). Trans -\(\ce{C=C}\) bonds are rare in some fatty acids. Research reports indicate that trans-fatty acids affect blood cholesterol like unsaturated fatty acids. Research reports also indicated that trans-fatty acids increase the level of low-density lipids (LDL), considered bad cholesterol, and decrease the level of high-density lipids (HDL), regarded as good cholesterol. Health organizations force the manufacturers to show the trans-fatty acid contents on the product labels. They are trying to educate the consumers about trans-fatty acids, as shown in Figure \(\PageIndex{4}\) from U.S. Food and Drug Administration.


8.5.4: Types of Modified Diets
The type of diet the patient has been prescribed can be found in the Care Plan. There are numerous reasons why a patient may be on a modified diet. Below are some of the more common types of modified diets and food suggestions that Home Health Aides/Personal Care Aides may encounter.
Low Sodium/Low Salt Diets
Patients who have conditions such as heart disease, heart failure, high blood pressure, coronary artery disease, and kidney disease may require a diet low in sodium/salt. This is because excess sodium causes the body to retain (hold in) more water. This excess water causes the heart to pump harder. For people who have heart or kidney problems, this makes their organs have to work harder than they are able. Excess sodium in their diets can cause a progression (worsening) of their disease. The excess fluids can also lead to edema (swelling) in their bodies. The Care Plan will specify how much sodium the patient should have in their diet.
Here are some tips for Home Health Aides / Personal Care Aides to help people who have to be on l
ow sodium diets :- Always read the food label for salt or sodium amounts.
- Avoid canned soups, stews, meats, pastas, and prepared foods that are high in sodium.
- Avoid frozen and prepackaged meals that are high in sodium.
- Avoid items such as soy sauce, ketchup, barbeque sauce, and other bottled sauces and marinades, which tend to be high in sodium.
- Don’t add salt to foods during food preparation.
- Use herbs, mustard, pepper, onion, garlic, and other spices the patient enjoys to enhance the flavor of food.
- Salt substitutes can often be used to add flavor to food. Check with the Care Plan if the patient can have a salt substitute, as these often have extra nutrients such as potassium which patients with kidney disease need to avoid.
1. Which of the following foods should a person on a low sodium diet avoid? Select all that apply .
a). Fresh fruit
b). Frozen dinners
c). Boxed dinners
d). Vegetables
e). Canned soups
f). Herbs and spices
- Answer
-
1. B, C, & E
FEEDBACK :
Low sodium diets may be required of a patient with heart or kidney disease or those with high blood pressure. Foods that are high in sodium include frozen and boxed dinners, many pre-prepared foods, and canned foods such as soups, stews, and chili.
Fluid-Restricted Diets
Patients who have heart and kidney disease may also need to watch their fluid intake. Just as in high sodium diets, excess water can make their heart and kidneys have to work harder. For patients with these types of problems, excess fluids can lead to a worsening of their disease. The excess fluids can also lead to edema (swelling) in their bodies.
Here are some tips for Home Health Aides / Personal Care Aides to help people who are on restricted fluids :
- Measure all fluids taken in accurately. Use measuring cups and document intake.
- Measure all urine output accurately. Use a graduated cylinder and document output.
- Don’t offer pitchers of water or other drinks. Instead, give one glass at a time.
- Count foods such as ice cream, sherbet, sorbet, frozen yogurt, popsicles, and soups as fluid intake. These foods have high fluid content.
- Juice, milk, coffee, tea, and soda, in addition to water, all count as fluid intake.
- Avoid foods with high sodium/salt content. They can lead to fluid retention.
High Potassium Diet
Patients who are taking medications such as diuretics may need to be on a high potassium diet (Leahy, Fuze & Grafe, 2013). Diuretics are medications that help the body to reduce fluid volume. Many people take diuretics to help lower blood pressure, or if they have heart disease. They help the heart to work less hard, as diuretics help remove water from the body. Too much water in the body makes the heart work harder, which weakens the heart and may cause or worsen heart failure. Some diuretics also remove potassium from the body. Other patients may have low potassium levels from other reasons such as excessive diarrhea, sweating, or vomiting and may need to eat diets higher in potassium to replace potassium that is lost. As we have discussed, potassium is necessary for a healthy body and heart. Low levels of potassium can cause an irregular heart rate, weakness, fatigue, muscle cramps, and constipation (Mayo Clinic, 2014). A list of foods high in potassium can be found at the National Kidney Foundation at: https :// www . kidney . org / atoz / content / potassium
According to the National Kidney Foundation (2015), here are some foods which are high in potassium:
- Bananas
- Potatoes (white and sweet)
- Tomatoes (and tomato sauce)
- Dried or fresh apricots, figs, dates, and prunes
- Avocados
- Nuts and peanut butter
- Oranges and orange juice
- Prune juice
- Squash
- Cantaloupe
- Honeydew
- Beans and lentils
- Dark greens such as broccoli and spinach
Low Potassium Diets
Some patients need to avoid taking in extra potassium in their diets. This may be due to kidney disease or from taking certain diuretics. While many diuretics deplete (get rid of) potassium, some diuretics help the body hold onto potassium. For these patients, they may be asked to not consume extra potassium. Just as not enough potassium can lead to problems, too much potassium can also lead to problems, such as irregular heart rhythms. If potassium levels are too high in the body, the patient may feel weakness, numbness, or tingling, and too high levels of potassium can cause an irregular heart rate or heart attack (National Kidney Foundation, 2015). Diets low in potassium may be prescribed for these patients.
Instead of encouraging the foods found on the high potassium list, Home Health Aides/Personal Care Aides should teach patients to avoid these foods. Also be cautious to avoid using salt substitutes, which are high in potassium. A list of foods low in potassium can be found at the National Kidney Foundation at: https :// www . kidney . org / atoz / content / potassium
1. Which of the following foods should a person on a low potassium diet avoid? Select all that apply .
a). Bananas
b). Potatoes
c). Steak
d). Chicken and turkey
e). Beans
f). Tomatoes
g). Peanut butter
h). Milk
i). Dried fruits
- Answer
-
1. A, B, E, F, G, I
FEEDBACK :
Bananas, potatoes, beans, tomatoes, peanut butter, and dried fruits are some foods that are high in potassium and should be avoided on a low potassium diet.
Low Fat/Low Cholesterol
Some patients may need to limit their fat or cholesterol intake. People with heart disease, high cholesterol levels, gallbladder disease, liver disease, and some digestive problems may have to limit how much fat or cholesterol is in their diets.
