Circles in the Coordinate Plane

Short Quiz in Math 10 graphs a circle and other geometric figures on the coordinate plane

Understanding Quantum Theory of Electrons in Atoms The goal of this section is to understand the electron orbitals (location of electrons in atoms), their different energies, and other properties. The use of quantum theory provides the best understanding to these topics. This knowledge is a precursor to chemical bonding. As was described previously, electrons in atoms can exist only on discrete energy levels but not between them. It is said that the energy of an electron in an atom is quantized, that is, it can be equal only to certain specific values and can jump from one energy level to another but not transition smoothly or stay between these levels. The energy levels are labeled with an n value, where n = 1, 2, 3, …. Generally speaking, the energy of an electron in an atom is greater for greater values of n. This number, n, is referred to as the principal quantum number. The principal quantum number defines the location of the energy level. It is essentially the same concept as the n in the Bohr atom description. Another name for the principal quantum number is the shell number. The shells of an atom can be thought of concentric circles radiating out from the nucleus. The electrons that belong to a specific shell are most likely to be found within the corresponding circular area. The further we proceed from the nucleus, the higher the shell number, and so the higher the energy level (Figure 9.4.1). The positively charged protons in the nucleus stabilize the electronic orbitals by electrostatic attraction between the positive charges of the protons and the negative charges of the electrons. So the further away the electron is from the nucleus, the greater the energy it has. This quantum mechanical model for where electrons reside in an atom can be used to look at electronic transitions, the events when an electron moves from one energy level to another. If the transition is to a higher energy level, energy is absorbed, and the energy change has a positive value. To obtain the amount of energy necessary for the transition to a higher energy level, a photon is absorbed by the atom. A transition to a lower energy level involves a release of energy, and the energy change is negative. This process is accompanied by emission of a photon by the atom. The following equation summarizes these relationships and is based on the hydrogen atom: The values nf and ni are the final and initial energy states of the electron. The principal quantum number is one of three quantum numbers used to characterize an orbital. An atomic orbital, which is distinct from an orbit, is a general region in an atom within which an electron is most probable to reside. The quantum mechanical model specifies the probability of finding an electron in the three-dimensional space around the nucleus and is based on solutions of the Schrödinger equation. In addition, the principal quantum number defines the energy of an electron in a hydrogen or hydrogen-like atom or an ion (an atom or an ion with only one electron) and the general region in which discrete energy levels of electrons in a multi-electron atoms and ions are located. Another quantum number is l, the angular momentum quantum number. It is an integer that defines the shape of the orbital, and takes on the values, l = 0, 1, 2, …, n – 1. This means that an orbital with n = 1 can have only one value of l, l = 0, whereas n = 2 permits l = 0 and l = 1, and so on. The principal quantum number defines the general size and energy of the orbital. The l value specifies the shape of the orbital. Orbitals with the same value of l form a subshell. In addition, the greater the angular momentum quantum number, the greater is the angular momentum of an electron at this orbital. Orbitals with l = 0 are called s orbitals (or the s subshells). The value l = 1 corresponds to the p orbitals. For a given n, p orbitals constitute a p subshell (e.g., 3p if n = 3). The orbitals with l = 2 are called the d orbitals, followed by the f-, g-, and h-orbitals for l = 3, 4, 5, and there are higher values we will not consider. There are certain distances from the nucleus at which the probability density of finding an electron located at a particular orbital is zero. In other words, the value of the wavefunction ψ is zero at this distance for this orbital. Such a value of radius r is called a radial node. The number of radial nodes in an orbital is n – l – 1. Consider the examples in Figure 9.4.2. The orbitals depicted are of the s type, thus l = 0 for all of them. It can be seen from the graphs of the probability densities that there are 1 – 0 – 1 = 0 places where the density is zero (nodes) for 1s (n = 1), 2 – 0 – 1 = 1 node for 2s, and 3 – 0 – 1 = 2 nodes for the 3s orbitals. The s subshell electron density distribution is spherical and the p subshell has a dumbbell shape. The d and f orbitals are more complex. These shapes represent the three-dimensional regions within which the electron is likely to be found. Principal quantum number (n) & Orbital angular momentum (l): The Orbital Subshell: https://youtu.be/ms7WR149fAY If an electron has an angular momentum (l ≠ 0), then this vector can point in different directions. In addition, the z component of the angular momentum can have more than one value. This means that if a magnetic field is applied in the z direction, orbitals with different values of the z component of the angular momentum will have different energies resulting from interacting with the field. The magnetic quantum number, called ml, specifies the z component of the angular momentum for a particular orbital. For example, for an s orbital, l = 0, and the only value of ml is zero. For p orbitals, l = 1, and ml can be equal to –1, 0, or +1. Generally speaking, ml can be equal to –l, –(l – 1), …, –1, 0, +1, …, (l – 1), l. The total number of possible orbitals with the same value of l (a subshell) is 2l + 1. Thus, there is one s-orbital for ml = 0, there are three p-orbitals for ml = 1, five d-orbitals for ml = 2, seven f-orbitals for ml = 3, and so forth. The principal quantum number defines the general value of the electronic energy. The angular momentum quantum number determines the shape of the orbital. And the magnetic quantum number specifies orientation of the orbital in space, as can be seen in Figure 9.4.3. Figure 9.4.4 illustrates the energy levels for various orbitals. The number before the orbital name (such as 2s, 3p, and so forth) stands for the principal quantum number, n. The letter in the orbital name defines the subshell with a specific angular momentum quantum number l = 0 for s orbitals, 1 for p orbitals, 2 for d orbitals. Finally, there are more than one possible orbitals for l ≥ 1, each corresponding to a specific value of ml. In the case of a hydrogen atom or a one-electron ion (such as He+, Li2+, and so on), energies of all the orbitals with the same n are the same. This is called a degeneracy, and the energy levels for the same principal quantum number, n, are called degenerate energy levels. However, in atoms with more than one electron, this degeneracy is eliminated by the electron–electron interactions, and orbitals that belong to different subshells have different energies. Orbitals within the same subshell (for example ns, np, nd, nf, such as 2p, 3s) are still degenerate and have the same energy. While the three quantum numbers discussed in the previous paragraphs work well for describing electron orbitals, some experiments showed that they were not sufficient to explain all observed results. It was demonstrated in the 1920s that when hydrogen-line spectra are examined at extremely high resolution, some lines are actually not single peaks but, rather, pairs of closely spaced lines. This is the so-called fine structure of the spectrum, and it implies that there are additional small differences in energies of electrons even when they are located in the same orbital. These observations led Samuel Goudsmit and George Uhlenbeck to propose that electrons have a fourth quantum number. They called this the spin quantum number, or ms. The other three quantum numbers, n, l, and ml, are properties of specific atomic orbitals that also define in what part of the space an electron is most likely to be located. Orbitals are a result of solving the Schrödinger equation for electrons in atoms. The electron spin is a different kind of property. It is a completely quantum phenomenon with no analogues in the classical realm. In addition, it cannot be derived from solving the Schrödinger equation and is not related to the normal spatial coordinates (such as the Cartesian x, y, and z). Electron spin describes an intrinsic electron “rotation” or “spinning.” Each electron acts as a tiny magnet or a tiny rotating object with an angular momentum, even though this rotation cannot be observed in terms of the spatial coordinates. The magnitude of the overall electron spin can only have one value, and an electron can only “spin” in one of two quantized states. One is termed the α state, with the z component of the spin being in the positive direction of the z axis. This corresponds to the spin quantum number ms=12. The other is called the β state, with the z component of the spin being negative and ms=−12. Any electron, regardless of the atomic orbital it is located in, can only have one of those two values of the spin quantum number. The energies of electrons having ms=−12 and ms=12 are different if an external magnetic field is applied. Figure 9.4.5 illustrates this phenomenon. An electron acts like a tiny magnet. Its moment is directed up (in the positive direction of the z axis) for the 12 spin quantum number and down (in the negative z direction) for the spin quantum number of −12. A magnet has a lower energy if its magnetic moment is aligned with the external magnetic field (the left electron) and a higher energy for the magnetic moment being opposite to the applied field. This is why an electron with ms=12 has a slightly lower energy in an external field in the positive z direction, and an electron with ms=−12 has a slightly higher energy in the same field. This is true even for an electron occupying the same orbital in an atom. A spectral line corresponding to a transition for electrons from the same orbital but with different spin quantum numbers has two possible values of energy; thus, the line in the spectrum will show a fine structure splitting. The Pauli Exclusion Principle An electron in an atom is completely described by four quantum numbers: n, l, ml, and ms. The first three quantum numbers define the orbital and the fourth quantum number describes the intrinsic electron property called spin. An Austrian physicist Wolfgang Pauli formulated a general principle that gives the last piece of information that we need to understand the general behavior of electrons in atoms. The Pauli exclusion principle can be formulated as follows: No two electrons in the same atom can have exactly the same set of all the four quantum numbers. What this means is that electrons can share the same orbital (the same set of the quantum numbers n, l, and ml), but only if their spin quantum numbers ms have different values. Since the spin quantum number can only have two values (±12), no more than two electrons can occupy the same orbital (and if two electrons are located in the same orbital, they must have opposite spins). Therefore, any atomic orbital can be populated by only zero, one, or two electrons. The properties and meaning of the quantum numbers of electrons in atoms are briefly

