Lecture 11 Atomic Structure

Transcription

1 Lecture 11 Atomic Structure Earlier in the semester, you read about the discoveries that lead to the proposal of the nuclear atom, an atom of atomic number Z, composed of a positively charged nucleus surrounded by Z electrons. When Rutherford proposed the nuclear atom, he expected to be able to use classical mechanics (Newton's laws of motion) to describe the properties of the sub-atomic particles. It soon became apparent that classical mechanics could not be used to study the electrons in atoms. New laws were developed in the early twentieth century to explain this unexpected behavior of electrons. These laws are known as quantum mechanics. This is a fascinating, but complex, area of chemistry. We will take a superficial look at the general principles. When salt solutions are heated, the flames have distinct and diagnostic colors. Shown below are flame tests of Na +, K + and Li + salts. Other salts have similar, characteristic colors. These experiments suggest that there are spectral (related to light) properties associated with atoms. When elemental gases are heated, they also produce radiation, with each element emitting only certain colors. In order to understand the connection between these spectral properties and the quantum mechanical structure of the atom, we first have to learn more about the nature of light, or more generally, electromagnetic radiation. The Wave Nature of Light All waves have a characteristic wavelength. In the figures shown below, the wavelength is the distance between the peaks or troughs of the wave. 1 second This is given the symbol "λ" in chemistry. As the wavelengths get smaller, there are more cycles of the wave in a given distance on the x-axis. The x-axis is a measure of time. In the first wave, there are two cycles per second, in the second, four cycles per second and in the third wave, eight cycles per second. The unit for frequency (cycles

2 per second, or s -1 ) is Hz (hertz) and frequency is given the symbol "ν". Frequency and wavelength are obviously related; the shorter the wavelength, the larger the frequency. They are related by the equation, u = λν, where u is the speed that the wave is traveling. Energy travels through space as electromagnetic radiation. It is comprised of an electrical field (pink) and magnetic field (purple) traveling perpendicular to each other and to the direction of propagation. Electromagnetic radiation has wave properties. (Image from Jim Hawkins Radio and Technology Page The speed of propagation, u, of electromagnetic radiation is the speed of light, x 10 8 meters per second (ms -1 ). This value is given its own symbol, c, the speed of light. The speed of light is equal to the wavelength of the radiation, times the frequency of the radiation, c = λν. There are no upper or lower limits in the electromagnetic spectrum. At the long wavelength, low frequency end are radio waves. They have wavelengths of the order of 100 meters. Their frequency can be calculated because, c = λν x 10 8 ms -1 = 100 m x ν ν 3 x 10 6 s -1 At the short wavelength, high frequency end are gamma rays. They have wavelengths of the order of m, and corresponding frequencies of about s -1. In the middle of the electromagnetic spectrum is the visible light spectrum, represented below. These are the colors that we can detect visually. They range from red (low frequency, long wavelength, 700 nm), to violet, (high frequency, short wavelength, 400 nm). The drawing to the right of the spectrum illustrates how the wavelength and frequency change in the visible region, in going from red, to yellow to green light. λ = 700 nm λ = 580 nm λ = 530 nm

3 Let's try to tie this in with Chemistry. When elements are heated, they produce electromagnetic radiation. However, each element emits light of only a few, specific frequencies. Quantum Physics This phenomenon could not be explained by classical physics, which would have predicted that a continuous spectrum would be emitted. These line emission spectra were first explained by the German scientist, Max Planck. He proposed that matter can absorb or emit radiation with energies that are some multiple of a frequency, ν. The equation that he proposed is: E = hν where E is energy, ν is wavelength, and h is Planck's constant, x J s. E is the smallest quantity of energy that can be emitted or absorbed, a quantum of energy. Classical physics describes energy as continuous. Quantum physics, describes energy as quantized. Going from energy level A to energy level B can be viewed two different ways: B B A A

4 Classical physics Quantum physics In Quantum physics, energy can only change by certain discrete amounts (steps): ΔE = nhν The symbol, ΔE, means, final energy minus initial energy. This explains why only a fixed number of emission lines are seen from the elements. These frequencies are associated with allowed quanta of energy leaving the atoms. For example, the yellow light emitted by sodium with a wavelength of 589 nm, corresponds to a frequency of: c = λν = x 10 8 m s -1 ν = x 10 8 m s x 10-9 m = 5.09 x s -1 Substituting this value into Planck's equation gives: E = hν = (6.626 x J s)(5.09 x s -1 ) = 3.37 x J This means that sodium (with a line at 589 nm) can absorb (or emit) 3.37 x joules of energy, or 6.74 x J, or x J of energy, but not 5.00 x J of energy. Planck couldn't explain why this was the case. The Photoelectric effect Five years after Planck proposed the quantum theory, Albert Einstein used this theory to explain another mystery of physics, the photoelectric effect. It was known that electrons are ejected from the surface of certain metals when they are exposed to light of a certain minimum frequency, called the threshold frequency. If a solid potassium surface is exposed to red light, no matter how intense, no electrons are ejected. But, if even a weak beam of violet light is used, electrons are given off. The number of electrons emitted is proportional to the intensity of the light, above the threshold frequency. The energy of the electrons given off is independent of the intensity of the light, but dependent on the frequency. What these results suggested to Einstein, is the light exists as particles. He called them, photons, each with an energy equal to hν. In each collision with a metal atom, the photon gives up its energy to an electron in the metal atom. There is a minimum energy needed to pull the electron from its atom. That is the threshold energy. So, no amount of red photons can dislodge electrons from potassium, since they do not possess enough energy. Any violet photon does possess this minimal energy, and will allow electrons to escape. The higher the intensity of the light, the more photons will hit the metal surface, and the more electrons will be ejected. The proposal that light has properties of a particle was a very startling proposition. The Bohr atom Niels Bohr, a Danish scientist, combined the ideas of quantized energy and the line emission spectra in an atomic model of hydrogen, H. He proposed that energy of the electron in the H atom is quantized, that it is limited to a discrete set of values. E 1 is the ground state energy. E 2 is the 1 st excited state energy. Hydrogen atoms radiate energy only when the electron "jumps" from one allowed energy level to another. ΔE = hν. The

