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**Quantum Theory and the Atom**

Section 5.2 Quantum Theory and the Atom

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**Objectives Compare the Bohr and quantum mechanical models of the atom.**

Explain the impact of De Broglie’s wave-particle duality and the Heisenberg uncertainty principle on the modern view of electrons in atoms. Identify the relationships among a hydrogen atom’s energy levels, sublevels, and atomic orbitals.

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**Quantum Theory and the Atom**

Einstein’s theory of light’s dual nature accounted for several unexplainable phenomena but it did not explain the atomic emission spectra of elements. In 1913, Niels Bohr (who was working for Rutherford) proposed a quantum model for the hydrogen atom that seemed to do that.

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**The Bohr Model of the Atom**

Bohr proposed that the hydrogen atom can have only certain allowable energy states. The lowest allowable energy state is called the ground state. When an atom gains energy, it is said to be in an excited state. Many ”excited” states are possible.

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Bohr’s Atomic Model Bohr said that hydrogen’s energy states were related to the motion of its electrons. He said hydrogen’s electron moved in certain allowed circular orbits around the nucleus. The closer the orbit was to the nucleus, the smaller the orbit was AND the lower the atom’s energy level.

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The Planetary Model Bohr’s model came to be known as the planetary model.

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**Bohr assigned a quantum number (n) to each orbit.**

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**Interpreting the Data Bohr was able to explain the line spectra**

of hydrogen this way: Hydrogen is in its ground state when its electron is in the n = 1 orbit. If enough energy is added, the electron can move to n = 2. It is then excited and unstable. Since the electron is unstable, it will not remain in n=2 but will drop back to n = 1.

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Interpreting the Data The energy that was absorbed is now released as a photon. The energy of the photon (Ephoton = hf) is equal to the energy difference between n = 2 and n = 1.

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In Other Words . . . ΔE (energy released) = Ehigher energy orbit - Elower energy orbit = hf Only certain amounts of energy are given off so only certain frequencies are emitted. That means these frequencies/spectral lines correspond to electron transfers between one specific n (or energy level) to another specific n.

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**Each spectral line in an AES will represent one possible electron transfer.**

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**Hydrogen’s Line Spectrum**

The AES that falls in the visible range of the EM spectrum is pictured. n=3 to n=2 produces a red line n=4 to n=2 produces a blue-green line n=5 to n=2 produces a blue line n=6 to n=2 produces a violet line Note that the energy levels are not evenly spaced from one another.

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Unanswered Questions . . . Bohr’s planetary model failed to explain the AES for elements other than hydrogen. It also failed to account for chemical behavior of atoms.

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**The Quantum Mechanical Model**

By the 1920’s, scientists were convinced that Bohr’s model was fundamentally incorrect. New explanations of how electrons were arranged in atoms were formed. Louis De Broglie ( )

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**Quantum Mechanical Model**

De Broglie knew that when light traveled through space, it behaved like a wave. He also knew that when light interacted with matter, its behavior was like that of a stream of particles. He thought that if energy had a dual nature then maybe matter did too…

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**Quantum Mechanical Model**

De Broglie believed that all moving particles of matter, like electrons, had wave characteristics. He referred to the wavelike behavior of particles as matter waves. He derived an equation to describe the matter waves: λ = h (m =mass) mv (v = velocity)

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**Quantum Mechanical Model**

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**Quantum Mechanical Model**

In 1927, Werner Heisenberg proposed his uncertainty principle: it is impossible to know both the position and velocity of a moving object at the same time. He believed any attempt to determine an object’s position would change its velocity and vice versa.

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For example Suppose you had to locate a helium-filled balloon in a dark room. To locate it, you would touch it with your hand. Such an act would cause a change in the velocity of the balloon. Hence, you cannot know the position and velocity at the same time.

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**Quantum Mechanical Model**

But what if you used a flashlight? You would locate the balloon when the light bounced off it and hit your eyes. The balloon is so much more massive then the photons that they will have “no effect” on the balloon’s position.

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**Quantum Mechanical Model**

What about locating electrons? Could they be hit with a photon (which would then bounce back to some detection device)? No - Heisenberg’s Uncertainty Principle applies. Because the electron has such a small mass, its collision with a photon would move it in some unpredictable way.

