Topic 7 Quantum Theory of the Atom

1. Topic 7 Quantum Theory of the Atom Towards the end of the 19th century, experiments involving light interacting with atoms and molecules could not be explained fully by classical physics; therefore, quantum mechanics and relativity were developed to address failures of classical physics in describing the current accepted model of the atom.

2. The Wave Nature of Light Light is known as electromagnetic radiation. Much of the behavior of light can be explained by thinking of it as a wave. Light can also be thought of as a stream of particles called photons. Awaveis a periodic (repeating) disturbance that transfers energy from one place to another. • The term electromagnetic means that the disturbance is due to oscillation of charged particles in electric and magnetic fields and exerts a force on any charged particle that is in its way. • Visible light, X rays, and radio waves are all forms of electromagnetic radiation.

3. The Wave Nature of Light • The wavelength,l (lambda), is the distance (m or nm) between any two adjacent identical points of a wave. • Thefrequency,n (nu), of a wave is the number of wavelengths that pass a fixed point in one second (1/s or s-1 or Hz). The amplitude is the height of the wave. A wave can be characterized by its wavelength and frequency. amplitude

4. The Wave Nature of Light The product of the frequency, n (waves/sec) and the wavelength, l (m/wave) would give the speed of the wave in m/s. • Electromagnetic waves travel at the speed of light, c, which is 3.00 x 108 m/s. Therefore, • So, given the frequency of light, its wavelength can be calculated, or vice versa. note: freq and wavelength are inversely proportional meaning the shorter the wavelength, the higher the frequency.

5. The Wave Nature of Light What is the wavelength of yellow light with a frequency of 5.09 x 1014 s-1? (Note: s-1, commonly referred to asHertz (Hz) is defined as “cycles or waves per second”.) We simply rearrange the equation to solve for the wavelength by multiplying both sides by lambda, l, and dividing both sides by nu, n, giving Next, we plug in the values into the equation and cancel units. l Note: s-1 = 1/s and 1 x 10-9 m = 1 nm

6. The Wave Nature of Light What is the frequency of violet light with a wavelength of 408 nm? We simply plug the values into the equation and cancel units. n Hz Note: 1/s = s-1 = Hz and 1 nm = 1 x 10-9 m

7. The range of frequencies or wavelengths of electromagnetic radiation is called the electromagnetic spectrum and is related to what we perceive as the color of light. low E and v, long l high E and v, short l • Visible light extends from the violet end of the spectrum at about 400 nm (short l, high n) to the red end with wavelengths about 800 nm (long l, low n) . • Beyond these extremes, electromagnetic radiation is not visible to the human eye. HW 57 code: wave

8. Quantum Effects and Photons By the end of 19th century, experiments dealing with light showed behaviors that are inconsistent with the notion of light as an electromagnetic wave. These inconsistencies are resolved by assuming that the energies of light and matter are quantized (meaning values are restricted). Thus, the theory eventually developed into what we call quantum theory. In the case of electromagnetic radiation, we have to think of energy as being carried by a stream of particles called photons.

9. Quantum Effects and Photons The energy carried by each photon is given by: where h is called Planck’s constant = 6.626 x 10-34 J s. Max Planck discovered this constant while trying to explain blackbody radiation. Planck’s constant turned out to be a universal constant used to explain other phenomena as well. Light energy is quantized because it is only available in multiples of hn. Note: energy is proportional to frequency and inversely proportional to wavelength.

10. Quantum Effects and Photons Photoelectric Effect is one phenomena that could not be explained with the notion that light is a wave. A photoelectron is an electron ejected from a metallic surface when the surface is exposed to light. It was found that: • there is a minimum frequency of light (no) needed to cause electrons to be ejected; the threshold frequency depends on the nature of the metallic surface. • the kinetic energy of the electrons ejected depends on the frequency, but not on the intensity of light; this was puzzling because the intensity of light is the amount of energy it delivers per unit time per unit area. • if electrons are ejected, more of them are ejected if the light is made intense.