Here are some tips for Home Health Aides / Personal Care Aides to help a patient on a low fat / low cholesterol diet :
- Eat only lean cuts of meats, fish, and poultry.
- Remove the skin from poultry before cooking.
- Limit the amount of egg yolks consumed. Use more egg whites when scrambling eggs or making an omelet.
- Avoid adding extra oils and butter to foods.
- Use light or diet margarine.
- Do not fry foods. Steam, bake, or roast foods.
- Use low-fat or fat-free milk and milk products.

Bland
Patients who have digestive problems such as Crohn’s disease or Irritable Bowel Syndrome, or ulcers in their digestive tract may be required to be on a bland diet . This means that foods that are irritating to the gastric mucosa (stomach lining) need to be eliminated. This is because certain foods make the stomach produce more acid. Increased acid can lead to irritation of the stomach or other parts of the digestive tract such as the small or large intestine. Foods high in acid content can also irritate the esophagus as they are being swallowed.
Here are some foods to avoid for a bland diet:
- Alcohol
- Caffeine (tea, cola, coffee)
- Spices (peppers such as black pepper, chili pepper, cayenne pepper)
- Citrus fruits and juices (oranges, grapefruit, lemons, limes)
Gluten-Free Diet
Some people may be required to be on a gluten– free diet . Gluten is a protein in wheat, rye, and barley, and helps to hold foods together (Celiac Disease Foundation, 2015c). Gluten can also be found in foods in which it may be unexpected that gluten would be so it is important that Home Health Aides/Personal Care Aides always read food labels when preparing food for patients on a gluten-free diet. People on a gluten-free diet may have digestive problems, such as celiac disease.
Celiac disease is an autoimmune disorder in which ingestion of gluten leads to damage to the small intestine. Eating gluten leads to damage on the villi , which are finger-like projections lining the small intestine. These villi are important to help us absorb nutrients. When they are damaged, malabsorption (difficulty absorbing nutrients) can occur. This can lead to many health problems. According to the Celiac Disease Foundation (2015b) about 1 in 100 people around the world have Celiac disease with 2.5 million Americans undiagnosed with this potentially health-complicating disease.
Some people have a gluten intolerance , which does not damage the villi but causes symptoms such as abdominal pain, gas, and diarrhea (Leahy, Fuze & Grafe, 2013). Other people may just prefer to be on a gluten-free diet.
Here is a list of foods to avoid for people on a gluten – free diet :
- Wheat (found in breads, baked goods, crackers, tortillas, soups, cereals, sauces, and salad dressings)
- Barley (found in malt, food coloring, soups, and beer)
- Rye (found in breads, cereals, and rye beer)
It can be frustrating for a person who cannot have gluten to find foods they can have. As a Home Health Aide/Personal Care Aide you can assist your patient to make healthy food selections to help them be compliant with their diet. Many foods are naturally gluten-free such as fruit, vegetables, meat, poultry, dairy, beans, legumes, and nuts (Celiac Disease Foundation, 2015a).
Here is a list of items according to the Celiac Disease Foundation that a person on a gluten– free diet can have :
- Fruits
- Vegetables
- Meat, poultry, and fish
- Beans, legumes, and peas
- Milk and milk products
- Rice
- Cornmeal
- Soy
- Potato
- Quinoa
- Tapioca
- Juices, soda, sport drinks
There are many products now found on the market that make bread, cereals, and pasta without gluten in them. These items may be more expensive. It is important to always read the label. Many foods are made with wheat in them, which is often used as a thickener.
1. Which of the following foods should a person on a gluten-free diet avoid? Select all .
a). Wheat cereals and breads
b). Corn
c). Potatoes
d). Vegetable soup with barley
e). Soy
f). Rye bread
- Answer
-
1. A, D, F
FEEDBACK :
Foods with wheat such as cereals and breads, foods with barley, and those containing rye should all be avoided by a person on a gluten-free diet. Gluten is a protein found in products with wheat, rye, and barley.
Vegetarian Diet
Some people may be on a vegetarian diet for health, personal, or religious reasons. There are many different types of vegetarian diets, which indicate what types of foods are avoided. According to the American Heart Association (2014a) there are several types of vegetarian diets, including lacto-vegetarian, ovo-vegetarian, vegan, and semi-vegetarian.
Lacto – vegetarian : excludes meat, fish, poultry, and eggs. This diet allows dairy products, along with fruits, vegetables, nuts, seeds, grains, legumes, lentils, and peas.
Ovo – vegetarian : excludes meat, fish, poultry, and dairy, but allows eggs, along with fruits, vegetables, nuts, seeds, grains, legumes, lentils, and peas.
Vegan : excludes all meat, fish, poultry, eggs, dairy products, and any food with these ingredients. Foods allowed include fruits, vegetables, nuts, seeds, grains, legumes, lentils, and peas.
Semi – vegetarian : excludes red meat, but may allow poultry, fish, eggs, and dairy products, along with fruits, vegetables, nuts, seeds, grains, legumes, lentils, and peas.
It is important to remember that people who eat vegetarian type diets need to consume enough nutrients in their diets for good health. While they do not get protein from meat, fish, and poultry, protein can be found in other food sources. Excellent alternatives to meat protein include soy proteins and protein from legumes, lentils, peas, and milk products.
Liquid Diet
Liquid diets are those that patients may be on for a short period of time for a specific reason. Some people may need to be on a liquid diet prior to a medical procedure, test, or surgery. Others may be on a liquid diet in order to help heal their stomach or intestines from a medical condition. Foods on a liquid diet must be in their liquid state. There are two types of liquid diets: clear liquid and full liquid . A rule of thumb for clear liquid diets is that you should be able to see through them. They may be colored, such as popsicles and jello, but you can see through these foods.
Clear liquid diet :
- Clear juices (apple, grape)
- Broth
- Gelatin
- Popsicles
- Coffee, tea (no milk or cream)
- Gelatin (jello)
- Sugar, honey
- Hard candies
Full liquid diet :
- All items on the clear liquid diet list
- All fruit and vegetable based drinks
- Cream based soups
- Milk and milk products
- Milk shakes
- Yogurt (no fruit or seeds)
- Pudding
- Ice cream, frozen yogurt
- Cream of wheat, cream of rice
1. A full liquid diet means the person can have all items on the clear liquid food list plus also items like cream of wheat, milk shakes, yogurt, and pudding. True or False? ________
2. Which of these items would not be allowed on a clear liquid diet?
a). Beef broth
b). Jello
c). Frozen yogurt
d). Apple juice
- Answer
-
1. True
2. C
FEEDBACK:
1. A person on a full liquid diet can have all foods on a clear liquid diet plus cream of wheat, milk shakes, frozen yogurt, yogurt, and pudding.
