Quantum Theory and the Electronic Structure of Atoms Chapter 7

Relationship of Quantum Theory and role of electrons In chemistry, the role of electrons in an atom is understood better by the quantum theory -How many electrons are present in a particular atom -What energies do individual electrons possess -Where in the atom can electrons be found

The Wave Nature of Light A wave can be characterized by its wavelength and frequency. The wavelength,  lambda), is the distance between any two adjacent identical points of a wave. The frequency,  (nu), of a wave is the number of wavelengths that pass a fixed point in one second.

Properties of Waves Wavelength ( ) is the distance between identical points on successive waves. Amplitude is the vertical distance from the midline of a wave to the peak or trough. Frequency ( ) is the number of waves that pass through a particular point in 1 second (Hz = 1 cycle/s). The speed (u) of the wave = x

So, given the frequency of light, its wavelength can be calculated, or vice versa. The Wave Nature of Light The product of the frequency, and the wavelength, would give the speed of the wave in m/s. In a vacuum, the speed of light, c, is 3.00 x 10 8 m/s. Therefore,

Maxwell (1873), proposed that visible light consists of electromagnetic waves. Electromagnetic radiation is the emission and transmission of energy in the form of electromagnetic waves. Speed of light (c) in vacuum = 3.00 x 10 8 m/s All electromagnetic radiation  c

If c =  then rearranging, we obtain  = c/ The Wave Nature of Light What is the wavelength of yellow light with a frequency of 5.09 x s -1 ? (Note: s -1, commonly referred to as Hertz (Hz) is defined as “cycles or waves per second”.)

If c =  then rearranging, we obtain = c/  The Wave Nature of Light What is the frequency of violet light with a wavelength of 408 nm?

Visible light extends from the violet end of the spectrum at about 400 nm to the red end with wavelengths about 800 nm. Beyond these extremes, electromagnetic radiation is not visible to the human eye. The Wave Nature of Light The range of frequencies or wavelengths of electromagnetic radiation is called the electromagnetic spectrum.

Electromagnetic Spectrum high energy short wavelength low energy long wavelength

x = c = c/ = 3.00 x 10 8 m/s / 6.0 x 10 4 Hz = 5.0 x 10 3 m Radio wave A photon has a frequency of 6.0 x 10 4 Hz. Convert this frequency into wavelength (nm). Does this frequency fall in the visible region? = 5.0 x nm

Quantum Effects and Photons Planck’s Quantization of Energy (1900) According to Max Planck, when solids are heated, they emit electromagnetic radiation over a wide range of wavelengths. where h (Planck’s constant) is assigned a value of 6.63 x J. s He proposed that an atom could emit or absorb energy only in discrete quantities, like small packages, and quantum is the smallest quantity of that energy for electromagnetic radiation. The energy E, of a single quantum of energy is given by,

Quantum Effects and Photons By the early part of twentieth century, the wave theory of light seemed to be well entrenched. In 1905, Albert Einstein proposed that light had both wave and particle properties as observed in the photoelectric effect Einstein based this idea on the work of Max Planck.

Quantum Effects and Photons Photoelectric Effect The energy of the photons proposed by Einstein would be proportional to the observed frequency, and the proportionality constant would be Planck’s constant. In 1905, Einstein used this concept to explain the “photoelectric effect.”

The photoelectric effect is the ejection of electrons from the surface of a metal when light shines on it. Quantum Effects and Photons Photoelectric Effect Electrons are ejected only if the light exceeds a certain “threshold” frequency. Violet light, for example, will cause potassium to eject electrons, but no amount of red light (which has a lower frequency) has any effect.

Einstein’s assumption that an electron is ejected when struck by a single photon implies that it behaves like a particle. Quantum Effects and Photons Photoelectric Effect When the photon hits the metal, its energy, h is taken up by the electron. The photon ceases to exist as a particle; it is said to be “absorbed.”

The “wave” and “particle” pictures of light should be regarded as complementary views of the same physical entity. Quantum Effects and Photons Photoelectric Effect This is called the wave-particle duality of light. The equation E = h displays this duality; E is the energy of the “particle” photon, and is the frequency of the associated “wave.”

