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Unit 3: Atomic Theory & Quantum Mechanics Section A.6 – A.7

Unit 3: Atomic Theory & Quantum Mechanics Section A.6 – A.7. In which you will learn about: The quantum mechanical model Heisenberg uncertainty principle Orbitals and their shapes Quantum numbers. A.6 The Quantum Mechanical Model of the Atom.

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Unit 3: Atomic Theory & Quantum Mechanics Section A.6 – A.7

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  1. Unit 3: Atomic Theory & Quantum MechanicsSection A.6 – A.7 In which you will learn about: The quantum mechanical model Heisenberg uncertainty principle Orbitals and their shapes Quantum numbers

  2. A.6 The Quantum Mechanical Model of the Atom • Scientists in the mid-1920s, by then convinced that the Bohr atomic model was incorrect, formulated new and innovative explanations of how electrons are arranged in atoms. • In 1924, a French graduate student in physics named Louis de Broglie (1892-1987) proposed an idea that eventually accounted for the fixed energy levels of Bohr’s model

  3. Electrons as Waves • De Broglie had been thinking that Bohr’s quantized electron orbits had characteristics similar to those of waves. • Imagine the path of an electron around a circle of fixed radius • Notice that in figure (a) there is an odd number of waves, and in (b) there is an even number of waves • (a) works out perfectly and (b) does not

  4. Electrons as Waves Cont’d • Only multiples of half-wavelengths are possible on a plucked guitar string because the string is fixed at both ends • These finite waves for musical instruments and circles led de Broglie to ask an interesting question • If light can act as both a wave and particle, can a particle (like an electron) act like a wave?

  5. De Broglie Equation • The de Broglie equation predicts that all moving particles have wave characteristics • It also explains why it is impossible to notice the wavelength of a fast-moving car – a car moving at 25 m/s and weighing 910 kg would have a wavelength of 2.9 x 10-38 m (too small to be seen or detected!) • For comparison, an electron moving at the same speed has a wavelength of 2.9 x 10-5 m (which can be easily measured). • De Broglie knew that if an electron has wavelike motion and is restricted to circular orbits of fixed radius, only certain wavelengths, frequencies and energies are possible. • This is summed up in the equation λ = h/mv, where λ is still wavelength, h is still Planck’s constant, m is mass and v is velocity (speed). • No, we will NOT be doing calculations with this equation.

  6. The Heisenberg Uncertainty Principle • Werner Heisenberg (1901-1976) showed that it is impossible to take any measurement of an object without disturbing the object • Scientists try to locate electrons by bombarding them with photons of light, but once the electron is hit, it moves to a new location with a new speed! • In other words, the act of observing the electron produces a significant, unavoidable uncertainty in the position and motion of the electron • The Heisenberg uncertainty principle states that it is fundamentally impossible to know precisely both the velocity and position of a particle at the same time.

  7. The Real Deal with Heisenberg • So we can’t know exactly where or how fast an electron is moving • Which means that it is impossible to assign fixed paths for electrons like the circular orbits in Bohr’s model. • The only quantity that can be known is the probability for an electron to occupy a certain region around the nucleus.

  8. The SchrödingerWave Equation • In 1926, Austrian physicist Erwin Shrödinger (1887-1961) furthered the wave-particle theory proposed by de Broglie. • Shrödinger derived an equation that treated the hydrogen atom’s electron as a wave • Shrödinger’s new model for the hydrogen atom seems to apply equally well to atoms of other elements • This is where Bohr’s model failed • The atomic model in which electrons are treated as waves is called the wave mechanical model of the atom, or the quantum mechanical model of the atom. • This model limits an electron’s energy to certain values (like Bohr) • Makes no attempt to describe the electron’s path around the nucleus

  9. Shrödinger is way too complicated • The Shrödinger wave equation is too complex to be considered here • Mrs. Pford didn’t deal with it until junior year of college and she had to use Calculus III-level math to solve it! • Solutions to the equation are called wave functions • Wave functions are related to the probability of finding the electron within a certain volume of space around the nucleus

  10. Electron’s probable location • The wave function predicts a three-dimensional region around the nucleus, called an atomic orbital, which describes the electron’s probable location. • An atomic orbital is like a fuzzy cloud in which the density at a given point is proportional to the probability of finding the electron at that point

