3.1.1: Definitions and Examples
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Symmetry in Chemistry
Symmetry is actually a concept of mathematics and not of chemistry. However, symmetry, and the underlying mathematical theory for symmetry, group theory, are of tremendous importance in chemistry because they can be applied to many chemistry problems. For example it helps us to classify the structures of molecules and crystals, understand chemical bonding, predict vibrational spectra, and determine the optical activity of compounds. We will therefore first discuss the general foundations of symmetry and group theory, and then apply them to chemical problems, in particular chemical bonding.
Let us first find a definition for symmetry. Symmetry is very familiar to us as we associate symmetry with beauty, but very familiar things are not necessarily easy to define scientifically. One common definition is that symmetry is the self-similarity of an object. The more similar parts it has the more symmetric it appears. For example, we would argue that the two wings of the butterfly depicted look similar. If the left wing was very different from the right wing the butterfly would look less symmetric.
How can we measure the self-similarity, or symmetry of an object quantitatively? We can do this using the concept of the symmetry operation. It is defined as a movement of an object into an equivalent indistinguishable orientation. The number and kind of symmetry operations that can be carried out defines the symmetry of the object.
Definition: Symmetry Operation
Movement of an object into an equivalent indistinguishable orientation
Symmetry operations are carried out around so-called symmetry elements. A symmetry element is a point, line, or plane about which a symmetry operation is carried out. Let us understand know what symmetry elements and operations exist.
Definition: Symmetry Element
A point, line or plane about which a symmetry operation is carried out
The Identity Operation ( E )
The most simple operation is the identity operation. It can be denoted by the Schoenflies symbol E . Schoenflies symbols are the most common symbols to denote a symmetry operation. The identity operation says that each object is self-similar to itself when you do not move it in any way. This is a trivial statement, but as we will see later, the identity operation is necessary to make the mathematical framework of symmetry, group theory, complete. The identity operation is present in any object. In the example of the depicted snail shell it is the only operation (Fig. 2.1.2).
The Proper Rotation Operation
The proper rotation operation is a counter-clockwise rotation about a proper rotational axis by an angle of 360° over an integer number n. After that rotation the object must be indistinguishable from its original form. That means that the object after the rotation must superimpose the original object before the rotation. For example, when n=4, then we rotate around 90°, and after that the object must superimpose its original form. The proper rotational axis is the symmetry element associated with the proper rotation operation. It’s Schoenflies symbol is C n , whereby n is called the order of the rotational axis. A proper rotational operation has the symbol C n m , whereby m counts the number of times the operation is carried out. Overall we rotate by and angle of 360°× m/n when we carry out an operation C n m . This means that when m=n, then we have rotated around 360° (Fig. 2.1.3). Then all points in the object are at their original position. It is as though we had done nothing with the object. We can also say we have reached the identity E . In mathematical form we can say that C n n =E. If we rotated one more time, C n (n+1) , then this would be equal to rotating only one time, and thus C n (n+1) =C n 1 . For example if n=4, then rotating four times around 90° will produce the identity. We have rotated around 4 x 90°=360° which is the same a though we had not rotated at all, because all points in the object are in their original position after a 360° rotation. If we rotated 5 times around 90°, it would the same as rotating only one time.
If an object has several axes with different order n, then the one with the highest order is called the principal axis. If there is more than one axis of the same order, then they get distinguished by primes if they are not conjugate. We will learn about the exact definition of conjugation somewhat later, we can however often see by inspection if two axes are conjugate. This is usually the case when they pass through the object in a equivalent way, and rotate the object in an equivalent way. Axes that pass through less many bonds, get less many primes. This is just a convention, but you have to follow it. An additional rule is that an axis which is in the same position as the principal axis gets the least number of primes. For example, a square planar molecule such as PtCl 4 2- has a C 4 principal axis standing perpendicular to the square plane of the molecule. There is an additional C 2 axis where the C 4 axis runs. It is present because one can also rotate around 180°, and not only around 90°. This C 2 axis does not get any prime. You an see that there are four additional C 2 axes, two of them are denoted C 2 ’, and two others are denoted C 2 ’’. You can see that the two axes which only have one prime pass through two Pt-Cl bonds, while the ones that have two primes, do not pass through any bonds. The two C 2 ’ axes are conjugate, meaning that they transform the object in an equivalent way. The two C 2 ’’ axes are also conjugate.
We can now think about how we can write out the axes and their associated proper rotational operations in a systematic way. Let us first look at the symmetry elements, the proper axes: We conventionally write the principal axis first, and then all other axes from their highest to their lowest order. When there are axes of the same order, those with the least number of primes get denoted first, and those with the highest number of primes last. If there are conjugate axes then their number is placed in front of their Schoenflies symbol. For the proper axes of the PtCl 4 2- the notation would therefore be: C 4 , C 2 , 2C 2 ', 2C 2 ’’. Now let us see how to denote the rotation operations that are associated with these symmetry elements. The notation follows the same rules as for the symmetry elements. In addition, we have to consider that we must not count identical operations twice, we also do not denote the operations that are the same as the identity. For the C 4 axis there are four operations until we reach the identity. C 4 1 , C 4 2 , C 4 3 , and C 4 4 . For each of the C 2 , C 2 ’, and C 2 ’’ elements there are two operations until the identity is reached. C 2 1 , C 2 2 , C 2 1 ’, C 2 2 ’, C 2 1 ’’, C 2 2 ’’. The C 4 4 , the C 2 2 , the C 2 2 ’, and the C 2 2 ’’are the same as the identity and therefore we do not consider them. In addition we can see that the C 4 2 is the same as the C 2 1 . This is because the C 4 and the C 2 axes are in the same location, and rotating two times around 90° is the same as rotating one time around 180°. By convention we eliminate the operation associated with the higher order, thus the C 4 2 . The overall notation would then be: C 4 1 , C 4 3 , C 2 1 , 2 C 2 1 ’ 2 C 2 1 ’’ (Fig. 2.1.5).