Using active voice of the verbs with this stimulus Bees Collecting Nectar Bees make honey to survive. It is their only essential food. If there are 60,000 bees in a hive about one third of them will be involved in gathering nectar which is then made into honey by the house bees. A small number of bees work as foragers or searchers. They find a source of nectar, then return to the hive to tell the other bees where it is. Foragers let the other bees know where the source of the nectar is by performing a dance which gives information about the direction and the distance the bees will need to fly. During this dance the bee shakes her abdomen from side to side while running in circles in the shape of a figure 8. The dance follows the pattern shown on the following diagram. MAKING HONEY When the bees arrive at the hive carrying nectar, they give this to the house bees. The house bees move the nectar around with their mandibles, exposing it to the warm dry air of the hive. When it is first gathered the nectar contains sugar and minerals mixed with about 80% water. After ten to twenty minutes, when much of the excess water has evaporated, the house bees put the nectar in a cell in the honeycomb where evaporation continues. After three days, the honey in the cells contains about 20% water. At this stage, the bees cover the cells with lids which they make out of beeswax. At any one time the bees in a hive usually gather nectar from the same type of blossom and from the same area. Some of the main sources of nectar are fruit trees, clover and flowering trees. Source: “Hum Sweet Hum”, National Foundation for Educational Research, 1993. GLOSSARY house bee a worker bee which works inside the hive. Mandible mouth-part

Alexander Hamilton, a key figure in the founding of the United States, served as the first Secretary of the Treasury under President George Washington. Aaron Burr, on the other hand, was the sitting Vice President under Thomas Jefferson at the time of the duel. The animosity between Hamilton and Burr was well-documented, stemming from political disagreements and personal slights over the years. The Duel Date and Location: The duel took place on July 11, 1804, in Weehawken, New Jersey, a common site for duels due to its less strict enforcement of anti-dueling laws compared to New York. Cause: The immediate cause of the duel was a series of letters and meetings between intermediaries after Hamilton allegedly insulted Burr at a dinner, which was later reported in a newspaper. Burr demanded an apology; Hamilton refused, leading to the challenge. The Event: On the morning of the duel, both men, along with their seconds and a doctor, rowed across the Hudson River to the dueling ground. The exact events are a matter of historical debate, but it is generally believed that Hamilton fired his shot into the air, adhering to a principle of honor without intent to kill. Burr, however, aimed directly at Hamilton, hitting him in the abdomen. Outcome: Hamilton was severely wounded and transported back to New York City, where he died the following day, surrounded by family and friends5. Aftermath Public Reaction: The news of Hamilton's death caused widespread grief and indignation. Angelica Schuyler Church, Hamilton's sister-in-law, expressed the communal sorrow in a letter, highlighting the shock and consternation that gripped the town5. Impact on Burr: Although Burr was never tried for the duel, his political career suffered greatly. He became a pariah in many circles and faced various legal and financial troubles in the years that followed. Historical Significance: The duel is often cited as a turning point in American political culture, highlighting the dangers of political rivalry and the need for civility in discourse. Conclusion The duel between Alexander Hamilton and Aaron Burr remains a poignant reminder of the intense personal and political conflicts that shaped the early years of the United States. It underscores the tragic potential of unchecked animosity and the importance of reconciliation and dialogue in a democratic society.

