Everything that can be invented has been invented.

Transcription

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2 Everything that can be invented has been invented. attributed to Charles H. Duell, Commissioner, U.S. patent office, 1899, in letter to President McKinley.

3 1835 philosopher Auguste Comte considered the limits of human knowledge. In his book Positive Philosophy, he wrote that we would never understand the stars: We see how we may determine their forms, their distances, their bulk, their motions, but we can never know anything of their chemical or mineralogical structure. Auguste Comte ( )

4 Spectral Lines Joseph von Fraunhofer ( )

5 Spectral Lines Joseph von Fraunhofer ( ) Identified black lines in Sunlight when dispersed by a prism. Regions with no emission. By 1814, Fraunhofer had cataloged over 475 of these lines.

6 Fraunhofer Lines 400nm 450nm 500nm 550nm 600nm 650nm 700nm 750nm H K h g G e f d c F b1+b2+b4 E D2 D1 a C B A Wavelength [nm] Name Atom K Ca H Ca h (Hδ) H g Ca G,Hγ H e Fe d Fe F (Hβ) H c Fe b4 Mg b2 Mg b1 Mg E Fe D2 Na D1 Na a O C (Hα) H B O A O2

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8 Spectral Lines Applications: Measure Elements and Velocities of Stars Fraunhofer identified one line in the Solar spectrum to correspond to a line of wavelength 590 nm observed when salt is burnt in a flame. Corresponds to Sodium in the Sun s atmosphere. Studies of absorption lines in the Sun s spectrum identified Helium (from Greek Helios for Sun) in 1868, not seen previously on Earth (and not discovered until 1895). Other studies of Sun showed that it contains a wide variety of elements. Other stars show similar (yet often different) absorption lines.

9 Stellar Classification Spectral Lines Ca H H Fe Na H

10 Measuring the shift because of speed Stationary Moving away Moving away faster Moving Toward Moving toward faster We generally measure the shift in the wavelengths of spectra of astronomical objects.

11 Spectral Lines Application: Measure Elements and Velocities of Stars Comparing Sun s spectrum to other stars. For non-relativistic speeds, the line-of-sight (called the recessional) velocity is λobs - λrest λrest = Δλ obs λrest = v r c In Vega, important Hydrogen line has λobs = nm compared to λrest = nm, measured in the lab. vr = c (λobs - λrest) / λrest = km s -1 Vega also has a measured proper motion (perpendicular to line-ofsight) of μ= yr -1. At r=7.76 pc, the proper motion (or transverse velocity) is v θ = rμ = 12.9 km s -1. The total velocity is v = vr 2 + v θ 2 = 19.0 km s -1

12 What does an atom look like?

13 What does an atom look like? In 19th century, people thought it looked something like this.

14 Rutherford Experiment Ernest Rutherford ( )

15 Rutherford Experiment Old Atomic Model Rutherford s Atomic Model

16 Structure of Atoms Series Name Symbol Transition Wavelength [nm] Balmer Hα 3 to Hβ 4 to Hγ 5 to Hδ 6 to Hε 7 to Hζ 8 to Hη 9 to H limit to When dense Hydrogen gas is heated, it shows emission lines at exact wavelengths. By 1885, 14 spectral lines in Hydrogen had been measured. After much trial and error, Johann Balmer ( ) found that the wavelengths of hydrogen corresponded to a pattern, now called the Balmer series made of the Balmer lines. 1 = RH ( 1/4-1/n 2 ), n=3, 4, 5,..., λ RH = x 10 7 m -1 is an experimentally measured quantity called the Rydberg constant for hydrogen.

17 Classical (incorrect) picture of atoms electron nucleus As electron accelerates around nucleus, it emits radiation, loses energy and spirals into nucleus. The frequency of the radiation increases to infinity as R goes to zero. For hydrogen, the electron should collide with the nucleus in less than one-billionth of a second.

18 Structure of Atoms Balmer realized that he could generalize this to: 1 = RH ( 1/m 2-1/n 2 ), m=2, n=3, 4, 5,..., λ And he predicted that other line series could be found with m < n (both integers). In 1906, Theodore Lyman confirmed the series for m=1, n=2, 3, 4,... as the Lyman Series. In 1908, Friedrich Paschen confirmed the series for m=3, n=4, 5, 6,... as the Paschen Series. Series Name Symbol n to m Wavelength [nm] Lyman Lyα 2 to Lyβ 3 to Lyγ 4 to Ly limit to Balmer Hα 3 to Hβ 4 to Hγ 5 to H limit to Paschen Paα 4 to Paβ 5 to Paγ 6 to Pa limit to

19 Bohr model of Atom Niels Bohr ( ) proposed new atomic model in When an electron is in one of the quantized orbits, it does not emit any electromagnetic radiation. The electron can make a discontinuous transition from one stationary state to another. It emits or absorbs radiation during these jumps. 2. Laws of classical mechanics apply to orbital motion of electrons in a stationary state, but not during a transition between states. 3. When an electron makes a transition, the energy difference is released as a single photon of frequency, ν = E/h. 4. Permitted orbits are characterized by quantized values of the orbital angular momentum with L = n h / 2π = n ħ (ħ is h bar ).