The Reflection Operations (σ)
Let us look at reflection operations which are carried out around reflection planes, or mirror planes. Mirror planes have the Schoenflies symbol σ. When we carry out a reflection operation, then we move any point of the object to the other side of the mirror plane. There are two types of mirror planes, so-called horizontal mirror planes and vertical mirror planes. A horizontal mirror plane always stands perpendicular to the principal axis. For example in the depicted BH 3 molecule there is a horizontal mirror plane that stands perpendicular to a C 3 principal axis (Fig. 2.1.6). A horizontal mirror plane has the Schoenflies symbol σ h .
A vertical mirror plane has the property that it contains the principal axis, this means that it is part of the principal axis. It is denoted σ v . The BH 3 molecule has three vertical axes that pass through the three B-H bonds. You can see that each of them contain the principal C 3 axis (Fig. 2.1.7). The three vertical mirror planes are conjugate, and therefore they are not distinguished by primes. We can write a coefficient 3 in front of the symbol σ v to indicate that there are three conjugate vertical mirror planes.
Now let us look at how many symmetry operations are associated with a particular mirror plane. Fortunately, things are simple here: There is only one reflection operation associated with one mirror plane. This because reflecting two times at a mirror plane produces the identity E: σ v 2 =E. More generally, when we reflect n times, and n is an even number then this is the same as the identity or, σ v n =E (n is even). When n is odd the reflecting n times is the same a reflecting only one time or, σ v n =σ v 1 (n=odd).
The horizontal mirror plane of the BH 3 molecule deserves an additional comment. Carrying out the horizontal reflection does not change the position of any atom. It is important to understand why the operation does exist despite the fact it does not change the position of any atom. The criterion is not whether the position of an atom is changed, but whether the position of the points in the object changes. In the case of the BH 3 molecule the part of the molecule that is located above the mirror plane will be located below the mirror plane after the reflection operation has been carried out. Vice versa any part of the molecule that was formerly below the mirror plane will be located above the mirror plane after the reflection operation has been carried out. For example the lower half of the B atom with be above the mirror plane, and the half above the plane will be below the plane after the execution of the reflection operation.
Like non-conjugate proper rotations are distinguished by primes, also non-conjugate vertical mirror planes must be distinguished by primes. The smaller the number of bonds the vertical mirror plane contains, the larger the number of primes.
For example in the water molecule (Fig. 2.1.8) there are two non-conjugate vertical mirror planes. One contains the two O-H bonds, the other stands perpendicular to the first mirror plane. It is easy to see that these two mirror planes will not move the points in the atom in an equivalent way, and therefore they are not conjugate. The first mirror plane does not change the position of any atom, while the second one swaps the positions of the two hydrogen atoms. Therefore, the second mirror plane not containing any O-H bonds gets one prime the one that contains the O-H bonds doe not get a prime.
Dihedral Reflection Planes (σ d )
A special case of a vertical mirror plane is a dihedral mirror plane, denoted σ d . A dihedral mirror plane bisects the angle between two conjugate C 2 axes.
For example, in the PtCl 4 2- anion (FIg. 2.1.9) there are two vertical mirror planes that bisect the angle between the two conjugate C 2 axes that pass through the Pt-Cl bonds. Therefore, these mirror planes are vertical mirror planes σ d .
The Inversion Operation ( i )
Let us look next at the inversion operation which is symbolized by a Schoenflies symbol i . The symmetry element associated with an inversion, is the inversion center, also called center of symmetry. It is a single point. When an inversion operation is performed, then each point of the object is moved through the inversion center to the other side. Each coordinate in the object (x,y,z) is inverted into the coordinates (-x,-y,-z).
For example, the octahedral molecule SF 6 has an inversion center in the center of the molecule (Fig. 2.1.10). When the inversion operation is carried out, then each fluorine atom is moved through the inversion center to the other side. This means that the fluorine atoms 1 and 2 swap up their positions, the fluorine atoms 3 and 5 swap up their positions, and so do the fluorine atoms 4 and 6. The sulfur atom does not change its position. There is only one inversion operation associated with each inversion center. Inverting two times, or more generally, an integer number of two times produces the identity. Inverting an odd number of times is the same as inverting one time.
The Rotation-Reflection Operation (S n )
The rotation-reflection operation S n is the most complex symmetry operation. It is carried out in two steps. First, a rotation around an improper axis is carried out. The angle is determined by the order n of the improper axis, and is 360°/n. This axis is called improper, because the object does not need to superimpose the original object after the rotation. Achieving superposition requires the second step which is the reflection at a mirror plane that stands perpendicular to the improper axis. Only after the second step the operation is complete. The presence of the rotation-reflection does not require a proper rotational axis or a regular mirror plane σ to exist, however it also do not preclude their existence.
An example of a molecule with an improper axis is the methane molecule (Fig. 2.1.11). It has an S 4 improper axis. The axis bisects the H-C-H tetrahedral bond angle. The order of the axis is four which requires that we rotate by 90° around this axis. You can see that after we carry out the rotation, the molecule does not superimpose the original molecule. Only after we reflect the rotated molecule at a mirror plane standing perpendicular to the improper axis the molecule superimposes the original molecule.
Properties of the Rotation-Reflection Operation (S n )
The rotation-reflection has a number of interesting properties. One of the them is that an S 1 operation is the same as a reflection. This is because the order 1 implies a rotation around 360° which produces the identity, and all points within the object are in their original position. This is the same as though we had not rotated at all. This means actually we only did the second step, the reflection, and therefore the S 1 is identical to a “regular” reflection. The second property is that an S 2 operation is the same as an inversion. When you rotate around 180° and then reflect perpendicular to the improper axis of rotation, the positions of the points in the object change exactly the same way as they do when you invert through an inversion center.
