6.5: The Early Classical Period
Marble Sculpture and Architecture in the Greek Early Classical Period
Early Classical Greek marble sculptures and temple decorations display new conventions to depict the body and severe style facial expressions.
Describe the characteristics of the Kritios Boy, Spear Bearer, and the Temple of Zeus
Key Points
- The sculpture found on the pediment and metopes at the Temple of Zeus at Olympia represent the style of relief and pedimental sculptural during the Early Classical period.
- The Severe style is an Early Classical style of sculpting where the body is depicted naturalistically and the face remains blank and expressionless. This style notes the artist’s understanding of the body’s musculature, while maintaining a screen between art and reality with the stoic face.
- Contrapposto is a weight shift depicted in the body that rotates the waist, hips, chest, shoulders, and sometimes even the neck and head of the figure. It increases that naturalism in the body since it correctly mimics the inner workings of human musculature.
- Kritios Boy is an early example of contrapposto and Severe style. This marble statue depicts a nude male youth, muscular and well built, with an air of naturalism that dissolves when examining his Severe style face.
- Polykleitos, an artist and art theorist, developed a canon for the creation of the perfect male body based on mathematical proportions. His Doryphoros (Spear Bearer) is believed to be a sculptural representation of his treatise. The figure stands in contrapposto, with a Severe-style face.
Key Terms
- Severe style : The dominant idiom of Greek sculpture in the period from 490 to 450 BCE. It marks the breakdown of the canonical forms of Archaic art and the transition to the greatly expanded vocabulary and expression of the classical movement of the late 5th century.
- Perserchutt : A German term meaning Persian debris or rubble, that refers to the location of ritually buried architectural and votive sculptures that were destroyed following the sack of Athens by the Persians. The area was first excavated by the Germans in the late 19th century.
- hexastyle : Describes a building with six columns in the front and back and 13 down each side.
Temple of Zeus at Olympia
The Temple of Zeus at Olympia is a colossal ruined temple in the center of the Greek capital Athens that was dedicated to Zeus, king of the Olympian gods. Its plan is similar to that of the Temple of Aphaia at Aegina.
It is hexastyle , with six columns across the front and back and 13 down each side. It has two columns directly connected to the walls of the temple, known as in antis, in front of both the entranceway ( pronaos ) and the inner shrine ( opisthodomos ). Like the Temple of Aphaia, there are two, two-story colonnades of seven columns on each side of the inner sanctuary (naos).
The pedimental figures are depicted in the developing Classical style with naturalistic yet overly muscular bodies. Most of the figures are shown with the expressionless faces of the Severe style.
The figures on the east pediment await the start of a chariot race, and the whole composition is still and static . A seer, however, watches it in horror as he foresees the death of Oenomaus. This level of emotion would never be present in Archaic statues and it breaks the Early Classical Severe style, allowing the viewer to sense the forbidding events about to happen.
Unlike the static composition of the eastern pediment, the Centauromachy on the western pediment depicts movement that radiates out from its center. The centaurs, fighting men, and abducted women struggle and fight against each other, creating tension in another example of an early portrayal of emotion. Most figures are depicted in the Severe style. However, some, including a centaur, have facial features that reflect their wrath and anger.
The twelve metopes over the pronaos and opisthodomos depict scenes from the twelve labors of Herakles. Like the development in pedimental sculpture, the reliefs on the metopes display the Early Classical understanding of the body. Herakles’ body is strong and idealized, yet it has a level of naturalism and plasticity that increases the liveliness of the reliefs.
The scenes depict varying types of compositions. Some are static with two or three figures standing rigidly, while others, such as Herakles and the Cretan Bull, convey a sense of liveliness through their diagonal composition and overlapping bodies.
Kritios Boy
A slightly smaller-than-life statue known as the Kritios Boy was dedicated to Athena by an athlete and found in the Perserchutt of the Athenian Acropolis. Its title derives from a famous artist to whom the sculpture was once attributed.
The marble statue is a prime example of the Early Classical sculptural style and demonstrates the shift away from the stiff style seen in Archaic kouroi. The torso depicts an understanding of the body and plasticity of the muscles and skin that allows the statue to come to life.
Part of this illusion is created by a stance known as contrapposto. This describes a person with his or her weight shifted onto one leg, which creates a shift in the hips, chest, and shoulders to create a stance that is more dramatic and naturalistic than a stiff, frontal pose. This contrapposto position animates the figure through the relationship of tense and relaxed limbs.
However, the face of the Kritios Boy is expressionless, which contradicts the naturalism seen in his body. This is known as the Severe style. The blank expressions allow the sculpture to appear less naturalistic, which creates a screen between the art and the viewer. This differs from the use of the Archaic smile (now gone), which was added to sculpture to increase their naturalism. However, the now empty eye sockets once held inlaid stone to give the sculpture a lifelike appearance.
Polykleitos
Polykleitos was a well-known Greek sculptor and art theorist during the early- to mid-fifth century BCE. He is most renowned for his treatise on the male nude, known as the Canon, which describes the ideal, aesthetic body based on mathematical proportions and Classical conventions such as contrapposto.
His Doryphoros, or Spear Bearer, is believed to be his representation of the Canon in sculpted form . The statue depicts a young, well-built soldier holding a spear in his left hand with a shield attached to his left wrist. Both military implements are now lost. The figure has a Severe-style face and a contrapposto stance. In another development away from the stiff and seemingly immobile Archaic style, the Doryphoros’ left heel is raised off the ground , implying an ability to walk.
This sculpture demonstrates how the use of contrapposto creates an S-shaped composition. The juxtaposition of a tension leg and tense arm and relaxed leg and relaxed arm, both across the body from each other, creates an S through the body.
The dynamic power of this composition shape places elements—in this case the figure’s limbs—in opposition to each other and emphasizes the tension this creates. The statue, as a visualization of Polykletios’ canon, also depicts the Greek sense of symmetria, the harmony of parts, seen here in the body’s proportions.
Bronze Sculpture in the Greek Early Classical Period
Surviving Greek bronze sculptures from the Early Classical period showcase the skill of Greek artists in representing the body and expressing motion.
Discuss characteristics and examples of Greek Bronze sculpture during the Early Classical Period
Key Points
- While bronze was a popular material for Greek sculptors, few Greek bronzes exist today. We know a majority of famous sculptors and sculptures only through marble Roman copies and the few bronzes that survived, often from shipwrecks.
- Early Classical bronzes are sculpted in the lost wax method of casting . The figures are created in the Severe style with naturalistic bodies and blank, expressionless faces. The sculptures’ lightweight appearance is due to their hollowness and contributes to their implied potential energy and movement.
- The Charioteer of Delphi, the Riace Warriors, and the Artemision Bronze all display the sculpting characteristics of the Early Classical Severe style while also demonstrating the characteristics of bronze sculpting, including the lightness of the material and liveliness that could be achieved.
Key Terms
- strut : A support rod.
- contrapposto : The position of a figure whose hips and legs are twisted away from the direction of the head and shoulders.
- lost wax : A method of casting in which a model of the sculpture is made from wax. The model is used to make a mold. When the mold has set, the wax is made to melt and is poured away, leaving the mold ready to be used to cast the sculpture.
Greek Bronze Sculpture
Bronze was a popular sculpting material for the Greeks. Composed of a metal alloy of copper and tin, it provides a strong and lightweight material for use in the ancient world, especially in the creation of weapons and art. The Greeks used bronze throughout their history.
Because bronze is a valuable material, throughout history bronze sculptures were melted down to forge weapons and ammunition or to create new sculptures. The Greek bronzes that we have today mainly survived because of shipwrecks, which kept the material from being reused, and the sculptures have since been recovered from the sea and restored.
The Greeks used bronze as a primary means of sculpting, but much of our knowledge of Greek sculpture comes from Roman copies. The Romans were very fond of Greek art, and collecting marble replicas of them was a sign of status, wealth, and intelligence in the Roman world.
Roman copies worked in marble had a few differences from the original bronze. Struts , or supports, were added to help buttress the weight of the marble as well as the hanging limbs that did not need support when the statue was originally made in the lighter and hollow bronze. The struts appeared either as rectangular blocks that connect an arm to the torso or as tree stumps against the leg, which supports the weight of the sculpture, as in this Roman copy of the Diadoumenos Atenas.
Lost Wax Technique
The lost wax technique, which is also known by its French name, cire perdue, is the process that ancient Greeks used to create their bronze statues. The first step of the process involves creating a full-scale clay model of the intended work of art. This would be the core of the model.
Once completed, a mold is made of the clay core and an additional wax mold is also created. The wax mold is then be placed between the clay core and the clay mold, creating a pocket, and the wax is melted out of the mold, after which the gap is filled with bronze. Once cooled, the exterior clay mold and interior clay coreis are carefully removed and the bronze statue is finished.
The multiple pieces are welded together, imperfections smoothed, and any additional elements, such as inlaid eyes and eyelashes, are then added. Because the clay mold must be broken when removing the figure, the lost wax method can be used only for making one-of-a-kind sculptures.
