SAT Physics Atomic and Quantum Phenomena - Development Of The Atomic Theory

SAT Physics Atomic and Quantum Phenomena - Development Of The Atomic Theory

DEVELOPMENT OF THE ATOMIC THEORY
The ancient Greeks first proposed the word "atom” as the name for an indivisible unit of matter. Several observations and prominent experiments have led to a greater understanding of atoms. The following sections include major highlights and scientists involved in development of the atomic theory.

J. J. Thomson
In 1897, J. J. Thomson discovered that even atoms themselves were divisible when he discovered the electron. Thomson knew that matter had an overall neutral charge, so he theorized that an atom of matter would be a mixture of positive and negative components.His model, often referred to as the “raisin cake model” or “plum-pudding model,” visualized the atom as containing positive and negative charges that were distributed throughout the interior of the atom.

Figure 20.1. Plum pudding model of an atom

Ernest Rutherford
Ernest Rutherford, a student of Thomson’s, began to experiment with what appeared to be charged rays emanating from crystals of uranium. He named these rays alpha and beta. He also determined that they consist of streams of particles. Rutherford determined that the alpha particle was, in fact, a doubly charged positive ion, which we now know to be a helium nucleus consisting of two protons and two neutrons. The beta ray consists of a negative particle.
    To investigate the interior of the atom, Rutherford used a radioactive source to fire alpha particles through a thin sheet of gold foil. A screen sensitive to alpha particles surrounded the gold foil to record the strikes of the alpha particles. If Thomson’s model was correct, the even distribution of positive and negative components of gold atoms should not greatly affect the path of the positive alpha particles. As a result, Rutherford expected the fairly heavy alpha particles to pass through the gold atoms with little deflection.
    After the experiment was concluded, Rutherford was surprised at the result. Although most of the alpha particles passed through the gold foil as expected, some of the alpha particles were deflected at extreme angles. In fact, a few nearly reversed direction completely. Rutherford likened this to shooting a cannonball at a piece of tissue paper and watching it bounce back. These results prompted Rutherford to propose that the positive region inside the gold atoms (protons were not yet discovered) was actually concentrated in a very tiny nucleus at its center, as illustrated in Figure 20.2.

Figure 20.2. Rutherford’s gold foil experiment

        Most alpha particles, like the one labeled A in Figure 20.2, passed through the gold foil without being deflected. A small number of positive alpha particles were deflected at extreme angles, such as particle B in Figure 20.2. These particles had to pass close to a dense positive region in order to be deflected in this manner. The most amazing results were those similar to Cin Figure 20.2. These alpha particles nearly bounced back and must have encountered a very dense positive region head on. Rutherford was able to use the particle traces to map the atom. He proposed that the atom was mostly empty space, allowing the majority of particles through the gold atoms without interference. The deflected traces revealed a very small but extremely positive region at the center of the atom. The results led Rutherford to propose the following characteristics for atoms.

• Atoms are mostly empty space.
• The majority of the mass of an atom is concentrated in a tiny central nucleus.
• The central nucleus is positively charged.
• The electrons orbit the nucleus in a manner similar to planets orbiting the Sun.

   Rutherford’s observations provided new insights into the structure of the atom. However, many questions needed to be resolved. For example, how could the nucleus stay together with the electrostatic repulsive forces that must be acting on the positive charges? How can an orbiting electron be sustained indefinitely around the nucleus of the atom? Theoretically, an electron should lose energy and then fall into the positive nucleus. Obviously, this wasn’t happening. If it did, then all matter would have disappeared long ago. More research was needed.

Spectroscopy
Spectroscopy is the study of the light spectrum emitted by luminous objects, such as the Sun, and the light emitted by gas discharge tubes. A gas discharge tube emits light by using a high voltage to shoot electrons through atoms in gaseous form. The electrons collide with the atoms, adding energy to the atoms. Eventually, the excess energy is lost. In the process, light with specific wavelengths is emitted. Each element emits a unique spectrum of light consisting of exact wavelengths. When viewed through a diffraction grating, the colors of the emitted light appear as a series of discrete lines. Each line in the pattern has a single wavelength and color. The patterns formed by these lines differ for each element, and they act like a fingerprint or bar code identifying each element.