Look for example at the SF 6 molecule again (Fig. 2.1.12) which has an inversion center in the center of the atom. We previously saw that when we carry out the inversion, the atoms 1 and 2 swap their position, and so do the atoms 3 and 5, as well as the atoms 4 and 6. Let us carry out the rotation-reflection, and see if the atoms change the same way. Firstly we rotate 180° around an axis that goes through the atoms 1 and 2. This leaves the positions of the atoms 1 and 2 unchanged, but swaps up the positions of the atoms 3 and 5, as well as the atoms 4 and 6. Next we must do a reflection at a plane that stands perpendicular to the improper axis. It is the plane defined by the atoms 3, 4, 5, and 6. Reflection at this plane does not change the positions of the atoms 3, 4, 5, and 6, but it swaps up the positions of the atoms 1 and 2, which lie above and below the plane, respectively. We can see that the positions of the atoms are the same as after the inversion.
The fact that we can express a reflection by an S 1 rotation-reflection, and an inversion by a S 2 rotation-reflection means that the reflection and the inversion are not independent symmetry operations, and we would not need them. The symmetry of an object could be fully described by the identity, proper rotations, and rotation-reflections. However, by convention, we use reflections and inversions instead of S 1 and S 2 , simply because it is easier for the human mind to perform 1-step operations, rather than 2-step operations.
Improper rotations have different properties depending on whether the order of the axis is even or odd. For even orders, the presence of an S n improper rotational axis implies that there must also be a proper rotational axis with an order n/2. For improper axes of even orders n the identity is produced after n rotation-reflections (Fig. 2.1.13).
For improper axes of odd order n, carrying out the rotation-reflection n times is the same as carrying out a reflection at a horizontal mirror plane. We need to do the rotation-reflection 2n times to reach the identity (Fig. 2.1.4).
Rotation-Reflections of PF 5 (an example with odd order)
Let us show these properties using two examples. Let us first look at an example with an improper rotation of odd order. The PF 5 molecule has trigonal pyramidal shape with three F atoms in the equatorial plane, and two additional F atoms above and below the plane. The F atoms have been numbered from 1 to 5 as indicated by the subscripts that follow the element symbol (Fig. 2.1.15). The PF 5 has an improper rotational axis S 3 that stands perpendicular to the equatorial plane going through the P-F 1 and P-F 2 bonds. Let us carry out the S 3 symmetry operations step by step and see how the atoms move.
We would expect that after we carry out the operation six times the identity will be produced. After three rotation-reflections we would expect that the object has been moved the same way a horizontal mirror plane would do. After the first rotation-reflection the atoms F 1 and F 2 have swapped their positions, and the three F atoms in the equatorial plane have been rotated by 120 degrees counter-clockwise. The F 5 atom now occupies the position of the F 4 , the F 4 has been rotated into the position of the F 3 , and the F 3 has been moved into the position of the F 5 .    Now let us compare the position of the atoms with the atom positions of the molecule we started with. We see that the position of the equatorial F atoms are same as in the beginning, but the position of the axial F 1 and F 2 atoms have been swapped up. This is equivalent to a reflection at a horizontal mirror plane located within the equatorial plane. This horizontal mirror plane would only swap up the axial F atoms, but would not move the equatorial ones. The S 3 4 operation again swaps up the position of the F 1 and the F 2 atoms, and rotates the equatorial F atoms by 120° counter-clockwise. Now, F 4 points toward us, F 3 points away, and F 5 is in the paper plane. After the fifth rotation-reflection F 1 and F 2 are again swapped so that F 1 is below and F 2 is above the equatorial plane. The atoms F 3 , F 4 and F 5 are rotated so that F 3 is in the paper plane, F 5 points toward us, and F 4 points away from us. The sixth rotation-reflection again swaps up F 1 and F 2 , and rotates the other F atoms around 120° counter-clockwise. We can now see that the produced molecule is the same as the one we started with, and the identity has been produced.
It is also noteworthy that the S 3 2 and the S 3 4 operation can be expressed by simpler operations, namely C 3 2 and C 3 1 proper rotation operations. We can understand this considering that an S 3 2 operation requires that we reflect two times, and reflecting two times is just like not reflecting at all. So effectively, we only have two rotations around 120° which is equivalent to C 3 2 . Similarly, the S 3 4 operation requires to reflect four times, which is the same as not reflecting at all. Thus, effectively, we only rotate four times around 120°. This is the same as rotating only one time around this angle, which is equivalent to a C 3 1 operation.
Using similar considerations, we can also understand why the S 3 3 operation is equivalent to a σ h . In this case, we rotate 3 x 120° = 360°, and rotating around 360° is the same as not rotating at all. In addition, we carry out a reflection three times. Reflecting an odd number of times is the same as reflecting only one time. Thus overall, we effectively carry out a single reflection only.
Overall, only the S 3 1 and the S 3 5 operations are unique, all others can be expressed by other, simpler operations.
Rotation-Reflections of CH 4 (an example with even order)
Let us illustrate the properties of an improper rotational axis with even order using the example methane (Fig. 2.1.16).
We previously saw that the methane molecule has an S 4 improper axis. Executing the symmetry operation once moves the H atoms 1 and 2 to the right, whereby atom 1 points to the back and atom 2 points to the front. After the second rotation-reflection the atoms 1 and 2 are reflected back to the left side, but because we have rotated two times by 90° atom 2 points up, and atom 1 points down. Similarly, atoms 3 and 4 are reflected back to the right side, but atom four points toward us and atom 3 points to the back. After three rotation-reflections, the atoms 1 and 2 are again on the right side with atom 2 pointing to the back and atom 1 pointing to the front. Atoms 3 and 4 are on the left with atom 4 pointing up, and atom 3 pointing downward. After the operation is carried out four times, all atoms are back in their original position, meaning the identity has been produced. This is what we expected for an improper rotation of even order. We can see in addition that the S 4 2 operation is the same as an C 2 operation. Rotating two times about 90° is the same as rotating about 180°. Reflecting two times is the same as not reflecting at all. Therefore, effectively, we only rotated by 180° which is the same as what a C 2 1 operation does. Because an S 4 2 can be expressed by a C 2 1 , and an S 4 4 is the same as the identity, only the S 4 1 and the S 4 3 are unique symmetry operations.