Charioteer of Delphi
The Charioteer of Delphi is an Early Classical bronze sculpture of a life-sized chariot driver. An inscription at the base tells us that the statues were originally dedicated by a man, named Polyzalus of Gela, to Apollo at the Sanctuary of Apollo at Delphi.
Polyzalus commissioned and dedicated the work in commemoration of his victorious chariot race during the Pythian Games. The Charioteer is the only remaining part of a large statue group that included the chariot, grooms, and horses.
While the commissioner was a tyrant of the Greek colonial city of Gela on Sicily, the statue is believed to have been made in Athens. It was made by the lost wax technique in multiple sections and then assembled.
The Charioteer stands tall, his right arm stretched out to grasp reins; his left arm is missing. He has a high waist, which probably looked more natural when he stood on his chariot. However, despite the high waist, the figure has a high degree of naturalism , on par with the marble sculptural developments of the Early Classical style. The arms, face, and feet are rendered with high plasticity, and the inlaid eyes and added copper of his lips and eyelashes all add a degree of naturalism.
When compared to Archaic sculptures, it appears very natural. However, as an Early Classical sculpture, the Charioteer has yet to achieve the full Classical style. The Archaic smile is gone, but his appears almost blank and expressionless, on par with other sculptures produced in the Severe stye of the Early Classical period.
Riace Warriors
The Riace Warriors are a set of two nude, bronze sculptures of male warriors that were recovered off the coast of Riace, Italy. They are a prime example of Early Classical sculpture and the transition between Archaic to Classical sculpting styles.
The figures are nude, unlike the Charioteer. Their bodies are idealized and appear dynamic, with freed limbs, a contrapposto shift in weight, and turned heads that imply movement. The muscles are modeled with a high degree of plasticity, which the bronze material amplifies through natural reflections of light. Additional elements, such as copper for the lips and nipples, silver teeth, and eyes inlaid with glass and bone, were added to the figures to increase their naturalism. Both figures originally held a shield and spear, which are now lost. Warrior B wears a helmet, and it appears that Warrior A once wore a wreath around his head.
Artemision Bronze
The Artemision Bronze represents either Zeus or Poseidon. Both gods were represented with full beards to signify maturity. However, it is impossible to identify the sculpture as one god or the other because it can either be a lightning bolt (symbolic of Zeus) or a trident (symbolic of Poseidon) in his raised right hand.
The figure stands in heroic nude, as would be expected with a god, with his arms outstretched, preparing to strike. The bronze is in the Severe style with an idealized, muscular body and an expressionless face.
Like the Charioteer and the Riace Warriors, the Artemision Bronze once held inlaid glass or stone in its now-vacant eye sockets to heighten its lifelikeness. The right heel of the figure rises off the ground , which anticipates the motion the figure is about to undertake.
The full potential of the god’s motion and energy, as well as the grace of the body, is reflected in the modeling of the bronze.
Ceramics in the Greek Early Classical Period
The ceramic art from Early Classical Greece displays important compositional developments and increased naturalism in the figures.
Examine the developments in ceramic art during the Early Classical period, including naturalism and white-ground painting
Key Points
- Red-figure painting continued during the Early Classical period. The changes in the depiction of the body and in the drapery of the figures began to change, reflecting the stylistic changes and increased naturalism seen in Classical sculpture .
- The style of red-figure painting also diversified as painters began to depict figures on multiple ground lines , show characters from a variety of perspectives (including three-quarter view), and utilize more naturalism (as seen by the work of the Niobid Painter).
- White-ground painting, developed in 500 BCE, became popular during the Classical period. White-ground pottery was coated in a white slip before being fired and painted, allowing for the use of polychromy .
- Early white-ground painting (Type I) resembled black-figure painting until it was supplanted by the more familiar outline paintings. White-ground painting became the primary style for lekythoi, vessels used to hold oils that had a funerary context. Due to this context, many of the scenes painted on lekythoi depicted scenes of funerary rites and rituals , or scenes that alluded to impending death.
Key Terms
- lekythos : A type of ancient Greek pottery used for storing oil and previous liquids. The body is narrow and has a single handle attached to the neck of the vessel. They typically stood 10 to 20 inches tall, but when used as grave markers could be much larger.
- polychromy : The art or practice of combining different colors, especially brilliant ones, in an artistic way.
- white-ground : A style of ancient Greek vase painting in which figures appear on a white background.
Classical Period Ceramics
The Classical period witnessed the continuation of red- and black-figure painting techniques on ceramic objects. While artists continued to produce black-figure paintings into the second century BCE, the technique became increasingly rare, overtaken around 520 BCE by red-figure painting.
Attic red-figure vases were exported throughout Greece and beyond, and for a long time dominated the market for fine ceramics. Only a few centers of pottery production could compete with Athens in terms of its innovation, quality, and production capacity.
Red-Figure Painting
Red-figure painting continued to flourish during the Early, High, and Late Classical periods. The naturalism of the figures in Early Classical vase painting continued to increase, as the figures became less stocky and less linear. Both the figures and their drapery began to appear more plastic, and the scenes often depicted a single moment within a mythical story or event. Furthermore, vase painting began to be influenced by the changes occurring in both sculpture and the large-scale painting of walls and panels.
The Mannerists
The Mannerists were a group of Attic red-figure painters known for their affected (emotive) subject matter. They were active from about 480 BCE until near the end of the fifth century BCE. Their main characteristic is that they maintained features of black-figure vase painting in the red-figure technique.
Their figures seem elongated and have small heads, the garment folds fall stiff and resemble stairs, and the images are framed with black-figure style ornamentations. The range of motifs is also influenced by previous periods. The figures gesticulate as if using a form of sign language—the hands often appear stiff and theatrical. We can see typical Mannerist small heads and affected gestures in the Pan Painter’s Herakles Fighting Busiris (c. 470 BCE).
The Niobid Painter’s red-figure krater of Artemis and Apollo slaying the children of Niobe, from 460 BCE, is believed to be a composition inspired by a panel painting. The side of the vessel depicting Artemis and Apollo relates to the myth of the twin god and goddess who slew Niobe’s fourteen children after she boasted that her ability to birth children exceeded Leto, the mother of Apollo and Artemis.
This story alludes to ancient Greek admonitions against hubris, or extreme pride. The scene is one of the first vase painting scenes to show the figures on different ground lines. Apollo and Artemis stand in the center of the vessel as Niobe’s children fall to ground around them. One child has even fallen behind a rock in the landscape.
On the other side of the vase is an image of gods and heroes, with Herakles at the center. All the figures stand and sit on various ground lines. The figures on both sides are depicted from multiple angles, including three-quarter view, and a profile eye is used for the figures in profile, a first in Greek vase painting.
The Berlin Painter is another well-known Early Classical vase painter. His unique style depicts figures, isolated from context, on a small ground line against a glossy black background. His figures also start in the middle of the vase and extend onto the vessel’s shoulder, stopping at the neck.
He pays particular attention to the details of the body and the drapery of each figure, and allows both figure and drapery to express emotion, space , and movement. His painting entitled Ganymede with a Hoop and Cock conveys a sense of dynamism by arranging the body through a series of diagonal outlines, using contour lines to mark the locations of muscles and tendons beneath the skin.
White-Ground Painting
White-ground painting developed around 500 BCE and gained popularity during the following century. The technique is based on the use of paints, instead of slip, to create polychrome vessels.
The vessels were first coated in a white slip before various colors of paint were added. The white background and firing techniques allowed for various colors to be used, including blue, yellow, red, brown, and green. Because the style is less durable than black- and red-figure painting, it was often used for votives and as grave offerings .
The common outline paintings of the white-ground technique would not dominate the style until approximately the mid-fifth century BCE. For the first 50 years of white-ground painting, known as Type I, the imagery resembled conventional black-figure painting, with the color of the background as the only difference. This depiction of Herakles fighting Geryon provides an example of Type I white-ground painting.
White-ground painting is often seen on a lekythos , a vessel used to hold oils, which were sometimes used for anointing the dead. Due to this funerary function, lekythoi were also used as grave offerings. As such, many of the scenes painted on white-ground lekythoi depict or allude to funerary scenes (such as funerary rites and rituals) or images of warriors departing their wives for battle and death. While the scene of Herakles fighting Geryon depicts a rather violent prelude to death, the imagery on later lekythoi is somewhat more sedate.
The Achilles Painter, a pupil of the Berlin Painter and creator of both red-figure and white-ground vessels, is one of the most well-known white ground painters.
Overall, in both white-ground and red-figure painting during the Early Classical period, the form of the body was perfected by the artisans. Painted vessels were now depicting figures on a two-dimensional plane, with the illusion of three-dimensional space. These figures were rendered in that space naturally, in terms of their movement and form.
Black-figure painting nearly disappeared in the Early Classical period and was primarily reserved for objects made to seem old or to recall antique styles, such as victory amphorae for the Panhellenic Games.
