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Quantum Numbers and Atomic Orbitals

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Quantum Numbers and Atomic Orbitals

We have discussed the duality of wave and particle in some length, and come to the conclusion that particles and waves are properties of the same identity. Quantum mechanics treats electrons as waves. Wave properties more than the particle properties of electrons.

Materials occupy spaces and have masses and their existence is very easy to perceive. Today, we take it for granted that materials are composed of molecules and atoms. Atoms are composed of subatomic particles such as electrons and nuclei. We used to think particles as hard solids, with a definite shapes, similar to sand or dust. Waves, on the other hand, are always on the move. They come and go, have no particular shape, and cannot be held by hand. Waves and particles are so different, that we find it difficult to accept the idea of treating particles as waves. Thus, adjustment must be made to perceive particles as waves in order to appreciate methods and results of quantum mechanics.

In order to treat particles as waves, you have to be familiar with wave motion. Some of the wave properties have been mentioned in the section on electromagnetic waves. However, in order to appreciate quantum numbers and energy states, we shall take another look at moving waves and standing waves.

Moving Waves

If you hold the end of a long rope on your hand and move the hand up and down once, you will see a wave moving away from you along the rope transmitting the disturbance or energy along the rope in the form of a wave. Every time you make an oscillation move, a wave propagates along the rope. A repetitive up-and-down motion causes a series of waves moving away. Due to friction encountered by the rope in its motion, the magnitudes of these waves diminish, and eventually disappear. If there is no friction or energy loss, the magnitudes of waves will not diminish, due to the principle of conservation of energy. Waves along a rope demonstrates the motion of transverse waves.

For example, when light is transmitted in a glass fibre, energy is not lost due to complete reflection. Thus, the light carries the message for a long distance.

As another example, transimitting a sound in a tube limits the spread of energy only along the tube, and there is less loss of energy as the wave propagates. Thus, a sound also goes much further in a tube as compared to open air.

Waves in a Confined Space

Violin, guitar, and piano strings are for making waves of different frequencies or wavelengths. These are determined by the tension, thickness, as well as by the lengths between fixed points. Waves on these strings are standing waves in that they last until all the energy is lost due to friction and air resistance. For a standing wave, the longest wavelength is twice the length of the string between fixed points. Of course, a wave with wavelength as long as the string can also be a standing wave. Actually, standing waves exist for the string as long as the string lengths L are multiples (n) of half of the wavelength (wl / 2).

L = n (wl /2)

Waves whose wavelengths do not meet these criteria cannot exist as standing waves.

The two fixed points are called boundary conditions. The boundary limits the waves in a region, and the waves are standing waves. The standing waves are restricted to certain wavelengths. This type of condition is analogous to the condition of quantization, and the number (n) of half-wave lengths is analogous to the quantum number. (for a one-dimensional space in this case)

A standing sound wave in a box is the so called resonance phenomena. The vibration of the wave continues until the energy is lost due to air resistance and friction. Waves whose wavelengths fit in the cavity exist as standing waves, and these wavelengths depend on the size and shape of the cavity. These are three-dimensional waves, and three quantum numbers are required to characterize these waves. When the shape of the cavity is not cubic or rectangular, the waves can be more complicated.

By way of analogy, we have just introduced the concept of quantum numbers.

Particles as Waves and Vice Versa

Max Planck's photon is a particle property of electromagnetic waves, and Einstein's photoelectric experiment confirmed the validity of these particle properties.

Do particles have wave properties? This has been investigated by de Broglie, and he has also derived a formula for the calculation of wavelength wl of a moving particle whose linear momentum being p,

wl = h / p

where h is the Planck's constant. Moving particles do have wave properties. What about particles confined in some space? Can they be considered as standing waves? Well, they have been treated as waves, and the results can be applied to interpret many natural phenomena.