The Symmetry of a Molecule
We have now discussed all types of symmetry elements and operations that can exist in an object. Next, let us think about how we can define the overall symmetry of an object. The overall symmetry of an object is defined as the sum of all the symmetry operations for this object. If two objects have exactly the same symmetry elements and operations then their symmetry is the same. Using the mathematical language of group theory, the mathematical theory for symmetry, we can say they belong to the same point group. The name point group comes from the fact, that it has at least one invariant point. The number of symmetry operations belonging to a point group is called the order of the point group. Let us see next what different point groups we know and how we can systematically classify them.
Dr. Kai Landskron ( Lehigh University ). If you like this textbook, please consider to make a donation to support the author's research at Lehigh University: Click Here to Donate .


1.3: What are Vaccines?
Vaccines are a product given orally, nasally, or by injection to prevent people from acquiring specific diseases. The types of diseases that are prevented by vaccines are called “vaccine-preventable diseases.” Examples of these diseases include chickenpox (varicella) and human papilloma virus (HPV) – though vaccine-preventable diseases will be discussed in detail in Chapter 2 .
As noted by the World Health Organization (n.d.), vaccines are a safe and proven tool to prevent life-threatening infectious diseases. “Vaccines work by taking advantage of antigen recognition and the antibody response” (Khan Academy, n.d., under Vaccines). “A vaccine contains the antigens of a [specific] pathogen that causes disease” (Khan Academy, n.d., under Vaccines). For example, the measles vaccine contains the antigens specific to measles. When a person is vaccinated against measles, the immune system responds by stimulating antibody-producing cells that make measles antibodies. As a result, if the body comes into contact with measles infection in the future, the body is prepared to fight it. See Video Clip 1.1 for a description of what vaccines are and how they work.
Video Clip 1.1: “ What are vaccines and how do they work?” by Oregan State University Ecampus. CC BY-NC-SA 4.0
Vaccines play an important role in healthy communities for several reasons:
- First, they stimulate a person’s immune system to develop antibodies against diseases for protection against that disease.
- Second, they protect others by limiting the spread of disease (community immunity). This is important in order to protect vulnerable individuals such as infants, older adults, and those who are ill and unable to receive vaccines for a variety of reasons, including weakened immune systems.
Points of Consideration
The act of administering a vaccine is called “vaccination” and the substance administered is called the “vaccine.” “Immunization” is the process of developing an immune response; it is what happens to a person’s body after a vaccination. When a vaccine is introduced into the body, it stimulates the body to develop antibodies against that disease and protects the body against the disease if exposed to it. “Immunity” involves being protected from a disease and a person is “immune” when they are protected from the disease. If a person does not have immunity and is not protected, then they would be deemed “susceptible”.
An interactive or media element has been excluded from this version of the text. You can view it online here:
https://ecampusontario.pressbooks.pub/immunizations/?p=29


12.11A: Pain Sensation
Pain is an unpleasant sensation caused by the activation of nociceptors by thermal, mechanical, chemical, or other stimuli.
- Describe the sensation of pain in the peripheral nervous system
Key Points
- Sleeping or silent nociceptors do not respond to these types of signals, but may respond during inflammation of the surrounding tissue.
- Nociceptors receive and send pain signals through myelinated fast Aδ fibers and nonmyelinated slow C fibers that are only activated with intense or prolonged input.
- Nociceptive pain may also be divided into visceral, deep somatic, and superficial somatic pain.
- Neuropathic pain is caused by damage to the nervous system.
- Phantom pain is pain perceived as from amputated or paralyzed limbs.
- Psychogenic pain is caused or exacerbated by mental, emotional, and behavioral factors.
Key Terms
- visceral : Relating to the internal organs.
- ischemia : A restriction in blood supply to tissues, causing a shortage of oxygen and glucose needed for cellular metabolism.
- pain : An ache or bodily suffering, or an instance of this; an unpleasant sensation, resulting from a derangement of functions, disease, or injury by violence; hurt.
- nociceptor : A sensory receptor that sends signals that cause the perception of pain in response to a potentially damaging stimulus.
Overview
Pain is an unpleasant feeling often caused by intense or damaging stimuli, such as stubbing a toe, burning a finger, putting alcohol on a cut, and bumping the funny bone. The International Association for the Study of Pain’s widely used definition states: “Pain is an unpleasant sensory and emotional experience associated with actual or potential tissue damage, or described in terms of such damage.”
Pain motivates the individual to withdraw from damaging situations, to protect a damaged body part while it heals, and to avoid similar experiences in the future. Most pain resolves promptly once the painful stimulus is removed and the body has healed, but sometimes pain persists despite removal of the stimulus and apparent healing of the body. Pain sometimes arises in the absence of any detectable stimulus, damage, or disease.
Nociceptors
A nociceptor is a sensory receptor that responds to potentially damaging stimuli by sending nerve signals to the spinal cord and brain. This process, called nociception, usually causes the perception of pain. There are several types and functions of nociceptors:
- Thermal nociceptors are activated by noxious heat or cold at various temperatures.
- Mechanical nociceptors respond to excess pressure or mechanical deformation. They also respond to incisions that break the skin surface. These mechanical nociceptors frequently have polymodal characteristics. So it is possible that some of the transducers for thermal stimuli are the same for mechanical stimuli.
- Chemical nociceptors respond to a wide variety of spices commonly used in cooking. The one that sees the most response and is very widely tested is capsaicin. Other chemical stimulants are environmental irritants like acrolein, a World War I chemical weapon and a component of cigarette smoke. Besides these external stimulants, chemical nociceptors have the capacity to detect endogenous ligands, and certain fatty acid amines that arise from changes in internal tissues.
