CHEM 1411 Concept Reviews: The Quantum Mechanical Model of the Atom

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Electromagnetic Radiation: radiation consisting of self-sustaining oscillating electric and magnetic fields at right angles to each other and to the direction of propagation. It does not require a supporting medium and travels through empty space at the speed of light. Visible light, radio waves, and X-rays are some examples.

Wavelength: The distance between adjacent peaks (or troughs) in any regularly repeating wave pattern.

Frequency: The number of complete wavelengths, or cycles, that pass a given point each second.

Amplitude: The vertical height of the crest (or depth of the trough) of a wave. The amplitude of a wave represents the power of the wave. For electromagnetic waves, this means the greater the amplitude, the greater the intensity or “brightness” of the light will be.

For electromagnetic radiation, the relationship between frequency and wavelength is given by the equation:

$LaTeX: c=\lambda\nu$ $c=\lambda\nu$

where ν is the frequency (in s^-1), λ is the wavelength (in meters), and c is the speed of light ( $LaTeX: 2.998\times10^8\:m/s$ $2.998\times10^8\:m/s$ )

Interference: The effect experienced when 2 or more waves from different sources interact with one another.

Constructive Interference: Interference where waves interact in phase (waves align with overlapping crests), resulting in increased amplitude (power)

Destructive Interference: Interference where waves interact out of phase (the crests of waves from one source overlap with the troughs of waves from another source), causing a partial or complete cancellation of the wave.

Quantum Theory

Quantum Theory: This theory states that atoms can only release or absorb energy in discrete “chunks” of some minimum size, which have been named quanta (quantum is the singular). Thus, electrons in atoms can only have specific and incremental energies. Since electromagnetic energy has been shown to behave as a particle as well as a wave, a quantum of this energy has been given the name of photon. The energy of a photon is directly proportional to its frequency and is given by the following equation:

$LaTeX: E=h\nu$ $E=h\nu$

where E is the energy of the photon (in J), h is Planck’s constant ( $LaTeX: 6.626\times10^{-34}\:J\cdot s$ $6.626\times10^{-34}\:J\cdot s$ ), and ν is the frequency (in s^-1).

The following equation which is a derived form of the Rydberg equation gives the relationship between the initial and final energy states in an electron of a hydrogen atom and the overall change in its energy:

$LaTeX: \Delta E=\left(-2.18\times10^{-18}\:J\right)\left(\frac{1}{n_{f\:}^2}-\frac{1}{n_{i\:}^2}\right)$ $\Delta E=\left(-2.18\times10^{-18}\:J\right)\left(\frac{1}{n_{f\:}^2}-\frac{1}{n_{i\:}^2}\right)$

where ∆E is the change in energy and n_f and n_i are the final and initial energy states (i.e.-levels) of the electron respectively.

If a photon was absorbed, $LaTeX: \Delta E_{electron}=E_{photon}=h\nu$ $\Delta E_{electron}=E_{photon}=h\nu$ : The wavelengths of the photons absorbed can be determined with an absorption spectrum. Absorption spectroscopy is an indispensable tool for identifying compounds.

If a photon was emitted, $LaTeX: \Delta E_{electron}=-E_{photon}=-h\nu$ $\Delta E_{electron}=-E_{photon}=-h\nu$ : The wavelengths of the photons emitted can be seen on an emission spectrum. The emission spectrum for each element is unique and can be used to identify their presence.

Photoelectric Effect: the phenomenon of metals ejecting electrons when light of sufficient energy is shined upon them.

Binding energy (φ): the energy with which an electron is bound to a metal.

For an electron to be ejected from a metal, the photon absorbed must have an energy greater than or equal to the binding energy ( $LaTeX: h\nu\ge\phi$ $h\nu\ge\phi$ ). The kinetic energy of the ejected electron is given by the following equation:

$LaTeX: KE=h\nu-\phi$ $KE=h\nu-\phi$

where “hν” is the energy of the photon absorbed and “φ” is the binding energy of the electron.

Matter Waves: Just as electromagnetic radiation has both wave and particle characteristics, matter has also been shown to have both wave and particle characteristics. The wavelength of an object can be solved for using the DeBroglie matter wave equation shown below:

$LaTeX: \lambda=\frac{h}{m\nu}$ $\lambda=\frac{h}{m\nu}$

where λ is the wavelength (in meters) of the matter, h is Planck’s constant ( $LaTeX: 6.626\times10^{-34}\:J\cdot s$ $6.626\times10^{-34}\:J\cdot s$ ), m is the mass (in kg), and v is the velocity (in m/s).

Heisenberg’s Uncertainty Principle: States that it is impossible to accurately determine both the position and velocity of an electron at the same time. The more accurately one is determined, the less accurate the other measurement will be.

In equation form, this principle is stated as follows:

$LaTeX: \Delta x\times m\Delta v\ge\frac{h}{4\pi}$ $\Delta x\times m\Delta v\ge\frac{h}{4\pi}$

where Δx is the uncertainty of the position (in meters), m is the mass (in kg), and Δv is the uncertainty of the velocity (in m/s)

Wave Function (ψ): a mathematical function that describes the wave nature of a given electron. A three dimensional plot of the wave function squared (ψ²) represents an orbital.

Orbital: A probability distribution map showing where an electron of a distinct energy is likely to be found. An orbital represents a designated 3-D area around the nucleus where up to 2 electrons can reside.

Probability Density: the probability (per unit volume) of finding an electron at a point of space. In other words, it is how likely you are to find an electron in a given area around the nucleus.

Radial Probability: The probability that an electron will be found at a given distance from the nucleus.

Total Radial Probability: the probability that an electron will be found within a radius of r from the nucleus.

Node: A point at which the probability of finding an electron is zero.

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Phase: The sign of the amplitude, positive or negative, of a wave. The phase determines how a wave will interfere with other waves. Three dimensional waves like orbitals also have phases and interfere with each other.