Cycle III AP Physics
John Dewey High School
Mr. Klimetz
Models of the Atom
Atomic Structure and Rutherford's Model. The unique properties of ores of uranium were the subject of intense study by Henri Becquerel during the waning years of the 19th century. He discovered that certain uranium-bearing rocks (such as uraninite [UO2]) would glow after being exposed to sunlight, and many physicists of the time were examining the light emitted from these same ores. Several days after inadvertently placing a sample of ore on a sheet of unexposed photographic paper kept in a sealed envelope, he discovered a pattern on the paper which could only be explained by the exposure of the paper to an energy form with all the properties of light but which could penetrate the envelope. Subsequent investigations on the part of Marie and Pierre Curie revealed that this energy, called radioactivity, originated from the uranium in the ore sample as a consequence of the instability of its atoms. Ernest Rutherford discovered that the radiation emitted from these unstable atoms consisted (at least in part) of particles that could be useful in further studies of the atom. When passed through a magnetic field, the emitted radiation split into three separate components, called alpha rays, beta rays, and gamma rays. Analysis of their respective trajectories led to the following conclusions: Alpha particles, equivalent to helium nuclei, are positively charged; Beta particles are electrons (which are negatively charged); and Gamma rays consist of photons with a neutral charge. Thomson, applying all available data, devised an electrically-neutral model of the atom. He suggested that an atom is essentially a positively charged entity associated with with electrons uniformly embedded within it as separate entities. A test of this model in 1911 was performed by Rutherford and colleagues employing alpha-particle scattering. As a result of the alpha-particle bombardment of a thin sheet of gold foil underlain by a sheet of zinc sulfide, the consequent scattering pattern produced on the underlying sheet revealed that most of the incident particles passed through the foil undeflected. However, there were several observed instances of strong deflection, a concluded result of the Coulomb force of repulsion between a positively charged nucleus and the positively-charged alpha particles. Some alpha particles were totally reflected, suggesting that a head-on collision of the alpha particles with a very dense mass. Rutherford therefore concluded that the atoms consisted mainly of empty space and that the nucleus was very small, dense, and positively charged. He likened the nucleus-orbiting electrons to planets orbiting the Sun. Among the deficiencies in Rutherford's model was that it could not account for the discrete emission line spectra. Rutherford's model suggested that the constantly-orbiting electrons should produce a continuous band of electromagnetic radiation which, however, was not observed. Additionally, he predicted orbital loss of energy would cause an atom to disintegrate in a very short period of time and thus break-apart, which, of course, does not occur.

Bohr's Model.  Danish physicist Niels Bohr's investigations of the spectral lines hydrogen atom in 1913 followed the work of Einstein expressed in his quantum theory of light, the concept of the photon, and an explanation of the photoelectric effect. Bohr observed that the Balmer formula for the hydrogen spectrum could be viewed in terms of energy differences. He then realized that these energy differences demanded a new set of postulates concerning atomic structure:
1.  Electrons do not emit EM radiation while occupying a particular orbit (energy state).
2.  Electrons cannot remain in any arbitrary energy state. They can remain only in discrete or quantized energy states.
3.  The angular momentum of an electron is quantized (by an integer index n) and is proportional to Planck's constant.
4.  Electrons emit EM energy when they make transitions from higher energy states to lower energy states. The lowest state is the ground state.
5.  Absorption spectra (dark lines appearing on an otherwise continuous-colored spectrum) occur due to electron transitions from lower to higher energy states as emitted light is absorbed by a cooler gas in front and reradiated.
The formula for the visible hydrogen spectrum is
1/L = R(1/4 - 1/nexp2)
where L is wavelength, n is orbital number, and R is the Rydberg constant. The energy of the emitted photon can be determined if we multiply both sides by hc:
Ep = hc/l = hcR/4 - hcR/nexp2
In this case, n is an integer.

For the UV hydrogen spectrum, the factor hcR/4 is replaced by hcR/1, and for the infrared spectrum, by the factor hcR/9. In thses cases, acceptable values for the principal quantum number, n, would also change.

Since electrons appear to be "bound" to a nucleus, (due to attractive Coulomb forces), forces designate the energy levels as increasingly negative as an electron gets closer to the nucleus, with n = 1 being the ground state. Thus we can quantitatively write
En = -(hcR/nexp2)
The energy states are given as E1 = -13.6 eV, E2 = -3.4 eV, E3 = -1.5 eV, E4 = -0.84 eV, etc.