Max Planck
In 1900, Max Planck attempted to explain the color spectrum seen when substances were heated to the point of glowing. The light emitted was due to atomic oscillations. The patterns seen could be explained only if Planck assumed that the atomic oscillations had very specific quantities of energy. He suggested that the oscillations were quantized (came in specific  quantities). He was able to develop a mathematical relationship for these oscillations. He determined that it was based on a constant, now known as Planck's constant, h.

h = 6.63 x 10^-34 joule • seconds = 4.14 × 10^-15 electron volt • seconds

Electron Volts
Planck’s constant may be given in units of joule-seconds and/or units of electron volt- seconds. An electron volt (eV) is an alternate unit of energy. Atoms are incredibly small, so working with joules of energy is not ideal. It is like measuring the length of a pen with a mile stick. The electron volt is a unit that is scaled to match the size of an atom. The conversion between joules and electron volts involves the same numerical value as the charge on an electron.

1 electron volt = 1.6 x 10^-19 joules

Albert Einstein
In 1905, Albert Einstein applied Planck’s idea of quantization to electromagnetic radiation. He suggested that light is quantized and that it consists of massless, particle-like packets that have a specific quantum (quantity) of energy. These packets of light came to be known as photons. Einstein clarified the relationship suggested by Max Planck. Einstein determined that the energy of a photon, E, is the product of frequency, f, and Planck’s constant, h.

E = hf

.
   Chapter 15, “Waves,” showed that wave speed was a function of wavelength and frequency and that the speed of light in a vacuum is c.

v = f λ
c = f λ

   The second formula can be rearranged to solve for frequency, f = c/λ. Then substitute this into the equation E = hf. The result is a useful equation that relates the energy of a photon of light to its wavelength.

Energy of Photons

How are the frequency and wavelength affected when the energy of photons is doubled?
 
WHAT'S THE TRICK?

The frequency of light is directly proportional to energy. High-frequency photons have more energy. The wavelength of light is inversely proportional to energy. Short wavelengths of light have more energy.

Doubling photon energy doubles frequency and halves wavelength.

Niels Bohr
In 1913, Niels Bohr used Einstein’s light quanta to suggest a model of the atom that explained why electrons do not fall into the nucleus and why the light emitted from excited atoms produces the observed emission spectra. Bohr theorized that the electrons of atoms could occupy only exact energy levels. An energy level is a specific energy state with an exact quantum (quantity) of energy.
    The absorption and emission of specific wavelengths of light are related to the energy difference between these energy levels. When atoms absorb light, the absorbed photons combine with the electrons. The energy of the photons adds to the energy of the electrons. Electrons with this added energy are said to be excited and must occupy a higher energy level. When atoms emit light, the excited electrons lose energy by emitting photons. The less energetic electrons must now occupy lower energy levels. The light that is absorbed and emitted is restricted by the energy levels in the atom. Only photons with quanta (quantities) of energy matching the exact difference between energy levels can be absorbed and emitted by an atom. Every atom has unique energy levels, and the difference between energy levels varies from atom to atom. Photons emitted from different elements experience different energy changes, resulting in unique wavelengths and colors. As a result, each element emits a unique color spectrum.
    Bohr borrowed elements of Einstein’s idea of light quanta and merged them with atomic spectra to propose a quantum mechanical model of atomic structure. The addition of exact energy levels provided the stability that the Rutherford model lacked. The next section details the absorption and emission of light according to the Bohr model of the atom.

ENERGY LEVEL TRANSITIONS
Niels Bohr’s research involved the simplest atom possible, the hydrogen atom. This atom consists of a single proton and a single electron. A partial energy level diagram of the hydrogen atom is shown in Figure 20.3.

Figure 20.3. Energy levels of an atom

Although the actual model of the atom is not as simple as that shown in Figure 20.3, the Bohr model is still used in energy level problems. The left side of the sketch shows a simplified Bohr model of a hydrogen atom. It shows the energy levels as circles similar to the orbits of planets in the solar system. The right side of the diagram shows the corresponding energy level diagram of the atom. In an energy level diagram, horizontal lines are used to portray the energy levels. The ground state is the lowest level an electron can occupy. It is numbered as the first energy level (n = 1). All the higher energy states are known collectively as the excited states. They are numbered from the ground state to the edge of the atom, n = 2,n = 3, etc. Be careful with the excited states. Since the ground state is n = 1, the first excited state is n = 2 and the second excited state is n = 3. The energy levels typically have negative values and they are often measured in electron volts. The edge of the atom has a value of zero electron volts. The ground state for hydrogen has an energy of -13.6 electron volts. Moving deeper into the atom is similar to taking an elevator ride below ground. The floor numbers become larger the farther down the elevator travels, and the elevator is moving in the negative direction.