  11. Density Maps • The density map can be thought of as a time-exposure photograph of the electron moving around the nucleus • The electron cloud = all the probably places ONE electron COULD be • The electron cloud ≠ all of the electrons in an atom surrounding the nucleus • To overcome the inherent uncertainty about the electron’s location, chemistry arbitrarily draw an orbital’s surface to contain 90% of the electron’s total probability distribution. • Simply, there is a 90% chance you will find an electron somewhere within the electron cloud

  12. A.7 Quantum Numbers • There are four quantum numbers that are used to describe the probable position of an electron. • Each quantum number is usually only referenced by name or variable, but there are also actual numbers, too • No two electrons can have the same exact set of four quantum numbers (more on this next time)

  13. Principal Quantum Number (1st) • The principal quantum number (n) indicates the relative size and energy of atomic orbitals • In other words, n = energy level • n can have whole-number values ranging from 1-7. • If quantum numbers were an address, this is like telling you what state the electron lives in (not very specific if I want to find it)

  14. Angular Momentum Quantum Number (2nd) • The angular momentum quantum number (l) specifies the shape of the orbital that the electron is in. • This is sometimes referred to as the sublevel, but I find this term to be confusing. I try to explain below (somewhat unsuccessfully?) • Sublevel = shape of orbital or orbital type (can be s, p, d, or f) • Orbital = specific orientation of sublevel (can be px, py, or pz depending on the axis the density map is on) • l can have whole number values ranging from 0 to n-1. • If l = 0 it’s an s orbital, if l = 1  p, if l = 2  d, if l = 3  f • See next slide for pictures of orbitals • In the address analogy, using this number helps specify which city the electron is in.

  15. Before We Move On… • Shapes of orbitals include:

  16. Number of Orbitals • There is only ONE type of s orbital • There are THREE types of p orbital • There are FIVE types of d orbital • There are SEVEN types of f orbital (not shown in previous slide)

  17. Hydrogen’s First Four Principal Energy Levels

  18. Magnetic Quantum Number (3rd) • The magnetic quantum number (m)specifies which orientation of the orbitals an electron is in • For example, if we know the electron is in energy level 2 and it is in the p-type orbital, we need to know exactly which p-orbital it is in (there are three possibilities) • m can have integer values going from –l to +l. • In the address analogy, this is like giving the street the electron is on.

  19. Spin Magnetic Quantum Number (4th) • The spin magnetic quantum number (ms) is an inherent property of electrons that separate them into individual positions. • Up until this point, two electrons can share the first 3 quantum numbers, but since no two electrons can share ALL four, we use spin to indentify which is which • The electrons aren’t actually spinning! We refer to them as spin up or spin down but these are just arbitrary terms. • ms can only be +1/2 or -1/2 • In the address analogy, this is like finally giving the house number where the electron is at.

  20. Using Quantum Numbers • If you’ve been following along with the rules… • n = 1, 2, 3, 4, 5, 6, or 7 • l = 0 up to n-1 in integers (0 =s, 1 = p, 2 = d, 3 = f) • m = - l up to +l in integers • ms = ±1/2 • Example Problem: Write the quantum numbers associated with the first electron added to the 4f sublevel. • ANSWER: n = 4 (given), l = 3 (known because it’s an f orbital), m = 3 (-l up to +l in integers and in this case l = 3 – I’m choosing 3, it could be -3, -2, -1, 0, 1, 2 or 3), and ms = +1/2 (again, I’m choosing + because it can only be for ONE electron).

  21. And now that you’re completely confused…Homework! • 1) Differentiate between the wavelength of visible light and the wavelength of a moving soccer ball. • 2) List the number and types of orbitals contained in the hydrogen atom’s first four energy levels. • 3) Explain why the location of an electron in an atom is uncertain using the Heisenberg uncertainty principle and de Broglie’s wave-particle duality. How is the location of electrons in the atom defined? • 4) Compare and contrast Bohr’s model and the quantum mechanical model of the atom. • 5) Write the numbers associated with each of the following: • A) the fifth energy level • B) the 6s sublevel • C) an orbital on the 3d level

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