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9.3: SEM and its Applications for Polymer Science
SEM and its Applications for Polymer Science
Introduction
The scanning electron microscope (SEM) is a very useful imaging technique that utilized a beam of electrons to acquire high magnification images of specimens. Very similar to the transmission electron microscope (TEM), the SEM maps the reflected electrons and allows imaging of thick (~mm) samples, whereas the TEM requires extremely thin specimens for imaging; however, the SEM has lower magnifications. Although both SEM and TEM use an electron beam, the image is formed very differently and users should be aware of when each microscope is advantageous.
Microscopy Physics
Image Formation
All microscopes serve to enlarge the size of an object and allow people to view smaller regions within the sample. Microscopes form optical images and although instruments like the SEM have extremely high magnifications, the physics of the image formation are very basic. The simplest magnification lens can be seen Figure \(\PageIndex{1}\). The formula for magnification is shown in \ref{1}, where M is magnification, f is focal length, u is the distance between object and lens, and v is distance from lens to the image.
\[ M\ =\ \frac{f}{u-f}\ = \frac{v-f}{f} \label{1} \]
Multistage microscopes can amplify the magnification of the original object even more as shown in Figure. Where magnification is now calculated from \ref{2}, where f1, f2 are focal distances with respect to the first and second lens and v1, v2are the distances from the lens to the magnified image of first and second lens, respectively.
\[M \ =\ \frac{(v_{1}\ -\ f_{1})(v_{2}\ -\ f_{2})}{f_{1}f_{2}} \label{2} \]
In reality, the objects we wish to magnify need to be illuminated. Whether or not the sample is thin enough to transmit light divides the microscope into two arenas. SEM is used for samples that do not transmit light, whereas the TEM (transmission electron microscope) requires transparent samples. Due to the many frequencies of light from the introduced source, a condenser system is added to control the brightness and narrow the range of viewing to reduce aberrations, which distort the magnified image.
Electron Microscopes
Microscope images can be formed instantaneous (as in the optical microscope or TEM) or by rastering (scanning) a beam across the sample and forming the image point-by-point. The latter is how SEM images are formed. It is important to understand the basic principles behind SEM that define properties and limitations of the image.
Resolution
The resolution of a microscope is defined as the smallest distance between two features that can be uniquely identified (also called resolving power). There are many limits to the maximum resolution of the SEM and other microscopes, such as imperfect lenses and diffraction effects. Each single beam of light, once passed through a lens, forms a series of cones called an airy ring (see Figure \(\PageIndex{3}\)). For a given wavelength of light, the central spot size is inversely proportional to the aperture size (i.e., large aperture yields small spot size) and high resolution demands a small spot size.
Aberrations distort the image and we try to minimize the effect as much as possible. Chromatic aberrations are caused by the multiple wavelengths present in white light. Spherical aberrations are formed by focusing inside and outside the ideal focal length and caused by the imperfections within the objective lenses. Astigmatism is because of further distortions in the lens. All aberrations decrease the overall resolution of the microscope.
Electrons
Electrons are charged particles and can interact with air molecules therefore the SEM and TEM instruments require extremely high vacuum to obtain images (10-7 atm). High vacuum ensures that very few air molecules are in the electron beam column. If the electron beam interacts with an air molecule, the air will become ionized and damage the beam filament, which is very costly to repair. The charge of the electron allows scanning and also inherently has a very small deflection angle off the source of the beam.
The electrons are generated with a thermionic filament. A tungsten (W) or LaB 6 filament is chosen based on the needs of the user. LaB 6 is much more expensive and tungsten filaments meet the needs of the average user. The microscope can be operated as field emission (tungsten filament).
Electron Scattering
To accurately interpret electron microscopy images, the user must be familiar with how high energy electrons can interact with the sample and how these interactions affect the image. The probability that a particular electron will be scattered in a certain way is either described by the cross section, σ, or mean free path, λ, which is the average distance which an electron travels before being scattered.
Elastic Scatter
Elastic scatter, or Rutherford scattering, is defined as a process which deflects an electron but does not decrease its energy. The wavelength of the scattered electron can be detected and is proportional to the atomic number. Elastically scattered electrons have significantly more energy that other types and provide mass contrast imaging. The mean free path, λ, is larger for smaller atoms meaning that the electron travels farther.
Inelastic Scatter
Any process that causes the incoming electron to lose a detectable amount of energy is considered inelastic scattering. The two most common types of inelastic scatter are phonon scattering and plasmon scattering. Phonon scattering occurs when a primary electron looses energy by exciting a phonon, atomic vibrations in a solid, and heats the sample a small amount. A Plasmon is an oscillation within the bulk electrons in the conduction band for metals. Plasmon scattering occurs when an electron interacts with the sample and produces plasmons, which typically have 5 - 30 eV energy loss and small λ.
Secondary Effects
A secondary effect is a term describing any event which may be detected outside the specimen and is essentially how images are formed. To form an image, the electron must interact with the sample in one of the aforementioned ways and escape from the sample and be detected. Secondary electrons (SE) are the most common electrons used for imaging due to high abundance and are defined, rather arbitrarily, as electrons with less than 50 eV energy after exiting the sample. Backscattered electrons (BSE) leave the sample quickly and retain a high amount of energy; however there is a much lower yield of BSE. Backscattered electrons are used in many different imaging modes. Refer to Figure \(\PageIndex{4}\) for a diagram of interaction depths corresponding to various electron interactions.
SEM Construction
The SEM is made of several main components: electron gun, condenser lens, scan coils, detectors, specimen, and lenses (see Figure \(\PageIndex{5}\)). Today, portable SEMs are available but the typical size is about 6 feet tall and contains the microscope column and the control console.
A special feature of the SEM and TEM is known as depth of focus, dv/du the range of positions (depths) at which the image can be viewed with good focus, see \ref{3}. This allows the user to see more than a singular plane of a specified height in focus and essentially allows a range of three dimensional imaging.
\[ \frac{dv}{du}\ =\ \frac{-v^{2}}{u^{2}}\ =\ -M^{2} \label{3} \]
Electron Detectors (image formation)
The secondary electron detector (SED) is the main source of SEM images since a large majority of the electrons emitted from the sample are less than 50 eV. These electrons form textural images but cannot determine composition. The SEM may also be equipped with a backscatter electron detector (BSED) which collects the higher energy BSE’s. Backscattered electrons are very sensitive to atomic number and can determine qualitative information about nuclei present (i.e., how much Fe is in the sample). Topographic images are taken by tilting the specimen 20 - 40° toward the detector. With the sample tilted, electrons are more likely to scatter off the top of the sample rather than interact within it, thus yielding information about the surface.
Sample Preparation
The most effective SEM sample will be at least as thick as the interaction volume; depending on the image technique you are using (typically at least 2 µm). For the best contrast, the sample must be conductive or the sample can be sputter-coated with a metal (such as Au, Pt, W, and Ti). Metals and other materials that are naturally conductive do not need to be coated and need very little sample preparation.
SEM of Polymers
As previously discussed, to view features that are smaller than the wavelength of light, an electron microscope must be used. The electron beam requires extremely high vacuum to protect the filament and electrons must be able to adequately interact with the sample. Polymers are typically long chains of repeating units composed primarily of “lighter” (low atomic number) elements such as carbon, hydrogen, nitrogen, and oxygen. These lighter elements have fewer interactions with the electron beam which yields poor contrast, so often times a stain or coating is required to view polymer samples. SEM imaging requires a conductive surface, so a large majority of polymer samples are sputter coated with metals, such as gold.
The decision to view a polymer sample with an SEM (versus a TEM for example) should be evaluated based on the feature size you expect the sample to have. Generally, if you expect the polymer sample to have features, or even individual molecules, over 100 nm in size you can safely choose SEM to view your sample. For much smaller features, the TEM may yield better results, but requires much different sample preparation than will be described here.
Polymer Sample Preparation Techniques
Sputter Coating
A sputter coater may be purchased that deposits single layers of gold, gold-palladium, tungsten, chromium, platinum, titanium, or other metals in a very controlled thickness pattern. It is possible, and desirable, to coat only a few nm’s of metal onto the sample surface.
Spin Coating
Many polymer films are depositing via a spin coater which spins a substrate (often ITO glass) and drops of polymer liquid are dispersed an even thickness on top of the substrate.
Staining
Another option for polymer sample preparation is staining the sample. Stains act in different ways, but typical stains for polymers are osmium tetroxide (OsO4), ruthenium tetroxide (RuO4) phosphotungstic acid (H3PW12O40), hydrazine (N2H4), and silver sulfide (Ag2S).
Examples
Comp-block Copolymer (Microstructure of Cast Film)
- Cast polymer film (see Figure \(\PageIndex{6}\)).
- To view interior structure, the film was cut with a microtome or razor blade after the film was frozen in liquid N 2 and fractured.
- Stained with RuO 4 vapor (after cutting).
- Structure measurements were averaged over a minimum of 25 measurements.
Polystyrene-polylactide Bottlebrush Copolymers (Lamellar Spacing)
- Pressed polymer samples into disks and annealed for 16 h at 170 °C.
- To determine ordered morphologies, the disk was fractured (see Figure \(\PageIndex{7}\)).
