Modal of Necessity and Probability

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

Modal Verbs of necessity and lack of necessity

Modals of Obligation and Necessity

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Land warfare is a complex domain that involves the application of military power on the ground to achieve political and strategic objectives. Modern military doctrine, such as that used by the U.S. Army and the Indian Army, categorizes these elements into Combat Power and the Principles of War. 1. The 8 Elements of Combat Power Combat power is the total means of destructive, constructive, and information capabilities that a military unit can apply. It is typically broken down into eight key elements: ElementDescriptionLeadershipThe "multiplier" of all other elements. It provides purpose, direction, and motivation to soldiers.InformationEnables commanders to make informed decisions and creates opportunities to achieve results.Mission CommandThe system used to integrate the other elements. It focuses on decentralized execution based on the commander's intent.Movement & ManeuverThe movement of forces to gain a positional advantage over the enemy to deliver lethal or non-lethal effects.IntelligenceThe understanding of the enemy, terrain, weather, and civil considerations.FiresThe use of weapon systems (artillery, mortars, air support) to create specific lethal or non-lethal effects.SustainmentThe logistics required to maintain operations, including ammunition, fuel, food, and medical support.ProtectionThe preservation of the force so that the commander can apply maximum combat power.2. The Principles of War These are the enduring "rules of thumb" that guide how land forces are employed strategically and tactically: Objective: Direct every operation toward a clearly defined and attainable goal. Offensive: Seize, retain, and exploit the initiative. You cannot win by defending alone. Mass: Concentrate the effects of combat power at the most advantageous place and time. Economy of Force: Allocate the minimum essential combat power to secondary efforts so you can "mass" elsewhere. Maneuver: Place the enemy in a position of disadvantage through flexible movement. Unity of Command: Ensure all forces operate under a single responsible commander toward a common objective. Security: Prevent the enemy from gaining an unexpected advantage. Surprise: Strike the enemy at a time, place, or in a manner for which they are unprepared. Simplicity: Prepare clear, uncomplicated plans to minimize confusion in the "fog of war." 3. The Modern Legal Framework Land warfare is also governed by the Law of Land Warfare (International Humanitarian Law), which rests on four pillars: Military Necessity: Actions must be necessary to achieve a legitimate military goal. Distinction: Forces must distinguish between combatants and non-combatants (civilians). Proportionality: The anticipated harm to civilians must not be excessive in relation to the concrete military advantage gained. Unnecessary Suffering: Weapons and methods must not cause gratuitous or superfluous injury. Note: Contemporary land warfare is increasingly "Multi-Domain," meaning land forces must now integrate with cyber, space, and electronic warfare to be effective. , While land warfare uses many tools, the two primary "philosophies" of how to win a war are Attrition and Maneuver. Most modern conflicts are a spectrum of both, but understanding the pure form of each helps explain military strategy. 1. Attrition Warfare: The "Sledgehammer" Attrition warfare is a strategy where one side attempts to win by wearing down the enemy to the point of collapse through continuous losses in personnel, equipment, and supplies. Core Logic: "I have more than you." It assumes that if you can destroy the enemy’s resources faster than they can replace them, you will eventually win. Focus: Firepower and mass. Success is measured by "body counts," equipment destroyed, and the steady seizing of terrain. Command Style: Usually centralized and methodical. It requires strict synchronization of massive resources (artillery, logistics, manpower). Historical Example: The Battle of Verdun (WWI). German Chief of Staff Erich von Falkenhayn famously stated his goal was to "bleed France white" by forcing them to defend a position they could not afford to lose, regardless of the cost in lives. 2. Maneuver Warfare: The "Scalpel" Maneuver warfare seeks to shatter the enemy’s moral and physical cohesion—their ability to act as a unified force—rather than simply destroying every soldier. Core Logic: "I am faster and more unpredictable than you." It aims to create a state of chaos where the enemy's leadership can no longer make effective decisions. Focus: Speed, surprise, and dislocation (forcing the enemy to be in the wrong place at the wrong time). The OODA Loop: Developed by Col. John Boyd, this is the heart of maneuver theory. It stands for Observe, Orient, Decide, Act. The goal is to cycle through these steps faster than the enemy, essentially "getting inside" their decision-making process until they collapse from confusion. Historical Example: The 1940 Invasion of France (Blitzkrieg). Instead of fighting a line-by-line battle of attrition, German forces used speed and concentrated armor to bypass strongpoints, cut communication lines, and cause a total systemic collapse of the French military in weeks. 3. Key Differences at a Glance FeatureAttrition WarfareManeuver WarfareObjectivePhysical destruction of the enemy army.Functional/Psychological collapse of the enemy.TargetThe enemy's strength (mass).The enemy's weakness (vulnerability).Primary ToolMassed Firepower.Movement and Tempo.Command"Command Push" (Top-down, rigid)."Recon Pull" (Decentralized, flexible).Success MetricExchange ratios (Kill counts).Disruption and loss of enemy control.4. The Modern Synthesis: "Schwerpunkt" In practice, no army is purely "maneuver" or "attrition." To maneuver successfully, you often need a period of attrition to punch a hole in the enemy's line. A critical concept here is the Schwerpunkt (Center of Gravity/Focus of Effort). A commander identifies the single most important place to strike and concentrates all available "elements of power" there. While the rest of the front might look like attrition, the Schwerpunkt is where the maneuver happens to achieve a breakthrough. Modern Reality: In high-intensity conflicts today (like the war in Ukraine), we see a "return to attrition" because modern sensors (drones, satellites) make it very difficult to achieve the surprise needed for pure maneuver warfare. When you can see everything, it's hard to be "unexpected."