2. Frozen yogurt would not be allowed on a clear liquid diet as it is made with dairy. A good rule of thumb is if you can’t see through it, it is not allowed on a clear liquid diet. Beef broth, jello, and apple juice would be allowed on a clear liquid diet.
Soft Diet
Soft diets are used for people who may have poor dentition , (this refers to the strength, number of, and arrangement of teeth in the mouth) who are recovering from a gastrointestinal surgery, and people who have difficulty with chewing and swallowing. When a patient is on a soft diet, the HHA/PCA must be concerned with the consistency of the diet. Soft foods require almost no chewing. They are easy to chew and swallow. Raw fruits and vegetables, hard-to-chew meats, and dry foods that can easily be choked on such as crackers and dry toast should be avoided.
Mechanically Altered Diets
A mechanically altered diet is a diet in which the texture (consistency) of the food is changed to help the person chew or swallow. A mechanically altered diet is given to a person who has dysphagia. Dysphagia means difficulty chewing or swallowing. A dysphagia diet or mechanically altered diet makes it easier to chew and swallow food and reduces the risk of aspiration (Ohio State University Wexner Medical Center, 2013). Dysphagia can occur due to many reasons, including reflux which can eventually cause the esophagus to narrow, making food difficult to pass through, neurological disorders such as from a stroke, cerebral palsy, multiple sclerosis, Alzheimer’s disease, and Parkinson’s disease (Jackson, 2015). For a comprehensive list of food suggestions on mechanically altered diet as well as recipes, visit http :// gicare . com / diets / dysphagia – diet /
Mechanical soft diet
A mechanical soft diet is a diet that consists of food that is made softer and easier to chew and swallow by changing the texture of the food. For example, by cooking chopped or diced carrots until soft, they can be mashed or pureed. Cooked foods can also be chopped, diced, or ground. This helps a person who has trouble chewing and swallowing to still take in the nutrients they need.
Preparing mechanically altered diets :
- Chopping food means to cut it into small pieces. Foods should be about ½ inch in size, which would be about the size of uncooked elbow macaroni or small croutons (Jackson, 2015). Food can be chopped by using a sharp knife on a cutting board.
- Grinding food means to cut it into even smaller pieces. Foods should be ¼ inch or less, which is about the size of a grain of rice (Jackson, 2015). Foods can be ground by cutting food that is chopped into even smaller pieces using a knife on a cutting board. Foods can also be ground in a food processor or blender. You may need to add a small amount of liquid to help the food become ground.
- Pureed means to put cooked and cut up foods into a blender or processor, while adding some liquid in order to make it into mashed potato-like consistency (Jackson, 2015). Foods can also be pureed by pushing soft, cooked foods through a colander or sieve with the back of a spoon.
- Meats, fish, and poultry can be cooked, finely chopped, and then ground or pureed using a food processor or blender. About an ounce of liquid such as gravy, milk, or water can be added to about 3 ounces of cooked meats to help them easily process in the blender (Jackson, 2015).
- Sauce and gravy can be added to help the person swallow, while also giving the food good taste.
- Cooked vegetables and fruits can also be blended by adding about an ounce of liquid to help them puree.
- Always cook foods such as meats prior to grinding or pureeing.
- Do not mix foods in the blender. Blend your meats, grains, vegetables, and fruits separately. Keep them separated on the serving plate just as you would for any other diet.
- Seasonings, sauces, gravies, spices, and herbs can be added for flavor, unless the Care Plan states otherwise.
Pureed diet
A diet that is pureed consists of food that is cooked and then chopped, blended, or ground into a thick paste that is the consistency of baby food or mashed potatoes. No chewing is required for this type of diet.
Mechanical soft diet f ood s uggestions :
- Soft, moist ground or finely diced meats, poultry, fish
- Meatloaf
- Egg salad, tuna salad, chicken salad (avoid adding chopped onions or celery)
- Sloppy Joe (without the bun)
- Chili (make sure the pieces are cut or diced)
- Cottage cheese
- Scrambled and soft cooked eggs
- Moist macaroni and cheese or small pastas
- Bananas, cut into small pieces
- Soft canned or cooked chopped or diced fruits (avoid seeds and skin)
- Pancakes or French toast with syrup, cut into small pieces
- Hot cooked cereal
- Applesauce
- Soft cooked, cut up vegetables
- Yogurt
- Pudding or custard
- Jello
- Ice cream, frozen yogurt
- Cream or broth based soups (thicken according to Care Plan)
Foods to a void :
- Dry, tough meats
- Raw fruits or vegetables
- Fruits or vegetables with skin or seeds
- Nuts, seeds
- Peanut butter
- Hard cooked eggs
- Bread and bread products
- Granola
- Popcorn, chips, or pretzels
- Crackers
- Dried fruits
- Salads
- Soups with meat, rice, corn, peas
- Gum, hard candy
1. Which of the following foods are safe to eat on a mechanical diet? Select all that apply .
a). Meatloaf
b). Applesauce
c). Gum
d). Popcorn
e). Cream of wheat
f). Yogurt
g). Granola
h). Tough meat
i). Crackers
j). Jello
k). Egg salad
l). Cottage cheese
m). Dried fruit
n). Salad
o). Pudding
- Answer
-
1. A, B, E, F, J, K, L, & O
FEEDBACK :
Meatloaf, applesauce, cream of wheat, yogurt, jello, egg salad, cottage cheese, and pudding are safe foods for a person on a mechanical diet to eat. Dried fruit, salad, crackers, granola, tough meats, gum, and popcorn are not safe foods for a person on a mechanical diet. Always follow the Care Plan. When in doubt, ask your supervisor.
Provide nutritious and attractive mechanically altered food
When food is mechanically altered (chopped, ground, or pureed) it may lose its appeal to the patient. It looks different than the food they are used to. It is important for Home Health Aides/Personal Care Aides to remember to maintain nutritious meal planning using ChooseMyPlate guidelines, and to try to present food in an appealing way.
Important things to keep in mind when preparing mechanically altered diet s :
- Use different colored foods for visual appeal.
