Chemical Bonding

LightThe wavelength () is the distance between any 2 adjacent identical points on a wave—commonly crest to crest or trough to trough.

LightFrequency is the number of wavelengths that pass a given point in a unit of time (commonly 1 second).The unit of frequency is, “per second,” or s-1, is called hertz, Hz, and is represented as , “new.”

The length of the wave per second is the speed of the wave: speed =

For light, c: c =

LightLight is very unique. Newton believed light consisted of a beam of particles. Thomas Young showed that light could be diffracted, indicating that it had wavelike properties.

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The Photoelectric EffectIn 1905, Einstein discovered the photoelectric effect postulating that light had both wave and particle properties. The work of Einstein was based on the work of Max Plank.

The Photoelectric EffectPlank’s experimental work was based on energy emitted from bodies he’d experimentally heated up.

As the heated solid increased in temperature, it began to glow red at 750°C, then white near 1200°C.From this, he discovered at the lower temperature, mostly red light is emitted; at higher temperatures, yellow and blue light were emitted along with the red—giving the characteristic white glow.

The Photoelectric EffectAccording to Plank, the atoms vibrate with a particluar frequency characteristic of the solid. This would only occur with glowing solids if a physical constant (now called Plank’s constant) was used. This constant relates energy and frequency of a vibrating solid and has the value of 6.63 x 10-34J • s.

The Photoelectric EffectWe use the following formula to calculate vibrational energy of the solids in question:E = nhv where n = 1, 2, 3, …

n is a quantum number, must be a whole number, and of which the possible energies are limited to certain values.This whole idea bothered Plank and he tried to eliminate it from his work.

Einstein extended the work and reasoned that as the vibrating atoms changed energy, from say 3h to 2h, they would decrease in energy which would be emitted as a quantum of light that had the energy of h.

The Photoelectric EffectEinstein’s hypothesis was that light consists of quanta, or particles of electromagnetic energy that have the energy, E, proportional to the observed frequency of light.E = hvThese quanta we now call ph0tons.

Einstein used this concept to explain the photoelectric effect—the ejection of electrons from the surface of a metal when light shines on it.

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The Photoelectric EffectThese electrons are only ejected when the frequency of light exceeds a certain threshold (characteristic of a certain metal).For example: violet light will cause potassium metal to emit electrons, red light, however will not.

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The Photoelectric EffectWhen a particle of light has enough energy to remove an electron from a metal, that photon will be absorbed and will cease to exist.

This wave and particle picture of light is referred to as wave-particle duality, E = h. Where E is the energy of the photon is the frequency of the associated waveh is Plank’s constant.

Ernest RutherfordRutherford hypothesized that the atom consisted of a central nucleus with most of the mass (p+ and n0), and was surrounded by electrons which made the atom neutral. This could not be possible due to the disagreement with the current theory.

Atomic ModelsIn John Dalton’s time (early 1800’s) until the early 1900’s, the atom was believed to be a solid, indivisible mass.

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Atomic ModelsThis didn’t make sense to J.J. Thompson after he discovered the electron, so he developed the “plum pudding” model explaining the electrons of the atom to be similar to a lump of dough with raisins stuck into it.

Atomic ModelsErnest Rutherford discovered, with his gold foil experiment, the nucleus of the atom. With this discovery, he found that the majority of the mass of the atom was confined to a small region in the center of the atom around which the electrons were located.

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Atomic ModelsAfter Rutherford’s discovery, Niels Bohr began experimenting and arrived at a new model. This model depicted the electrons whirling around the nucleus in much the same was that planets revolve around the sun.

Each electron existed in an energy level and could be moved to a higher or lower energy level depending on whether it was gaining or losing energy.

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Atomic ModelsFinally, Erwin Schrödinger and other physicists have taken the atomic models one step further and developed the quantum mechanical model of the atom.

This model is highly mathematical and assigns electrons to various orbitals within a principle energy level.

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Neils BohrNeils Bohr applied a new theory to hydrogen. He did so using the spectral line pattern of hydrogen.

Spectral line patterns are characteristic of each element and show certain colors of a given wavelength of light.

These colors can be discerned with a prism.