The peripheral terminal of the mature nociceptor is where the noxious stimuli are detected and transduced into electrical energy. When the electrical energy reaches a threshold value, an action potential is induced and driven towards the central nervous system. This leads to the train of events that allows for the conscious awareness of pain.
Gray matter in the spinal cord : A delta fibers (Aδ fibers), a type of sensory fiber, are associated with the sensation of cold and pressure. Aδ fibers are thinly myelinated; therefore, they conduct signals more rapidly than unmyelinated C fibers, but more slowly than other, more thickly myelinated “A” class fibers. Aδ fibers terminate at Rexed laminae I and V (labeled I and V in the diagram). C fibers respond to thermal, mechanical, and chemical stimuli and terminate at the Rexed lamina II (labeled II in the diagram).
Nociceptors have two different types of axons.
- The Aδ fiber axons are myelinated and can allow an action potential to travel at a rate of about 20 meters/second towards the central nervous system.
- The other type is the more slowly conducting C fiber axons. These only conduct at speeds of around 2 meters/second. This is due to the light or nonmyelination of the axon.
Phases of Pain
Pain comes in two phases. The first phase is mediated by the fast-conducting Aδ fibers, and the second part is due to C fibers. The pain associated with the Aδ fibers can be associated to an initial extremely sharp pain.
The second phase is a more prolonged and slightly less intense feeling of pain as a result of the damage. If there is massive or prolonged input to a C fiber, there is a progressive build up in the spinal cord dorsal horn; this phenomenon is similar to tetanus in muscles but is called wind-up. If wind-up occurs, there is a probability of increased sensitivity to pain.
Although each nociceptor can have a variety of possible threshold levels, some do not respond at all to chemical, thermal, or mechanical stimuli unless injury actually has occurred. These are typically referred to as silent or sleeping nociceptors since their response comes only at the onset of inflammation to the surrounding tissue.
Types of Nociceptive Pain
Nociceptive pain can be divided into visceral, deep somatic and superficial somatic pain.
- Visceral structures are highly sensitive to stretch, ischemia, and inflammation, but relatively insensitive to other stimuli that normally evoke pain in other structures, such as burning and cutting. Visceral pain is diffuse, difficult to locate, and often referred to a distant, usually superficial, structure. It may be accompanied by nausea and vomiting and may be described as sickening, deep, squeezing, and dull.
- Deep somatic pain is initiated by stimulation of nociceptors in ligaments, tendons, bones, blood vessels, fasciae and muscles, and is a dull, aching, poorly localized pain. Examples include sprains and broken bones.
- Superficial pain is initiated by the activation of nociceptors in the skin or other superficial tissue, and is sharp, well-defined, and clearly located. Examples of injuries that produce superficial somatic pain include minor wounds and minor (first degree) burns.


23.5G: Transforming Chemical Energy
Cellular respiration is the process of transforming chemical energy into forms usable by the cell or organism.
- Discuss the importance of cellular respiration
Key Points
- Organisms ingest organic molecules like the carbohydrate glucose to obtain the energy needed for cellular functions.
- The energy in glucose can be extracted in a series of chemical reactions known as cellular respiration.
- Cellular respiration produces energy in the form of ATP, which is the universal energy currency for cells.
Key Terms
- aerobic respiration : the process of converting the biochemical energy in nutrients to ATP in the presence of oxygen
- adenosine triphosphate : a multifunctional nucleoside triphosphate used in cells as a coenzyme, often called the “molecular unit of energy currency” in intracellular energy transfer
- cellular respiration : the set of the metabolic reactions and processes that take place in the cells of organisms to convert biochemical energy from nutrients into adenosine triphosphate (ATP)
- catabolism : the breakdown of large molecules into smaller ones usually accompanied by the release of energy
Introduction: Cellular Respiration
An electrical energy plant converts energy from one form to another form that can be more easily used. For example, geothermal energy plants start with underground thermal energy (heat) and transform it into electrical energy that will be transported to homes and factories.
Energy Plant : This geothermal energy plant transforms thermal energy from deep in the ground into electrical energy, which can be easily used.
Like a generating plant, living organisms must take in energy from their environment and convert it into to a form their cells can use. Organisms ingest large molecules, like carbohydrates, proteins, and fats, and convert them into smaller molecules like carbon dioxide and water. This process is called cellular respiration, a form of catabolism, and makes energy available for the cell to use. The energy released by cellular respiration is temporarily captured by the formation of adenosine triphosphate (ATP) within the cell. ATP is the principle form of stored energy used for cellular functions and is frequently referred to as the energy currency of the cell.
The nutrients broken down through cellular respiration lose electrons throughout the process and are said to be oxidized. When oxygen is used to help drive the oxidation of nutrients the process is called aerobic respiration. Aerobic respiration is common among the eukaryotes, including humans, and takes place mostly within the mitochondria. Respiration occurs within the cytoplasm of prokaryotes. Several prokaryotes and a few eukaryotes use an inorganic molecule other than oxygen to drive the oxidation of their nutrients in a process called anaerobic respiration. Electron acceptors for anaerobic respiration include nitrate, sulfate, carbon dioxide, and several metal ions.
The energy released during cellular respiration is then used in other biological processes. These processes build larger molecules that are essential to an organism’s survival, such as amino acids, DNA, and proteins. Because they synthesize new molecules, these processes are examples of anabolism.