- Put sauces and gravies on top of food as you normally would, even if it looks different.
- Keep foods separated on the serving plate. The plate should look like any other plate. It should not be a mix of all the foods into one pile.
- Tell patients what the food is as it will look different than the food they are used to. Even though it looks different, remind them that it can still taste delicious.
- Use special molds to make the food look attractive. Many food molds are on the market to make food look like what it is. For example, fish shaped molds help indicate the food is fish, even if it does not look it.
- Use food thickeners as indicated and according to package instructions.
- Encourage the use of adaptive eating equipment such as cups, plates, and eating utensils to allow independence during meal time.
- If a patient has a weak side, feed to the stronger side.
- Encourage or feed only small bites at a time. Allow the patient to chew/swallow completely before serving the next bite. If necessary, check the inside of their mouth for pocketed food.
- Be cautious about patient positioning during feeding, especially for patients who have trouble chewing or swallowing.
- Always sit a patient in an upright position to avoid choking or aspiration (when food enters areas of the respiratory tract such as the lungs where it does not belong, which could lead to pneumonia).
- Encourage the patient to tuck their chin down and bend their body forward while swallowing. This encourages the food to move down the esophagus .
- Keep the patient in an upright position for 30-60 minutes after eating.
True or False
1. It is very important, especially for a patient with a swallowing problem to be in an upright position during mealtime and to be kept upright for 30-60 minutes after eating to prevent choking. True or False? ______
2. It is okay to put all the food items together in the blender instead of blending each separately when preparing a pureed diet for a patient, as this saves time.
True or False? ________
- Answer
-
1. True
2. False
FEEDBACK :
1. Keep patients in an upright position during a meal and for 30-60 minutes after a meal to prevent aspiration.
2. Never put all the foods together in a blender. Keep foods separately on a plate just as you would when serving any patient, whether the food is pureed or not. This helps to keep food attractive and appealing for a patient, which aids in helping them maintain good nutrition.
Follow safe food preparation and storage guidelines
Always use proper food safety and storage precautions. This applies to preparing any type of food, even foods that are mechanically altered.
Keep the following important points in mind :
- Wash your hands before preparing foods.
- Keep your preparation space clean.
- Wash equipment such as blenders and food processors thoroughly and allow to air-dry before storing.
- Use safe knife skills while chopping and grinding foods.
- Only use a sharp knife as a dull one increases your chance of getting cut.
- When leaving your preparation area, ensure knives are placed away from the counter edge to avoid someone accidentally getting cut.
- Follow proper food storage guidelines.
- Keep hot foods hot and cold foods cold.
- Refrigerate or freeze leftovers immediately to avoid spoilage.


22: Magnetism
Magnetism is a class of physical phenomena that are mediated by magnetic fields. Electric currents and the magnetic moments of elementary particles give rise to a magnetic field, which acts on other currents and magnetic moments. Every material is influenced to some extent by a magnetic field.
-
- 22.1: Magnets
- Magnetism is a subject that includes the properties of magnets, the effect of the magnetic force on moving charges and currents, and the creation of magnetic fields by currents. There are two types of magnetic poles, called the north magnetic pole and south magnetic pole. North magnetic poles are those that are attracted toward the Earth’s geographic north pole. Like poles repel and unlike poles attract. Magnetic poles always occur in pairs of north and south.
-
- 22.2: Ferromagnets and Electromagnets
- All magnetism is created by electric current. Ferromagnetic materials, such as iron, are those that exhibit strong magnetic effects. The atoms in ferromagnetic materials act like small magnets (due to currents within the atoms) and can be aligned, usually in millimeter-sized regions called domains. Domains can grow and align on a larger scale, producing permanent magnets. Such a material is magnetized, or induced to be magnetic.
-
- 22.3: Magnetic Fields and Magnetic Field Lines
- Magnetic fields can be pictorially represented by magnetic field lines, the properties of which are as follows: The field is tangent to the magnetic field line. Field strength is proportional to the line density. Field lines cannot cross. Field lines are continuous loops.
-
- 22.4: Magnetic Field Strength- Force on a Moving Charge in a Magnetic Field
- Magnetic fields exert a force on a moving charge q. The SI unit for magnetic field strength B is the tesla (T). The direction of the force on a moving charge is given by right hand rule 1: Point the thumb of the right hand in the direction of v, the fingers in the direction of B, and a perpendicular to the palm points in the direction of F. The force is perpendicular to the plane formed by \(mathbf{v}\) and \mathbf{B}.
-
- 22.5: Force on a Moving Charge in a Magnetic Field- Examples and Applications
- Magnetic force can supply centripetal force and cause a charged particle to move in a circular path of radius \[r = \frac{mv}{qB},\] where \(v\) is the component of the velocity perpendicular to \(B\) for a charged particle with mass \(m\) and charge \(q\).
-
- 22.6: The Hall Effect
- We have seen effects of a magnetic field on free-moving charges. The magnetic field also affects charges moving in a conductor. One result is the Hall effect, which has important implications and applications. The Hall effect is the creation of voltage εε\varepsilon, known as the Hall emf, across a current-carrying conductor by a magnetic field.
-
- 22.7: Magnetic Force on a Current-Carrying Conductor
- The magnetic force on current-carrying conductors is given by \[F = \pi B sin \theta,\] where \(\) is the current, \(l\) is the length of a straight conductor in a uniform magnetic field \(B\), and \(\theta\) is the angle between \(I\) and \(B\). The force follows RHR-1 with the thumb in the direction of \(I\).\
-
- 22.8: Torque on a Current Loop - Motors and Meters
- The torque \(\tau\) on a current-carrying loop of any shape in a uniform magnetic field. is \[\tau = NIABsin\theta,\] where \(N\) is the number of turns, \(I\) is the current, \(A\) is the area of the loop, \(B\) is the magnetic field strength, and \(\theta\) is the angle between the perpendicular to the loop and the magnetic field.
-
- 22.9: Magnetic Fields Produced by Currents- Ampere’s Law
- The strength of the magnetic field created by current in a long straight wire is given by \[B = \frac{\mu_{0}I}{2 \pi r} \left(long \quad straight \quad wire\right),\tag{22.10.1}\] where \(I\) is the current, \(r\) is the shortest distance to the wire, and the constant \(\mu_{0} = 4\pi \times 10^{-7} T \cdot m/a\) is the permeability of free space. The direction of the magnetic field created by a long straight wire is given by right hand rule 2 (RHR-2): Point the thumb of the right hand in the d
-
- 22.10: Magnetic Force between Two Parallel Conductors
- The force between two parallel currents \(I_{1}\) and \(I_{2}\) separated by a distance \(r\), has a magnitude per unit length given by \[\frac{F}{l} = \frac{\mu_{0}I_{1}I_{2}}{2\pi r}.\] The force is attractive if the currents are in the same direction, repulsive if they are in opposite directions.