Absorption
Absorption occurs when a photon of light with the correct amount of energy enters an atom and is absorbed by an electron. The energy of the photon adds to the energy of the electron, creating an excited electron called a photoelectron. The high-energy photoelectron must move to a higher energy level. If the hydrogen atom is radiated by light with 12.1 electron volts of energy, the electron will absorb the photon of light and their energies will add.

E_{electron initial} + E_photon = E_{electron final} = (-13.6 eV) + (12.1 eV) = -1.5 eV

   Absorptions are indicated with upwardly drawn arrows in energy level diagrams. The absorption calculated above is depicted in Figure 20.4. The photoelectron formed in this absorption moves upward 12.1 electron volts from the ground state to the third energy level, n = 3 (the second excited state).
    To be absorbed, a photon must have an energy corresponding to the exact difference in energy levels in an atom. For example, suppose photons with 11.0 electron volts of energy radiate a gas consisting of hydrogen atoms. Adding the electron and photon energies should move the electron to an energy level of -2.6 electron volts (-13.6 eV+11 eV=-2.6 eV). However, this energy level does not exist. The absorption cannot take place, and these mismatched photons pass through the hydrogen atoms.

Figure 20.4. Energy-level diagram

Emission
Nature prefers low-energy states. So electrons in high-energy levels will spontaneously move to lower energy levels until they finally reach the ground state. When electrons drop to a lower energy level, they lose energy. The energy they lose is given off as a photon of light. An emission is the light given off by an atom when electrons drop to lower energy levels. If an electron in the hydrogen atom at energy level two (n = 2) drops to the ground state (n = 1), it will lose 10.2 electron volts of energy.

E_photon = E₂ - E₁  = (-3.4 eV) - (-13.6 eV) = 10.2 eV

   The transitions to lower energy levels are haphazard. An electron in the third energy level might return all the way to the ground state in a single step. In a different atom, another electron in the third energy level might first drop to the second energy level and then drop to the ground state. Energy level problems involve samples that contain vast quantities of atoms. All the possible drops between the energy levels take place in many different atoms simultaneously. Emissions of light are pictured as downward arrows in energy level diagrams. Figure 20.5(a) shows all the possible emissions due to energy level drops from energy level three. Figure 20.5(b) shows the possible emissions due to energy drops from energy level four.

Figure 20.5. Emission of light

Think of the energy level diagram as a series of stairs and the electron as a ball on the stairs. When the electron is given enough energy to move to the high stair, such as the fourth energy level (n = 4), it can then fall to the bottom floor (n = 1) in a variety of ways. The ball can fall from the fourth stair to the first without hitting any of the other stairs. Instead, it can hit any combination of the stairs along the way. Each drop releases a photon with an amount of energy equal to the energy difference between the stairs.
    To find the energy of a photon, simply subtract the energy levels between which the electron is moving. Once the energy of the photon (E), is found, it can be used in the following formulas to determine the frequency (f), and wavelength (λ), of the emitted light.

   Each wavelength of light has a unique color, and this produces the variety of colors seen in color spectra. The Sun is composed of many types of atoms with electrons dropping through all kinds of energy differences. As a result, a continuous spectrum of light is emitted from the Sun. However, an excited gas made of one type of atom will emit a discrete spectrum containing light matching only its energy level differences. Keep in mind that many emitted wavelengths of light are not visible to the human eye. The hydrogen atom has two well- known light series that are produced when electrons drop down to a specific energy level.

• Balmer series: Transitions to n = 2, creating visible light.
• Lyman series: Transitions to n = 1, creating ultraviolet light (not visible)..

Energy Level Diagrams

(A) The energy level diagram above shows a sample of atoms initially in the ground state. The atoms are radiated by photons having 8 electron volts of energy. Determine the energy level of the electrons after they absorb the 8 electron volt photons.
 
WHAT'S THE TRICK?

Add the energy of the photons to the initial energy of the electrons in the ground state.

E_{electron initial} + E_photon = E_{electron final} = (-10 eV) + (8 eV) = -2 eV

   The electrons move to energy level three, n = 3.
(B) Subsequently, the resulting excited electrons drop to lower energy levels, emitting photons of light. Determine all the possible energies of the emitted photons.
 
WHAT'S THE TRICK?

Determine every possible drop that can occur from n = 3.

Now subtract the different energy levels for each drop.

E_photon = E₃ - E₁  = (-2 eV) - (-10 eV)  = 10.2 eV
E_photon = E₃ - E₂ = (-2 eV) - (-5 eV)     = 10.2 eV
E_photon = E₂ - E₁  = (-5 eV) - (-10 eV)  = 10.2 eV