Electrons in an Atom

Electrons under the influence of positive charges of atomic nuclei are confined to the neighborhood of the atomic nuclei. The motion of these electrons have been treated as waves. In other words, equations that describe the wave motions have been applied to represent the electrons. Solutions of these equations lead to wavefunctions. Due to the confinement of the electron in the neighborhood of the atomic nuclei, these solutions are associated with some integers called quantum numbers.

For the descriptions of electrons in an atom, we rely on quantum numbers and wavefunctions. Wavefunctions are used to describe the shape of the electron cloud in chemical bonding later. In the next section, the significance of quantum numbers is described.

The Quantum Number Rules

The solution of the 3-dimensional wave equation leads to three quantum numbers: n, l and m.

The principle quantum number n most affects the energy of the state of the electron. For hydrogen-like atoms, the energy is related to n by the equation:

        - 2 p2 m Z2 e4
E = ----------------
            h2     n2
where p2 = 3.1416, m is the mass of the electron, Z is the atomic number or number of protons in the nucleus, e is the charge of the electron, h is the Planck's constant, and n is the principle quantum number.

The angular momentum quantum number l is related to the directional property of the wavefunction. For a spherical wave, l = 0, the number of nodal planes is equal to l.

The angular momentum is further quantized by the quantum number -m based on the orientation of the angular momentum. Since the orientation is related to l, m has values ranging from -l, -(l-1), ... 0, ..., l-1, and l. A short summary of the rules is given below.

The three quantum numbers (q.n.) n, l, m follow the rules:


 for the principal q.n.  n
         azimuthal q.n.  l = 0, 1, 2, ..., n-1
          magnetic q.n.  m = -l, -(l-1), ..., (l-1), l
Rules are algorithms, by which we generate possible quantum numbers. The lowest value of n is 1 (NOT zero). For n = 1, the only possible value for quantum number l is 0, and m = 0. Each set of quantum numbers is called a state. Thus, for n = 1, there is only one state (1,0,0). The states are represented by symbols, and special symbols have been used to represent the quantum number l as follows:

           l = 0, 1, 2, 3, 4, ...
      symbol = s, p, d, f, g, ...
Using symbols, the valid quantum states can be listed in the following manner:

1s
2s 2p
3s 3p 3d
4s 4p 4d 4f
5s 5p 5d 5f 5g
6s 6p 6d 6f 6g 7h
7s 7p 7d 7f 7g 7h 8i

For hydrogen-like atoms, that is atoms or ions with one electron, the energy level is solely determined by the principle quantum number n. and the energy levels of the subshells np and nd etc. are the same as the ns. For these species, the energy levels have the reverse order as the list given earlier:


 = 7s 7p--- 7d----- ....  ]  These are very close together!
 = 6s 6p--- 6d----- ....  ]
 = 5s 5p--- 5d----- 5f------- 5g---------
 - 4s 4p--- 4d----- 4f-------
- 3s 3p--- 3d-----
- 2s 2p---
- 1s

Confidence Building Questions

  1. How many possible orbitals are there if n = 3?

    Discussion:
    3s, 3 3p, 5 3d; total = 1 + 3 + 5 = ?

  2. How many possible orbitals are there in the subshell [n=5, l=4]?

    Discussion:
    For 5g, l = 4; ml = -4, -3, -2, -1, 0, 1, 2, 3, 4. A total of 9.

  3. How many electrons can be accommodated in the subshell 4f?

    Discussion:
    Each of the 7 4f orbitals accommodates a pair of electrons. There are 14 elements in the lanthanides, the 4f block elements.

  4. How many atomic orbitals are there for the subshell with [n = 3, l = 2]?

    Discussion:
    The magnetic q.n. = -2, 1, 0, 1, 2 The number of orbitals associated with a given value of l is (2l + 1).

  5. What is the symbol representing the set of orbitals in the subshell with [n = 3, l = 2]?

    Discussion:
    For symbols, consider the following

    Symbol = s  p  d  f  g  correspond to
         l = 0  1  2  3  4
    
    The d-block in each period of the periodic table has 10 elements.

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