Thumbnail: Magnetic field of an ideal cylindrical magnet with its axis of symmetry inside the image plane. The magnetic field is represented by magnetic field lines, which show the direction of the field at different points. (CC-SA-BY-3.0; Geek3).


4.1: Compost Facility Field Trip
Learning Objectives
- Discuss vermicomposting and review how worm farms are managed.
- Visit the K-State composting operation at the Agronomy North Farm.
- Learn about managing a thermophilic composting operation.
- Discuss how organic waste from the KSU campus is turned into compost.
- Conduct a test for compost maturity
Composting is essentially decomposition under controlled conditions. It uses soil organisms to break down organic matter, resulting in CO 2 being released to the atmosphere, and ultimately resulting in organic matter that’s highly concentrated in nutrients. It is performed at various scales for many reasons, such as municipalities composting food and yard waste to reduce landfill expenses, farmers composting deceased animals to reduce rendering costs, gardeners turning their autumn leaves into valuable mulch, and a coffee drinker feeding coffee grounds and filters to earthworms. In this lab we will examine the Soils Teaching Lab worm farm, and visit a model industrial scale composting facility at the Agronomy North Farm where dining center food waste and campus greenhouse waste is turned into compost to be later used for agricultural research experiments.
Materials
- Active worm farm
- Compost thermometer
- Basic Compost Maturity Test, Woods End Laboratories, Inc.
- Optional: Digital Color Reader – Multi-Mode Unit, Woods End Laboratories, Inc.
- Compost Maturity Test Directions, Woods End Laboratories, Inc.
Recommended Reading & Viewing
- Agronomy Compost Facility (Moorberg, 2020) ( Note: this is a video tour of the compost facility, and is an abbreviated version of the in-person tour. )
- Types of Composting and Understanding the Process (US EPA, 2015)
- Solvita Compost Test: How to Perform It (Woods End Laboratories, 2009)
Prelab Assignment
Using the recommended reading and the introduction to this lab, consider the questions listed below. These definitions/questions will provide a concise summary of the major concepts addressed in the lab. They will also serve as the basis for the post-lab quiz and are useful study notes for exams.
- Define vermicomposting.
- Define thermophilic composting and describe how it results in the building up heat.
- Name and describe the three stages of thermophilic composting.
- Organic matter is decomposed during composting, resulting in a decrease in the overall C content and relatively low C:N ratios. Describe how the C:N ratio changes over time, and identify the gas that is released as C is emitted to the atmosphere.
- What is the range of C:N ratios that would be expected in mature compost?
- Why does compost need to be turned during thermophilic composting?
Introduction
Composting is the practice of creating soil organic material from decomposing organic matter under aerobic conditions outside the soil. The product, compost, is highly desirable, commonly used in potting soil mixes, as mulch, as slow-release fertilizer, and as a soil amendment. Compost can be produced under ambient temperatures through vermicomposting (composting using litter-dwelling worms) or via slow decomposition.
Vermicomposting is growing in popularity.  Worm farms can be as small and simple as a plastic bin tucked under a desk, or large, industrial-scale operations. The Soils Teaching Lab worm farm is built with multiple levels. Note the increasing level of decay as you go from the top tray to the bottom. Common food wastes and some paper wastes make great substrates for worms. We will discuss what foods are ideal for worms, and what things should be avoided later.
A more common method of composting is thermophilic decomposition. Decomposing organic matter in thermophilic composting undergoes a dramatic buildup in heat due to both the mass of the decomposing material and the insulating properties of the pile itself. Thermophilic composting involves three distinct stages: the mesophilic, thermophilic, and curing stages. During the mesophilic stage, easy-to-decompose organic materials are metabolized, with a resulting gradual increase in temperature to more than 40°C. The next stage is the thermophilic stage, in which the temperature increases to between 50 and 75°C. Materials like cellulose that decompose more slowly and under more difficult conditions decompose during this stage. Regular mixing is required during this stage so that aerobic decomposition can be maintained. Without adequate oxygen, the humification process will slow, and the overall time required for decomposition will increase. The final stage is the curing stage. During this stage, the temperature drops to near-ambient conditions. As decomposition and humification progresses, carbon is removed from the organic matter and released to the atmosphere as CO 2 . This removal of carbon decreases the C:N ratios and increases the percentages of N, P, and other nutrients in the compost.
At the K-State compost facility at the Agronomy North Farm, food waste from the campus dining halls and plant and soil waste from the campus greenhouses are broken down into compost that is then used for erosion, greenhouse, and field experiments. Figure 11.1 shows the C:N ratios, nitrogen content, and phosphorus content of both fresh and cured compost. The figure shows cattle feedlot manure for comparison. As carbon is removed through respiration and released to the atmosphere as CO 2 , the C:N ratio decreases, and nitrogen and phosphorus content increase to levels comparable to the manure.
Composting using thermophilic decomposition organic waste is becoming very popular because of its many benefits. Some examples include:
- Organic materials are safely stored with minimal odor release.
- Compost is easier to handle than raw materials because of a 30 to 60% reduction in volume
- Nitrate depression during the decomposition of organic materials with high initial C:N ratios is limited to the compost pile and thus avoids any effect on plant growth.
- High temperatures during the thermophilic stage kills most weed seeds and pathogens.
- Toxic organic compounds like pesticides are destroyed by the time the compost is mature.
- Many soil-borne plant diseases are suppressed through microbial antagonism
- Because compost is made from recently photosynthesized carbon, it is carbon-neutral and thus a much more environmentally friendly choice than peat for potting mixes.
Activity 1: Vermicomposting Demonstration.
We will start with a group discussion of vermicomposting, and a demonstration of the Soils Teaching Lab worm farm. Participate in the discussion, and answer the following questions.
Why are redworms preferred over nightcrawlers?
What can foods and substrates can be added to worm farms? Why is gritty material needed?
What should not be added to worm farms?
Activity 2: Compost Facility Field Trip.