THE STRATEGIC PLAN OF RICHARD BLAND COLLEGE OF WILLIAM & MARY 2020-2025 “The dogmas of the quiet past are inadequate to the stormy present. The occasion is piled high with difficulty, and we must rise with the occasion. As our case is new, so we must think anew and act anew.” – Abraham Lincoln What is the role of a selective, two-year, residential, liberal arts transfer institution within the higher education landscape of the Commonwealth of Virginia? This is a key question that must be answered to ensure the success of Richard Bland College (RBC) and the constituency that the College serves. The 2020 RBC strategic plan’s primary objective is to answer that very question so that the College, the community and the Commonwealth can engage successfully within this identity and purpose to the benefit of all. RBC has long been identified as the hidden gem of higher education in Virginia. The hidden adjective is based both on its relative obscurity—few are aware of RBC outside the Tri-Cities region—and its rural setting featuring 750+ acres of wetlands, bucolic forest, and the state’s oldest and largest pecan grove. Additionally, on average, a student of Richard Bland College travels a mere 36 miles to campus. This keeps the knowledge of RBC in a tightly focused radius. The gem moniker refers both to the College’s reputation for excellence and the undeniable sensation that the campus often elicits in its students, visitors, faculty and staff, the feeling of a warm and palpable embrace of care, compassion and support. That sensation is where we start. According the State Council of Higher Education for Virginia (SCHEV), 99% of the 11.5 million new jobs created since the great recession require workers to have more than a high-school education. Students with a bachelor’s degree have an earning potential almost double that of people with only a high school education, and yet only 17% of residents in the Petersburg area have a bachelor’s degree, 15% below the national average. The obstacles in the way of education have been exhaustively researched and include financial challenges, academic under-preparedness, low self-esteem, slow college assimilation and immature levels of self-efficacy. To combat this growing problem, Richard Bland College initiated a pilot program to determine the viability of a data-driven approach to improve retention and graduation rates. The program ultimately effected a cultural, organizational and operational shift at RBC, resulting in a personalized model of student support, the Exceptional Student Experience (ESE@RBC). Originally many of the practices that RBC used as the basis of ESE@RBC were adapted from the four key principles found in the American Association of Community Colleges (AACC) Pathways Project: 1) map pathways to student end goals; 2) help students choose and enter a program pathway; 3) keep students on path; and 4) ensure that students are learning. Unfortunately, limited resources made it necessary to skip some primary elements of guided pathways and instead to focus on a specific, high-priority project that was immediately available for implementation, dedicated student support. This strategic framework reimagines the way that RBC serves students, faculty and staff within the context of our existing culture, the principles of guided pathways and a hybrid work-college experience. Rather than thinking of a two-year college as a pipeline to a four-year university, this vision describes a more expansive menu of well-defined pathways to high-demand fields, all radiating from a curriculum constructed around the development of soft skills that define the liberal arts experience: critical thinking, written communication, analytical reasoning, civic engagement and oral communication. Furthermore, the impact of meaningful work is a resonating theme, providing avenues to participate in career-focused internships and jobs that develop important life & work skills, confidence, and character. Richard Bland has tested its entrepreneurial mettle and its capacity for transformation in recent years. The College was among a select few Competency-Based Education sites established by the U.S. Department of Education. We were ahead of the curve using predictive analytics to improve student retention and success rates, and online enrollment now makes up nearly 20 percent of course offerings. It may be counter-intuitive, but these and other deep-level institutional changes still to come will ensure that Richard Bland College remains true to its original mission. We prepare our students for a lifetime of endless potential.

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