Chapter 7 The Quantum- Mechanical Model of the Atom Roy Kennedy Massachusetts Bay Community College Wellesley Hills, MA Principles of Chemistry: A Molecular Approach, 1 st Ed. Nivaldo Tro

2 The Behavior of the Very Small Electrons are incredibly small. A single speck of dust has more electrons than the number of people who have ever lived on Earth. Electron behavior determines much of the behavior of atoms. Directly observing electrons in the atom is impossible—the electron is so small that observing it changes its behavior. Tro, Principles of Chemistry: A Molecular Approach

3 A Theory that Explains Electron Behavior The quantum-mechanical model explains how electrons exist and behave in atoms. It helps us understand and predict the properties of atoms that are directly related to the behavior of the electrons. why some elements are metals while others are nonmetals why some elements gain one electron when forming an anion, while others gain two why some elements are very reactive while others are practically inert and other periodic patterns we see in the properties of the elements

Tro, Principles of Chemistry: A Molecular Approach4 The Nature of Light: Its Wave Nature Light is a form of electromagnetic radiation. composed of perpendicular oscillating waves, one for the electric field and one for the magnetic field  An electric field is a region where an electrically charged particle experiences a force.  A magnetic field is a region where a magnetized particle experiences a force. All electromagnetic waves move through space at the same, constant speed x 10 8 m/s in a vacuum = the speed of light, c

Tro, Principles of Chemistry: A Molecular Approach5 Electromagnetic Radiation

Tro, Principles of Chemistry: A Molecular Approach6 Characterizing Waves The amplitude is the height of the wave. the distance from node to crest  or from node to trough The amplitude is a measure of how intense the light is—the larger the amplitude, the brighter the light The wavelength ( ) is a measure of the distance covered by the wave. the distance from one crest to the next  or the distance from one trough to the next, or the distance between alternate nodes

Tro, Principles of Chemistry: A Molecular Approach7 Wave Characteristics

Tro, Principles of Chemistry: A Molecular Approach8 Characterizing Waves The frequency ( ) is the number of waves that pass a point in a given period of time. the number of waves = number of cycles units are hertz (Hz) or cycles/s = s −1  1 Hz = 1 s −1 The total energy is proportional to the amplitude of the waves and the frequency. The larger the amplitude, the more force it has. The more frequently the waves strike, the more total force there is.

Tro, Principles of Chemistry: A Molecular Approach9 The Relationship between Wavelength and Frequency For waves traveling at the same speed, the shorter the wavelength, the more frequently they pass. This means that the wavelength and frequency of electromagnetic waves are inversely proportional. Since the speed of light is constant, if we know wavelength we can find the frequency, and vice versa.

Tro, Principles of Chemistry: A Molecular Approach10 Color The color of light is determined by its wavelength. or frequency White light is a mixture of all the colors of visible light. a spectrum RedOrangeYellowGreenBlueViolet When an object absorbs some of the wavelengths of white light while reflecting others, it appears colored. The observed color is predominantly the colors reflected.

Tro, Principles of Chemistry: A Molecular Approach11 Amplitude and Wavelength

Tro, Principles of Chemistry: A Molecular Approach12 The Electromagnetic Spectrum Visible light comprises only a small fraction of all the wavelengths of light—called the electromagnetic spectrum. Short-wavelength (high-frequency) light has high energy. Radio wave light has the lowest energy. Gamma ray light has the highest energy. High-energy electromagnetic radiation can potentially damage biological molecules. ionizing radiation

Tro, Principles of Chemistry: A Molecular Approach13 Types of Electromagnetic Radiation Electromagnetic waves are classified by their wavelength. radio waves = > 0.01 m microwaves = 10 −4 m < < 10 −2 m infrared (IR) far = 10 −5 m < < 10 −4 m middle = 2 x 10 −6 m < < 10 −5 m near = 8 x 10 −7 m < < 2 x 10 −6 m visible = 4 x 10 −7 m < < 8 x 10 −7 m ROYGBIV ultraviolet (UV) near = 2 x 10 −7 m < < 4 x 10 −7 m far = 1 x 10 −8 m < < 2 x 10 −7 m X-rays = 10 −10 m < < 10 −8 m gamma rays = < 10 −10 m low frequency and energy high frequency and energy