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- OpenStax College, Carbohydrate Metabolism. November 10, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m46451/latest/ . License : CC BY: Attribution
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8.7.2: Regulation of Gene Expression
-
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- Ying Liu
- City College of San Francisco
Learning Objectives
By the end of this section, you will be able to do the following:
- Discuss why every cell does not express all of its genes all of the time
- Describe how prokaryotic gene regulation occurs at the transcriptional level
- Discuss how eukaryotic gene regulation occurs at the epigenetic, transcriptional, post-transcriptional, translational, and post-translational levels
For a cell to function properly, necessary proteins must be synthesized at the proper time and place. All cells control or regulate the synthesis of proteins from information encoded in their DNA. The process of turning on a gene to produce RNA and protein is called gene expression . Whether in a simple unicellular organism or a complex multi-cellular organism, each cell controls when and how its genes are expressed. For this to occur, there must be internal chemical mechanisms that control when a gene is expressed to make RNA and protein, how much of the protein is made, and when it is time to stop making that protein because it is no longer needed.
The regulation of gene expression conserves energy and space. It would require a significant amount of energy for an organism to express every gene at all times, so it is more energy efficient to turn on the genes only when they are required. In addition, only expressing a subset of genes in each cell saves space because DNA must be unwound from its tightly coiled structure to transcribe and translate the DNA. Cells would have to be enormous if every protein were expressed in every cell all the time.
The control of gene expression is extremely complex. Malfunctions in this process are detrimental to the cell and can lead to the development of many diseases, including cancer.
Prokaryotic versus Eukaryotic Gene Expression
To understand how gene expression is regulated, we must first understand how a gene codes for a functional protein in a cell. The process occurs in both prokaryotic and eukaryotic cells, just in slightly different manners.
Prokaryotic organisms are single-celled organisms that lack a cell nucleus, and their DNA therefore floats freely in the cell cytoplasm. To synthesize a protein, the processes of transcription and translation occur almost simultaneously. When the resulting protein is no longer needed, transcription stops. As a result, the primary method to control what type of protein and how much of each protein is expressed in a prokaryotic cell is the regulation of DNA transcription. All of the subsequent steps occur automatically. When more protein is required, more transcription occurs. Therefore, in prokaryotic cells, the control of gene expression is mostly at the transcriptional level.
Eukaryotic cells, in contrast, have intracellular organelles that add to their complexity. In eukaryotic cells, the DNA is contained inside the cell’s nucleus and there it is transcribed into RNA. The newly synthesized RNA is then transported out of the nucleus into the cytoplasm, where ribosomes translate the RNA into protein. The processes of transcription and translation are physically separated by the nuclear membrane; transcription occurs only within the nucleus, and translation occurs only outside the nucleus in the cytoplasm. The regulation of gene expression can occur at all stages of the process (Figure 16.3). Regulation may occur when the DNA is uncoiled and loosened from nucleosomes to bind transcription factors ( epigenetic level), when the RNA is transcribed ( transcriptional level), when the RNA is processed and exported to the cytoplasm after it is transcribed ( post-transcriptional level), when the RNA is translated into protein ( translational level), or after the protein has been made ( post-translational level).
The differences in the regulation of gene expression between prokaryotes and eukaryotes are summarized in Table 16.1. The regulation of gene expression is discussed in detail in subsequent modules.
| Prokaryotic organisms | Eukaryotic organisms |
|---|---|
| Lack a membrane-bound nucleus | Contain nucleus |
| DNA is found in the cytoplasm | DNA is confined to the nuclear compartment |
| RNA transcription and protein formation occur almost simultaneously | RNA transcription occurs prior to protein formation, and it takes place in the nucleus. Translation of RNA to protein occurs in the cytoplasm. |
| Gene expression is regulated primarily at the transcriptional level | Gene expression is regulated at many levels (epigenetic, transcriptional, nuclear shuttling, post-transcriptional, translational, and post-translational) |
Evolution Connection
Evolution Connection
Evolution of Gene Regulation Prokaryotic cells can only regulate gene expression by controlling the amount of transcription. As eukaryotic cells evolved, the complexity of the control of gene expression increased. For example, with the evolution of eukaryotic cells came compartmentalization of important cellular components and cellular processes. A nuclear region that contains the DNA was formed. Transcription and translation were physically separated into two different cellular compartments. It therefore became possible to control gene expression by regulating transcription in the nucleus, and also by controlling the RNA levels and protein translation present outside the nucleus.
Most gene regulation is done to conserve cell resources. However, other regulatory processes may be defensive. Cellular processes such as gene silencing developed to protect the cell from viral or parasitic infections. If the cell could quickly shut off gene expression for a short period of time, it would be able to survive an infection when other organisms could not. Therefore, the organism evolved a new process that helped it survive, and it was able to pass this new development to offspring.


9.11: Skills Checklist- Application of Compression Stockings (TED Hose)
1. Gather Supplies: Compression stockings
2. Routine Pre-Procedure Steps:
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- Knock on the client’s door.
- Perform hand hygiene.
- Introduce yourself and identify the resident.
- Maintain respectful, courteous, and professional communication at all times.
- Provide for privacy.
- Explain the procedure to the client.
3. Procedure Steps:
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- Raise the bed height if needed.
- Position the resident in the supine position (bed flat).
- Expose only the leg you will be placing the stocking on.
- Roll, gather, or turn the stocking inside out to the heel.
- Place the stocking over the resident’s toes, foot, and heel.
- Check the placement of heel marking.
- Roll or gently pull the stocking up their leg.
- Check the resident’s toes for possible pressure from stocking.
- Adjust stocking as needed; stocking should be wrinkle-free to the knee or hip, depending on length of stocking.
4. Post-Procedure Steps:
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- Perform hand hygiene.
- Check on resident comfort and ask if anything else is needed.
- Ensure the bed is low and locked. Check the brakes.
- Place the call light or signaling device within reach of the resident.
- Open the door and privacy curtain.
- Perform hand hygiene.
- Document and report application of TED hose and any skin issues, pain with movement, or other changes noted with the resident.
View a YouTube video [1] of an instructor demonstration of a pplication of compression stockings (TED hose):
- Chippewa Valley Technical College. (2022, December 3). Application of Compression Stockings (TED hose). [Video]. YouTube. Video licensed under CC BY 4.0 . https://youtu.be/MeV0Kitfy0I ↵


2.3: READ- HSA Health Savings Accounts
HSA like a Pro!