20 Bohr model of Atom Coulomb s Law Proton has charge of q = +e = x C Electron has charge of q = -e This is a two-body problem, for which we can model as a reduced mass, μ, orbiting a total mass M. For this problem:

21 Bohr model of Atom Coulomb s Law Proton has charge of q = +e = x C Electron has charge of q = -e Total Energy: Satisfies Virial Theorem! Electron-Proton system is a Bound Orbit.

22 Bohr model of Atom Bohr s quantization of angular momentum gives Rewrite top equation in terms of angular momentum and then solve for r : a0 = Bohr Radius = nm Insert eqn. for rn into equation for Energy gives: n is the principle quantum number

23 Emission Lines in Hydrogen Ephoton = Ehigh - Elow En=3 - En=2 = ev -(-3.40 ev) = 1.90 ev Wavelength of that photon is E = hc / λ, or λ = hc / E = nm.

24 Absorption Lines in Hydrogen Ephoton = Ehigh - Elow En=3 - En=2 = ev -(-3.40 ev) = 1.90 ev Wavelength of that photon is E = hc / λ, or λ = hc / E = nm.

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26 What are the three basic types of spectra? Continuous Spectrum Continuous Spectrum Emission Line Spectrum Emission Line Spectrum Absorption Line Spectrum Absorption Line Spectrum UV blue green red infrared Wavelength Astrophysical objects have spectra that are usually composed of these three basic types.

27 Kirchhoff s Laws: A hot, dense gas or hot solid object produces a continuous Spectrum with no dark spectral lines. A hot, diffuse (low density) gas produces bright spectral lines (emission lines). A cool, diffuse gas in front of a source of continuous spectrum produces dark spectral lines (absorption lines) in the continuous spectrum.

28 Continuous Spectrum The spectrum shows a smooth continuous rainbow of light. Hot Light Source PRISM The light bulb produces light of all visible wavelengths. Any object with temperature above T > 0 K will emit a continuous spectrum. A graph of the spectrum is also continuous, notice that intensity varies at different wavelengths.

29 Emission Line Spectrum We see bright emission lines at specific wavelengths (colors), but no other light. Gas Cloud PRISM The atoms in a warm gas cloud emit light only at specific wavelengths (colors) determined by the cloud s atoms and temperature. The graph shows an upward spike at the wavelength of each emission line.

30 Absorption Line Spectrum We see dark absorption lines where the cloud has absorbed light of specific wavelengths (colors). Hot Light Source Gas Cloud PRISM If light from a hot source passes through a cooler gas cloud, atoms in the cloud absorb light at wavelengths determined by the clouds atoms and temperature. The graph shows a dip in intensity at the wavelength of each absorption line.

31 Kirchhoff s Laws, Restated. A hot, dense gas or hot solid object produces a continuous spectrum with no dark spectral lines. This is blackbody radiation emitted at any temperature, T > 0 K, with a spectrum described by the Planck function B λ (T) and B ν (T). The wavelength at which the Planck Function B λ (T) reaches its maximum is given by Wien s displacement law. A hot, diffuse (low density) gas produces bright emission lines. Emission lines are produced when an electron makes a downward transition from a higher orbit to a lower orbit. The energy lost by then electron is carried away by a single photon. A cool, diffuse gas in front a source of continuous spectrum produces dark absorption lines in the continuous spectrum. Absorption lines are produced when an electron absorbs a photon with enough energy equal to the energy difference between the two transitions.

32 What are the three basic types of spectra? Continuous Spectrum Continuous Spectrum Emission Line Spectrum Emission Line Spectrum Absorption Line Spectrum Absorption Line Spectrum UV blue green red infrared Wavelength Astrophysical objects have spectra that are usually composed of these three basic types.

33 Emission Line Galaxy

34 Galaxy with Absorption and Emission Lines

35 Transition to Quantum Mechanical Picture of Atoms de Broglie asked the question, if photons sometimes act as particles, do particles sometimes act as waves? Answer led him to state: ν = E / h λ = h / p = h / (m v) Louis de Broglie ( )

36 Transition to Quantum Mechanical Picture of Atoms Examples: λ = h / p = h / (m v) 1. electron moving at 1% the speed of light (3x10 6 m/s): λ = h / (me v) = (6.63x10-34 J s) / (9.11x10-31 kg x 3 x 10 6 m/s) = 2.42 x m = nm 2. jogging person with m = 80 kg and v = 2 m/s: Louis de Broglie ( ) λ = h / (m v) = (6.63x10-34 J s) / (80 kg x 2 m/s) = 4.14 x m = 4.14 x nm Roughly 1 billionth of a billionth of the size of the nucleus of an atom!