11. Quantum Effects and Photons Albert Einstein explained the observations as follows: • the kinetic energy of the ejected electron is the difference between the energy of the photon and the energy needed to dislodge the electron from the metal. • if the individual photons do not have sufficient energy to dislodge electrons, no photoelectrons will be observed regardless of how intense the radiation is. • if the photon energies are sufficient, then a higher intensity would mean more photons and, consequently, more photoelectrons produced. • but the kinetic energy of the electrons depend only on the photon energy (hence, on the frequency, not on the intensity).

12. Quantum Effects and Photons Based on the data, Einstein deduced: • the photon energy is hn, where h is the same constant that Planck earlier discovered when trying to explain blackbody radiation. • the minimum photon energy needed to dislodge the electron, called the work function of the metal, hno.

13. Radio Wave Energy What is the energy of a photon corresponding to radio waves of frequency 1.255 x 10 6 Hz? We simply plug the values into the energy equation and cancel units. Hz HW 58 code: energy

14. Atomic and Molecular Spectra When light is passed through a prism, the prism causes the waves corresponding to different wavelengths to bend at different angles. If the light comes from a source where there is a very high concentration of particles, we tend to get a continuous band of colors (the familiar rainbow of colors). Visible Light Spectrum If we pass that light through a sample that has a very low concentration of particles before we pass it through the prism, we are likely to see a pattern of dark lines interspersed within what would otherwise be a continuous band of colors. We say that the light waves (dark lines) were l’sabsorbed by the particles in the sample. Absorption Spectrum of Hydrogen

15. Atomic and Molecular Spectra If we were to examine light from a source where the concentration of particles is very low, we are not likely going to get a continuous band of colors. If we project the light that passes through the prism onto a screen, we would see a pattern of bright colored lines with dark regions in between. The pattern depends on what atoms or molecules are producing the light, and is called the atom’s or molecule’s emission spectrum. Emission spectrum of hydrogen Each atom or molecule has a characteristic spectrum that can serve as its “fingerprint” (unique) for identification purposes. Spectra from atoms tend to have sharp lines and are referred to as line spectra.

16. Figure: Emission (line) spectra of some elements.

17. Atomic and Molecular Spectra How come atoms and molecules do not absorb or emit a continuous band of colors and give line spectra instead? Niels Bohr provided the following explanation: • Energies of atoms and molecules are quantized. They cannot have just any amount of energy; there are particular energies available to be absorbed or emitted for each species. • Transitions between allowed energy levels can occur when a photon is absorbed or released. • The energy of photon must be equal to the difference in energy between two allowed levels (known as Bohr frequency condition) and is associated with the wavelengths absorbed or emitted in the spectrum thereby causing the line spectrum of a species.

18. The Bohr Theory of the Hydrogen Atom Atomic Line Spectra • In 1885, J. J. Balmer showed that the allowed transition wavelengths, l, in the visible spectrum of hydrogen could be reproduced by a simple formula. • The known wavelengths of the four visible lines for hydrogen correspond to values of n = 3, n = 4, n = 5, and n = 6 (electrons jump down from higher level to second level).

19. Figure : Transitions of the electron in the hydrogen atom. There are names for certain transitions: Paschen series: transition from higher level to the n=3. Balmer series: Transition from higher level to n=2 Lyman series: Transition from higher level to n=1 Represents the four lines of hydrogen: transitions from 6 to 2, 5 to 2, 4 to 2, and 3 to 2.

20. The Bohr Theory of the Hydrogen Atom Prior to the work of Niels Bohr,the stability of the atom could not be explained using the then-current theories. Classical mechanics predict that a electron will crash into the nucleus indicating that the nuclear structure is not stable. • In 1913, using the work of Einstein and Planck, Bohr applied a new theory to the simplest atom, hydrogen, which involved the transition of electrons between allowed energy levels.

21. The Bohr Theory of the Hydrogen Atom Bohr’s Postulates Bohr set down postulates to account for (1) the stability of the hydrogen atom and (2) the line spectrum of the atom. • 1.Energy level postulate An electron can have only specific energy levels in an atom. • 2. Transitions between energy levels An electron in an atom can change energy levels by undergoing a “transition” from one energy level to another by either absorbing or emitting a photon equal to the energy difference between the two allowed energy levels. • 3. Angular momentum keeps electrons in orbit.