In 1885, J.J. Balmer showed that the wavelength of light could be reproduced using the following equation:1/ = 1.097 x 107/m(1/22 – 1/n2) here, n is a whole number greater than 1.

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Atomic Model Summary

Bohr’s PostulatesBohr’s postulates, which account for the stability of the hydrogen atom and the line spectrum:1. Energy-level Postulate—an electron can only have specific energy values in an atom called energy levels.

Bohr devised a rule that could only be applied to the motion of an electron in an atom.

From this, Bohr derived the following formula to explain the energy levels of the electron in the hydrogen atom.E = -RH/n2 n = 1,2,3, … ∞ and is the principle quantum number; RH is a constant with a value of 2.179 x 10-18J

Bohr’s Postulates2. Tranisitions between energy levels. An electron in an atom can change energy only by going from one energy level to another.When this happens, an electron undergoes a “transition.”

With these 2 postulates, Bohr explained Balmer’s formula for the wavelength in the spectrum of the hydrogen atom.

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Bohr’s PostulatesBohr was also able to apply his formula to explain the energy lost by an electron that had been excited in the hydrogen atom when they go from some initial value to some final value.

E = Ef – Ei

Ei = -RH/ni2; Ef = -RH/nf

2

E = -(RH/nf2) – (-RH/ni

2) = -RH(1/nf2 – 1/ni

2)

When an electron goes from ni = 4 to nf = 2E = -2.179 x 10-18J (1/22 – 1/42) = -4.086 x 10-19J

∆E = -4.086 x 10-19JThe negative sign indicates energy is given off.

In general, the energy emitted by a photon, h, equals the positive energy lost by the atom.

Energy emitted by a photon, h = -E = -(Ef – Ei)

That is, h = RH (1/nf2 – 1/ni

2)Recall, = c/

1/ = RH/hc(1/nf2 – 1/ni

2)RH = 2.179 x 10-18 Jh = 6.626 x 10-34 J• sc = 2.998 x 108 m/s

Bohr’s PostulatesRH/hc = 1.097 x 107/m = the constant given in Balmer’s formula.According to Bohr, the emission of light from an atom occurs when an electron undergoes a transition from an upper energy level to a lower one.

To get to the upper energy level, an electron must somehow get excited, that is, gain energy.

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Bohr’s PostulatesObjects appear colored because they absorb and emit light. The absorption of light exicites electrons, and when they travel back to their transition, state, they emit light.

Bohr’s model was good for establishing energy levels within atoms, but it was not good for explaining the atomic details of other atoms—as any good model should.

The principles of quantum mechanics gave rise to our current understanding of the atom.

Light’s PropertiesRecall that Einstein stated light has both wave-like and particle-like behavior.

So, not only can we characterize light by its frequency and wavelength, we can also show that light has momentum.

The energy of light, E = hMomentum is the product of mass and velocity (the velocity of light, in this case), mc. So the relationship is mc = h/ or = h/mc

Light’s PropertiesIn 1923, Louis de Broglie reasoned that if light exhibits particle aspects, then perhaps the particles of matter show characteristics of waves under the proper circumstances.

Thus, if a particle of matter of mass m and velocity v has an associated wavelength, by analogy with light. = h/mv The de Broglie relation.

We don’t observe this in “regular” sized matter because it is too big and the is too small.For example, a baseball, 0.145kg, 60mi/hr, = 10-34mFor an electron, = 10-12m which is much easier to observe.

Ernst RuskaThese wave properties of electrons gave Ernst Ruska the tools he needed to construct the first electron microscope in 1933.

This incredibly small allows for high resolving power. In other words, we can see a tremendous amount of detail with a small resolving power.RP = /2NA

The numerical aperature, NA, is the condensing and collecting ability of the device being used.

Louis de BroglieThe de Broglie equation applies quantitatively only to particles in a force free environment and cannot be directly applied to electrons in an atom due to the attractions in the nucleus.

In 1926, Erwin Schrödinger built upon de Broglie’s work and came up with a theory to explain the wave properties of electrons in atoms and molecules. This mathematical explanation of the wave properties of subatomic particles is called quantum mechanics.

