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Transcript
Topic 7.1 Extended C –The Bohr theory of the hydrogen atom When a gas in a tube is subjected to a voltage, the gas ionizes, and emits light. We can analyze that light by looking at it through a spectroscope. A spectroscope acts similar to a prism, in that it separates the incident light into its constituent wavelengths. For example, barium gas in a gas discharge tube will produce an emission spectrum that looks like this: 4000 4500 5000 5500 6000 6500 7000 The emission spectrum is really an elemental fingerprint - it uniquely identifies the element producing it. FYI: The wavelengths are given in angstroms Å. 1 Å = 10-10 m. 7500 Extended FYI: Note the possibleTopic fingerprint7.1 for calcium as an absorbing constituent of the atmosphere. C Sun's –Theouter Bohr theory of the hydrogen atom Calcium gas produces this spectrum: 4000 4500 5000 5500 6000 6500 7000 7500 Not only do glowing gases emit spectral lines, but cool gases absorb light in the same wavelengths and produce what is called an absorption spectrum. For example, the Sun produces a continuous spectrum that looks like this... 4000 4500 5000 5500 6000 6500 7000 7500 ...with characteristic absorption spectral lines, revealing what non-glowing elements are in the Sun's atmosphere. FYI: Balmer studied the middle series, which has parts in the visible spectrum. Obviously, the series is in his name. Topic 7.1 Extended C –The Bohr theory of the hydrogen atom In the late 1800s a Swedish physicist by the name of J.J. Balmer observed the spectrum of hydrogen - the simplest of all the elements: Emission Spectra of Hydrogen 0 200 Lyman Series UV 400 600 Balmer Series 800 1000 1200 1400 1600 1800 2000 Paschen Series IR The general spectral signature is divided up into natural groups of spectral lines called series, falling roughly in the UV (ultraviolet), Visible, and IR (infrared) ranges of wavelengths. FYI: Each series is characterized by spectral line "bunching" at the smaller wavelengths, and "spreading" at the larger wavelengths. Topic 7.1 Extended C –The Bohr theory of the hydrogen atom In fact, Balmer found an Balmer Series empirical formula that predicted the allowed spectral wavelengths for the Balmer series of the hydrogen atom: 1 = R 1 - 1 n2 22 for n = 3,4,5,... Balmer Series where R = 1.09710-2 nm-1 is called the Rydberg constant. FYI: The visible spectrum for hydrogen was found to fit this formula, but it was NOT understood why. FYI: Similar formulas were found to fit the other two series. FYI: Note that we are using the relation U = -kqQ / r for two point charges. Recall that this energyTopic is negative sinceExtended q and Q are oppositely charged. 7.1 FYI: to this point in Bohr's derivation, mechanics has atom been C Up–The Bohr theory of classical the hydrogen used. An explanation was finally given in 1913 by the Danish physicist Niels Bohr. r Bohr postulated that the single electron was held in a circular + orbit about the single proton in the hydrogen nucleus by the Coulomb force: Fc = FE ke2 ke2 mv2 1 2 2mv = = 2r r2 r The total mechanical E energy of the hydrogen atom is given by E = K + U so that ke2 1 2 E = mv 2 r 2 ke2 2 ke2 ke Energy in E = E = H Atom 2r 2r 2r FYI: We subscript the r to indicate that the electron can only orbit the Topic 7.1 Extended nucleus at certain quantized radii, determined by the principal quantum number. C –The Bohr theory of the hydrogen atom Recall that the angular momentum l of a point mass moving in a circle of radius r is given by l = mvr Bohr then postulated the radical idea that the angular momentum of the electron was quantized, just like light. He stated that the angular momentum of the electron can only carry the discrete values given by r + h Principal Quantum for n = 1,2,3,... Number - H Atom 2 2 ke 1 and the previous equation we can From mv2 = 2 2r eliminate v, solving for r: mvr = n h2 rn = n2 for n = 1,2,3,... 