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. Topic 7.1 Extended C – The Bohr theory of the hydrogen atom 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: 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 = m.
Calcium gas produces this spectrum: Topic 7.1 Extended C – The Bohr theory of the hydrogen atom 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 ... with characteristic absorption spectral lines, revealing what non-glowing elements are in the Sun's atmosphere. FYI: Note the possible fingerprint for calcium as an absorbing constituent of the Sun's outer atmosphere.
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: Topic 7.1 Extended C – The Bohr theory of the hydrogen atom 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 Emission Spectra of Hydrogen Lyman Series UV Balmer Series Paschen Series IR FYI: Each series is characterized by spectral line "bunching" at the smaller wavelengths, and "spreading" at the larger wavelengths. FYI: Balmer studied the middle series, which has parts in the visible spectrum. Obviously, the series is in his name.
In fact, Balmer found an empirical formula that predicted the allowed spectral wavelengths for the Balmer series of the hydrogen atom: Topic 7.1 Extended C – The Bohr theory of the hydrogen atom 1 Balmer Series = R n21n2 for n = 3,4,5,... Balmer Series where R = 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.
An explanation was finally given in 1913 by the Danish physicist Niels Bohr. Topic 7.1 Extended C – The Bohr theory of the hydrogen atom Bohr postulated that the single electron was held in a circular orbit about the single proton in the hydrogen nucleus by the Coulomb force: + r F c = F E = mv 2 r ke 2 r 2 The total mechanical E energy of the hydrogen atom is given by E = K + U E = mv ke 2 r so that FYI: Note that we are using the relation U = -kqQ / r for two point charges. Recall that this energy is negative since q and Q are oppositely charged. mv 2 = 1212 ke 2 2r E = - ke 2 r ke 2 2r 2 E = - ke 2 2r Energy in H Atom FYI: Up to this point in Bohr's derivation, classical mechanics has been used.
Recall that the angular momentum l of a point mass moving in a circle of radius r is given by Topic 7.1 Extended C – The Bohr theory of the hydrogen atom l = mvr + r 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 mvr = n h2h2 for n = 1,2,3,... Principal Quantum Number - H Atom mv 2 = 1212 ke 2 2r From and the previous equation we can eliminate v, solving for r: r = n 2 h 2 4 2 ke 2 m for n = 1,2,3,... Allowed Radii - H Atom n FYI: We subscript the r to indicate that the electron can only orbit the nucleus at certain quantized radii, determined by the principal quantum number.
We can then take our energy equation for the hydrogen atom and substitute our allowed values for r: Topic 7.1 Extended C – The Bohr theory of the hydrogen atom E = - ke 2 2r E n = - 22k2e4mh222k2e4mh2 1n21n2 for n = 1,2,3,... Everything in the parentheses is a constant whose value we know. We can then rewrite both r n and E n like this: Energy Quantization in the Bohr Hydrogen Atom E n = eV n 2 r n = n 2 nm The Bohr Hydrogen Atom
What are the orbital radius and energy of an electron in a hydrogen atom characterized by principal quantum number 3? Topic 7.1 Extended C – The Bohr theory of the hydrogen atom r n = n 2 nm r 3 = 3 2 nm r 3 = nm E n = eV n 2 E 3 = eV E 3 = eV What is the change in energy if the electron "drops" to the energy characterized by principal quantum number 2? E 2 = eV E 2 = -3.4 eV E = ( ) eV E = 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. FYI: We call the lowest energy level (n = 1) the GROUND STATE. We call the next highest energy level (n = 2) the 1 ST EXCITED STATE. We call the next highest energy level (n = 3) the 2 ND EXCITED STATE. Et cetera.
What is the orbital velocity of an electron in the second excited state (n = 3)? Topic 7.1 Extended C – The Bohr theory of the hydrogen atom From the previous slide r 3 = nm. mvr = n h2h2 Then v = nh 2 mr v = 3(6.63 ) 2 (9.11 )( ) v = 7.30 10 5 m/s What then is the centripetal acceleration of the electron? a c = v2rv2r = (7.30 10 5 ) = 1.12 m s -2 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."
Consider a plot of energies for n = 1 to : Topic 7.1 Extended C – The Bohr theory of the hydrogen atom E n = eV n 2 n = eV n = eV n = eV n = eV n = eV n = 0.00 eV Excited States 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 allowed states shown here. EXAMPLE: If an electron at r 3 suddenly "drops" to the ground state, the hydrogen atom LOSES energy having a value of E = ( ) = eV. To conserve energy, a PHOTON having an energy of E = eV is released. Ground State EXAMPLE: If a photon having an energy of E = eV is absorbed by a hydrogen atom in its ground state, the electron will "jump" up to the second excited state (n = 3) since E = ( ) = eV.
So how do the three series of hydrogen spectra relate to the Bohr model? Topic 7.1 Extended C – The Bohr theory of the hydrogen atom n = eV n = eV n = eV n = eV n = eV n = 0.00 eV First Excited State Ground State Second Excited State Consider the following transi- tions of hydrogen from higher to lower states: Lyman Series UV Balmer Series Visible Paschen Series IR Each transition gives off a photon of a different wavelength Lyman Series UV Balmer Series Paschen Series IR FYI: The Lyman Series has as its final state the GROUND STATE. The Balmer Series has as its final state the FIRST EXCITED STATE. The Paschen Series has as its final state the SECOND EXCITED STATE.