The Bohr Model vs. The Quantum Mechanical ModelIn Bohr’s model of the atom, we thought of the electons moving around the nucleus like planets around the sun.

Quantum mechanics is different. There is no precise orbit of the electrons in the atom. Moreover, we can’t say where the electron is around the atoms nucleus and how fast it is moving, nor what its momentum is—only any one of these three independently of the others.

The Heisenberg Uncertainty PrincipleThis led to the Heisenberg uncertainty principle, which states that the product of the uncertainty in position and uncertainty in momentum of a particle can be no smaller than Plank’s constant divided by 4.

In other words, the uncertainty in the x coordinate x, and the uncertainty in momentum of the particle on the x coordinate, x, gives us:

(x)(x) ≥ h/4m

The Heisenberg Uncertainty PrincipleSimilar things can be said of the y and z directions. In other words, the uncertainty principle says the more precisely you know the position, the less well you know the momentum.

This principle is only significant for particles of vanishingly small mass—electrons for instance.

Taking into account = mv, the above relation becomes:

(x)(vx) ≥ h/4m

The Heisenberg Uncertainty Principle(x)(vx) ≥ h/4mWhen m is large—baseballs—the term on the right becomes nearly zero—hence the uncertainty is small—we know where the object is and how fast it is moving. For small objects—electrons, the uncertainty becomes huge. Thus, the electron’s orbit is “unpredictable.”

Quantum MechanicsQuantum mechanics gives us the ability to statistically determine where the electron is likely to be—but not its exact spot.

The information about a particle’s given energy level is contained in the mathematical expression called a wave function—.

Squaring the solution to the Schrödinger equation (2) gives us the probability of finding the particle within a given region of space—again, not the exact spot.

Quantum MechanicsThe value of 2 is high near the nucleus, and smaller the further away we get.

It’s never zero, however, and never has a fixed boundary as suggested with the Bohr model.

With quantum mechanics, each electron is described by 4 different quantum numbers—3 of which specify the wave function that gives the probability of finding an electron at various points in space (n, l, ml).

Quantum MechanicsA wave function for an electron in an atom is called an atomic orbital.

The quantitative picture of an orbital describes the region of space where there is a high probability of finding electrons.

The 4th quantum number, ms, refers to the spin of the electrons.

Quantum Numbers1. n—the number upon which the energy of the electron depends. It has any positive value.

The larger n, the larger the size of the orbital.

This is the number in front of the s, p, etc., for instance 1s2.

Letter K L M N

n 1 2 3 4

Quantum Numbers2. Angular momentum quantum number. This number distingushes orbitals of given n having different shapes.

It has any value from 0 to n-1.For example, when n = 3, possible l values are o, 1, 2. In other words, when n = 3 there are 3 kinds of orbitals, each having a different shape for the region where the electron is most likely to be found.

Quantum NumbersOrbitals of the same n, but different l, belong to different subshells.

Thus, 2p means n = 2, l = 1.

Letter

s p d f g …

l 0 1 2 3 4 …

Magnetic Quantum Number, ml3. The magnetic quantum number, ml, distinguishes orbitals of given n and l. It can have values from –l to +l.

When l = 0 (s subshell), ml = 0 (there is only one orbital in the s subshell).

When l = 1 (p subshell) ml = -1, 0, +1—there are 3 orbitals in the p subshell.

Spin Quantum NumberThe spin quantum number, ms, refers to the 2 possible orientations of the spin axis of an electron. Either +½ or -½.

Orbital Shapes: s orbitalsThe s-orbital is spherical in shape.

The color shading shows the probability distribution of finding an electron.

The 1s orbital has one sphere.

The 2s orbital consists of 2 spheres.

99% contour.

Orbital Shapes: s orbitals

Animations Courtesy of Oleg Alexandrov

Orbital Shapes: p orbitalsThere are three p orbitals in each p subshell.

Each is lobe-shaped along a particular axis and they run perpendicular to each other.

Orbital Shapes: p orbitals

Animations Courtesy of Oleg Alexandrov

Orbital Shapes: d orbitalsThere are 5 d orbitals.They have complicated shapes which exist along the x, y, and z axes.

Orbital Shapes: d orbitals

Animations Courtesy of Oleg Alexandrov