Light has both: 1.wave nature 2.particle nature h = KE + W Photon is a “particle” of light KE = h - W h KE e -

E = h x E = 6.63 x (J s) x 3.00 x 10 8 (m/s) / x (m) E = 1.29 x J E = h x c /  When copper is bombarded with high-energy electrons, X rays are emitted. Calculate the energy (in joules) associated with the photons if the wavelength of the X rays is nm.

(6.626 x J. s) x (1.255 x 10 6 s -1 ) = x J Radio Wave Energy Solve for E, using E = h, and four significant figures for h. What is the energy of a photon corresponding to radio waves of frequency x 10 6 s -1 ?

The Bohr Theory of the Hydrogen Atom Prior to the work of Niels Bohr, the stability of the atom could not be explained using the then- current theories. In 1913, using the work of Einstein and Planck, he applied a new theory to the simplest atom, hydrogen. Before looking at Bohr’s theory, we must first examine the “line spectra” of atoms.

The Bohr Theory of the Hydrogen Atom Atomic Line Spectra When a heated metal filament emits light, we can use a prism to spread out the light to give a continuous spectrum-that is, a spectrum containing light of all wavelengths. The light emitted by a heated gas, such as hydrogen, results in a line spectrum-a spectrum showing only specific wavelengths of light.

Line Emission Spectrum of Hydrogen Atoms

emission spectrum - light emitted when a substance is excited by an energy source. The emission spectrum of hydrogen lead to the modern understanding of the electronic structure of the atom.

1.e - can only have specific (quantized) energy values 2.light is emitted as e - moves from one energy level to a lower energy level Bohr’s Model of the Atom (1913) E n = -R H ( ) 1 n2n2 n (principal quantum number) = 1,2,3,… R H (Rydberg constant) = 2.18 x J

The Bohr Atom Electrons exist in fixed energy levels surrounding the nucleus. Quantization of energy Initial understanding of the atom by Niels Bohr Promotion of electron occurs as it absorbs energy Excited State Energy is released as the electron travels back to lower levels. Relaxation 8

Orbit - what Bohr called the fixed energy levels. Ground state - the lowest possible energy state.

The orbits are also identified using “quantum numbers” –n = 1, 2, 3, … n=1 n=2 n=3 When the electron relaxes (c) the energy released is observed as a single wavelength of light.

E = h

The Bohr Theory of the Hydrogen Atom Bohr’s Postulates Bohr set down postulates to account for (1) the stability of the hydrogen atom and (2) the line spectrum of the atom. 1. Energy level postulate An electron can have only specific energy levels in an atom. 2. Transitions between energy levels An electron in an atom can change energy levels by undergoing a “transition” from one energy level to another.

The Bohr Theory of the Hydrogen Atom Bohr’s Postulates Bohr derived the following formula for the energy levels of the electron in the hydrogen atom. R h is a constant (expressed in energy units) with a value of 2.18 x J.

The Bohr Theory of the Hydrogen Atom Bohr’s Postulates When an electron undergoes a transition from a higher energy level to a lower one, the energy is emitted as a photon. –From Postulate 1,

The Bohr Theory of the Hydrogen Atom Bohr’s Postulates If we make a substitution into the previous equation that states the energy of the emitted photon, h, equals E i - E f, Rearranging, we obtain

The Bohr Theory of the Hydrogen Atom Bohr’s Postulates Bohr’s theory explains not only the emission of light, but also the absorbtion of light. When an electron falls from n = 3 to n = 2 energy level, a photon of red light (wavelength, 685 nm) is emitted. When red light of this same wavelength shines on a hydrogen atom in the n = 2 level, the energy is gained by the electron that undergoes a transition to n = 3.

A Problem to Consider Calculate the energy of a photon of light emitted from a hydrogen atom when an electron falls from level n = 3 to level n = 1.

E photon =  E = E f - E i E f = -R H ( ) 1 n2n2 f E i = -R H ( ) 1 n2n2 i i f  E = R H ( ) 1 n2n2 1 n2n2 n f = 1 n i = 2 n f = 1 n i = 3 n f = 2 n i = 3