Chapter 8 The Quantum Mechanical Matter of the Atom (2)

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Chapter 8
The Quantum Mechanical Model of the Atom
The Wave Nature of Light 
Electromagnetic radiation carries energy = radiant energy
	some forms are visible light, x rays, and radio waves
Wavelength ( λ) is the distance between any 2 adjacent identical points of a wave
Frequency ( ν) is the number of wavelengths of that wave that pass a fixed point in one unit of time
 
					
c = ν λ
where the speed of light = c = 3.00 x 108 m/s
Example:
What is the wavelength of the yellow light given off by a sodium vapor lamp used for public lighting has a wavelength of 589 nm. What is the frequency of this radiation?
What is the wavelength of the violet light, which has a frequency of 7.35 x 1014s-1?
Electromagnetic spectrum - the range of wavelengths of electromagnetic radiation.
 
Max Plank (1900) described the intensity of light of various frequencies. 
He determined that an atom could have only certain energies of vibration, E, 
				E = n h ν n = 1, 2, 3, ….
where h = Plank's constant = 6.63 x 10-34 J.s
 n = quantum numbers
Planck believed that energy can be released or absorbed by atoms only in "chunks" of some minimum size called quantum (fixed amount). These chunks are emitted or absorbed in whole number multiples of hν, 2hν, 3hν …
Albert Einstein (1905) used Planck's theory to explain photoelectric effect. He postulated that light consists of 'quanta" or particles of electromagnetic energy, with E proportional to the observed frequency of the light - now known as a photon. 
					E = hν
Photoelectric effect is the ejection of electrons from the surface of a metal or from another material when light shines on it.
Calculating the Energy of a Photon
The red spectral line of lithium occurs at 671 nm. Calculate the energy of one photon of this light.
Niels Bohr (1920s) applied a new theory to the simplest atom, hydrogen, using Planck's and Einstein's theories and J.J. Balmer's equation. The visible spectrum of hydrogen could be reproduced by a simple formula:
				 = (1.097 x 107/m) ( 2- 2)
Bohr's postulates to account for the stability of the hydrogen atom and the line spectrum of the atom were:
1. Energy level Postulate - an electron can have only specific energy values in an atom which are called energy levels.
				E = - 	
where RH is an energy constant = 2.179 x 10 -18J and n is an integer and is called the principle quantum number		
2. Transitions between energy levels - an electron can change energy levels by going from one energy level to the another energy level.
				∆E = Ef - Ei
				hν = -∆E = - (Ef - Ei)
If ∆E is + when nf > ni which means radiant energy is absorbed
If ∆E is - when nf < ni which means radiant energy is emitted
Calculate the wavelength that corresponds to the transition of the electron from n=4 to n=2 state of the hydrogen atom. Is the light absorbed or emitted by the atom?
Quantum Mechanics, or wave mechanics, is a branch of physics that mathematically describes the wave properties of submicroscopic particles.
de Broglie (1923) considered that if radiant energy behaved like a stream of particles, could matter?
Electrons in an orbit could be thought of as a wavelength. Given these particles had a mass, m, and a speed of ʋ.
					λ = h / mʋ
Calculate the wavelength of an electron with velocity of 5.9 x 106 m/s . The mass of electron is 9.11 x 10-28 g 
Heisenberg (1927) showed from quantum mechanics that it is impossible to know simultaneously both the position and the momentum of a particle such as an electron.
Uncertainty Principle is a relationship that states that the product of the uncertainty in position and the uncertainty in the momentum of a particle can be no smaller than Planck's constant divided by 4π.
				(∆x) (∆px) h / 4 π
According to quantum mechanics, each electron in an atom is described by four different quantum numbers, three of which (n, l and ml) specify the wave function that gives the probability of finding the electron at various points in space.
A wave function for an electron in an atom is called an atomic orbital.
Each orbital describes a specific distribution of electron density in space.
1. Principal quantum Number (n)
 this is the one on which the energy of an electron in an atom principally depends; it can have any positive value
2. Angular Momentum Quantum Number (l) 
	this one distinguishes orbitals of a given n having different shapes; it can have any integer value from 0 to n-1
3. Magnetic Quantum Number (ml)
	this one distinguishes orbitals of given n and l - of given energy and shape but having a different orientation in space; the allowed values are the integers -l to +l
4. Spin Quantum Number (ms)
	this one refers to the two possible orientations on the spin axis of an electron; possible values are +1/2 and -1/2 
Quantum Mechanics and Atomic Orbitals
Erwin Schrodinger, 1926, proposed an equation that incorporated both the wave-like behavior of the electron and its particle –like behavior , known as a wave equation. 
Wave functions emphasized orbital energies. It also provided information about an electron’s probable location in space. 
			 
Recall that the l quantum number for the s orbital is 0; therefore the ml quantum number must be 0. For each value of n there is only one s orbital.
	
Radial probability function – the probability that we will find the electron at a specific distance from the nucleus. 
Comparing 1s, 2s and 3s orbitals reveals 3 trends:
1. The number of peaks increases with increasing n, with the outermost peak being larger than the inner ones.
2. The number of nodes increases with increasing n.
3. The electron density becomes more spread out with increasing n.
The distribution of electron density in the p orbitals is not distributed spherically, instead the density is concentrated in 2 regions on either side of the nucleus separated by a node at the nucleus. 
	 
Where n= 2 shell
Each shell has 3 p orbitals. Quantum number l = 1, therefore the magnetic quantum numbers, ml can have three possible values - -1, 0, and +1.
The d and f orbitals begin when n= 3 or greater where l = 2. There are 5 3d orbitals due to the 5 possible ml value : -2, -1, 0, +1 and +2. 
		
In a many electron atom , for a given value of n, the energy of an orbital increases with increasing value of l. All orbitals of a given subshell have the same energy as one another. Orbitals with the same energy are said to be degenerate.
Studying the line spectra of many electron atoms showed what they originally thought were single lines were actually closely paired lines. Therefore there were twice as many energy levels than there were supposed to be. 
Uhlenbeck and Goudsmit, 1925, postulated that electrons have an intrinsic property, called electron spin, that cause each electron to behave as if it were a tiny sphere spinning on its own axis that is quantized. 
Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers n, l, ml and ms.
Therefore we can conclude that an orbital can hold two electrons.