These energy states have been derived qualitatively on the basis of the appearance of emission line spectra. Bohr's gifted insight enabled him to to begin with the electrical Coulomb forces and the quantized angular momentum and then to derive theoretically the same relationship for the hydrogen spectrum as just described.

Bohr realized that via Rutherford's experiments that the average radius of a hydrogen atom is 5.3 x 10exp-11 m. The circumference of this orbit would therefore be equal to
C = 2(p)r = (6.28)(5.3 x 10exp-11m) = 3.3 x 10exp-10 m

To better understand how angular momentum is quantized, let us use an argument that was proposed ten years before Bohr's theory. Recall that an electron possesses a wavelength (L) inversely proportional to its momentum (L = h/mv). Physicists suspected, on the basis of Rutherford's model, that the orbital velocity of an electron is approximately 2.2 x 10exp6 m/s. The wavelength associated with this velocity is given by
L = h/mv = (6.63 x 10exp-34)/(9.1 x 10exp-31)(2.2 x 10exp6) = 3.3 x 10exp-10 m
This wavelength is exactly the same as the circumference previously calculated.

If we state that the wavelength corresponds to a closed standing wave that is a multiple of the circumference (a two-dimensional standing wave), we can write
nL = 2(p)r = nh/mv
Transposing, we can write
L = mvr = nh/2(p)

The quantity mvr, designated by the letter L is, in a classical sense, the angular momentum of the electron. The expression above is Bohr's statement about the quantization of the angular momentum. The angular momentum is proportional to Planck's constant divided by 2(p), and the units of Planck's constant are consistent with those for angular momentum.

Bohr was able to derive the fact that the energy levels are determined by
En = -2(p)exp2kexp2meexp4/nexp2hexp2 = -13/6eV/nexp2
where k is Coulomb's constant, m is electron mass, e is electron charge, h is Planck's constant, and n is the principal quantum number, equal to the energy level (ground level and excited states).

Bohr's expression completely matches the empirical and qualititative energy-level values suggestd by the emission-line spectra in the absence of any quantum theory (except that E = hf for photons). According to the Bohr theory, the energy of a photon is equal to the energy difference between two energy states. The different spectral series arise from transitions from higher energy states  to specified lower energy states (see figure below). The

ultraviolet Lyman series occurs when electrons fall to level 1; the visible Balmer series, when electrons fall to level 2, the infrared Paschen series, when electrons fall to level 3. Thus we can write that the energy, in joules, for an emitted photon is given by
f = Einitial- Efinal/h

The energy in a given state represents the amount of energy, called the ionization energy, needed to free an electron from an atom. A charged atom is termed an ion. Electron ionization energies have been confirmed by direct measurement.

Although we designate the levels mathematically as negative so that the ionization level will have a value of zero, the energies are positive quantities. Even though hydrogen has only one electron, the many atoms of hydrogen in a sample of this gas, coupled with the different probabilities of electrons being in any one particular state, produce the multiple lines that appear in the visible spectrum when the gas is "excited."
Quantum Mechanics and the Electron Cloud Model. The success of the Bohr model was limited to the hydrogen atom and certain hydrogen-like atoms (atoms with a single electron in the outermost shell). The wave nature of matter, proposed by deBroglie, led to a problem in predicting the behavior of an electron in a strict Bohr energy state. Erwin Schroedinger developed a new mechanics of the atom in 1926, then called wave mechanics, but later termed quantum mechanics. In his wave mechanics, the position of an electron was determined by a mathematical function called the wave function. This wave function was related to the probability of finding an electron in any one energy state. A wave equation, called the Schroedinger equation, was derived using ideas about time-dependent motion in classical mechanics. The energy states of Bohr were now replaced by a probability "cloud." The electrons are not necessarily limited to to specified orbits in this model. A cloud of uncertainty is produced, with the densest regions corresponding to the highest probability of an electron being in a given state. The use of probability to explain the structure of the atom led to much debate between proponents and its fiercest opponent, Einstein, defending that fact that God would not "play dice" with the Universe! At about the same time that Schroedinger's wave mechanics appeared, Heisenberg's matrix mechanics for the atom presented itself. Instead of using wave analogy, Heisenberg developed a purely abstract mathematical theory using matrices. Alongside was Pauli, who advanced an exclusion principle whereby two electrons having the same spin orientations cannot occupy the same quantum state at the same time.
1.   Calculate the energy required to ionize a hydrogen atom in the n = 4 energy state.
2.   Calculate the frequency of a photon emitted in an electron transition from n = 3 to n = 1 in a hydrogen atom.
3.   How many different photon frequencies can be emitted if an electron is in an excited state n = 4 in a hydrogen atom?