37 Transition to Quantum Mechanical Picture of Atoms Heisenberg s Uncertainty principle. λ x x One wave, well known wavelength, can determine momentum exactly by, p = h/λ. But, because wave has infinite number of peaks, x location is unknown. Superposition of many waves with a combination of wavelengths. Ψ = Σ sin(2πx/(nλ)) n λ not exactly known, p has uncertainty. But, x can be determined more accurately.

38 Transition to Quantum Mechanical Picture of Atoms Heisenberg s Uncertainty principle. formally: Δx Δp ħ / 2 ΔE Δt ħ / 2 in practice: Δx Δp ħ ΔE Δt ħ

39 Radioactivity In 1896 Antoine Henri Becquerel discovered that Uranium would fog a photographic plate, and that the radiation emitted by Uranium could be deflected in a magnetic field. (Unlike Marie Curie s discovery of X-ray radiation.) Rutherford subsequently identified two kinds of radiation from Uranium, alpha, and beta radiation. Paul Villard discovered that another element, radium, gave off a third type of radiation particle with no charge, named gamma. Today, we understand these radiation originate from an unstable nucleus that emits a Helium nucleus (alpha particle), an electron (beta particle) or a positron (positive beta particle), or a high-energy photon (gamma particle). Becquerel shared the Nobel Prize with Marie Curie in 1903 for their discoveries of emission. Edward Rutherford ( ) Antoine Henri Becquerel ( )

40 Radioactivity Notation for the number of nucleons of a particular atom: X is the chemical symbol (H, He, C, O, N, Fe,...) Z is the number of protons (and total charge) A is the number of nucleons (neutrons + protons) Helium with 2 protons and 2 neutrons is : aa Alpha Decay : Beta Decay : Gamma Radiation :

41 Age-dating Rocks Find the oldest rocks and determine their age:

42 Radioactivity Radioactive Decay is a Statistical process, so it must be proportional to the number of atoms in a sample Carbon dating measures the ratio of Carbon-14 in a substance to that in the Earth s atmosphere. Carbon-14 is unstable and has a half-life of 5730 years. It decays by the following:

43 Radioactivity Radioactive Decay is a Statistical process, so it must be proportional to the number of atoms in a sample: One can show that if then N0 is the original number in a sample of atoms. λ is related to the half-life, τ1/2, by The half-life is the time it takes for the number of atoms you have to be reduced by a factor of 2. Alternatively, there is a 50% chance that any individual atom will decay in a time τ1/2.

44 Radioactivity Radioactive Dating: You can use radioactive isotopes and compare them to stable isotopes to derive the age of that sample of atoms. This has been done with Earth Rocks, Moon Rocks, Asteroids. (Also Carbondating, used to date age of fossils, for example.) Suppose an isotope type A decays into isotope B (which is stable) then after some time t where NA,i is the initial number and NA,f is the final number. The total number of atoms of A and B must remain constant, so : Solving for NA,i yields a formula for the change in the number of atoms of B: In practice you study the ratio of B and A to a third (stable and constant) element C:

45 Radioactivity One plots the relative abundances of the stable product versus the relative abundances of the radioactive isotope in the sequence at various locations in the rock. The slope m=(e λt - 1) is directly related to the age of the sample. Data from the Moon looks at the decay of rubidium-87 to strontium-87. Using the plot m = (e λt - 1) = We know that λ= x 10-9 yr -1 for rubidium-87, so we can solve for t = 4.39 x 10 9 yr.

46 Radioactivity An example is the natural decay chain of uranium-238 which is as follows: decays via α-emission, with a half-life of 4.5 billion years to thorium-234 decays via β-emission, with a half-life of 24 days to protactinium-234 decays via β-emission, with a half-life of 1.2 minutes to uranium-234 decays via α-emission, with a half-life of 240 thousand years to thorium-230 decays via α-emission, with a half-life of 77 thousand years to radium-226 decays via α-emission, with a half-life of 1.6 thousand years to radon-222 decays via α-emission, with a half-life of 3.8 days to polonium-218 decays via α-emission, with a half-life of 3.1 minutes to lead-214 decays via β-emission, with a half-life of 27 minutes to bismuth-214 decays via β-emission, with a half-life of 20 minutes to polonium-214 decays via α-emission, with a half-life of 160 microseconds to lead-210 decays via β-emission, with a half-life of 22 years to bismuth-210 decays via β-emission, with a half-life of 5 days to polonium-210 decays via α-emission, with a half-life of 140 days to lead-206, which is a stable nuclide.