Quantum Numbers In Chemistry Philosophy Essay

From early models of the atom such as Rutherford’s nuclear atom proposed in 1911[1] to the improved model Bohr put forth in 1913, applying recently developed quantum physics to introduce the idea of quantised energy levels with use of a quantum number. To the current accepted theory of atomic structure, the quantum atomic model stemming from the development of quantum mechanics. Our understanding of atomic structure today and our use of orbitals and four quantum numbers in explaining the arrangement and behaviour of electrons in atoms has been due to the work of many scientists building upon the work of others and linking once seemingly unrelated ideas that have led to scientific breakthrough, such as Rutherford’s discovery of the nucleus, or Bohrs explanation of the Balmer series.

Note; information according to reference number [#], see reference list on page 14

The changing atomic model, Rutherford and Bohr

Rutherford’s nuclear atom- the discovery of the nucleus

After conducting experiments in which alpha (α) particles were fired at a leaf of thin gold foil (beaten to around 400 atoms thick [18] ) in a vacuum, New Zealand born physicist, Ernest Rutherford, in 1911[12] proposed his theory of a nuclear atom. Upon bombarding the thin leaf of gold foil with alpha particles, Rutherford observed that most of the alpha particles travelled undeflected straight though the gold foil ( which he was able detect on a screen coated in zinc sulfide. As when an alpha particles struck the screen they would cause it to fluoresce at that point) However, Rutherford also observed that some of the alpha particles, around 1 in 8000 were deflected through large angles (i.e >90˚) as they passed the foil and some were even reflected. From these experimental findings, Rutherford concluded that there must be a dense, positively charged mass in the centre of the atoms , the nucleus, in which most the mass was contained. And it was this concentrated positively charged nuclei that were causing these large deflections and scattering of the alpha particles, by electrostatic repulsion. By finding the diameter of the atoms to be around 10-10m, he estimated the nucleus must be smaller than one hundredth the radius of the atom, so the nucleus must be less than 10-13m in diameter. Rutherford also reasoned that there must be enough negatively charged electrons to balance the charge of this positive nucleus and that they must move around the outside, in large orbits (maintained by electrostatic attraction with the nucleus) occupying otherwise empty space. This accounted for the vast majority of the alpha particles that passed undeflected through the foil. Later in 1919 Rutherford inferred that the nucleus contained positively charged protons, which possessed a relative charge of +1, and electrons a relative charge of -1 [18].

Figure 1: Apparatus used in the experiment on alpha- particle bombardment of gold foil.

Niels Bohr and the hydrogen atom

The inability of previous atomic models to successfully explain the constancy of the hydrogen spectral lines observed through experimentation, led Danish Physicist, Niels Bohr, in 1913 to propose a new model of the hydrogen atom that accurately predicted the hydrogen spectra. Bohr’s innovative model showed a clear link between the Balmer equation, Planck’s theory of quanta and Rutherford’s nuclear atom.

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Bohr’s model incorporated Planck’s ideas of quantum theory and contained electrons orbiting the central nucleus in discrete, quantized and stable energy levels, or shells which are represented by the principal quantum number (n), where n=1,2,3..etc, n ≠0 and can take any integer value. This overcame the major downfall of Rutherford’s model in which the orbiting electrons would emit electromagnetic radiation, thus loosing energy and consequently spiral into the nucleus.

1.2.1 Bohr’s postulates

When Bohr proposed his atomic theory, he set out certain postulates, these are as follows;

Electrons orbit the nucleus in stable, stationary states, i.e. while electrons occupy these stationary states no electromagnetic radiation is emitted.

Electrons could make the transition to a different stationary state, upon the absorption or emission of a photon with the correct energy constant, i.e. the difference in energy (∆E) between these stationary states [Ef (final energy state) & Ei (initial energy state)] corresponds to the energy of the photon (hf) that has been absorbed or emitted. This relationship if given by;

Figure 2: emission of photon as an electron returns to a lower stationary state, ‘the ground state, n=1’

Ef – Ei= hf = ∆H

The Balmer formula led Bohr to hypothesise that angular momentum was quantised, and that an electron in a stationary state has angular momentum (Ln=mvrn) of [3]. Where n is the principle quantum number, corresponding to the energy level or stationary state.

