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Study Guides Big Picture The quantum mechanical model is now the modern and accepted model of the atom. Unlike previous models, the quantum mechanical model of the atom does not predict the path that an electron takes around the nucleus. Due to the Heisenberg uncertainty principle, the position of an electron cannot be precisely known. Instead, electrons occupy orbitals, regions of space where the electrons have the highest probability of existing. To understand the properties of an atom, there are four quantum numbers that describe an electron’s energy, orbital shape, orientation, and spin state. No two electrons in a given atom can have the same four quantum numbers. Key Terms Chemistry Quantum Mechanical Model Orbital: A region of space within the atom where an electron is likely to be found. Orbital Energy: The amount of energy associated with an electron in a particular orbital. Quantum Number: A number describing a property of an electron. Principal (n): Describes the principal energy level of the electron. Aizmuthal (l): Describes the shape of the electron orbital (s: l=0, p: l=1, d: l=2, etc). This number must be between 0 and n-1. Magnetic (ml): Describes the orientation of the electron orbital. This number must be between +n and -n. Spin (ms): Describes the spin state of the electron (±½). Pauli Exclusion Principle: Electrons cannot have the same four quantum numbers within the same atom. Shell: A set of electrons with the same principal quantum number (n). Subshell: A set of electrons with the same azimuthal quantum number (l). Quantum Mechanics In addition to the quantum theory that energy is quantized, the quantum mechanical model of the atom accounts for two features of quantum mechanics: the wave-particle duality and the Heisenberg uncertainty principle. Wave-Particle Duality All moving objects have wavelike behavior. Louis de Broglie (1892-1987) derived an equation to find the wavelength of the associated matter wave for a moving object. • For the matter wave to have a measurable wavelength, the particle needs to atomic or subatomic in size. Heisenberg Uncertainty Principle The Heisenberg uncertainty principle (proposed by Werner Heisenberg) states: It is impossible to know both the velocity and the position of a particle at the same time. The process of making a measurement actually changes what is being measured. • For larger objects, this effect is very small and can be ignored. Schrödinger Equation Edwin Schrödinger treated the electron as a wave when developing his mathematical equation. • Solutions to the Schrödinger equation are called wave functions. • The wave functions can be used to calculate the probability of finding an electron at a given point. • The solutions naturally supported the idea that the energy of the electrons are quantized (electrons can only be found in certain energy levels). An atomic orbital describes the region of space where the electron is likely to be found. Typically it is the region of space where there is a 90% probability of finding the electron. • The electron is not contained solely within this region but rather has a high probability of being found within it. • The uncertainty principle tells us there is no way to know exactly where the electron is, and we cannot know the path the electron takes. • Orbitals have different sizes and shapes. • The orbital energy is the energy of an electron in a particular orbital. Why do we draw orbitals where there is less than 100% probability of finding the electron? The region of space where there is a 100% probability of finding the electron would be as large as the universe! This guide was created by Steven Lai, Rory Runser, and Jin Yu. To learn more about the student authors, visit http://www.ck12.org/about/about-us/team/ interns. Page 1 of 3 v1.1.12.2012 Disclaimer: this study guide was not created to replace your textbook and is for classroom or individual use only. Atomic Orbitals Chemistry Quantum Mechanical Model cont . Quantum Numbers You are not expected to solve the Schrödinger equation. The equation is so complex, it is impossible to solve the equation for atoms and ions with more than one electron. Powerful computers are needed to produce very close approximations to the solutions. Instead, you need to understand the four quantum numbers that describe the atomic orbitals and the electrons in the orbitals. The numbers come from the Schrödiner equation. The four numbers have special names. They are: • Principal quantum number (n) • Aizmuthal quantum number (l) (also known as the angular momentum quantum number) • Magnetic quantum number (ml) • Spin quantum number (ms) No two electrons within the same atom can have the same four numbers! This is known as the Pauli exclusion principle. Principal Quantum Number The principal quantum number is the main energy level occupied by the electron. • Can have the integer values n = 1, 2, 3, 4, and so on • Determines the size of the orbital (larger n, larger orbital) • As n increases, the energy of the electron increases, and the electron is more likely to be found at a distance farther away from the nucleus • n = 1 is lowest in energy, and the electron is most likely close to the nucleus • Electrons with the same value of n in an atom belong to the same shell • The maximum number of electrons in each shell is 2n2 • The total number of allowable orbitals in each shell is equal to n2 • Contains one or more sublevels, or subshells Aizmuthal Quantum Number The aizmuthal quantum number indicates the overall shape of the orbital. • Also referred to as energy sublevels • Can have integer values between 0 and n–1 • Each nth principal energy level has n sublevels available • Can also be designated by the letters s, p, d, f • s orbital is spherical • p orbital is dumbbell-shaped • d and f orbitals are more complex • Affects orbital energies (larger l, higher energy) • Electrons with the same value of l in an atom belong to the same subshell Electron Arrangement Within Energy Levels Principal Quantum Number (n) Allowable Sublevels Number of Orbitals per Sublevel Number of Orbitals per Principal Energy Level Number of Electrons per Sublevel Number of Electrons per Principal Energy Level 1 s 1 1 2 2 2 s p 1 3 4 2 6 8 3 s p d 1 3 5 9 2 6 10 18 4 s p d f 1 3 5 7 16 2 6 10 14 32 Page 2 of 3 cont . Chemistry Quantum Mechanical Model Quantum Numbers (cont.) Magnetic Quantum Number The magnetic quantum number indicates the orientation of the orbital. • Can have integer values between – l and + l • The number of possible ml values for a subshell is equal to the number of orbitals within the subshell • s orbital: l = 0, so ml = 0 • It is spherical, so it has only one possible orientation • p orbital: l = 1, so ml can equal -1, 0, and 1 • The p orbitals have three possible orientation: px aligned along the x-axis, py aligned along the y-axis, and pz aligned along the z-axis Image Credit: CK-12 Foundation, CC-BY-NC-SA 3.0 • The d orbitals have 5 possible orientations Image Credit: CK-12 Foundation, CC-BY-NC-SA 3.0 • The f orbitals have 7 possible orbitals Image Credit: CK-12 Foundation, CC-BY-NC-SA 3.0 Spin Quantum Number The spin quantum number indicates the orientation of the magnetic field. • Electrons spin on their own internal axis, and the spinning creates a magnetic field • The orientation depends on the direction the electron is spinning • There are two possible spin quantum numbers: +½ and -½ Each orbital holds a maximum of two electrons and have opposite spins. 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