- Used SEM to verify lamellar spacing from USAXS.
SWNTs in Ultrahigh Molecular Weight Polyethylene
- Dispersed SWNTs in interactive polymer.
- Samples were sputter-coated in gold to enhance contrast.
- The films were solution-crystallized and the cross-section was imaged.
- Environmental SEM (ESEM) was used to show morphologies of composite materials.
- WD = 7 mm.
- Study was conducted to image sample before and after drawing of film.
- Images confirmed the uniform distribution of SWNT in PE (Figure \(\PageIndex{8}\)).
- M W = 10,000 Dalton.
- Study performed to compare transparency before and after UV irradiation.
Nanostructures in Conjugated Polymers (Nanoporous Films)
- Polymer and NP were processed into thin films and heated to crosslink.
- SEM was used to characterize morphology and crystalline structure (Figure \(\PageIndex{9}\)).
- SEM was used to determine porosity and pore size.
- Magnified orders of 200 nm - 1 μm.
- WD = 8 mm.
- M W = 23,000 Daltons
- Sample prep: spin coating a solution of poly-(thiophene ester) with copper NPs suspended on to ITO coated glass slides. Ziess, Supra 35
Cryo-SEM Colloid Polystyrene Latex Particles (Fracture Patterns)
- Used cryogenic SEM (cryo-SEM) to visualize the microstructure of particles (Figure \(\PageIndex{10}\))
- Particles were immobilized by fast-freezing in liquid N 2 at –196 °C.
- Sample is fractured (-196 °C) to expose cross section.
- 3 nm sputter coated with platinum.
- Shapes of the nanoparticles after fracture were evaluated as a function of crosslink density.


21.6: Lumped-Element Model
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It is possible to ascertain the relevant behaviors of a transmission line using elementary circuit theory applied to a differential-length lumped-element model of the transmission line. The concept is illustrated in Figure \(\PageIndex{1}\), which shows a generic transmission line aligned with its length along the \(z\) axis. The transmission line is divided into segments having small but finite length \(\Delta z\). Each segment is modeled as an identical two-port having the equivalent circuit representation shown in Figure \(\PageIndex{2}\). The equivalent circuit consists of 4 components as follows:
- The resistance \(R'\Delta z\) represents the series-combined ohmic resistance of the two conductors. This should account for both conductors since the current in the actual transmission line must flow through both conductors. The prime notation reminds us that \(R'\) is resistance per unit length ; i.e., \(\Omega\)/m, and it is only after multiplying by length that we get a resistance in \(\Omega\).
- The conductance \(G'\Delta z\) represents the leakage of current directly from one conductor to the other. When \(G'\Delta z>0\), the resistance between the conductors is less than infinite, and therefore, current may flow between the conductors. This amounts to a loss of power separate from the loss associated with \(R'\) above. \(G'\) has units of S/m. Further note that \(G'\) is not equal to \(1/R'\) as defined above. \(G'\) and \(R'\) are describing entirely different physical mechanisms (and in principle either could be defined as either a resistance or a conductance).
- The capacitance \(C'\Delta z\) represents the capacitance of the transmission line structure. Capacitance is the tendency to store energy in electric fields and depends on the cross-sectional geometry and the media separating the conductors. \(C'\) has units of F/m.
- The inductance \(L'\Delta z\) represents the inductance of the transmission line structure. Inductance is the tendency to store energy in magnetic fields, and (like capacitance) depends on the cross-sectional geometry and the media separating the conductors. \(L'\) has units of H/m.
In order to use the model, one must have values for \(R'\), \(G'\), \(C'\), and \(L'\). Methods for computing these parameters are addressed elsewhere in this book.


7.3: Navigating Cultural Differences
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- Krischa Esquivel, Emily Elam, Jennifer Paris, & Maricela Tafoya
- College of the Canyons
Cultural differences can lead to conflicts. For example, people may disagree on practices for handling a baby, responding to crying, or feeding. Home visit staff may be concerned over how and when to intervene in family arguments. Staff and families in early childhood education programs may differ about how programs should support children’s home or native language. Given the wide range of cultural ideas, it is not surprising that adults can have differences that are rooted in the core of their being.
Gonzalez-Mena (1992, 2001, 2008) indicated that these disagreements may be when adults from different cultural backgrounds may find that their familiar ways of working with children are different or when adults within the same culture can disagree. In both of these situations of conflict between program staff and families, Gonzalez-Mena identified four possible outcomes:
- All sides gain understanding, negotiate, and/or compromise, leading to resolution of the conflict.
- Program staff understand the families’ perspective(s) and change their practices.
- Families take on the perspective of the program staff and change their practices.
- No resolution is reached (here, the conflict may continue or intensify; or both sides can cope with the differences).
Of course, conflicts can occur over numerous issues. To help program staff make progress, Gonzalez-Mena challenges them to question their own assumptions about child development practices (e.g., “My way of thinking about X is not the only way to think about it. My way of doing Practice Y is not the only way to work with the child.”). Once this commitment to test one’s own assumptions is in place, two goals for a conflict situation are: (1) to minimize (or eliminate) extreme differences in practices; and (2) to resolve the situation for the benefit of the child. Program staff are encouraged to take a child centered look at any situation of conflicting practices. [86]
The process for each varies greatly not only based on cultures, but even within similar cultures. These practices can often times go against program policy and best practice as we have been taught in the field. [87]
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Pin it! Differences in Feeding Practices The following is a scenario from the NAEYC’s publication, Diversity and Infant/Toddler Caregiving : “Junior, who is new to the center, is excited when he sees a bowl of food. The baby makes happy sounds, kicks his legs, and waves his arms. But when Helen puts Junior in the high chair and places the bowl in front of him, he just sits there and makes no attempt to feed himself. He looks confused and becomes distressed. Finally he slumps over, a glazed look in his eyes. His mother explains later that she has taught Junior not to touch his food. In fact, her son has never been in a high chair; he has always been fed on his mother’s lap, wrapped up tightly in a blanket to discourage him from interfering with her.” [88] |
Gonzalez-Mena and Bhavnagri suggest that when the family and program do not agree about a practice or policy, early childhood educators should ask themselves:
- What is the family’s cultural perspective on the issue?
- How do the family’s child care practices relate to their cultural perspective?
- What are the family’s goals for the child? How has the family’s culture influenced these goals?
- In review of these goals, is the family’s practice in the child’s best interest?
- Is there any sound research that shows that the family’s practice is doing actual harm?
- Is the program’s practice or policy universally applicable, or is it better suited to a particular culture?
- Did the family choose the program because of its particular philosophy, even if it is based on a different culture from their own?
- Have program staff members attempted to fully understand the family’s rationale for its practices, the complexity of the issues, and other contributing factors?
- Have staff members attempted to fully explain the rationale for program practices? Have they looked at how their own culture influences their perspective?
- What are some creative resolutions that address the concerns of both partners and the program? [89]
The point is to begin and continue to dialogue with families and to exchange information with the goal of resolving the conflict for the benefit of the child. The “bottom line” is really: What is in the best interest of the child? As stated in the National Association for the Education of Young Children’s Code of Ethical Conduct , our first and foremost ethical responsibility to children is to do no harm.
By learning more about the goals that families have for their children, and about the types of behaviors or practices that families prioritize and implement as they raise their children, program staff can more easily match the learning experiences of the classroom to those of the home. For example, if a teacher is concerned that a 3-year-old in her class is not skilled with using a fork, she should first find out if this is a goal of the family. Do they scoop their food at home using spoons? Do they use chopsticks? Do they feed the child or allow the child to self-feed? It is best for the teacher to check what the family practices and goals are before they misjudge what this child needs from them in terms of support and understanding.
According to National Association for the Education of Young Children’s developmentally appropriate practice one of the tenets is to be culturally responsive. In preparing an environment that supports children and families, we need to ensure that we have considered the beliefs, values, and needs of the family to deliver curriculum that addresses the child both individually and as a group. [90]
These exercises can be done individually and at a staff/team meeting. It’s important to involve the family and all caregivers as a way to ensure all perspectives are heard. Now that questions have been identified, having the conversation to gather the information is next; however, this comes with challenges. As you enter into the conversation, it’s important to put personal biases and beliefs aside and be ready to actively listen to hear and learn about the family and their point of view. [91]


3.2: Boolean Algebra
3.2.1 Terminology
It is possible to process logic concepts using arithmetic and algebra concepts. To do so the following arithmetic definitions are needed. The first table defines addition the second defines multiplication and the third defines complement.
Checkpoint \(\PageIndex{2}\)
Compare these three operations to the logical operations \(\neg\), \(\vee\), and \(\wedge\).
Functions can be defined using boolean algebra as well. Consider the examples in Table \(\PageIndex{3}\) .
Table \(\PageIndex{3}\) Boolean functions
| \(x\) | \(y\) |
\(f_{1}(x,y)=x+y\) |
\(f_{2}(x,y)=xy\) |
\(f_{3}(x,y)=x+x\) |
\(f_4(x,y)=\overline{x}+y\) |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 |
Practice
Checkpoint \(\PageIndex{4}\)
Evaluate the following boolean functions.