22. The Bohr Theory of the Hydrogen Atom Bohr’s Postulates Bohr derived the following formula for the energy levels of the electron in the hydrogen atom. • Rh is a constant (expressed in energy units) with a value of 2.18 x 10-18 J.

23. From Postulate 1, The Bohr Theory of the Hydrogen Atom Bohr’s Postulates When an electron undergoes a transition from a higher energy level to a lower one, the energy is emitted as a photon. Energy difference between two energy levels. Energy for initial level Energy for final level

24. The Bohr Theory of the Hydrogen Atom Bohr’s Postulates If we make a substitution into the previous equation that states the energy of the emitted photon, hn, equals Ei- Ef, • Rearranging, we obtain Energy of photon emitted or absorbed for electrons changing energy levels

25. The Bohr Theory of the Hydrogen Atom Bohr’s Postulates • Bohr’s theory explains not only the emission of light, but also the absorption of light. • When an electron falls from n = 3 to n = 2 energy level, a photon is emitted equal to the energy difference between the levels (wavelength, 685 nm). • When light of this same wavelength shines on a hydrogen atom in the n = 2 level, the energy is gained by the electron and undergoes a transition to n = 3.

26. A Problem to Consider Calculate the energy of a photon of light emitted from a hydrogen atom when an electron falls from level n = 3 to level n = 1. This problem involves plugging the correct values into Bohr’s equation where ni = 3 (initial energy level) and nf = 1 (final energy level): = 1.94 HW 59 code: bohr

27. Quantum Mechanics Bohr’s theory established the concept of atomic energy levels but did not thoroughly explain the “wave-like” behavior of the electron. • Current ideas about atomic structure depend on the principles of quantum mechanics, a theory that applies to subatomic particles such as electrons.

28. Matter Waves Light exhibits wave-like behavior, as well as particle-like behavior. The term wave-particle duality is used to describe this dual nature of light. • Wave-Particle Duality of light-the “wave” and “particle” pictures of light should be regarded as complementary views of the same physical entity. • The equation E = hndisplays this duality; E is the energy of the “particle” photon, andnis the frequency of the associated “wave.”

29. Matter Waves The first clue in the development of wave-particle duality came with the discovery of the de Broglie relation. • In 1923, Louis de Broglie reasoned that if light exhibits particle aspects, perhaps particles of matter show characteristics of waves. • He postulated that a particle with massmand a velocityvhas an associated wavelength. Momentum is the product of mass and speed. • The equationl = h/mv is called the de Broglie wavelength. • It is now generally accepted that wave characteristics apply to all matter; however, for large masses, the De Broglie wavelength is too small to detect.

30. Quantum Mechanics • In 1927, Werner Heisenberg showed (from quantum mechanics) that it is impossible to know both (position & velocity) simultaneously – Heisenberg’s Uncertainty Principle. • Electrons are moving; therefore, if we try to locate it by bouncing a photon off it, the electron’s location would be affected as well. • Wave-particle duality and the uncertainty principle resolve the problem regarding the stability of the nuclear atom. Classical mechanics predict that the electron will crash into the nucleus indicating that the nuclear structure is not stable. A consequence of the wave-particle duality is the Uncertainty Principle.

31. Quantum Mechanics • Therefore, classical mechanics could not explain experimental results involving the nuclear structure. • We could no longer think of an electron as having a precise orbit (spherical) in an atom. • To describe such an orbit would require knowing its exact position and velocity. • It became evident that classical mechanics is just an approximation on a macroscopic level; a more general theory was needed in order to account for observations on a microscopic level which lead to the birth of Quantum Mechanics.

32. Quantum Mechanics Quantum mechanics is the branch of physics that mathematically describes the wave properties of submicroscopic particles. This is a probabilistic theory as opposed to a deterministic one. In quantum mechanics, we do not try to locate particles; we only try to calculate probabilities or likelihood of finding particles in whatever region we are interested in. Erwin Schrodinger defined this probability in a mathematical expression called a wavefunction.