You will be traveling to the Agronomy North Farm compost operation. The compost operation is managed by Dr. DeAnn Pressley and Brent Wehmeyer. They will lead a tour of the compost facility. During the tour, pay close attention, and answer the following questions.
What are the relative proportions of food waste and greenhouse waste in these compost piles?
How often is the fresh compost pile turned? How often are the mature compost piles turned?
List the temperatures for each of the different compost piles, from the freshest pile (closest to the dumpsters) to the most mature pile (furthest from the dumpsters).
What practices are used to control unwanted animals on the compost pile, such as crows?
How long does it take for the compost to reach maturity?
How many pounds of waste are typically composted each year?
Activity 3: Testing for Compost Maturity.
The last thing you’ll do at compost facility is collect samples from the compost piles to test for maturity. Collect one sample from a compost pile selected by your instructor using a clean trowel, then place it in a Solvita ® test jar. Gently tap the jar on the ground or a bench top to allow it to compact to the compost’s natural density. Add enough compost to reach the fill line. Then open the foil packs for a CO 2 test probe and a NH 4 + test probe and place those probes in the jar so that the color can be viewed from outside the jar. Close and tighten the lid to allow the CO 2 and NH 4 + gasses to react with the test probes. Mark the start time of the test on the jar lid, then allow the test to proceed for four hours at room temperature (20-25°C or 65-75°F). Your instructor will complete the test for you, and will send you a picture of the final test probes next to the color chart for reading the test probes. Your instructor may also provide you with more precise results analyzed using a Digital Color Reader to measure the color of the test probes.
What is the number on the color chart for the color most closely corresponding to the color on the CO 2 test probe?
What is the number on the color chart for the color most closely corresponding to the color on the NH 4 + test probe?
You will be provided a copy of the directions for the Solvita Compost Maturity Test. Based on the information provided in those directions, was the compost pile your lab group sampled mature?
Compare the maturity of the various compost piles using the results from this maturity test?
Assignment: Online Quiz
A quiz for this lab will be available online. Please access it as directed by your instructor.
Solvita ® is a trademark of Woods End Laboratories, Inc.


27.E: Active Galaxies, Quasars, and Supermassive Black Holes (Exercises)
For Further Exploration
Articles
Bartusiak, M. “A Beast in the Core.” Astronomy (July 1998): 42. On supermassive black holes at the centers of galaxies.
Disney, M. “A New Look at Quasars.” Scientific American (June 1998): 52.
Djorgovski, S. “Fires at Cosmic Dawn.” Astronomy (September 1995): 36. On quasars and what we can learn from them.
Ford, H., & Tsvetanov, Z. “Massive Black Holes at the Hearts of Galaxies.” Sky & Telescope (June 1996): 28. Nice overview.
Irion, R. “A Quasar in Every Galaxy?” Sky & Telescope (July 2006): 40. Discusses how supermassive black holes powering the centers of galaxies may be more common than thought.
Kormendy, J. “Why Are There so Many Black Holes?” Astronomy (August 2016): 26. Discussion of why supermassive black holes are so common in the universe.
Kruesi, L. “Secrets of the Brightest Objects in the Universe.” Astronomy (July 2013): 24. Review of our current understanding of quasars and how they help us learn about black holes.
Miller, M., et al. “Supermassive Black Holes: Shaping their Surroundings.” Sky & Telescope (April 2005): 42.
Nadis, S. “Exploring the Galaxy–Black Hole Connection.” Astronomy (May 2010): 28. Overview.
Nadis, S. “Here, There, and Everywhere.” Astronomy (February 2001): 34. On Hubble observations showing how common supermassive black holes are in galaxies.
Nadis, S. “Peering inside a Monster Galaxy.” Astronomy (May 2014): 24. What X-ray observations tell us about the mechanism that powers the active galaxy M87.
Olson, S. “Black Hole Hunters.” Astronomy (May 1999): 48. Profiles four astronomers who search for “hungry” black holes at the centers of active galaxies.
Peterson, B. “Solving the Quasar Puzzle.” Sky & Telescope (September 2013): 24. A review article on how we figured out that black holes were the power source for quasars, and how we view them today.
Tucker, W., et al. “Black Hole Blowback.” Scientific American (March 2007): 42.
Voit, G. “The Rise and Fall of Quasars.” Sky & Telescope (May 1999): 40. Good overview of how quasars fit into cosmic history.
Wanjek, C. “How Black Holes Helped Build the Universe.” Sky & Telescope (January 2007): 42. On the energy and outflow from disks around supermassive black holes; nice introduction.
Websites
Monsters in Galactic Nuclei: http://chandra.as.utexas.edu/stardate.html . An article on supermassive black holes by John Kormendy, from StarDate magazine.
Quasar Astronomy Forty Years On: http://www.astr.ua.edu/keel/agn/quasar40.html . A 2003 popular article by William Keel.
Quasars and Active Galactic Nuclei: www.astr.ua.edu/keel/agn/. An annotated gallery of images showing the wide range of activity in galaxies. There is also an introduction, a glossary, and background information. Also by William Keel.
Quasars: “The Light Fantastic”: http://hubblesite.org/newscenter/arc...35/background/ . This brief “backgrounder” from the public information office at the HubbleSite gives a bit of the history of the discovery and understanding of quasars.
Videos
Active Galaxies: https://www.youtube.com/watch?v=Y_HgsFmwCeg . Part of the Astronomy: Observations and Theories series; half-hour introduction to quasars and related objects (27:28).
Black Hole Chaos: The Environments of the Most Supermassive Black Holes in the Universe: https://www.youtube.com/watch?v=hzSgU-3d8QY . May 2013 lecture by Dr. Belinda Wilkes and Dr. Francesca Civano of the Center for Astrophysics in the CfA Observatory Nights Lecture Series (50:14).
Hubble and Black Holes: http://www.spacetelescope.org/videos/hubblecast43a/ . Hubblecast on black holes and active galactic nuclei (9:10).
Monster Black Holes: https://www.youtube.com/watch?v=LN9oYjNKBm8 . May 2013 lecture by Professor Chung-Pei Ma of the University of California, Berkeley; part of the Silicon Valley Astronomy Lecture Series (1:18:03).