- \(\displaystyle f_1(x,y)=1+x\)
- \(\displaystyle f_2(x,y)=\overline{xy}\)
- \(\displaystyle f_3(x,y)=\overline{x}+\overline{y}\)
- \(\displaystyle f_4(x,y)=xy+y\)
Checkpoint \(\PageIndex{5}\)
A boolean function with two inputs, such as those above, has four input/output pairs. How many different boolean functions of two variables (inputs) exist? How many for a boolean function of three variables?
Checkpoint \(\PageIndex{6}\)
Is the representation of a boolean function unique? This means for each of the functions counted above is there only one expression that produces that result?
Checkpoint \(\PageIndex{7}\)
Find a function with the same input/output pattern as if \(x\) then \(y\).
Checkpoint \(\PageIndex{8}\)
Figure out the value for \(f(x)=\overbrace{x+x+x+\ldots}^{\shortstack{n}}\) for all integers \(n\).
Representation
Terminology
In complicated statements such as those that arise in complicated circuits, it is possible to optimize some statements by finding equivalent sets of operations. Boolean function equivalence can be used for this purpose.
First we will consider how to represent arbitrary boolean combinations as boolean functions. First, consider the expression in \(x\), \(y\), and \(z\) with the value 1 when \(x = y = 1\) and \(z=0\). Because boolean multiplication is only 1 when all factors are one, this expression can be written as \(xy\overline{z}\). Likewise an expression that is 1 when \(x = y = z = 0\) is \(\overline{x}\overline{y}\overline{z}\).
Next consider a function \(f(x,y,z)\) defined by the following input/output table.
| x | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| y | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| z | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| f(x,y,z) | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
Note that this function ff is 1 exactly when one of the two expressions above (\(xy\overline{z}\) and \(\overline{x}\overline{y}\overline{z}\)) is one. Because \(u+v\) is 1 when either is one, \(f(x,y,z)= (xy\overline{z} + \overline{x}\overline{y}\overline{z})\) is one way to represent this function.
While the mechanism presented above can be used to produce a boolean function or equivalent circuit for any desired combination of inputs and outputs, it does not necessarily produce the most efficient solution. The following example demonstrates how a boolean function can be simplified using the table of identities in Table \(\PageIndex{10}\)
| \(x+x\) | = | \(x\) |
| \(x \cdot x\) | = | \(x\) |
| \(x+0\) | = | \(x\) |
| \(x+1\) | = | \(1\) |
| \(x+\overline{x}\) | = | \(1\) |
| \(x\overline{x}\) | = | \(0\) |
| \(x \cdot 0\) | = | \(0\) |
| \(x \cdot 1\) | = | \(x\) |
| \(x+y\) | = | \(y+x\) |
| \(xy\) | = | \(yx\) |
| \(x+(y+z)\) | = | \((x+y)+z\) |
| \(x(yz)\) | = | \((xy)z\) |
| \(x(y+z)\) | = | \(xy+xz\) |
| \(x+yz\) | = | \((x+y)(x+z)\) |
| \(x+xy\) | = | \(x\) |
| \(x(x+y)\) | = | \(x\) |
| \(\overline{x+y}\) | = | \(\overline{x} \cdot \overline{y}\) |
| \(\overline{xy}\) | = | \(\overline{x} + \overline{y}\) |
The following example demonstrates one way to simplify this boolean function.
\begin{align*}
f(x,y) & = & x\overline{y} + xy + y\overline{x}\\
& = & x\overline{y} + yx + y\overline{x}\\
& = & x\overline{y} + y \cdot (x+\overline{x})\\
& = & x\overline{y} + y \cdot 1\\
& = & x\overline{y}+y\\
& = & y+x\overline{y}\\
& = & (y+x)(y+\overline{y})\\
& = & (y+x) \cdot 1\\
& = & y+x.
\end{align*}
Practice
Checkpoint \(\PageIndex{11}\)
Find an expression that is 1 when \(x=z=1\) and \(y=0\). Use the technique above.
Checkpoint \(\PageIndex{12}\)
Find a representation of the boolean function in Table \(\PageIndex{13}\)
| x | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| y | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| z | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| f(x,y,z) | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 |
Checkpoint \(\PageIndex{14}\)
Remember that boolean functions can be represented as logic statements (and/or/negate) which can be represented as electrical circuits. Design the electrical circuit needed for a light with two switches using the following steps. Note the following supposes that the light is off when both switches are turned down (typical off).
- Using 0 for down and 1 for up, determine which combinations of up and down results in the light being off (0) and on (1).
- Write a boolean function for this using the technique above.
- Convert this boolean function into a logic statement (using \(\neg\), \(\vee\), and \(\wedge\)).
- Convert this boolean function into a circuit diagram.
Reduce the boolean function \(f(x,y,z)=xy\overline{z}+x\overline{y}\overline{z}+\overline{x}y\overline{z}\).


2.1: Combination Series and Parallel Circuits
With simple series circuits , all components are connected end-to-end to form only one path for electrons to flow through the circuit:
With simple parallel circuits , all components are connected between the same two sets of electrically common points, creating multiple paths for electrons to flow from one end of the battery to the other:
With each of these two basic circuit configurations, we have specific sets of rules describing voltage, current, and resistance relationships.
- Series Circuits:
- Voltage drops add to equal total voltage.
- All components share the same (equal) current.
- Resistances add to equal total resistance.
- Parallel Circuits:
- All components share the same (equal) voltage.
- Branch currents add to equal total current.
- Resistances diminish to equal total resistance.
However, if circuit components are series-connected in some parts and parallel in others, we won’t be able to apply a single set of rules to every part of that circuit. Instead, we will have to identify which parts of that circuit are series and which parts are parallel, then selectively apply series and parallel rules as necessary to determine what is happening. Take the following circuit, for instance:
This circuit is neither simple series nor simple parallel. Rather, it contains elements of both. The current exits the bottom of the battery, splits up to travel through R 3 and R 4 , rejoins, then splits up again to travel through R 1 and R 2 , then rejoins again to return to the top of the battery. There exists more than one path for current to travel (not series), yet there are more than two sets of electrically common points in the circuit (not parallel).
Because the circuit is a combination of both series and parallel, we cannot apply the rules for voltage, current, and resistance “across the table” to begin analysis like we could when the circuits were one way or the other. For instance, if the above circuit were simple series, we could just add up R 1 through R 4 to arrive at a total resistance, solve for total current, and then solve for all voltage drops. Likewise, if the above circuit were simple parallel, we could just solve for branch currents, add up branch currents to figure the total current, and then calculate total resistance from total voltage and total current. However, this circuit’s solution will be more complex.
The table will still help us manage the different values for series-parallel combination circuits, but we’ll have to be careful how and where we apply the different rules for series and parallel. Ohm’s Law , of course, still works just the same for determining values within a vertical column in the table.
If we are able to identify which parts of the circuit are series and which parts are parallel, we can analyze it in stages, approaching each part one at a time, using the appropriate rules to determine the relationships of voltage, current, and resistance. The rest of this chapter will be devoted to showing you techniques for doing this.
Process of Series-Parallel Resistor Circuit Analysis
The goal of series-parallel resistor circuit analysis is to be able to determine all voltage drops, currents, and power dissipations in a circuit. The general strategy to accomplish this goal is as follows:
- Step 1: Assess which resistors in a circuit are connected together in simple series or simple parallel.
- Step 2: Re-draw the circuit, replacing each of those series or parallel resistor combinations identified in step 1 with a single, equivalent-value resistor. If using a table to manage variables, make a new table column for each resistance equivalent.
- Step 3: Repeat steps 1 and 2 until the entire circuit is reduced to one equivalent resistor.
- Step 4: Calculate total current from total voltage and total resistance (I=E/R).
- Step 5: Taking total voltage and total current values, go back to last step in the circuit reduction process and insert those values where applicable.
- Step 6: From known resistances and total voltage / total current values from step 5, use Ohm’s Law to calculate unknown values (voltage or current) (E=IR or I=E/R).
- Step 7: Repeat steps 5 and 6 until all values for voltage and current are known in the original circuit configuration. Essentially, you will proceed step-by-step from the simplified version of the circuit back into its original, complex form, plugging in values of voltage and current where appropriate until all values of voltage and current are known.
- Step 8: Calculate power dissipations from known voltage, current, and/or resistance values.
This may sound like an intimidating process, but its much easier understood through example than through description.
In the example circuit above, R 1 and R 2 are connected in a simple parallel arrangement, as are R 3 and R 4 . Having been identified, these sections need to be converted into equivalent single resistors , and the circuit re-drawn:
The double slash (//) symbols represent “parallel” to show that the equivalent resistor values were calculated using the 1/(1/R) formula. The 71.429 Ω resistor at the top of the circuit is the equivalent of R 1 and R 2 in parallel with each other. The 127.27 Ω resistor at the bottom is the equivalent of R 3 and R 4 in parallel with each other.