33. Quantum Numbers and Atomic Orbitals An orbital is a wavefunction that gives us information about an electron; a description of the region around the nucleus where we are most likely to find the electron. For each atom, ion, or molecule, quantum theory gives us an infinite set of orbitals that we can use to describe the electrons. It is convenient to refer to these orbitals by a set of numbers called quantum numbers that is unique for each orbital.

34. Quantum Numbers and Atomic Orbitals According to quantum mechanics, each orbital describing electrons has four quantum numbers: • Principal quantum number (n) • Can only be a positive integer: n = 1, 2, 3,….. • Orbital Angular momentum quantum number (l) • Can only be a positive integer that is less than n • l = 0, 1, …, n-1 • Magnetic quantum number (ml or m) • Can only be an integer from –l to + l • Spin quantum number (ms) • Can only be -½ or + ½

35. Quantum Numbers and Atomic Orbitals Orbitals can be classified into shells or levels. The principal quantum number (n) represents the “shell number or energy level” in which an electron “resides.” • The smaller n is, the smaller the orbital. • The smaller n is, the lower the energy of the electron. • May be an integer from 1 to , although greater than 7 as far as today, are unimportant. • n determines the size and energy of the orbital. The effective volume of space electron is moving about.

36. Quantum Numbers and Atomic Orbitals Orbitals can also be classified into subshells or sublevels. The orbitalangular momentum quantum number (l) distinguishes “sub shells” within a given shell that have different shapes (shape of region (volume) that electron occupies). • Each main “shell - n” is subdivided into “sub shells.” Within each shell of quantum number n, there are “l” sub shells, each with a distinctive shape. • l can have any integer value from 0 to (n - 1) • The different subshells are denoted by letters. • Letter s p d f g … • l = 0 1 2 3 4 ….

37. Quantum Numbers and Atomic Orbitals In general, the number of orbitals that belong to the same subshell is equal to 2l + 1: If l = 0, 2(0) + 1 = 1; there is only one orbital in an s subshell. If l = 1, 2(1) + 1 = 3; there are three orbitals in an psubshell. If l = 2, 2(2) + 1 = 5; there are five orbitals in an dsubshell. If l = 3, 2(3) + 1 = 7; there are seven orbitals in an fsubshell. We can say that s orbitals comes in a set of 1, p orbitals come in sets of 3, d orbitals come in sets of 5, forbitals come in sets of 7, etc. Each orbital in a subshell has a unique ml quantum number.

38. Quantum Numbers and Atomic Orbitals The magnetic quantum number (ml or m) distinguishes orbitals within a given sub-shell that have different shapes and magnetic orientations in space. It is the magnetism caused by the orbital motion of an electron around the nucleus that affects the orientation. • Each subshell (s, p, d, f) is subdivided into “orbitals,” each capable of holding a pair of electrons (max 2 electrons/orbital orientation). • ml can have any integer value from -lto 0 to +l. • Each orbital within a given subshell has the same energy (all “nl” have same energy for all orientations but not each “l” – 4s<4p<4d<4f). • So, when we say “3d orbital”, we mean any one of the five orbitals in the 3d subshell (-2, -1, 0, +1, or +2) • d: l = 2 and ml = -2 to 0 to +2 meaning -2, -1, 0, +1, +2

39. Figure : Cutaway diagrams showing the spherical shape of S orbitals. n = 1 n = 2 l = 0 are referred to as “s” subshells and are spherical in shape. There is only 1 magnetic orientation (ml: 0) for the “s” subshell.

40. Figure : The 2p orbitals. l = 1 are referred to as “p” subshells and are dumbbell in shape. There are 3 magnetic orientations (ml: -1, 0, +1) for the “p” subshell.

41. Figure : The five 3d orbitals. l = 2 are referred to as “d” subshells and are cloverleaf in shape. There are 5 magnetic orientations (ml: -2, -1, 0, +1, +2) for the “d” subshell.