Tro, Principles of Chemistry: A Molecular Approach14

Tro, Principles of Chemistry: A Molecular Approach15 Continuous Spectrum

Tro, Principles of Chemistry: A Molecular Approach16 Thermal Imaging Using Infrared Light

Tro, Principles of Chemistry: A Molecular Approach17 Using High-Energy Radiation to See Inside a Person

Tro, Principles of Chemistry: A Molecular Approach18 The Photoelectric Effect It was observed that many metals emit electrons when a light shines on their surface. This is called the photoelectric effect. Classic wave theory attributed this effect to the light energy being transferred to the electron. According to this theory, if the wavelength of light is made shorter, or the light wave’s intensity made stronger, more electrons should be ejected. Remember: the energy of a wave is directly proportional to its amplitude and its frequency. If a dim light was used, there would be a lag time before electrons were emitted.  to give the electrons time to absorb enough energy

Tro, Principles of Chemistry: A Molecular Approach19

Tro, Principles of Chemistry: A Molecular Approach20 The Photoelectric Effect— The Problem In experiments with the photoelectric effect, it was observed that there was a maximum wavelength for electrons to be emitted. called the threshold frequency regardless of the intensity It was also observed that short-wavelength light with a dim source caused electron emission without any lag time.

Tro, Principles of Chemistry: A Molecular Approach21 Einstein’s Explanation Einstein proposed that the light energy was delivered to the atoms in packets, called quanta or photons. The energy of a photon of light was directly proportional to its frequency. inversely proportional to its wavelength The proportionality constant is called Planck’s constant (h) and has the value x J∙s.

Tro, Principles of Chemistry: A Molecular Approach22 Example 7.2 Calculate the number of photons in a laser pulse with wavelength 337 nm and total energy 3.83 mJ. Solve: E = hc/, 1 nm = m, 1 mJ = J, E total = E photon  # photons Conceptual Plan: Relationships: = 337 nm, E pulse = 3.83 mJ number of photons Given: Find: (nm) (m) E photon number photons

Tro, Principles of Chemistry: A Molecular Approach23 Practice—What is the frequency of radiation required to supply 1.0 x 10 2 J of energy from 8.5 x photons?

Tro, Principles of Chemistry: A Molecular Approach24 Practice — What is the frequency of radiation required to supply 1.0 x 10 2 J of energy from 8.5 x photons? Solve: E = h, E total = E photon ∙# photons Conceptual Plan: Relationships: E total = 1.0 x 10 2 J, number of photons = 8.5 x Given: Find: (s -1 ) E photon number photons

Tro, Principles of Chemistry: A Molecular Approach25 Practice—Order the following types of electromagnetic radiation: microwaves, gamma rays, green light, red light, ultraviolet light. by wavelength (short to long) by frequency (low to high) by energy (least to most) gamma < UV < green < red < microwaves microwaves < red < green < UV < gamma

Tro, Principles of Chemistry: A Molecular Approach26 Spectra When atoms or molecules absorb energy, that energy is often released as light energy. fireworks, neon lights, etc. When that emitted light is passed through a prism, a pattern of particular wavelengths of light is seen that is unique to that type of atom or molecule. The pattern is called an emission spectrum. not continuous can be used to identify the material  flame tests

Tro, Principles of Chemistry: A Molecular Approach27 Exciting Gas Atoms to Emit Light with Electrical Energy HgHeH

Tro, Principles of Chemistry: A Molecular Approach28

Tro, Principles of Chemistry: A Molecular Approach29 The Bohr Model of the Atom Bohr developed a model of the atom to explain how the structure of the atom changes when it undergoes energy transitions. Bohr’s major idea was that the energy of the atom is quantized, and that the amount of energy in the atom is related to the electron’s position in the atom. Quantized means that the atom can only have very specific amounts of energy. Neils Bohr (1885–1962)

Tro, Principles of Chemistry: A Molecular Approach30 Bohr’s Model The electrons travel in orbits that are at a fixed distance from the nucleus. stationary states Therefore, the energy of the electron is proportional to the distance of the orbit from the nucleus. Electrons emit radiation when they “jump” from an orbit with higher energy down to an orbit with lower energy. The emitted radiation is a photon of light. The distance between the orbits determines the energy of the photon of light produced.