It should be apparent now that the circuit has been reduced to a simple series configuration with only two (equivalent) resistances. The final step in reduction is to add these two resistances to come up with a total circuit resistance. When we add those two equivalent resistances, we get a resistance of 198.70 Ω. Now, we can re-draw the circuit as a single equivalent resistance and add the total resistance figure to the rightmost column of our table. Note that the “Total” column has been relabeled (R 1 //R 2 —R 3 //R 4 ) to indicate how it relates electrically to the other columns of figures. The “—” symbol is used here to represent “series,” just as the “//” symbol is used to represent “parallel.”
Now, total circuit current can be determined by applying Ohm’s Law (I=E/R) to the “Total” column in the table:
Back to our equivalent circuit drawing, our total current value of 120.78 milliamps is shown as the only current here:
Now we start to work backwards in our progression of circuit re-drawings to the original configuration. The next step is to go to the circuit where R 1 //R 2 and R 3 //R 4 are in series:
Since R 1 //R 2 and R 3 //R 4 are in series with each other, the current through those two sets of equivalent resistances must be the same. Furthermore, the current through them must be the same as the total current, so we can fill in our table with the appropriate current values, simply copying the current figure from the Total column to the R 1 //R 2 and R 3 //R 4 columns:
Now, knowing the current through the equivalent resistors R 1 //R 2 and R 3 //R 4 , we can apply Ohm’s Law (E=IR) to the two right vertical columns to find voltage drops across them:
Because we know R 1 //R 2 and R 3 //R 4 are parallel resistor equivalents, and we know that voltage drops in parallel circuits are the same, we can transfer the respective voltage drops to the appropriate columns on the table for those individual resistors. In other words, we take another step backwards in our drawing sequence to the original configuration, and complete the table accordingly:
Finally, the original section of the table (columns R 1 through R 4 ) is complete with enough values to finish. Applying Ohm’s Law to the remaining vertical columns (I=E/R), we can determine the currents through R 1 , R 2 , R 3 , and R 4 individually:
Placing Voltage and Current Values into Diagrams
Having found all voltage and current values for this circuit, we can show those values in the schematic diagram as such:
As a final check of our work, we can see if the calculated current values add up as they should to the total.  Likewise, since R 3 and R 4 are in parallel, their combined currents should also add up to the total of 120.78 mA. You can check for yourself to verify that these figures do add up as expected.


8.18: Assignment- Hypothesis Testing for the Population Proportion p
The objectives of this activity are:
1. To give you guided practice in carrying out a hypothesis test about a population proportion. (Note: This hypothesis test is also called a z-test for the population proportion.)
2. To learn how to use statistical software to help you carry out the test.
Background: This activity is based on the results of a recent study on the safety of airplane drinking water that was conducted by the U.S. Environmental Protection Agency (EPA). A study found that out of a random sample of 316 airplanes tested, 40 had coliform bacteria in the drinking water drawn from restrooms and kitchens. As a benchmark comparison, in 2003 the EPA found that about 3.5% of the U.S. population have coliform bacteria-infected drinking water. The question of interest is whether, based on the results of this study, we can conclude that drinking water on airplanes is more contaminated than drinking water in general
Question 1:
Let p be the proportion of contaminated drinking water in airplanes. Write down the appropriate null and alternative hypotheses.
Question 2:
Based on the collected data, is it safe to use the z-test for p in this scenario? Explain.
Use the following instructions to conduct the z-test for the population proportion:
Instructions
Click on the link corresponding to your statistical package to see instructions for completing the activity, and then answer the questions below.
R | StatCrunch | Minitab | Excel | TI Calculator
Question 3:
Now that we have established that it is safe to use the Z-test for p for our problem, go ahead and carry out the test. Paste the output below.
Question 4:
Note that, according to the output, the test statistic for this test is 8.86. Make sure you understand how this was calculated, and give an interpretation of its value.
Question 5:
We calculated a P-value of 0 in this test. Interpret what that means, and draw your conclusions.
Contributors and Attributions
- Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution


5.3.2: Franz Schubert, “Elf King”
For our next example, we will return to the Viennese composer Franz Schubert. We have already examined his approach to musical storytelling using the song cycle as a vehicle. Now we will see how he tells a complex story within a single song. Our example will be perhaps his most famous song, “Elf King” (1815; German: “Erlkönig”). This ballad was inspired by centuries of folk tradition, but also exhibits Schubert’s thorough rejection of folk style.
Seeing as Schubert’s career was addressed in the previous section, we will start right in with the ballad itself. Before we can discuss the music, however, we must examine the text. And before we can examine the text, we must know something about the famous and influential poet who wrote it, Johann Wolfgang von Goethe (1749-1832).
Johann Wolfgang von Goethe
For not being a musician himself, Goethe had an enormous influence on German song composers of the era. To begin with, he wrote hundreds of poetic texts that were intended to be set to music—and indeed, countless composers set his texts thousands of times. However, Goethe also had very strong opinions about how his texts should be set to music. These were informed by his fascination with folk culture, which—following the ideas of Herder—he esteemed highly.
Goethe desired that composers setting his texts do their best to imitate folk style, which we have seen well represented in the ballad “Pretty Polly.” Specifically, he thought that all such songs should be strophic, that they should be simple and accessible, and that the composer should make no effort to interpret the text through music. The purpose of the musical setting, in Goethe’s mind, was simply to allow the text to be sung. The listener’s focus should be entirely on the words, which the singer would interpret through subtle variations from verse to verse.
Goethe studied folk poetry closely and imitated a number of folk forms in his own song texts. One of these was the ballad. Goethe was particularly influenced by the ballad forms of northern Europe, which had a unique set of characteristics. The stories told in such ballads were preoccupied with supernatural forces and often included dialogue between human and non-human characters. They also ended in disaster. (All of this is true of “The Gosport Tragedy,” which concludes with a ghost taking her revenge, although the supernatural element is absent in “Pretty Polly.”)
“Elf King”
Goethe and other German poets who took their inspiration from European folk traditions often began by translating folk ballads into their own language. Such was the origin of “Elf King,” which began life as a Danish folk ballad and was first translated by Herder. (The German title of the ballad, “Erlkönig,” in fact represents a mistranslation from the Dutch original; “Erlkönig” means “King of the Alder Trees,” while the original title, “Ellerkonge,” means “King of the Elves.”) Goethe’s version of the ballad tells the same story as the Dutch original, but the specific words are entirely his own.
Here is the text to Goethe’s poem. This literal translation does not capture the meter or rhyme scheme of the original, but it tells the story as Goethe intended:
Who rides so late through night and wind? It is the father with his child.
He has the little one well in the arm
He holds him secure, he holds him warm.
“My son, why hide your face in fear?” “See you not, Father, the Elf King?
The Elf King with crown and flowing cloak?” “My son, it is a wisp of fog.”
“You sweet child, come along with me! Such wonderful games I’ll play with you; Many lovely flowers are at the shore,
My mother has many golden garments.”
“My father, my father, and do you not hear, What the Elf King quietly promises to me?” “Be calm, stay calm, my child;
The wind is rustling the dry leaves.”
“Won’t you come along with me, my fine boy? My daughters shall attend to you so nicely; My daughters do their nightly dance,
And they will rock you and dance you and sing you to sleep.”
“My father, my father, do you not see there, Elf King’s daughters in that dark place?” “My son, my son, I see it definitely
It is the willow trees looking so grey.”
“I love you; I’m charmed by your beautiful shape; And if you are not willing, then I will use force.” “My father, my father, now he has taken hold of me! Elf King has hurt me!”
The father shudders, he rides swiftly, He holds in arm the groaning child,
He reaches the farmhouse with effort and urgency; In his arms, the child was dead.
translation from CPDL
In writing his version of the ballad, Goethe used simple language that reflected the speech of ordinary Germans. The structure of his poem, however, is fairly sophisticated. The first and last stanzas are spoken in the voice of an unidentified narrator. The first sets the scene, while the last delivers the tragic conclusion. The internal stanzas consist entirely of speech from the three characters: father, son, and Elf King. The father and son converse in stanzas two, four, and six, the son expressing his fear and the father offering assurances that there is no real danger. The Elf King seeks to tempt the child away in stanzas three and five. In stanza seven, the Elf King changes strategies and takes the child by force, provoking a frenzied response. The poem can be read literally or as a metaphor for childhood illness and death.