Stretch your dollars further and put more money in your pocket.
What is a Health Savings Account?
HSAs are tax-advantaged member-owned accounts that let you save pre-tax¹ dollars for future qualified medical expenses. You can invest in mutual funds tax-free—and funds never expire.
Am I eligible for a Health Savings Account?
HSAs are available exclusively to those with a qualifying health plan.
- You are eligible if you have a High-deductible health plan
- You are not eligible if you have Healthcare coverage beyond qualified health plans (including Medicare enrollment)
- You are not eligible if you are being claimed as a dependent on someone’s tax returns.
- You are not eligible if you are Receiving Veterans Affairs benefits within the past three months
How do I sign up?
You can enroll in an HSA-qualified health plan and sign up for an account during your organization’s annual open enrollment. If you have a high-deductible health plan on your own—not offered through an employer—you can sign up right now.
Save
- Put more money in your pocket.
- Enjoy lower health plan premiums
- Qualified health plans offer lower premiums, enabling you to save potentially thousands every year.
- Keep your premium savings
- Just put the extra money you would’ve paid toward traditional premiums into your account. Voila!
- Long-term savings.
| FSA | HSA | |||
|---|---|---|---|---|
| Traditional health plan | HSA-qualified health plan | |||
| Higher premiums | Lower premiums | |||
| Lower deductibles | Higher deductibles | |||
| Doesn't cover premium payments | Cover premium payments | |||
| Funds expire | Funds don't expire |
Unlike Flexible Spending Accounts (FSA), you own your HSA. That means your entire balance rolls over every year—even if you change health plans, retire, or leave your employer.
HSA vs FSA comparison
The more you contribute, the more you save
Each contribution potentially reduces your annual tax bill. Link a bank account and make tax-deductible contributions anytime.
HSA Annual Contribution Limits
Tax year Individual coverage limit Family coverage limit
2024 $4,150 $8,300
2025 $4,300 $8,550
At age 55, members can contribute an additional $1,000 beyond IRS limits.
Spend
Stretch your dollars further.
HSA qualified medical expenses
Because of the tax savings on contributions, you can save an average of 30 percent on qualified medical expenses, including but not limited to:
- Acne Medicine
- Braces
- Crutches
- Dentures
- Eyeglasses
- Eye Surgery
- Flu Shot
- Hearing Aids
- Motorized Wheelchair
- Prescriptions
- X-Rays
- Allergy Medicines
- Contact Lenses
- Dental Cleanings
- Doctor Fees
- Eye Exams
- Face Masks
- Hand Sanitizer
- Ibuprofen
- Orthodontia
- Sanitizing Wipes
Your Savings Can Add Up Fast
Here’s an example based on $4,000 annual spending and a 30 percent effective tax rate.
Medical expenses
$3,000
+
Vision expenses
$500
+
Dental expenses
$500
Annual tax savings
=
$1,200
Consider generics
Generic medications cost 20 to 70 percent less than branded medications.
Prefer urgent care
Emergency room visits cost up to 5X more than urgent care.⁸ Unless it’s a life-threatening event, consider urgent care instead.
Comparison shop
Whether you need a simple procedure or even major surgery, be sure to get prices from several healthcare providers.
Invest
Build long-term health savings.
HSA investing
Invest your money just like a 401(k)
- Access liquid funds anytime
- Enjoy lower fees and transparent pricing
| 401(k) | HSA |
|---|---|
| FICA taxed contributions | 100% tax deductible contributions |
| Tax-free earnings | Tax-free savings |
| Medical expenses taxed as ordinary income | Tax-free distribution for medical expenses |
| Regular expenses taxed as ordinary income | Regular expenses taxes as ordinary income |
| Minimum distributions required | No required minimum distributions |
HSA vs. 401(k)
Both accounts let you make pre-tax contributions and grow tax-free earnings. But only an HSA lets you take tax-free distributions for qualified medical expenses. After age 65 you can use your health savings account for any expense, you’ll simply pay ordinary income taxes— just like a 401(k).
HealthEquity does not provide legal, tax or financial advice. Always consult a professional when making life-changing decisions.
Source: "HSA like a pro." HSA Guide. HealthEquity.


21.5: Exercises
Show that each of the following numbers is algebraic over \({\mathbb Q}\) by finding the minimal polynomial of the number over \({\mathbb Q}\text{.}\)
- \(\displaystyle \sqrt{ 1/3 + \sqrt{7} }\)
- \(\displaystyle \sqrt{ 3} + \sqrt[3]{5}\)
- \(\displaystyle \sqrt{3} + \sqrt{2}\, i\)
- \(\cos \theta + i \sin \theta\) for \(\theta = 2 \pi /n\) with \(n \in {\mathbb N}\)
- \(\displaystyle \sqrt{ \sqrt[3]{2} - i }\)
Find a basis for each of the following field extensions. What is the degree of each extension?
- \({\mathbb Q}( \sqrt{3}, \sqrt{6}\, )\) over \({\mathbb Q}\)
- \({\mathbb Q}( \sqrt[3]{2}, \sqrt[3]{3}\, )\) over \({\mathbb Q}\)
- \({\mathbb Q}( \sqrt{2}, i)\) over \({\mathbb Q}\)
- \({\mathbb Q}( \sqrt{3}, \sqrt{5}, \sqrt{7}\, )\) over \({\mathbb Q}\)
- \({\mathbb Q}( \sqrt{2}, \root 3 \of{2}\, )\) over \({\mathbb Q}\)
- \({\mathbb Q}( \sqrt{8}\, )\) over \({\mathbb Q}(\sqrt{2}\, )\)
- \({\mathbb Q}(i, \sqrt{2} +i, \sqrt{3} + i )\) over \({\mathbb Q}\)
- \({\mathbb Q}( \sqrt{2} + \sqrt{5}\, )\) over \({\mathbb Q} ( \sqrt{5}\, )\)
- \({\mathbb Q}( \sqrt{2}, \sqrt{6} + \sqrt{10}\, )\) over \({\mathbb Q} ( \sqrt{3} + \sqrt{5}\, )\)
Find the splitting field for each of the following polynomials.
