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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 brieflyWhat is a Hurricane, Typhoon, or Tropical Cyclone? The terms "hurricane" and "typhoon" are regionally specific names for a strong "tropical cyclone". A tropical cyclone is the generic term for a non-frontal synoptic scale low-pressure system over tropical or sub-tropical waters with organized convection (i.e. thunderstorm activity) and definite cyclonic surface wind circulation (Holland 1993). Tropical cyclones with maximum sustained surface winds of less than 17 m/s (34 kt, 39 mph) are usually called "tropical depressions" (This is not to be confused with the condition mid-latitude people get during a long, cold and grey winter wishing they could be closer to the equator). Once the tropical cyclone reaches winds of at least 17 m/s (34 kt, 39 mph) they are typically called a "tropical storm" or in Australia a Category 1 cyclone and are assigned a name. If winds reach 33 m/s (64 kt, 74 mph), then they are called: "hurricane" (the North Atlantic Ocean, the Northeast Pacific Ocean east of the dateline, or the South Pacific Ocean east of 160E) "typhoon" (the Northwest Pacific Ocean west of the dateline) "severe tropical cyclone" or "Category 3 cyclone" and above (the Southwest Pacific Ocean west of 160°E or Southeast Indian Ocean east of 90°E) "very severe cyclonic storm" (the North Indian Ocean) "tropical cyclone" (the Southwest Indian Ocean) Coriolis Effect The Coriolis Effect—the deflection of an object moving on or near the surface caused by the planet’s spin—is important to fields, such as meteorology and oceanography. Storm Approaching Southeast Asia Because of the Coriolis Effect, hurricanes spin counterclockwise in the Northern Hemisphere, while these types of storms spin clockwise in the Southern Hemisphere. This Northern Hemisphere storm, approaching Southeast Asia, is spinning counterclockwise. Earth is a spinning planet, and its rotation affects climate, weather, and the ocean through the Coriolis Effect. Named after the French mathematician Gaspard Gustave de Coriolis (born in 1792), the Coriolis Effect refers to the curved path that objects moving on Earth’s surface appear to follow because of the spinning of the planet. As Earth turns, points near the equator—countries like Ecuador and Kenya—are moving much faster than places near the planet’s poles. This is because Earth is shaped like a marble: Its circumference is larger near its middle (the equator) than near its top and bottom. All places on Earth experience a day that is about 24 hours long, but points near the equator have to travel longer distances in the same period of time, which means that those places move faster. Scientists say these points have more “angular momentum.” This is why rockets are usually launched from places near the equator, like Cape Canaveral, Florida, United States. Such locations give rockets a large initial speed, which helps them get into orbit using the least possible amount of fuel. The Coriolis Effect influences wind patterns, which in turn dictate how ocean currents move. Imagine wind near the equator flowing to the north. That wind starts with a certain speed due to Earth’s rotation (near the equator, Earth rotates at a speed of roughly 1,600 kilometers per hour (1,000 miles per hour) from west to east). As the wind travels north toward the North Pole, it moves over parts of Earth that are rotating progressively more slowly. Since the wind retains its angular momentum, it keeps moving from west to east, overtaking the part of Earth turning more slowly below it. As a result, the wind appears to bend to the east (that is, to the right). This is the Coriolis Effect in action. Wind flowing south from the equator would likewise bend to the east. This effect is responsible for many meteorological and oceanographic phenomena. For instance, due to the Coriolis Effect, hurricanes in the Northern Hemisphere spin in a counterclockwise direction, while hurricanes in the Southern Hemisphere (known as cyclones) spin in a clockwise direction. Ocean-circling currents known as “gyres” also spin in spiral patterns thanks to the Coriolis Effect. There is an urban legend that water in toilets spins in opposite directions in the Northern and Southern Hemispheres because of the Coriolis Effect. But that isn't true—a toilet bowl is too small for the effect to be observed. Instead, other factors like the shape of the toilet bowl and the direction that the water enters are largely responsible for how the flushing water moves.8.10.C Explain the Effect of Translations, Reflections & Rotations Applied to 2-D ShapesRotation of the EarthQuarter 4 Science 6 Rotation of the EarthPS1.1h Revolution and rotation of the Earth Based on the provided sources, here is a comprehensive extraction of the information regarding the water cycle, energy