Goethe first created his ballad for use in a play entitled The Fisherwoman (1782). In the first scene of the play, the title character is seen washing dishes and singing a simple folk song (“Elf King”), as befits her lowly social status. The music was composed by the actress herself, Corona Schröter. In setting the poem, Schröter adhered closely to Goethe’s preferences. She created a strophic setting in which every verse of the ballad is sung to the same melody. That melody, in turn, is charming in its simplicity. Schröter makes no effort to capture the drama or terror of the text in her music. In short, it is easy to hear her version8 as being a “real” folk song.
|
Time |
Form |
What to listen for |
|---|---|---|
|
0’00” |
Introduction |
The piano accompaniment seems to echo the beating of the horse’s hooves and the blowing of the wind; use of the minor mode sets a serious tone |
|
0’23” |
Stanza 1 |
This stanza is delivered entirely in the narrator’s voice, which is in the middle of the singer’s range |
|
0’56” |
Stanza 2 |
This stanza is split between the father’s voice (low range) and son’s voice (high range) |
|
1’29” |
Stanza 3 |
This stanza is delivered in the Elf King’s voice (high range); the music changes to the major mode and the volume decreases |
|
1’52” |
Stanza 4 |
This stanza begins in the son’s voice and concludes in the father’s voice; the music returns to the minor mode and the volume increases |
|
2’14” |
Stanza 5 |
This stanza is delivered in the Elf King’s voice (high range); the music again changes to the major mode and the volume decreases |
|
2’32” |
Stanza 6 |
This stanza begins in the son’s voice and concludes in the father’s voice; the music returns to the minor mode and the volume increases |
|
3’00” |
Stanza 7 |
This stanza begins in the Elf King’s voice and concludes in the son’s voice; the music begins in the major mode but quickly returns to minor |
|
3’24” |
Stanza 8 |
This stanza is delivered in the narrator’s voice |
| 3’39” | The hoofbeats heard in the piano slow to a halt as the father arrives at his destination | |
| 3’47” | The final line is delivered without accompaniment; the piano provides a concluding cadence |
This first setting of Goethe’s poem is quite unlike Schubert’s “Elf King.”
By the time Schubert took on Goethe’s ballad, the text was well known and had been set to music by many composers. Schubert was unknown at the time, but his innovative and compelling version of “Elf King” would attract a great deal of attention and establish him as an important composer of songs.
Schubert completely ignored Goethe’s instructions. Instead of taking a folk-like approach, he went to great lengths to capture the characters and events depicted in the text. His setting begins with a turbulent piano introduction, in which we hear both the pounding hooves and raging storm. At the end of the song, we hear in the piano the gradual slowing of the horse as it comes to a clattering stop in front of the farmhouse. All of these are examples of text painting , a technique by which composers translate dramatic elements into sound. Text painting, of course, went against the folk tradition, as did using the piano to set a mood. In addition, Schubert’s piano part is much too difficult to be genuine folk music. Only a highly accomplished player could ever hope to execute it well.
Schubert’s disobedience does not stop there. His setting of the ballad is not strophic but through-composed, with each passage of music carefully designed to express the associated text. Schubert creates a sort of miniature, one-man opera with his setting. Although “Elf King” is performed by a solo singer, that singer assumes all four characters (narrator, father, son, and Elf King) and portrays each in their respective dramatic roles. This is done using a variety of means. To begin with, the voice of each character is heard in a unique range. The narrator sings in a neutral, middle range, while the father sings in the low range, with a deep, booming voice. The son sings in a high, childlike range, while the Elf King sings quietly in an even higher range.
 While the father and son repeat the same melodies and rhythms, the Elf King constantly introduces new musical material as he tries various approaches to tempting the child away. The Elf King is also the only character who sings in the major mode—a musical embodiment of his charming speech. He does not shift to the minor mode until his final threatening words: “then I will use force.”
The greatest musical disruption takes place with the final line of text. In order to maximize the impact of the story’s tragic conclusion, Schubert silences the piano and has the singer deliver the news out of time and without accompaniment. Following the final two words, “was dead,” the piano executes an abrupt cadence and the song is over.


14.1: Brønsted-Lowry Acids and Bases
- Identify acids, bases, and conjugate acid-base pairs according to the Brønsted-Lowry definition
- Write equations for acid and base ionization reactions
- Use the ion-product constant for water to calculate hydronium and hydroxide ion concentrations
- Describe the acid-base behavior of amphiprotic substances
Acids and bases have been known for a long time. When Robert Boyle characterized them in 1680, he noted that acids dissolve many substances, change the color of certain natural dyes (for example, they change litmus from blue to red), and lose these characteristic properties after coming into contact with alkalis (bases). In the eighteenth century, it was recognized that acids have a sour taste, react with limestone to liberate a gaseous substance (now known to be CO 2 ), and interact with alkalis to form neutral substances. In 1815, Humphry Davy contributed greatly to the development of the modern acid-base concept by demonstrating that hydrogen is the essential constituent of acids. Around that same time, Joseph Louis Gay-Lussac concluded that acids are substances that can neutralize bases and that these two classes of substances can be defined only in terms of each other. The significance of hydrogen was reemphasized in 1884 when Svante Arrhenius defined an acid as a compound that dissolves in water to yield hydrogen cations (now recognized to be hydronium ions) and a base as a compound that dissolves in water to yield hydroxide anions.
In an earlier chapter on chemical reactions, we defined acids and bases as Arrhenius did: We identified an acid as a compound that dissolves in water to yield hydronium ions (H 3 O + ) and a base as a compound that dissolves in water to yield hydroxide ions (\(\ce{OH-}\)). This definition is not wrong; it is simply limited.
Later, we extended the definition of an acid or a base using the more general definition proposed in 1923 by the Danish chemist Johannes Brønsted and the English chemist Thomas Lowry. Their definition centers on the proton, \(\ce{H^+}\).  A compound that donates a proton to another compound is called a Brønsted-Lowry acid, and a compound that accepts a proton is called a Brønsted-Lowry base. An acid-base reaction is the transfer of a proton from a proton donor (acid) to a proton acceptor (base). In a subsequent chapter of this text we will introduce the most general model of acid-base behavior introduced by the American chemist G. N. Lewis.
Acids may be compounds such as HCl or H 2 SO 4 , organic acids like acetic acid (\(\ce{CH_3COOH}\)) or ascorbic acid (vitamin C), or H 2 O. Anions (such as \(\ce{ HSO _4^-}\), \(\ce{H_2PO_4^-}\), \(\ce{HS^-}\), and \(\ce{HCO_3^-}\)) and cations (such as \(\ce{H_3O^+}\), \(\ce{NH_4^+}\), and \(\ce{[Al(H_2O)_6]^{3+}}\)) may also act as acids. Bases fall into the same three categories. Bases may be neutral molecules (such as \(\ce{H_2O}\), \(\ce{NH_3}\), and \(\ce{CH_3NH_2}\)), anions (such as \(\ce{OH^-}\), \(\ce{HS^-}\), \(\ce{HCO_3^-}\), \(\ce{CO_3^{2−}}\), \(\ce{F^-}\), and \(\ce{PO_4^{3−}}\)), or cations (such as \(\ce{[Al(H_2O)_5OH]^{2+}}\)). The most familiar bases are ionic compounds such as \(\ce{NaOH}\) and \(\ce{Ca(OH)_2}\), which contain the hydroxide ion, \(\ce{OH^-}\). The hydroxide ion in these compounds accepts a proton from acids to form water:
\[\ce{H^+ + OH^- \rightarrow H_2O} \label{14.11} \]
We call the product that remains after an acid donates a proton the conjugate base of the acid. This species is a base because it can accept a proton (to re-form the acid):
\[\text{acid} \rightleftharpoons \text{proton} + \text{conjugate base}\label{14.12a} \]
\[\ce{HF \rightleftharpoons H^+ + F^-} \label{14.12b} \]
\[\ce{H_2SO_4 \rightleftharpoons H^+ + HSO_4^{−}}\label{14.12c} \]
\[\ce{H_2O \rightleftharpoons H^+ + OH^-}\label{14.12d} \]
\[\ce{HSO_4^- \rightleftharpoons H^+ + SO_4^{2−}}\label{14.12e} \]
\[\ce{NH_4^+ \rightleftharpoons H^+ + NH_3} \label{14.12f} \]
We call the product that results when a base accepts a proton the base’s conjugate acid. This species is an acid because it can give up a proton (and thus re-form the base):
\[\text{base} + \text{proton} \rightleftharpoons \text{conjugate acid} \label{14.13a} \]
\[\ce{OH^- +H^+ \rightleftharpoons H2O}\label{14.13b} \]
\[\ce{H_2O + H^+ \rightleftharpoons H3O+}\label{14.13c} \]
\[\ce{NH_3 +H^+ \rightleftharpoons NH4+}\label{14.13d} \]
\[\ce{S^{2-} +H^+ \rightleftharpoons HS-}\label{14.13e} \]
\[\ce{CO_3^{2-} +H^+ \rightleftharpoons HCO3-}\label{14.13f} \]
\[\ce{F^- +H^+ \rightleftharpoons HF} \label{14.13g} \]
In these two sets of equations, the behaviors of acids as proton donors and bases as proton acceptors are represented in isolation. In reality, all acid-base reactions involve the transfer of protons between acids and bases. For example, consider the acid-base reaction that takes place when ammonia is dissolved in water. A water molecule (functioning as an acid) transfers a proton to an ammonia molecule (functioning as a base), yielding the conjugate base of water, \(\ce{OH^-}\), and the conjugate acid of ammonia, \(\ce{NH4+}\):
The reaction between a Brønsted-Lowry acid and water is called acid ionization . For example, when hydrogen fluoride dissolves in water and ionizes, protons are transferred from hydrogen fluoride molecules to water molecules, yielding hydronium ions and fluoride ions:
When we add a base to water, a base ionization reaction occurs in which protons are transferred from water molecules to base molecules. For example, adding pyridine to water yields hydroxide ions and pyridinium ions:
Notice that both these ionization reactions are represented as equilibrium processes. The relative extent to which these acid and base ionization reactions proceed is an important topic treated in a later section of this chapter. In the preceding paragraphs we saw that water can function as either an acid or a base, depending on the nature of the solute dissolved in it. In fact, in pure water or in any aqueous solution, water acts both as an acid and a base. A very small fraction of water molecules donate protons to other water molecules to form hydronium ions and hydroxide ions:
This type of reaction, in which a substance ionizes when one molecule of the substance reacts with another molecule of the same substance, is referred to as autoionization . Pure water undergoes autoionization to a very slight extent. Only about two out of every \(10^9\) molecules in a sample of pure water are ionized at 25 °C. The equilibrium constant for the ionization of water is called the ion-product constant for water ( K w ):
\[\ce{H_2O}_{(l)}+\ce{H_2O}_{(l)} \rightleftharpoons \ce{H_3O^+}_{(aq)}+\ce{OH^-}_{(aq)}\;\;\; K_\ce{w}=\ce{[H_3O^+][OH^- ]} \label{14.14} \]
The slight ionization of pure water is reflected in the small value of the equilibrium constant; at 25 °C, Kw has a value of \(1.0 \times 10^{−14}\). The process is endothermic, and so the extent of ionization and the resulting concentrations of hydronium ion and hydroxide ion increase with temperature. For example, at 100 °C, the value for \(K_\ce{w}\) is approximately \(5.1 \times 10^{−13}\), roughly 50-times larger than the value at 25 °C.
