Chapter 7 The Quantum-Mechanical Model of the Atom Chemistry: A Molecular Approach, 1st Ed. Nivaldo Tro Chapter 7 The Quantum-Mechanical Model of the Atom Roy Kennedy Massachusetts Bay Community College Wellesley Hills, MA 2007, Prentice Hall

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, Chemistry: A Molecular Approach

A Theory that Explains Electron Behavior the quantum-mechanical model explains the manner electrons exist and behave in atoms 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 1 electron when forming an anion, while others gain 2 why some elements are very reactive while others are practically inert and other Periodic patterns we see in the properties of the elements Tro, Chemistry: A Molecular Approach

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 an magnetized particle experiences a force all electromagnetic waves move through space at the same, constant speed 3.00 x 108 m/s in a vacuum = the speed of light, c Tro, Chemistry: A Molecular Approach

Speed of Energy Transmission Tro, Chemistry: A Molecular Approach

Electromagnetic Radiation Tro, Chemistry: A Molecular Approach

Characterizing Waves the amplitude is the height of the wave the distance from node to crest or node to trough the amplitude is a measure of how intense the light is – the larger the amplitude, the brighter the light the wavelength, (l) 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, Chemistry: A Molecular Approach

Wave Characteristics Tro, Chemistry: A Molecular Approach

Characterizing Waves the frequency, (n) 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 and frequency of the waves the larger the wave amplitude, the more force it has the more frequently the waves strike, the more total force there is Tro, Chemistry: A Molecular Approach

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 visa versa Tro, Chemistry: A Molecular Approach

Example 7.1- Calculate the wavelength of red light with a frequency of 4.62 x 1014 s-1 Given: Find: n = 4.62 x 1014 s-1 l, (nm) Concept Plan: Relationships: l∙n = c, 1 nm = 10-9 m n (s-1) l (m) l (nm) Solve: Check: the unit is correct, the wavelength is appropriate for red light Tro, Chemistry: A Molecular Approach

Practice – Calculate the wavelength of a radio signal with a frequency of 100.7 MHz Tro, Chemistry: A Molecular Approach

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, Chemistry: A Molecular Approach

Amplitude & Wavelength

Electromagnetic Spectrum Tro, Chemistry: A Molecular Approach

Continuous Spectrum Tro, Chemistry: A Molecular Approach

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 radiowave light has the lowest energy gamma ray light has the highest energy high energy electromagnetic radiation can potentially damage biological molecules ionizing radiation Tro, Chemistry: A Molecular Approach

Thermal Imaging using Infrared Light Tro, Chemistry: A Molecular Approach

Interference the interaction between waves is called interference when waves interact so that they add to make a larger wave it is called constructive interference waves are in-phase when waves interact so they cancel each other it is called destructive interference waves are out-of-phase Tro, Chemistry: A Molecular Approach

Interference Tro, Chemistry: A Molecular Approach

Diffraction when traveling waves encounter an obstacle or opening in a barrier that is about the same size as the wavelength, they bend around it – this is called diffraction traveling particles do not diffract the diffraction of light through two slits separated by a distance comparable to the wavelength results in an interference pattern of the diffracted waves an interference pattern is a characteristic of all light waves Tro, Chemistry: A Molecular Approach

Diffraction Tro, Chemistry: A Molecular Approach

2-Slit Interference Tro, Chemistry: A Molecular Approach

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 waves intensity made brighter, 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, Chemistry: A Molecular Approach

The Photoelectric Effect Tro, Chemistry: A Molecular Approach

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 high frequency light with a dim source caused electron emission without any lag time Tro, Chemistry: A Molecular Approach

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 it wavelength the proportionality constant is called Planck’s Constant, (h) and has the value 6.626 x 10-34 J∙s Tro, Chemistry: A Molecular Approach

Example 7.2- Calculate the number of photons in a laser pulse with wavelength 337 nm and total energy 3.83 mJ Given: Find: l = 337 nm, Epulse = 3.83 mJ number of photons Concept Plan: Relationships: E=hc/l, 1 nm = 10-9 m, 1 mJ = 10-3 J, Epulse/Ephoton = # photons l(nm) l (m) Ephoton number photons Solve: Tro, Chemistry: A Molecular Approach