42. In general, the nth shell has n2 orbitals: For n = 1, n2= 1; there is only one orbital; 1 in the 1s (n = 1; l = 0; ml = 0) For n = 2, n2 = 4; there are 4 orbitals; 1 in the 2s (n= 2; l = 0; ml = 0) 3 in the 2p (n = 2; l = 1; ml = -1, 0, +1) For n = 3, n2 = 9; there are 9 orbitals; 1 in the 3s (n = 3; l = 0; ml = 0) 3 in the 3p (n = 3; l = 1; ml = -1, 0, +1) 5 in the 3d (n = 3; l = 2; ml = -2, -1, 0, +1, +2) For n = 4, n2 = 16; there are 16 orbitals; 1 in the 4s (n = 4; l = 0; ml = 0) 3 in the 4p (n = 4; l = 1; ml = -1, 0, +1) 5 in the 4d (n = 4; l = 2; ml = -2, -1, 0, +1, +2) 7 in the 4f (n = 4; l = 3; ml = -3, -2, -1, 0, +1, +2, +3)

43. Energy level shape orientations 0 to (n-1) -l…0…+l Max #e- s s p s p d s p d f 2 2 6 2 6 10 2 6 10 14 px, py, pz

44. Quantum Numbers and Atomic Orbitals The magnetic properties of atoms suggest that electrons have an intrinsic magnetism due to their spinning motion in addition to the magnetism that is generated by their motion around the nucleus. The motion of an electron around the nucleus, ml, is analogous to that of a planet revolving around the sun, while electron spin is analogous to the rotation of the planet around its own axis, ms. The spin quantum number,ms, refers to the two possible spin orientations (spin up, +1/2, and spin down, -1/2 ) of the electrons residing within a given orbital. • Each orbital can hold only two electrons whose spins must oppose one another.

45. Quantum Numbers and Atomic Orbitals Thus, an electron in an atom is completely described by specifying four quantum numbers (n, l, ml, ms). The quantum numbers n and l tell us the shell and subshell, the quantum number ml tells us which orbital, and ms tells us the spin. No two electrons can have the same four quantum numbers; If two electrons have the same n, l, ml(same orbital), they must have opposite spins. • Pauli Exclusion Principle – no two electrons may have the same set of 4 quantum numbers. Where two electrons occupy the same orbital, they must have opposite spins: ms= +1/2, ms= –1/2. • The maximum number of electrons that we can assign to an orbital is 2; meaning one with spin up, +1/2, and the other with spin down, -1/2 .

46. Questions How many electrons can be assigned to a 4p orbital? The key to this question is that we are referring to “a” single 4p orbital. For any single orbital, the limit is 2 electrons; one electron with spin up (ms = +1/2) and the other with spin down (ms = -1/2). How many electrons can be assigned to 4p orbitals (meaning 4p subshells)? The key to this question is that we are referring to “all” the 4p orbitals or subshells. The 4p orbitals come in a set of 3 (ml: -1, 0, 1) with a maximum of 2 electrons in each orbital. All 6 electrons will have n = 4, l = 1. They will differ in ml and ms values; the six possible (ml, ms) values: (-1, +1/2), (0, +1/2), (1, +1/2) spin up (-1, -1/2), (0, -1/2), (1, -1/2) spin down

47. Questions How many electrons can be assigned to the n = 3 shell? In the n = 3 shell, there are 3 subshells (3s, 3p, 3d). The 3s subshell consists of 1 orbital with 2 e-. The 3p subshell consists of 3 orbitals with 2 electrons in each giving 6 e-. The 3d subshell consists of 5 orbitals with 2 electrons in each giving 10 e-. The total number of e- that can be assigned in the n = 3 shell is 2 + 6 + 10 = 18. In general, since there are n2 orbitals in the nth shell, the number of electrons that can be assigned to a shell is 2n2. Which of the following can accommodate the most number of e-? A. a 4f orbital B. the 3d subshell C. the n =2 shell D. a 3d orbital a 4f orbital is a single orbital accommodating a maximum of 2e- 2e- the 3d subshell has 5 orbitals (l = 2, ml = -2, -1, 0, +1, +2) with 2e- each  10e- the n =2 shell has 4 orbitals (l = 0, ml = 0; l = 1, ml = -1, 0, +1) with 2e- each  8e- D. a 3d orbital is a single orbital accommodating a maximum of 2e- 2e- The correct answer is B, the 3d subshell. HW 60 code: quantum