What are the hydronium ion concentration and the hydroxide ion concentration in pure water at 25 °C?
Solution
The autoionization of water yields the same number of hydronium and hydroxide ions. Therefore, in pure water, \(\ce{[H_3O^+]} = \ce{[OH^- ]}\). At 25 °C:
\[K_\ce{w}=\ce{[H_3O^+][OH^- ]}=\ce{[H_3O^+]^2}=\ce{[OH^- ]^2}=1.0 \times 10^{−14} \nonumber \]
So:
\[\ce{[H_3O^+]}=\ce{[OH^- ]}=\sqrt{1.0 \times 10^{−14}} =1.0 \times 10^{−7}\; M \nonumber \]
The hydronium ion concentration and the hydroxide ion concentration are the same, and we find that both equal \(1.0 \times 10^{−7}\; M\).
The ion product of water at 80 °C is \(2.4 \times 10^{−13}\). What are the concentrations of hydronium and hydroxide ions in pure water at 80 °C?
- Answer
-
\(\ce{[H_3O^+]} = \ce{[OH^- ]} = 4.9 \times 10^{−7}\; M\)
It is important to realize that the autoionization equilibrium for water is established in all aqueous solutions. Adding an acid or base to water will not change the position of the equilibrium. Example 14.12 demonstrates the quantitative aspects of this relation between hydronium and hydroxide ion concentrations.
The Inverse Proportionality of [H 3 O + ] and [OH - ] A solution of carbon dioxide in water has a hydronium ion concentration of \(2.0 \times 10^{−6}\; M\). What is the concentration of hydroxide ion at 25 °C?
Solution
We know the value of the ion-product constant for water at 25 °C:
\[\ce{2 H_2O}_{(l)} \rightleftharpoons \ce{H_3O^+}_{(aq)} + \ce{OH^-}_{(aq)} \nonumber \]
\[K_\ce{w}=\ce{[H3O+][OH^- ]}=1.0 \times 10^{−14} \nonumber \]
Thus, we can calculate the missing equilibrium concentration.
Rearrangement of the K w expression yields that \([\ce{OH^- }]\) is directly proportional to the inverse of [H 3 O + ]:
\[[\ce{OH^- }]=\dfrac{K_{\ce w}}{[\ce{H_3O^+}]}=\dfrac{1.0 \times 10^{−14}}{2.0 \times 10^{−6}}=5.0 \times 10^{−9} \nonumber \]
The hydroxide ion concentration in water is reduced to \(5.0 \times 10^{−9}\: M\) as the hydrogen ion concentration increases to \(2.0 \times 10^{−6}\; M\). This is expected from Le Chatelier’s principle; the autoionization reaction shifts to the left to reduce the stress of the increased hydronium ion concentration and the \(\ce{[OH^- ]}\) is reduced relative to that in pure water.
A check of these concentrations confirms that our arithmetic is correct:
\[\begin{align*} K_\ce{w} &=\ce{[H_3O^+][OH^- ]} \\[4pt] &=(2.0 \times 10^{−6})(5.0 \times 10^{−9})\\[4pt] &=1.0 \times 10^{−14} \end{align*} \nonumber \]
What is the hydronium ion concentration in an aqueous solution with a hydroxide ion concentration of 0.001 M at 25 °C?
- Answer
-
\[\ce{[H3O+]} = 1 \times 10^{−11} M \nonumber \]
Amphiprotic Species
Like water, many molecules and ions may either gain or lose a proton under the appropriate conditions. Such species are said to be amphiprotic . Another term used to describe such species is amphoteric , which is a more general term for a species that may act either as an acid or a base by any definition (not just the Brønsted-Lowry one). Consider for example the bicarbonate ion, which may either donate or accept a proton as shown here:
\[\ce{HCO^-}_{3(aq)} + \ce{H_2O}_{(l)} \rightleftharpoons \ce{CO^{2-}}_{3(aq)} + \ce{H_3O^+}_{(aq)} \label{14.15a} \]
\[ \ce{HCO^-}_{3(aq)} + \ce{H_2O}_{(l)} \rightleftharpoons \ce{H_2CO}_{3(aq)} + \ce{OH^-}_{(aq)} \label{14.15b} \]
Write separate equations representing the reaction of \(\ce{HSO3-}\)
- as an acid with \(\ce{OH^-}\)
- as a base with HI
Solution
- \(\ce{HSO3-}(aq)+ \ce{OH^-}(aq)\rightleftharpoons \ce{SO3^2-}(aq)+ \ce{H_2O}_{(l)} \)
- \(\ce{HSO3-}(aq)+\ce{HI}(aq)\rightleftharpoons \ce{H2SO3}(aq)+\ce{I-}(aq)\)
Write separate equations representing the reaction of \(\ce{H2PO4-}\)
- as a base with HBr
- as an acid with \(\ce{OH^-}\)
- Answer a
-
\(\ce{H2PO4-}(aq)+\ce{HBr}(aq)\rightleftharpoons \ce{H3PO4}(aq)+\ce{Br-}(aq)\)
- Answer b
-
\(\ce{H2PO4-}(aq)+\ce{OH^-} (aq)\rightleftharpoons \ce{HPO4^2-}(aq)+ \ce{H_2O}_{(l)} \)
Summary
A compound that can donate a proton (a hydrogen ion) to another compound is called a Brønsted-Lowry acid. The compound that accepts the proton is called a Brønsted-Lowry base. The species remaining after a Brønsted-Lowry acid has lost a proton is the conjugate base of the acid. The species formed when a Brønsted-Lowry base gains a proton is the conjugate acid of the base. Thus, an acid-base reaction occurs when a proton is transferred from an acid to a base, with formation of the conjugate base of the reactant acid and formation of the conjugate acid of the reactant base. Amphiprotic species can act as both proton donors and proton acceptors. Water is the most important amphiprotic species. It can form both the hydronium ion, H 3 O + , and the hydroxide ion, \(\ce{OH^-}\) when it undergoes autoionization:
\[\ce{2 H_2O}_{(l)} \rightleftharpoons \ce{H_3O^+}(aq)+\ce{OH^-} (aq) \nonumber \]
The ion product of water, K w is the equilibrium constant for the autoionization reaction:
\[K_\ce{w}=\mathrm{[H_3O^+][OH^- ]=1.0 \times 10^{−14} \; at\; 25°C} \nonumber \]
Key Equations
- \[K_{\ce w} = \ce{[H3O+][OH^- ]} = 1.0 \times 10^{−14}\textrm{ (at 25 °C)} \nonumber \]
Glossary
- acid ionization
- reaction involving the transfer of a proton from an acid to water, yielding hydronium ions and the conjugate base of the acid
- amphiprotic
- species that may either gain or lose a proton in a reaction
- amphoteric
- species that can act as either an acid or a base
- autoionization
- reaction between identical species yielding ionic products; for water, this reaction involves transfer of protons to yield hydronium and hydroxide ions
- base ionization
- reaction involving the transfer of a proton from water to a base, yielding hydroxide ions and the conjugate acid of the base
- Brønsted-Lowry acid
- proton donor
- Brønsted-Lowry base
- proton acceptor
- conjugate acid
- substance formed when a base gains a proton
- conjugate base
- substance formed when an acid loses a proton
- ion-product constant for water ( K w )
- equilibrium constant for the autoionization of water