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**Quantum Mechanical Model**

What we know so far The energy of electrons is quantized. (Electrons can only have certain amounts of energy.) Electrons exhibit wavelike characteristics and behavior. We cannot experiment with electrons to determine their nature - position and velocity of an electron are impossible to know at the same time.

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**Quantum Mechanical Model**

In 1926, Erwin Schrodinger furthered the wave-particle theory of de Broglie. He deriving a mathematical equation that described hydrogen atom’s electron as a wave.

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**Quantum Mechanical Model**

This new model seemed to apply equally well to atoms of other elements. The atomic model in which the electron is treated as a wave is called the quantum mechanical model of the atom.

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**Quantum Mechanical Model**

An electron’s energy is limited to certain values. An electron’s path around the nucleus is not circular but is described in terms of probability. The probability of finding an electron in various locations around the nucleus can be pictured in terms of a blurry cloud of negative charge. The Schrodinger wave equation is too complex to be considered here, however each solution to the equation is known as a wave function. The wave function is related to the probability of finding the electron within a particular volume of space around the nucleus

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**Quantum Mechanical Model**

The cloud is most dense where the probability of finding the electron is highest. The boundary of the “electron cloud” encloses the area that has a 90% probability of containing electrons. The probability of finding electrons in certain regions of an atom is described by orbitals.

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**Quantum Mechanical Model**

Because electrons have different energies, they are found in different probable locations around the nucleus. An atomic orbital is a 3-d region around the nucleus of an atom where an electron with a given energy is likely to be found. Orbitals (not orbits) have characteristic shapes, sizes and energies.

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**Quantum Mechanical Model**

A principle quantum number (n) is assigned to indicate the relative SIZE & ENERGY of atomic orbitals. As n increases, the orbital becomes larger and is further away from the nucleus. An atom’s principal energy levels are specified by n.

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**Quantum Mechanical Model**

Each principal energy level consists of one or more sublevels . . . As n increases, the # of sublevels increases as does their distance from the nucleus. Just like your home address consists of a state, a city and a street name within the city, the “address” of an electron consists of its principal energy level, its sublevel and its orbital within that sublevel.

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**Quantum Mechanical Model**

Sublevels are labeled s, p, d, or f, according to the shapes of their orbitals. For n=1, there is one sublevel. It is called “s”, specifically “1s” For n=2, there are 2 sublevels. They are called “s” and “p” (or 2s ,2p). For n=3, there are 3 sublevels. They are called ? Just as proposed in the Bohr model, the energy of the electron increases as n increases. Unlike the Bohr model, however, each principal energy level is divided into one or more sublevels. The number of sublevels in each principal energy = the quantum number n for that energy level. There is one sublevel when n=1, two sublevels when n=2 and so on. These sublevels are labeled with a number that is the value of the quantum number n and a letter (s, p, d or f) that corresponds to the type of sublevel. As we go on keep in mind the definition of an orbital – region in which an electron with a particular energy is likely to be found.

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**Quantum Mechanical Model**

Each type of sublevel consists of one or more orbitals. There is 1 “s” orbital There are 3 “p” orbitals There are 5 “d” orbitals There are 7 “f” orbitals

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**Quantum Mechanical Model**

All s orbitals are spherical. They will differ in size.

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**Quantum Mechanical Model**

All p orbitals are dumbbell-shaped. There are 3 p orbitals because the dumbbell shape can be oriented in 3 different ways in space. d and f orbitals are very complex in shape. See pg. 133. P orbitals are dumbbell shaped.

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**Quantum Mechanical Model**

Review The energy level or principal quantum number is designated by n. The number of sublevels always equals the quantum number n. For n = 1, the one sublevel is s. For n = 2, the two sublevels are s and p. For n = 3, the three sublevels are s, p, and d. For n = 4, the four sublevels are s, p, d, and f.

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**Quantum Mechanical Model**

The number of orbitals in each sublevel is always an odd number: s has 1 orbital; p has 3 orbitals; d has 5 orbitals; f has 7 orbitals. The total number of orbitals in each energy level = n2 (In n= 3, there are 9 orbitals: s, 3 p , and 5 d.) Each orbital may contain at most 2 electrons. The maximum number of electrons in each energy level = 2n2 See the summary - pg. 134

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