Tro, Principles of Chemistry: A Molecular Approach31 Bohr Model of H Atoms

Tro, Principles of Chemistry: A Molecular Approach32 Wave Behavior of Electrons de Broglie proposed that particles could have a wavelike character. de Broglie predicted that the wavelength of a particle is inversely proportional to its momentum. Because it is so small, the wave character of electrons is significant. = h (kg.m 2 s -2. s) / m (kg).v(m.s -1 ) Louis de Broglie (1892–1987)

Tro, Principles of Chemistry: A Molecular Approach33 Solve: Example 7.3 Calculate the wavelength of an electron traveling at 2.65 x 10 6 m/s.  = h/mv Conceptual Plan: Relationships: v = 2.65 x 10 6 m/s, m = 9.11 x kg (back leaf)  m  Given: Find: m, v  (m)

Tro, Principles of Chemistry: A Molecular Approach34 Complementary Properties When you try to observe the wave nature of the electron, you cannot observe its particle nature— and vice versa. wave nature particle nature The wave and particle nature of nature of the electron are complementary properties. As you know more about one, you know less about the other.

Tro, Principles of Chemistry: A Molecular Approach35 Uncertainty Principle Heisenberg stated that the product of the uncertainties in both the position and speed of a particle is inversely proportional to its mass. x = position,  x = uncertainty in position v = velocity,  v = uncertainty in velocity m = mass This means that the more accurately you know the position of a small particle, like an electron, the less you know about its speed. and vice versa

Tro, Principles of Chemistry: A Molecular Approach36 Determinacy vs. Indeterminacy According to classical physics, particles move in a path determined by the particle’s velocity, position, and forces acting on it. determinacy = definite, predictable future Because we cannot know both the position and velocity of an electron, we cannot predict the path it will follow. indeterminacy = indefinite future, can only predict probability The best we can do is describe the probability an electron will be found in a particular region using statistical functions.

Tro, Principles of Chemistry: A Molecular Approach37 Trajectory vs. Probability

Tro, Principles of Chemistry: A Molecular Approach38 Electron Energy Electron energy and position are complementary. because KE = ½mv 2 For an electron with a given energy, the best we can do is describe a region of the atom with a high probability of finding it, called an orbital. a probability distribution map of a region where the electron is likely to be found distance vs.  2 Many of the properties of atoms are related to the energies of the electrons.

Tro, Principles of Chemistry: A Molecular Approach39 Wave Function,  Calculations show that the size, shape, and orientation in space of an orbital are determined by three integer terms in the wave function. added to quantize the energy of the electron These integers are called quantum numbers. principal quantum number, n angular momentum quantum number, l magnetic quantum number, m l

Tro, Principles of Chemistry: A Molecular Approach40 Principal Quantum Number, n n characterizes the energy of the electron in a particular orbital. corresponds to Bohr’s energy level n can be any integer equal to or greater than 1. The larger the value of n, the more energy the orbital has. Energies are defined as being negative. An electron has E = 0 when it just escapes the atom. The larger the value of n, the larger the orbital. As n gets larger, the amount of energy between orbitals gets smaller.

Tro, Principles of Chemistry: A Molecular Approach41 Principal Energy Levels in Hydrogen

Tro, Principles of Chemistry: A Molecular Approach42 Electron Transitions In order to transition to a higher energy state, the electron must gain the exact amount of energy corresponding to the difference in energy between the final and initial states. Electrons in high-energy states are unstable and tend to lose energy and transition to lower energy states. energy released as a photon of light Each line in the emission spectrum corresponds to the difference in energy between two energy states.

Tro, Principles of Chemistry: A Molecular Approach43 Quantum Leaps

Tro, Principles of Chemistry: A Molecular Approach44 Predicting the Spectrum of Hydrogen The wavelengths of lines in the emission spectrum of hydrogen can be predicted by calculating the difference in energy between any two states. For an electron in energy state n, there are (n – 1) energy states it can transition to, and therefore (n – 1) lines it can generate. Both the Bohr and quantum mechanical models can predict these lines very accurately for a one-electron system.