Collaborative Group Activities
- When quasars were first discovered and the source of their great energy was unknown, some astronomers searched for evidence that quasars are much nearer to us than their redshifts imply. (That way, they would not have to produce so much energy to look as bright as they do.) One way was to find a “mismatched pair”—a quasar and a galaxy with different redshifts that lie in very nearly the same direction in the sky. Suppose you do find one and only one galaxy with a quasar very close by, and the redshift of the quasar is six times larger than that of the galaxy. Have your group discuss whether you could then conclude that the two objects are at the same distance and that redshift is not a reliable indicator of distance. Why? Suppose you found three such pairs, each with different mismatched redshifts? Suppose every galaxy has a nearby quasar with a different redshift. How would your answer change and why?
- Large ground-based telescopes typically can grant time to only one out of every four astronomers who apply for observing time. One prominent astronomer tried for several years to establish that the redshifts of quasars do not indicate their distances. At first, he was given time on the world’s largest telescope, but eventually it became clearer that quasars were just the centers of active galaxies and that their redshifts really did indicate distance. At that point, he was denied observing time by the committee of astronomers who reviewed such proposals. Suppose your group had been the committee. What decision would you have made? Why? (In general, what criteria should astronomers have for allowing astronomers whose views completely disagree with the prevailing opinion to be able to pursue their research?)
- Based on the information in this chapter and in Black Holes and Curved Spacetime, have your group discuss what it would be like near the event horizon of a supermassive black hole in a quasar or active galaxy. Make a list of all the reasons a trip to that region would not be good for your health. Be specific.
- Before we understood that the energy of quasars comes from supermassive black holes, astronomers were baffled by how such small regions could give off so much energy. A variety of models were suggested, some involving new physics or pretty “far out” ideas from current physics. Can your group come up with some areas of astronomy that you have studied in this course where we don’t yet have an explanation for something happening in the cosmos?
Review Questions
- Describe some differences between quasars and normal galaxies.
- Describe the arguments supporting the idea that quasars are at the distances indicated by their redshifts.
- In what ways are active galaxies like quasars but different from normal galaxies?
- Why could the concentration of matter at the center of an active galaxy like M87 not be made of stars?
- Describe the process by which the action of a black hole can explain the energy radiated by quasars.
- Describe the observations that convinced astronomers that M87 is an active galaxy.
- Why do astronomers believe that quasars represent an early stage in the evolution of galaxies?
- Why were quasars and active galaxies not initially recognized as being “special” in some way?
- What do we now understand to be the primary difference between normal galaxies and active galaxies?
- What is the typical structure we observe in a quasar at radio frequencies?
- What evidence do we have that the luminous central region of a quasar is small and compact?
Thought Questions
- Suppose you observe a star-like object in the sky.
- Why don’t any of the methods for establishing distances to galaxies, described in Galaxies (other than Hubble’s law itself), work for quasars?
- One of the early hypotheses to explain the high redshifts of quasars was that these objects had been ejected at very high speeds from other galaxies. This idea was rejected, because no quasars with large blueshifts have been found. Explain why we would expect to see quasars with both blueshifted and redshifted lines if they were ejected from nearby galaxies.
- A friend of yours who has watched many Star Trek episodes and movies says, “I thought that black holes pulled everything into them. Why then do astronomers think that black holes can explain the great outpouring of energy from quasars?” How would you respond?
- Could the Milky Way ever become an active galaxy? Is it likely to ever be as luminous as a quasar?
- Why are quasars generally so much more luminous (why do they put out so much more energy) than active galaxies?
- Suppose we detect a powerful radio source with a radio telescope. How could we determine whether or not this was a newly discovered quasar and not some nearby radio transmission?
- A friend tries to convince you that she can easily see a quasar in her backyard telescope. Would you believe her claim?
Figuring for Yourself
- Show that no matter how big a redshift (\(z\)) we measure, \(v/c\) will never be greater than 1. (In other words, no galaxy we observe can be moving away faster than the speed of light.)
- If a quasar has a redshift of 3.3, at what fraction of the speed of light is it moving away from us?
- If a quasar is moving away from us at \(v/c\) = 0.8, what is the measured redshift?
- In the chapter, we discussed that the largest redshifts found so far are greater than 6. Suppose we find a quasar with a redshift of 6.1. With what fraction of the speed of light is it moving away from us?
- Rapid variability in quasars indicates that the region in which the energy is generated must be small. You can show why this is true. Suppose, for example, that the region in which the energy is generated is a transparent sphere 1 light-year in diameter. Suppose that in 1 s this region brightens by a factor of 10 and remains bright for two years, after which it returns to its original luminosity. Draw its light curve (a graph of its brightness over time) as viewed from Earth.
- Large redshifts move the positions of spectral lines to longer wavelengths and change what can be observed from the ground. For example, suppose a quasar has a redshift of \(\frac{\Delta \lambda}{\lambda} = 4.1\). At what wavelength would you make observations in order to detect its Lyman line of hydrogen, which has a laboratory or rest wavelength of 121.6 nm? Would this line be observable with a ground-based telescope in a quasar with zero redshift? Would it be observable from the ground in a quasar with a redshift of \(\frac{\Delta \lambda}{\lambda} = 4.1\)?
- Once again in this chapter, we see the use of Kepler’s third law to estimate the mass of supermassive black holes. In the case of NGC 4261, this chapter supplied the result of the calculation of the mass of the black hole in NGC 4261. In order to get this answer, astronomers had to measure the velocity of particles in the ring of dust and gas that surrounds the black hole. How high were these velocities? Turn Kepler’s third law around and use the information given in this chapter about the galaxy NGC 4261—the mass of the black hole at its center and the diameter of the surrounding ring of dust and gas—to calculate how long it would take a dust particle in the ring to complete a single orbit around the black hole. Assume that the only force acting on the dust particle is the gravitational force exerted by the black hole. Calculate the velocity of the dust particle in km/s.
- In the Check Your Learning section of Example 27.1.1 in Section 27.1, you were told that several lines of hydrogen absorption in the visible spectrum have rest wavelengths of 410 nm, 434 nm, 486 nm, and 656 nm. In a spectrum of a distant galaxy, these same lines are observed to have wavelengths of 492 nm, 521 nm, 583 nm, and 787 nm, respectively. The example demonstrated that z = 0.20 for the 410 nm line. Show that you will obtain the same redshift regardless of which absorption line you measure.
- In the Check Your Learning section of Example 27.1.1 in Section 27.1, the author commented that even at \(z = 0.2\), there is already an 11% deviation between the relativistic and the classical solution. What is the percentage difference between the classical and relativistic results at \(z = 0.1\)? What is it for \(z = 0.5\)? What is it for \(z = 1\)?