- \(x^4 - 10 x^2 + 21\) over \({\mathbb Q}\)
- \(x^4 + 1\) over \({\mathbb Q}\)
- \(x^3 + 2x + 2\) over \({\mathbb Z}_3\)
- \(x^3 - 3\) over \({\mathbb Q}\)
Consider the field extension \({\mathbb Q}( \sqrt[4]{3}, i )\) over \(\mathbb Q\text{.}\)
- Find a basis for the field extension \({\mathbb Q}( \sqrt[4]{3}, i )\) over \(\mathbb Q\text{.}\) Conclude that \([{\mathbb Q}( \sqrt[4]{3}, i ): \mathbb Q] = 8\text{.}\)
- Find all subfields \(F\) of \({\mathbb Q}( \sqrt[4]{3}, i )\) such that \([F:\mathbb Q] = 2\text{.}\)
- Find all subfields \(F\) of \({\mathbb Q}( \sqrt[4]{3}, i )\) such that \([F:\mathbb Q] = 4\text{.}\)
Show that \({\mathbb Z}_2[x] / \langle x^3 + x + 1 \rangle\) is a field with eight elements. Construct a multiplication table for the multiplicative group of the field.
Show that the regular \(9\)-gon is not constructible with a straightedge and compass, but that the regular \(20\)-gon is constructible.
Prove that the cosine of one degree (\(\cos 1^\circ\)) is algebraic over \({\mathbb Q}\) but not constructible.
Can a cube be constructed with three times the volume of a given cube?
Prove that \({\mathbb Q}(\sqrt{3}, \sqrt[4]{3}, \sqrt[8]{3}, \ldots )\) is an algebraic extension of \({\mathbb Q}\) but not a finite extension.
Prove or disprove: \(\pi\) is algebraic over \({\mathbb Q}(\pi^3)\text{.}\)
Let \(p(x)\) be a nonconstant polynomial of degree \(n\) in \(F[x]\text{.}\) Prove that there exists a splitting field \(E\) for \(p(x)\) such that \([E : F] \leq n!\text{.}\)
Prove or disprove: \({\mathbb Q}( \sqrt{2}\, ) \cong {\mathbb Q}( \sqrt{3}\, )\text{.}\)
Prove that the fields \({\mathbb Q}(\sqrt[4]{3}\, )\) and \({\mathbb Q}(\sqrt[4]{3}\, i)\) are isomorphic but not equal.
Let \(K\) be an algebraic extension of \(E\text{,}\) and \(E\) an algebraic extension of \(F\text{.}\) Prove that \(K\) is algebraic over \(F\text{.}\) [ Caution : Do not assume that the extensions are finite.]
Prove or disprove: \({\mathbb Z}[x] / \langle x^3 -2 \rangle\) is a field.
Let \(F\) be a field of characteristic \(p\text{.}\) Prove that \(p(x) = x^p - a\) either is irreducible over \(F\) or splits in \(F\text{.}\)
Let \(E\) be the algebraic closure of a field \(F\text{.}\) Prove that every polynomial \(p(x)\) in \(F[x]\) splits in \(E\text{.}\)
If every irreducible polynomial \(p(x)\) in \(F[x]\) is linear, show that \(F\) is an algebraically closed field.
Prove that if \(\alpha\) and \(\beta\) are constructible numbers such that \(\beta \neq 0\text{,}\) then so is \(\alpha / \beta\text{.}\)
Show that the set of all elements in \({\mathbb R}\) that are algebraic over \({\mathbb Q}\) form a field extension of \({\mathbb Q}\) that is not finite.
Let \(E\) be an algebraic extension of a field \(F\text{,}\) and let \(\sigma\) be an automorphism of \(E\) leaving \(F\) fixed. Let \(\alpha \in E\text{.}\) Show that \(\sigma\) induces a permutation of the set of all zeros of the minimal polynomial of \(\alpha\) that are in \(E\text{.}\)
Show that \({\mathbb Q}( \sqrt{3}, \sqrt{7}\, ) = {\mathbb Q}( \sqrt{3} + \sqrt{7}\, )\text{.}\) Extend your proof to show that \({\mathbb Q}( \sqrt{a}, \sqrt{b}\, ) = {\mathbb Q}( \sqrt{a} + \sqrt{b}\, )\text{,}\) where \(a \neq b\) and neither \(a\) nor \(b\) is a perfect square.
Let \(E\) be a finite extension of a field \(F\text{.}\) If \([E:F] = 2\text{,}\) show that \(E\) is a splitting field of \(F\) for some polynomial \(f(x) \in F[x]\text{.}\)
Prove or disprove: Given a polynomial \(p(x)\) in \({\mathbb Z}_6[x]\text{,}\) it is possible to construct a ring \(R\) such that \(p(x)\) has a root in \(R\text{.}\)
Let \(E\) be a field extension of \(F\) and \(\alpha \in E\text{.}\) Determine \([F(\alpha): F(\alpha^3)]\text{.}\)
Let \(\alpha, \beta\) be transcendental over \({\mathbb Q}\text{.}\) Prove that either \(\alpha \beta\) or \(\alpha + \beta\) is also transcendental.
Let \(E\) be an extension field of \(F\) and \(\alpha \in E\) be transcendental over \(F\text{.}\) Prove that every element in \(F(\alpha)\) that is not in \(F\) is also transcendental over \(F\text{.}\)
Let \(\alpha\) be a root of an irreducible monic polynomial \(p(x) \in F[x]\text{,}\) with \(\deg p = n\text{.}\) Prove that \([F(\alpha) : F] = n\text{.}\)