5 diagram below shows the correspondence between energy levels (n = 1,2,3, etc.) and the observed emission lines of H. For the transition between n = 2 and n = 1, the wavelength corresponds to a line in the emission spectrum. A number of scientists (Balmer, Lyman, Rydberg) had developed mathematical equations to describe the placement of the lines in the emission spectra. Bohr applied their equations to the hydrogen atom and derived an equation to predict the allowed energy levels in hydrogen. E n = -R H ( Z 2 /n 2 ) where n = principal quantum number and R H = the Rydberg constant = 2.18 x J and Z is the nuclear charge. By using this equation, we should be able to predict the wavelength of any of the electronic transitions in the H atom. For example, what wavelength of light will be emitted when an electrons falls from the n = 2 (initial energy level) to n = 1 (final energy level)? Hydrogen has an atomic charge of +1, so Z = 1. It will be given by: E 1 - E 2 = -R H (1/1 2-1/2 2 ) = x J This number is negative, because energy is being given off by the atom. We would expect this, since the electron is moving from a higher energy state to a lower energy state. This can be used to calculate the frequency of the energy being emitted.. ΔE = hν = 1.63 x J = (6.626 x J s) ν ν = 2.45 x s -1 c = λν = x 10 8 m s -1 λ = 121 nm This falls outside of the visible spectrum, into the ultraviolet region. The red, green and blue lines in the hydrogen emission spectrum (shown above) correspond to the transitions between n = 3, 4 and 5 and n = 2. The green line appears at about 490 nm. This gives it a frequency of (2.998 x 10 8 m s -1 ) (490 x 10-9 m) = 6.12 x s -1. The energy of this photon would be ΔE = (6.626 x Js) (6.12 x s -1 ) = 4.05 x J x J = (2.18 x J) (1/2 2-1/n 2 i ) = (1/2 2-1/n 2 2 i ) = /n i

6 1/n i 2 = n i 2 = 16 n = 4 So, the green line corresponds to the transition between n = 4 and n = 2. (The energy levels are expressed in electron volts (ev) instead of joules in the diagram). Allowed energy levels in hydrogen Bohr's model was important because it intoduced the idea of quantized states in atoms. It can only be applied to systems with a single electron, like H, He +, Li 2+, etc. After the introduction of this model, the dual nature of light became quite well accepted. Light has both wavelike and particle like properties. What is not explained by this model, is why electrons seem to rotate around the nucleus at certain fixed distances. The dual nature of matter The French scientist and prince, Louis de Broglie, reasoned, that if light waves can behave like particles, then why shouldn't particles, like electrons, behave like light waves? He proposed that all matter has a dual nature, possessing both wavelike and particle like properties and that electrons should behave like "standing waves". A particle of mass, m, traveling at a velocity, v, has linear momentum, p. p = mv. By combining this with Einstein's theory of relativity, and definition of a quantum of energy, De Broglie derived the expression: p = h/λ This ties together wavelike properties (λ, the wavelength) and particle like properties, p, momentum. Since p = mv = h/λ

7 λ = h/mv With this equation, it is possible to calculate the wavelength of any particle. The pictures below illustrate the "standing waves" of different frequencies possible in a vibrating string. De Broglie envisioned a standing wave associated with an electron around the nucleus. Only "quantized" values of the wave (i.e., n = 4, 5 or some other multiple of the wavelength) could exist about the nucleus. Non-integer values, like 4 ½ would lead to destructive interference. It had already been shown that X-rays form a diffraction pattern when they are passed through crystals (X-ray crystallography). De Broglie's model predicted that accelerated electrons (with increased velocities, v), should have wavelengths very similar to X-rays, and should also give diffraction patterns with crystals. There was no experimental basis for making this very bold prediction. Within three years, groups observed exactly this phenomenon, electron diffraction. Electrons were accelerated to about 6 x 10 6 m/s. With this velocity and with the mass of an electron, his equation predicted that the wavelength of accelerated electrons should be about nm, in the same spectral range as X-rays. Shown below are two diffraction patterns. On the left is the diffraction patterns of X-rays passing through aluminum foil. On the right is the diffraction pattern of a beam of electrons passing through the same target.

8 This proved the wavelike nature of electrons. Does this apply to all matter? One can actually compute the wavelength of a 6.0 x 10-2 kg tennis ball, going 85 ms -1 (190 mph, a good tennis serve). λ = h/mv λ = (6.63 x Js) (6.0 x 10-2 g)(62 ms -1 ) λ = 1.8 x m This is smaller than any atom, and could never be measured with existing devices.