Explaining the Balmer Series

Through experimentation of the emission spectra of hydrogen, it was observed that in the visible spectrum (Balmer series) only certain colours were observed meaning that only certain photons were being emitted and indicating that only certain energy changes were occurring in the hydrogen atom. This also meant that there must be discrete, quantised energy levels in the hydrogen atoms for these discrete energy changes to occur. This experimental evidence was explained by Bohr by use of his postulates (specifically his 2nd postulate) which allowed him accurately, to predict the wavelengths of the spectral lines for the hydrogen atom, as electrons excited to high values of n would emit a photon of light as they returned back to lower stationary states (down to n=2), resulting in the observed Balmer series for hydrogen. By use of his postulates, Bohr was also able to derive Rydbergs equation thus providing a theoretical basis for Balmer’s empirical equation, originally proposed in 1885[3] by Swiss physics teacher J.J Balmer, and later modified by Janne Rydberg;

=R ( -)

Where R is Rydbergs constant, a value of 1.0974×10-7 m-1

Figure 3(above); Bohr’s model in explaining the Balmer series.

Figure 4(above): Balmer series emission spectra for hydrogen. Spectral line of λ 656nm (red) is named the Hα line, 486nm (green) is named the Hβ line, 434nm(blue) the Hγ line and 410nm(violet) the Hδ line.

The limitations of Bohrs model

Even though Bohr’s model was revolutionary at the time, accurately explaining the hydrogen spectra it had major downfalls that were soon realised. Bohr himself explained his model was mathematical and he believed that electrons did not move in such a manner as described in his model. The major limitations of Bohr’s model are as follows;

The model could not be applied to atoms that possessed more than one electron, i.e it could not explain the spectra of these larger atoms. It could only be correctly applied to the hydrogen atom, the helium ion (He+) and the lithium ion (Le2+), as all these species contained only one electron.

Bohr’s theory combined laws of classical and quantum physics, assuming that some laws of classical physics applied while others did not.

The model could not account for the why some spectral lines had a greater intensity than others, or why this was occurring.

With the advancement of spectroscopes that allowed spectral lines to be examined with greater detail, it was found that certain spectral lines previously thought to be a continuous, or solid line were actually made up of numerous hyperfine spectral lines. The cause of this observation could not be explained by Bohr’s model.

It was later observed, in 1896[13] by Dutch physicist Pieter Zeeman that when the sample being observed was placed in an external magnetic field, the spectral lines would split. This effect was fittingly termed the Zeeman Effect, and could not be explained by the Bohr model.

Modern atomic theory

2.1 Outline of modern atomic theory.

Our current theory of atomic structure, the wave mechanical, or quantum model has stemmed from the development of quantum mechanics and the work of numerous scientists such as Louis Victor de Broglie, Erwin Schrodinger, Wolfgang Pauli and Werner Heisenberg. The two dimensional orbits of Bohrs model have been replaced by complex three dimensional orbitals. The electron is now viewed not just as a particle, but as possessing a wave like nature that can be explained by the use of four quantum numbers, including the principal quantum number, introduced in Bohr’s model.

2.2 Introduction to quantum numbers.

Quantum numbers allow for the full description of the wave function, Ynlms of any electron in an atom. There are four quantum numbers; the principle quantum number (n), angular momentum quantum number (l), magnetic quantum number (ml ) and spin quantum number (ms ) which allow for this. The first three quantum numbers are the possible integral solutions to Schrödinger’s wave equation for hydrogen, and as these solutions are only possible with whole numbers, they are shown to be quantised. The fourth quantum number (ms), however, comes from the theory of relativity, and was shown to be quantised by the Stern-Gerlach experiment in 1922. Quantum numbers have allowed us to explain the different arrangements of electrons in atoms, showing us that they can occupy three dimensions (thus requiring 3 co-ordinates, the solutions to the Schrödinger equation). Each set of quantum numbers that describes an electrons wave function in a given atom is unique, in accordance with the Pauli Exclusion Principle, proposed in 1925 by Wolfgang Pauli which states this fact.

Principle quantum number (n)

The principle quantum number describes the electron shell and its distance from the nucleus. The principle quantum number was introduced by Bohr in his theory. It can take integer values of 1, 2, 3…etc, but cannot equal 0. Thus, by only taking these whole number values it expresses the quantisation of energy. As the value of n increases the distance of the shell from the nucleus increases, and hence the size of the shell is greater (since shells are concentric[4]). As the electrons are attracted to the positive nucleus and greater values of n correspond to a greater distance of electrons from the nucleus, electrons in the shells with a greater value of n possess greater energy, so n consequently describes the energy of the shell.