Tro, Principles of Chemistry: A Molecular Approach45 Energy Transitions in Hydrogen The energy of a photon released is equal to the difference in energy between the two levels the electron is jumping between. It can be calculated by subtracting the energy of the initial state from the energy of the final state. E emitted photon = −  E electron  E electron = E final state − E initial state

Tro, Principles of Chemistry: A Molecular Approach46 Hydrogen Energy Transitions

Tro, Principles of Chemistry: A Molecular Approach47 Solve: Example 7.7 Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 6 to n = 5. E = hc/  E n = −2.18 x J (1/n 2 ) Conceptual Plan: Relationships: n i = 6, n f = 5, R H = 2.18 x10 −18 J  m  Given: Find: n i, n f  E atom E photon  E atom = −E photon E photon = −(− x J) = x J Units are correct; the wavelength is in the infrared, which is appropriate because it's less energy than 4→2 (in the visible). Check:

Tro, Principles of Chemistry: A Molecular Approach48 Practice—Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 2 to n = 1.

Tro, Principles of Chemistry: A Molecular Approach49 Solve: Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 2 to n = 1. E = hc/  E n = −2.18 x J (1/n 2 ) Conceptual Plan: Relationships: n i = 2, n f = 1, R H = 2.18 x 10 −18 J  m  Given: Find: n i, n f  E atom E photon  E atom = −E photon E photon = − ( − 1.64 x J) = 1.64 x J Units are correct; the wavelength is in the UV, which is appropriate since it's more energy than 3→2 (in the visible). Check:

Tro, Principles of Chemistry: A Molecular Approach50 Probability and Radial Distribution Functions  2 is the probability density. the probability of finding an electron at a particular point in space for s orbital maximum at the nucleus? decreases as you move away from the nucleus The radial distribution function represents the total probability at a certain distance from the nucleus. maximum at most probable radius Nodes in the functions are points where the probability drops to 0.

Tro, Principles of Chemistry: A Molecular Approach51 Probability Density Function The probability density function represents the total probability of finding an electron at a particular point in space.

Tro, Principles of Chemistry: A Molecular Approach52 Radial Distribution Function The radial distribution function represents the total probability of finding an electron within a thin spherical shell at a distance r from the nucleus. The net result is a plot that indicates that the most probable distance of the electron in a 1s orbital of H is 52.9 pm.

Tro, Principles of Chemistry: A Molecular Approach53 The Shapes of Atomic Orbitals The l quantum number primarily determines the shape of the orbital. l can have integer values from 0 to (n – 1). Each value of l is called by a particular letter that designates the shape of the orbital. s orbitals are spherical. p orbitals are like two balloons tied at the knots. d orbitals are mainly like four balloons tied at the knots. f orbitals are mainly like eight balloons tied at the knots.

Tro, Principles of Chemistry: A Molecular Approach54 l = 0, s orbital Each principal energy state has 1 s orbital. lowest energy orbital in a principal energy state spherical number of nodes = (n – 1)

Tro, Principles of Chemistry: A Molecular Approach55 2s and 3s 2s n = 2, l = 0 3s n = 3, l = 0

Tro, Principles of Chemistry: A Molecular Approach56 l = 1, p orbitals Each principal energy state above n = 1 has 3 p orbitals. m l = −1, 0, +1 Each of the three orbitals points along a different axis p x, p y, p z second lowest energy orbitals in a principal energy state two-lobed node at the nucleus; total of n nodes

Tro, Principles of Chemistry: A Molecular Approach57 p orbitals

Tro, Principles of Chemistry: A Molecular Approach58 l = 2, d orbitals Each principal energy state above n = 2 has 5 d orbitals. m l = −2, −1, 0, +1, +2 Four of the five orbitals are aligned in a different plane. The fifth is aligned with the z axis, d z squared. d xy, d yz, d xz, d x squared – y squared third lowest energy orbitals in a principal energy state mainly four-lobed One is two-lobed with a toroid. planar nodes Higher principal levels also have spherical nodes.

Tro, Principles of Chemistry: A Molecular Approach59 d orbitals

Tro, Principles of Chemistry: A Molecular Approach60 l = 3, f orbitals Each principal energy state above n = 3 has 7 d orbitals. m l = − 3, − 2, − 1, 0, +1, +2, +3 fourth lowest energy orbitals in a principal energy state mainly eight-lobed some two-lobed with a toroid planar nodes Higher principal levels also have spherical nodes.

Tro, Principles of Chemistry: A Molecular Approach61 Energy Shells and Subshells