Practice – What is the frequency of radiation required to supply 1 Practice – What is the frequency of radiation required to supply 1.0 x 102 J of energy from 8.5 x 1027 photons? Tro, Chemistry: A Molecular Approach

Kinetic Energy = Ephoton – Ebinding Ejected Electrons 1 photon at the threshold frequency has just enough energy for an electron to escape the atom binding energy, f for higher frequencies, the electron absorbs more energy than is necessary to escape this excess energy becomes kinetic energy of the ejected electron Kinetic Energy = Ephoton – Ebinding KE = hn - f Tro, Chemistry: A Molecular Approach

Spectra when atoms or molecules absorb energy, that energy is often released as light energy fireworks, neon lights, etc. when that light is passed through a prism, a pattern is seen that is unique to that type of atom or molecule – the pattern is called an emission spectrum non-continuous can be used to identify the material flame tests Rydberg analyzed the spectrum of hydrogen and found that it could be described with an equation that involved an inverse square of integers Tro, Chemistry: A Molecular Approach

Emission Spectra Tro, Chemistry: A Molecular Approach

Exciting Gas Atoms to Emit Light with Electrical Energy Hg He H Tro, Chemistry: A Molecular Approach

Examples of Spectra Oxygen spectrum Neon spectrum Tro, Chemistry: A Molecular Approach

Identifying Elements with Flame Tests Na K Li Ba Tro, Chemistry: A Molecular Approach

Emission vs. Absorption Spectra Spectra of Mercury Tro, Chemistry: A Molecular Approach

Bohr’s Model Neils Bohr proposed that the electrons could only have very specific amounts of energy fixed amounts = quantized the electrons traveled in orbits that were a fixed distance from the nucleus stationary states therefore the energy of the electron was proportional the distance the orbital was from the nucleus electrons emitted radiation when they “jumped” from an orbit with higher energy down to an orbit with lower energy the distance between the orbits determined the energy of the photon of light produced Tro, Chemistry: A Molecular Approach

Bohr Model of H Atoms Tro, Chemistry: A Molecular Approach

Wave Behavior of Electrons de Broglie proposed that particles could have wave-like character because it is so small, the wave character of electrons is significant electron beams shot at slits show an interference pattern the electron interferes with its own wave de Broglie predicted that the wavelength of a particle was inversely proportional to its momentum (mass x velocity) Tro, Chemistry: A Molecular Approach

Electron Diffraction if electrons behave like particles, there should only be two bright spots on the target however, electrons actually present an interference pattern, demonstrating the behave like waves Tro, Chemistry: A Molecular Approach

Example 7. 3- Calculate the wavelength of an electron traveling at 2 Example 7.3- Calculate the wavelength of an electron traveling at 2.65 x 106 m/s Given: Find: v = 2.65 x 106 m/s, m = 9.11 x 10-31 kg (back leaf) l, m Concept Plan: Relationships: l=h/mv m, v l (m) Solve: Tro, Chemistry: A Molecular Approach

Practice - Determine the wavelength of a neutron traveling at 1 Practice - Determine the wavelength of a neutron traveling at 1.00 x 102 m/s (Massneutron = 1.675 x 10-24 g) Tro, Chemistry: A Molecular Approach

Complimentary Properties when you try to observe the wave nature of the electron, you cannot observe its particle nature – and visa versa wave nature = interference pattern particle nature = position, which slit it is passing through the wave and particle nature of the electron are complimentary properties as you know more about one you know less about the other Tro, Chemistry: A Molecular Approach

Uncertainty Principle Heisenberg stated that the product of the uncertainties in both the position and speed of a particle was inversely proportional to its mass x = position, Dx = uncertainty in position v = velocity, Dv = uncertainty in velocity m = mass the means that the more accurately you know the position of a small particle, like an electron, the less you know about its speed and visa-versa Tro, Chemistry: A Molecular Approach

Uncertainty Principle Demonstration any experiment designed to observe the electron results in detection of a single electron particle and no interference pattern Tro, Chemistry: A Molecular Approach