transfer, and Earth's wind systems, organized into key points: The Water Cycle and Its Reservoirs • Definition: The water cycle is the continuous movement of water among various reservoirs on Earth. • Water Reservoirs: These are storage locations for water and include: ◦ Oceans, seas, and lakes. ◦ Rivers, glaciers, soil, and rocks. ◦ The atmosphere and living organisms. • Total Volume: The total amount of water on Earth does not change, even when it changes state, because it is constantly being replaced or recycled through the cycle. Main Processes and Energy Transfer The movement of water through the cycle is driven by energy (thermal energy from the Sun) and force (gravity and wind). • Energy Gain (Absorption): ◦ Melting: Water changes from a solid state (ice) to a liquid state and gains energy. ◦ Evaporation: Liquid water changes into a gas state (water vapor) by gaining thermal energy. ◦ Transpiration: A specialized type of evaporation occurring in plants where water vapor is released through tiny holes in leaves called stomata. Approximately 10% of water vapor in the air comes from transpiration. • Energy Loss (Release): ◦ Condensation: Water vapor (gas) cools down and changes back into liquid water, releasing energy. ◦ Freezing: Liquid water changes into a solid state (ice) and loses energy. • Other Key Steps: ◦ Precipitation: Water falls back to Earth as rain, snow, sleet, or hail (snow pellets). ◦ Runoff: Water flows over Earth's surface into streams, rivers, and eventually larger bodies of water like oceans. ◦ Collection: Rainwater is collected in different water bodies to start the cycle again. Forces Driving Water Movement • Gravity: The main force that pulls water downward. It is responsible for: ◦ Bringing precipitation (rain and snow) from clouds to the surface. ◦ Moving ice in glaciers from higher to lower elevations. ◦ Causing liquid water to flow downhill into rivers and seas. ◦ Leakage: Pulling liquid water down into the ground to reach groundwater reservoirs. • Wind: Another force that affects water movement and transports water to different locations on Earth. Atmospheric Processes • Cloud Formation: Water vapor attaches to particles such as dust or smoke in the air and condenses into tiny droplets. When millions of these droplets join, they become heavy and fall as rain. • Convection: The transfer of heat in liquids and gases. ◦ Warm air/liquid: Becomes less dense, lighter, and rises upward. ◦ Cold air/liquid: Is more dense, heavier, and moves downward to replace the warm fluid. ◦ This process leads to convection currents, which help determine regional climates and drive wind and ocean currents. Solar Radiation and Climate The amount of solar energy reaching Earth differs from place to place, which affects the weather: • Hottest Regions (Equator): Sun rays fall perpendicular (vertical). Heat is concentrated on a small area, making the weather hot. • Moderate Regions: Sun rays fall semi-inclined. Heat is distributed over a larger area, making the weather warm. • Coolest Regions (Poles): Sun rays fall very slanted (inclined). Heat is spread over a very large area, making the weather very cold. Earth's Wind System • Wind Formation: Wind is generated when warm air (heated by the Sun) rises and is replaced by cooler air flowing from nearby areas. • Factors Affecting Wind: The amount of solar radiation and the rotation of Earth determine global wind directions. • Global Wind Cycle: Unequal heating between the equator and the poles generates a constant wind system. Warm air rises at the equator and moves toward the poles, while cold air from the poles moves toward the equator. • Importance: If there were no wind, the equator would become extremely hot, the poles would freeze solid, and many ecosystems would disappear. Practical Examples • Turkey’s Salt Lake: High evaporation in the summer can turn this large lake into a small puddle or dry it up completely. It is a critical site for flamingos, which migrate there to breed and feed on algae in the shallow, warm water.6.2 The student will investigate and understand that the solar system is organized and the various bodies in the solar system interact. Key ideas include matter is distributed throughout the solar system; planets have different sizes and orbit at different distances from the sun; gravity contributes to orbital motion; and the understanding of the solar system has developed over time. 6.3 The student will investigate and understand that there is a relationship between the sun, Earth, and the moon. Key ideas include Earth has unique properties; the rotation of Earth in relationship to the sun causes day and night; the movement of Earth and the moon in relationship to the sun causes phases of the moon; Earth’s tilt as it revolves around the sun causes the seasons; and the relationship between Earth and the moon is the primary cause of tides.