The Ship of Shapes One day, Elder Decagon saw that the shapes on Shape Island had become lazy. They sat in their huts, fanning themselves until it was time to eat. The different shapes didn't like to spend time with each other. The Rectangles stayed with the Rectangles, the Circles with the Circles, and so on. Elder Decagon came up with a plan. "Oh, oh, oh!" she exclaimed. Worried, the shapes gathered around her. "Big Scary Fire Mountain just spoke," she said. "It will erupt soon, and all our pants will be on fire." "We must leave the island!" The shapes were confused and scared. "Didn't you hear me?" "Pants will be on fire!" Elder Decagon yelled. "What should we do?" the shapes asked. "You should build a ship," she said very slowly. The shapes cheered for the great idea and hurried off to begin. The next day, Elder Decagon was surprised to see many ships on the beach. Each ship was meant for only one kind of shape. "None of these ships are shipshape, she said. "The Triangles' boat will tip in the water." "The Ovals' ship will float, but it won't move." "The Squares' ship will move, but too slowly." "What should we do?" the shapes asked. "You should build one big ship," Elder Decagon said very slowly. This time, the shapes didn't cheer. They weren't sure how to work together. Elder Decagon picked up a stick and started to draw. She showed them how the Squares sails would move the Ovals' boat. The Triangles' bottom would keep it from tipping. The Stars' propeller and the Hearts oars would help the ship move faster. In the end, all of the shapes went into the ship. The shapes stared at the drawing, but no one moved. "Pants will be on fire!" Elder Decagon yelled. The shapes went to work. When it was finished, all the shapes climbed onto the ship. They waited for Big Scary Fire Mountain to erupt, but it never did. The shapes asked Elder Decagon why it didn't. She just said, "Look at this wonderful, shipshape ship." "It shows that if you work hard together, you can go anywhere and do anything.' After some thought, the shapes agreed. They decided to work together to make Shape Island a better place. They also decided to explore the seas in their shipshape ship.

LARGE CARBON MOLECULES Many carbon compounds are built up from smaller, simpler molecules known as monomers (MAH-ne-mers), such as the ones shown in Figure 3-3. As you can also see in Figure 3-3, monomers can bond to one another to form polymers (PAWL-eh-mer). A polymer is a molecule that consists of repeated, linked units. The units may be identical or structurally related to each other. Large polymers are called macromolecules. There are many types of macromolecules, such as carbohydrates, lipids, proteins and nucleic acids. Monomers link to form polymers through a chemical reaction called a condensation reaction. Each time a monomer is added to a polymer, a water molecule is released. In the condensation reac- tion shown in Figure 3-4, two sugar molecules, glucose and fruc- tose, combine to form the sugar sucrose, which is common table sugar. The two sugar monomers become linked by a C—O—C bridge. In the formation of that bridge, the glucose molecule releases a hydrogen ion, H, and the fructose molecule releases a hydroxide ion, OH. The OH and H ions that are released then combine to produce a water molecule, H2O. In addition to building polymers through condensation reac- tions, living organisms also have to break them down. The break- down of some complex molecules, such as polymers, occurs through a process known as hydrolysis (hie-DRAHL-i-sis). In a hydrolysis reaction, water is used to break down a polymer. The water molecule breaks the bond linking each monomer. Hydrolysis is the reverse of a condensation reaction. The addition of water to some complex molecules, including polymers, under certain con- ditions can break the bonds that hold them together. For example, in Figure 3-4 reversing the reaction will result in sucrose breaking down into fructose and glucose. 2H2O Monomers Polymer C C O H OH C OH H CH2OH C H CH2OH C HO H C O H C OH H C CH2OH H C H OH O Sucrose C C O H OH C OH H CH2OH C H CH2OH C HO H C OH OH H C OH H C CH2OH H C H OH O Glucose Fructose H2O The condensation reaction below shows how glucose links with fructose to form sucrose. One water molecule is produced each time two monomers form a covalent bond. FIGURE 3-4 monomer from the Greek mono, meaning “single or alone,” and meros, meaning “a part” Word Roots and Origins A polymer is the result of bonding between monomers. In this example, each monomer is a six-sided carbon ring. The starch in potatoes is an example of a molecule that is a polymer. FIGURE 3-3 Copyright © by Holt, Rinehart and Winston. All rights reserved. 54 CHAPTER 3 ENERGY CURRENCY Life processes require a constant supply of energy. This energy is available to cells in the form of certain compounds that store a large amount of energy in their overall structure. One of these com- pounds is adenosine (uh-DEN-uh-SEEN) triphosphate, more commonly referred to by its abbreviation, ATP. The left side of Figure 3-5 shows a simplified ATP molecule struc- ture. The 5-carbon sugar, ribose, is represented by the blue carbon ring. The nitrogen-containing compound, adenine, is represented by the 2 orange rings. The three linked phosphate groups, —PO4 , are represented by the blue circles with a “P.” The phospate groups are attached to each other by covalent bonds. The covalent bonds between the phosphate groups are more unstable than the other bonds in the ATP molecule because the phosphate groups are close together and have negative charges. Thus, the negative charges make the bonds easier to break. When a bond between the phosphate groups is broken, energy is released. This hydrolysis of ATP is used by the cell to provide the energy needed to drive the chemical reactions that enable an organism to function.

Descriptors: 1) Circles True and False statementsappropriately 2)Match the pictures with the words correctly 3) Put the letters in correct order to make a word

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