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 to describe the probability an electron will be found in a particular region using statistical functions Tro, Chemistry: A Molecular Approach

Trajectory vs. Probability

Electron Energy The mathematical derivation of the energies and orbitals for electrons comes from solving the Shrodinger equation . The general form is shown above. The symbol H stands for the Hamiltonian operator, a set of mathematical operations that represent the total energy (kinetic and potential) of the electron within the atom. The symbol E is the actual energy of the electron. The symbol is the wave function, a mathematical function that describes the wavelike nature of the electron. Tro, Chemistry: A Molecular Approach

Electron Energy electron energy and position are complimentary for an electron with a given energy, the best we can do is describe a region in the atom of 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. y2 many of the properties of atoms are related to the energies of the electrons Tro, Chemistry: A Molecular Approach

Wave Function, y calculations show that the size, shape and orientation in space of an orbital are determined be 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, ml Tro, Chemistry: A Molecular Approach

Principal Quantum Number, n characterizes the energy of the electron in a particular orbital corresponds to Bohr’s energy level n can be any integer ³ 1 the larger the value of n, the more energy the orbital has energies are defined as being negative an electron would have 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 for an electron in H Tro, Chemistry: A Molecular Approach

Principal Energy Levels in Hydrogen Tro, Chemistry: A Molecular Approach

Electron Transitions in order to transition to a higher energy state, the electron must gain the correct 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

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, therefore (n – 1) lines it can generate both the Bohr and Quantum Mechanical Models can predict these lines very accurately Tro, Chemistry: A Molecular Approach

Hydrogen Energy Transitions

Example 7.7- Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 6 to n = 5 Given: Find: ni = 6, nf = 5 l, m Concept Plan: Relationships: E=hc/l, En = -2.18 x 10-18 J (1/n2) ni, nf DEatom Ephoton l DEatom = -Ephoton Solve: Ephoton = -(-2.6644 x 10-20 J) = 2.6644 x 10-20 J Check: the unit is correct, the wavelength is in the infrared, which is appropriate because less energy than 4→2 (in the visible)

Practice – Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 2 to n = 1 Tro, Chemistry: A Molecular Approach

Probability & Radial Distribution Functions y2 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 where the probability drops to 0

Probability Density Function Tro, Chemistry: A Molecular Approach

Radial Distribution Function Tro, Chemistry: A Molecular Approach

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 4 balloons tied at the knot f orbitals are mainly like 8 balloons tied at the knot Tro, Chemistry: A Molecular Approach

l = 0, the 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, Chemistry: A Molecular Approach

2s and 3s 2s n = 2, l = 0 3s n = 3, l = 0

l = 1, p orbitals each principal energy state above n = 1 has 3 p orbitals ml = -1, 0, +1 each of the 3 orbitals point along a different axis px, py, pz 2nd lowest energy orbitals in a principal energy state two-lobed node at the nucleus, total of n nodes Tro, Chemistry: A Molecular Approach

p orbitals Tro, Chemistry: A Molecular Approach

l = 2, d orbitals each principal energy state above n = 2 has 5 d orbitals ml = -2, -1, 0, +1, +2 4 of the 5 orbitals are aligned in a different plane the fifth is aligned with the z axis, dz squared dxy, dyz, dxz, dx squared – y squared 3rd lowest energy orbitals in a principal energy state mainly 4-lobed one is two-lobed with a toroid planar nodes higher principal levels also have spherical nodes Tro, Chemistry: A Molecular Approach

d orbitals Tro, Chemistry: A Molecular Approach

l = 3, f orbitals each principal energy state above n = 3 has 7 d orbitals ml = -3, -2, -1, 0, +1, +2, +3 4th lowest energy orbitals in a principal energy state mainly 8-lobed some 2-lobed with a toroid planar nodes higher principal levels also have spherical nodes Tro, Chemistry: A Molecular Approach

f orbitals Tro, Chemistry: A Molecular Approach

Why are Atoms Spherical? Tro, Chemistry: A Molecular Approach

Energy Shells and Subshells Tro, Chemistry: A Molecular Approach