Angular momentum quantum number (l)

The angular momentum quantum number describes the subshell shape. l can take integer values of 0,1,2…n-1, i.e. when n=1, l=0 and when n=2, l has allowed values of 0 and 1. Thus like n, l is also quantised. The values of l corresponding to the subshells* are named and represented (particularly in chemistry) by letters, i.e. l=0 corresponds to the s orbital* (sharp), l=1corresponds to p orbital (principle), l=2 to d orbital (diffuse) and l=3 relates to f orbital (fundamental).

Principle quantum number (n)

Angular momentum (l)

Magnetic quantum number (ml)

1

0 (s)

0

2

0 (s)

0

1(p)

-1,0,1

3

0 (s)

0

1 (p)

-1, 0, 1

2 (d)

-2,-1,0, 1,2

4

0 (s)

0

1 (p)

-1,0,1

2 (d)

-2,-1,0,1,2

3 (f)

-3,-2,-1,0,1,2,3

Magnetic quantum number (ml)

The magnetic quantum number describes the orientation of the orbital* in space, i.e describes the orbitals in a subshell, so for every value of l there are corresponding values of ml. These range from -l…0… l, and the total number of values will equal 2l+1. For example, when l=2 (d) the allowed values of ml will be -2,-1, 0, 1, 2, i.e. there are five d orbitals.

Spin quantum number (ms)

The spin quantum number describes the intrinsic angular momentum of an electron in an orbital and is required by the theory of relativity. It was shown to take one of two discrete values, either – ½ or + ½ i.e ‘spin up’ or ‘spin down’, by the Stern-Gerlach experiment in 1922.

In the Stern-Gerlach experiment a beam of hot silver atoms was passed through an inhomogeneous magnetic field, directed at a photographic emulsion screen. As the beam passed the magnetic field it split in two, as the electrons from the silver atoms were deflected either up or down in the z-direction by a constant amount, thus providing experimental evidence to show the quantisation of spin in the z direction. This also showed that electrons produced magnetic fields as they spun on their axis in a clockwise or counter-clockwise direction, which was independent of their orbital motion.

Figure 5: The Stern-Gerlach experiment

*note; subshells and orbitals will be discussed further in following section ( 2.3)

2.3 Shells, subshells and orbitals

An electron shell is region around a nucleus where electrons with approximately the same energy spend most of their time, given by n. An electron shell in an atom is the summation of the orbitals in that atom that possess the same principle quantum number and the number of orbitals in a shell is given by the square of the value of n ( i.e # orbitals= n2). For example if n=2, there will be 4 orbitals within that 2nd shell. An electron shell is further divided into subshells or sublevels;

Shell (n)

Number of Subshells

Subshells (l)

Number of orbitals

1

1

1s

1

2

2

2s, 2p

4

3

3

3s, 3p, 3d

9

4

4

4s, 4p, 4d, 4f.

16

An orbital, sometimes called an ‘electron cloud’ or ‘probability map’ is a region of high probability of finding an electron in an atom with a specific amount of energy. As we cannot exactly known where an electron will be at any given time, it is only where an electron is most likely to be found. This is due to Heisenberg’s uncertainty principle which states that the energy and position of an electron cannot be known simultaneously. Electrons move about the nucleus in these orbitals and each orbital will only hold two electrons. The shape of these orbitals is dependent on the energy (influenced by value of n) and the angular momentum of the electron.

As the values of n, l and ml describe one orbital in an atom, the ms value will identify one of the electrons in that orbital as if one electron in the orbital has ‘spin up’ (+ ½), the other will be orientated ‘spin down’ (- ½ ).As for two electrons to occupy one orbital they must have opposite spins. This is essential in order to comply with Pauli’s Exclusion Principle.

Figure 6: showing relationships between shells, subshell, orbitals and how electrons of each shell are held. For example, the 18 electrons contained in shell 3 are distributed between 3 subshells (3s, 3p, 3d) and 9 orbitals. The number of electrons in shell is given by 2n2

2.3.1 Orbital shape and orientation.

Orbitals within the same subshell have the same shape, but will differ in their orientation (specified by the different values of ml they can take). For example, considering the 2nd shell (n=2), we have two subshells as l=0 and l=1 (2s and 2p), so the values of ml are 0 and -1, 0, 1 respectively. Thus, there is one s orbital in the 2s subshell and three p orbitals in the 2p subshell, 2px, 2py, and 2px, which are orientated at 90Ëš to each around the nucleus on the x-, y – and z- axis of the Cartesian coordinate system[14] .