2 2 4 ke m Allowed Radii - H Atom Topic 7.1 Extended C –The Bohr theory of the hydrogen atom We can then take our energy equation for the hydrogen atom and substitute our allowed values for r: ke2 E = 2r Energy Quantization 22k2e4m 1 for n = 1,2,3,... En = in the Bohr 2 h2 n Hydrogen Atom Everything in the parentheses is a constant whose value we know. We can then rewrite both rn and En like this: -13.6 The Bohr En = eV n2 Hydrogen 2 rn = 0.0529n nm Atom FYI: We call the lowest energy level (n = 1) the GROUND STATE. Topic 7.1(n =Extended We call the next highest energy level 2) the 1ST EXCITED STATE. WeCcall–The the nextBohr highest energy level of (n = 3) the 2ND EXCITED STATE. theory the hydrogen atom Et cetera. What are the orbital radius and energy of an electron in a hydrogen atom characterized by principal quantum number 3? -13.6 E = eV n rn = 0.0529n2 nm n2 -13.6 r3 = 0.052932 nm E3 = eV 32 r3 = 0.4761 nm E3 = -1.51 eV What is the change in energy if the electron "drops" to the energy characterized by principal quantum number 2? -13.6 E = (-3.4 - -1.51) eV E2 = eV 22 E = -1.89 eV E2 = -3.4 eV FYI: In general, if an electron "drops" from a higher quantum state to a lower one, the hydrogen atom experiences a net loss of energy. Topic 7.1 Extended C –The Bohr theory of the hydrogen atom What is the orbital velocity of an electron in the second excited state (n = 3)? From the previous slide r3 = 0.4761 nm. h Then mvr = n 2 nh v = 2mr 3(6.6310-34) v = 2(9.1110-31)(0.476110-9) v = 7.30105 m/s What then is the centripetal acceleration of the electron? (7.30105)2 v2 = 1.12 1021 m s-2 = ac = -9 0.476110 r FYI: Classical theory predicts that electromagnetic radiation is created by accelerating charges. Since the hydrogen atom only radiates when its electron "drops" from one excited state to a less energetic state, Bohr postulated that "the hydrogen electron does NOT radiate energy when it is in one of its bound states (allowed by n). It only does so when "dropping" from a higher state to a lower state." EXAMPLE: If a photon having an energy of E = +12.09 eV is absorbed by a Topic 7.1 Extended hydrogen atom in its ground state, the electron will "jump" up to the second excited state (nBohr = 3) since E = (-1.51 12.09 eV. C –The theory of- -13.6) the= +hydrogen atom Consider a plot of energies for n = 1 to : Excited States -13.6 En = eV n2 FYI: Bohr's theory only allows electrons in the hydrogen atom to absorb or emit photons having energies equal to the difference between any two of the Ground allowed states shown here. State n= 0.00 eV n=5 n=4 -0.544 eV -0.850 eV n=3 -1.51 eV n=2 -3.40 eV n=1 -13.6 eV EXAMPLE: If an electron at r3 suddenly "drops" to the ground state, the hydrogen atom LOSES energy having a value of E = (-13.6 - -1.51) = -12.09 eV. To conserve energy, a PHOTON having an energy of E = +12.09 eV is released. FYI: The Lyman Series has as its final state the GROUND STATE. Topic 7.1 The Balmer Series has as its final state Extended the FIRST EXCITED STATE. The as its final state SECOND EXCITEDatom STATE. C Paschen –The Series Bohrhastheory of the the hydrogen 0.00 eV n= n=5 -0.544 eV So how do the three -0.850 eV series of hydrogen Second n = 4 spectra relate to Excited n = 3 -1.51 eV State the Bohr model? Paschen Consider the First n=2 -3.40 eV Series Excited following transiIR State tions of hydrogen Balmer from higher to lower Series states: Visible Each transition Lyman gives off a photon Ground n = 1 -13.6 eV Series State of a different UV wavelength. 0 200 Lyman Series UV 400 600 Balmer Series 800 1000 1200 1400 Paschen Series IR 1600 1800 2000