- The quasar that appears the brightest in our sky, 3C 273, is located at a distance of 2.4 billion light-years. The Sun would have to be viewed from a distance of 1300 light-years to have the same apparent magnitude as 3C 273. Using the inverse square law for light, estimate the luminosity of 3C 273 in solar units.


8.19B: Protists as Primary Producers, Food Sources, and Symbionts
- Give examples of how protists act as primary producers
Primary Producers/Food Sources
Protists function in various ecological niches. Some protist species are essential components of the food chain and are generators of biomass.
Protists are essential sources of nutrition for many other organisms. In some cases, as in plankton, protists are consumed directly. Alternatively, photosynthetic protists serve as producers of nutrition for other organisms. For instance, photosynthetic dinoflagellates called zooxanthellae use sunlight to fix inorganic carbon. In this symbiotic relationship, these protists provide nutrients for the coral polyps that house them, giving corals a boost of energy to secrete a calcium carbonate skeleton. In turn, the corals provide the protists with a protected environment and the compounds needed for photosynthesis. This type of symbiotic relationship is important in nutrient-poor environments. Without dinoflagellate symbionts, corals lose algal pigments in a process called coral bleaching and they eventually die. This explains why reef-building corals do not reside in waters deeper than 20 meters: insufficient light reaches those depths for dinoflagellates to photosynthesize.
The protists themselves and their products of photosynthesis are essential, directly or indirectly, to the survival of organisms ranging from bacteria to mammals. As primary producers, protists feed a large proportion of the world’s aquatic species. (On land, terrestrial plants serve as primary producers. ) In fact, approximately one-quarter of the world’s photosynthesis is conducted by protists, particularly dinoflagellates, diatoms, and multicellular algae.
Protists do not only create food sources for sea-dwelling organisms. Certain anaerobic parabasalid species exist in the digestive tracts of termites and wood-eating cockroaches where they contribute an essential step in the digestion of cellulose ingested by these insects as they bore through wood.
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Key Points
- Photosynthetic protists serve as producers of nutrition for other organisms.
- Protists like zooxanthellae have a symbiotic relationship with coral reefs; the protists act as a food source for coral and the coral provides shelter and compounds for photosynthesis for the protists.
- Protists feed a large portion of the world’s aquatic species and conduct a quarter of the world’s photosynthesis.
- Protists help land-dwelling animals such as cockroaches and termites digest cellulose.
Key Terms
- zooxanthella : an animal of the genus Symbiodinium, a yellow dinoflagellate, notably found in coral reefs
- primary producer : an autotroph organism that produces complex organic matter using photosynthesis or chemosynthesis


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1.4.1: Introduction
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If \(a\), \(b\) and \(c\) are all distinct, then the set \(\{a, b, c\}\) is intuitively larger than \(\{a, b\}\). But what about infinite sets? Are they all as large as each other?
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1.4.2: Enumerations and Countable Sets
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We’ve already given examples of sets by listing their elements. Let’s discuss in more general terms how and when we can list the elements of a set, even if that set is infinite.
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1.4.3: Cantor’s Zig-Zag Method
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Cantor’s zig-zag method shows that the sets of pairs of elements of countably infinite sets is also countable.
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1.4.4: Pairing Functions and Codes
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We can use pairing functions to represent each pair of elements using a single number. Using the inverse of the pairing function, we can decode the number, i.e., find out which pair it represents.
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1.4.5: An Alternative Pairing Function
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There are other enumerations of \(\mathbb{N}^2\) that make it easier to figure out what their inverses are. Here is one.
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1.4.6: Uncountable Sets
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One way of showing that a set is uncountable is to give a diagonal argument. We assume that the set \(A\) in question is countable, and use a hypothetical enumeration to define an element of \(A\) which, by the very way we define it, is guaranteed to be different from every element in the enumeration. So the enumeration can’t be an enumeration of all of \(A\) after all, and we’ve shown that no enumeration of \(A\) can exist.
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1.4.7: Reduction
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A reduction shows that \(A\) is uncountable by associating every element of \(A\) with an element of some known uncountable set \(B\) in a surjective way. If this is possible, then a hypothetical enumeration of \(A\) would yield an enumeration of \(B\). Since \(B\) is uncountable, no enumeration of \(A\) can exist.
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1.4.8: Equinumerosity
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In general, infinite sets can be compared sizewise: \(A\) and \(B\) are the same size, or equinumerous, if there is a bijection between them.
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1.4.9: Sets of Different Sizes, and Cantor’s Theorem
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We can define that \(A\) is no larger than \(B\) (\({A}\preceq{B}\)) if there is an injective function from \(A\) to \(B\).
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1.4.10: The Notion of Size, and Schröder-Bernstein
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Here is an intuitive thought: if \(A\) is no larger than \(B\) and \(B\) is no larger than \(A\), then \(A\) and \(B\) are equinumerous. This is justified by the Schröder-Bernstein Theorem.
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1.4.11: Summary
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4.1: Introduction to the Integumentary System
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The integumentary system refers to the skin and its accessory structures, and it is responsible for much more than simply lending to your outward appearance. In the adult human body, the skin makes up about 16 percent of body weight and covers an area of 1.5 to 2 square meters. In fact, the skin and accessory structures are the largest organ system in the human body.
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4.2: Layers of the Skin
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The skin is composed of two structurally distinct layers. These layers of differing cells and tissues are held to underlying structures by connective tissue. Together these layers provide that structure necessary for the overall functions of the integumentary system.
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4.3: Accessory Structures of the Skin
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Accessory structures of the skin include hair, nails, sweat glands, and sebaceous glands. These structures embryologically originate from the epidermis and can extend down through the dermis into the hypodermis.
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4.4: Functions of the Integumentary System
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The skin and accessory structures perform a variety of essential functions, such as protecting the body from invasion by microorganisms, chemicals, and other environmental factors; preventing dehydration; acting as a sensory organ; modulating body temperature and electrolyte balance; and synthesizing vitamin D. The underlying hypodermis has important roles in storing fats, forming a “cushion” over underlying structures, and providing insulation from cold temperatures.
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4.5: Diseases, Disorders, and Injuries of the Integumentary System
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The integumentary system is susceptible to a variety of diseases, disorders, and injuries. These range from annoying but relatively benign bacterial or fungal infections that are categorized as disorders, to skin cancer and severe burns, which can be fatal.