Spherically shaped s orbitals (l=0) can only take one value of ml, thus there is one orientation in space of these orbitals. The relative size of these orbitals increases in successive shells (n), due to the increasing energy in these shells. Relatively in a multi-electron atom, in each shell, electrons that occupy these s orbitals have the lowest energy.

Figure 7: spherically shaped s orbitals have only one orientation.

Figure 8: the relative size of the s orbital, increases with each successive shell (n)

Dumbbell shaped, or polar p orbitals (l=1)can take three values of ml, and so have three different orientations in space around the nucleus (px, py, pz ).P orbitals are found from the 2nd shell (n=2) onwards.

Figure 9: showing the 3 different p orbitals, with probable electron density. (a) Px, (b) Pz, (c) Py

The d orbitals (l=2) have two zero density planes [4], and as ml = -2,-1, 0, 1, 2 there are five d orbitals. d orbitals are found from the 3rd shell (n=3) onwards.

Figure 10: showing the five different d orbital.

The seven f orbitals (l=3), ml=-3,-2,-1, 0, 1, 2, 3, have complex shapes in their orientations around the nucleus. f orbitals are found from the 4th shell (n=4) onwards.

Figure 11: Showing the seven different f orbitals.

Degenerate orbitals

Orbitals are classified as degenerate when they have the same energy. This is dependent upon both the orbitals size (due to principle quantum number, n) and shape (angular momentum quantum number, l). Therefore, the three 2p orbitals, 2px, 2py 2pz are said to be degenerate, as the orientation in space does not influence the energy of an orbital.

Filling order, electron configuration and orbital diagrams

Electron filling order

In the hydrogen atom the energies of the subshells in a given shell are the same. For example in an excited hydrogen atom the 2s and 2p subshells have the same energy and this is why the Bohr model, worked for the hydrogen atom. However, In atoms with more than one electron, within each shell, electrons in the s orbital have the lowest energy followed by the electrons in the p orbitals, d orbitals and f orbitals; i.e the relative energies of subshells for a given shell are; sFigure 12: the energy of subshells in a multi-electron atom

Electrons will fill orbitals and shells complying with the Aufbau Principle, which states that an electron will fill the lowest energy orbital available, before filling the next, i.e 2s will fill before 2p. How electrons fill orbitals is also influenced by;

Hund’s Rule which tells us an electron will be added to each degenerate orbital in a subshell before two electrons are added to any orbital in the subshell [5], i.e. all orbitals in a subshell will be half filled with one electron, before electrons pair up, completely filling the orbital with two electrons.

Pauli Exclusion Principle which tells that electrons which occupy the same orbital will have a different value of spin quantum number.

Combined these give us a set of conditions we can use to predict electron filling order.

Figure 13: electron filling order, showing overlaps in energy in subshells of different shells

As the energies of subshells begin to overlap with an increasing shell number, as seen in figure 13, electrons will start to fill the subshells of a larger shell, before filling all the subshells of the previous shell, adhering to the Aufbau Principle. This pattern can be observed in figures 13 and 14, and the relative energies and filling order is given below;

1s<2s<2p<3s<3p<4s<3d<4p<5s<4d<5p<6s<4f<5d...

Figure 14

3.2 Electron configuration & orbital diagrams

Electron configuration is a statement of how many electrons an atom has in each of its subshells. It indicates which subshells electrons occupy. Electron configuration is written in shorthand notation, with the subshell (nl) and the number of electrons as a superscript. Below is the electron configuration of the first five elements, to demonstrate.

Atomic number (z)

Element (symbol)

Electron configuration

1

H

1s1

2

He

1s2

3

Li

1s22s1

4

Be

1s22s2

5

B

1s22s22p1

Orbital diagrams or box diagrams are used to provide more information about electron configuration they provide information about how many electrons an atom has in each of its orbitals. Boxes are used to indicate an orbital, and arrows for electron spin orientation;

spin up spin down

Examples;

Hydrogen orbital diagram;

1s

Nitrogen orbital diagram;

1s 2s 2p

3 p orbitals

Conclusion; Our understanding of atomic structure today and our use of orbitals and quantum numbers, has stemmed from the amalgamation of the work of numerous scientists. Even though we cannot precisely know where an electron will be within an atom at any given time, hence our use of orbitals. It is by far more accurate, than assuming we can predict where an electron will be within an atom, as is previous models such as Bohrs.