AP : ELECTRONIC STRUCTURE OF ATOMS 1
————————————————————————————————————————————————–
THE ELECTRONIC STRUCTURE OF ATOMS
A. BACKGROUND TO THE DEVELOPMENT OF MODERN ATOMIC THEORY
THE WAVE BEHAVIOUR OF LIGHT
———————————————————————————————————————————————
Definition:An ELECTROMAGNETIC WAVE is an oscillating electric and magnetic field caused by a moving charged particle.
A CYCLE is one complete oscillation of a wave.
The WAVELENGTH (“lambda”) of a wave is the distance between two successive peaks or valleys.
The FREQUENCY (“nu”) of a wave is the number of cycles passing by a point in a given time.
The frequency and wavelength of a light wave are related by the equation
• = cwhere: c = the velocity of light in a vacuum = 2.997 924 58 x 108
and wavelength is usually expressed in “nanometres” (nm) [ 1 nm = 10–9 m ]
frequency is expressed in “Hertz” (Hz). [ 1 Hz = 1 s–1 = 1 ]
EXAMPLE:Calculate the frequency of red light having a wavelength of 680 nm.
= = x = 4.41 x 1014 Hz
EXAMPLE:Calculate the wavelength of a radio wave having a frequency of 610 kHz.
= = x = 492 m
EXERCISES:
1.Calculate the frequency of a gamma ray with a wavelength of 1.25 x 10–3 nm.
2.Calculate the wavelength of the microwaves in a household microwave oven having a frequency of 2450 MHz.
3.Calculate the frequency of ultraviolet light having a wavelength of 95.0 nm.
4.Calculate the wavelength of radio waves emitted by an FM radio station broadcasting at 97.5 MHz.
THE BIRTH OF THE QUANTUM
———————————————————————————————————————————————
In 1900, the German physicist Max Planck studied BLACK BODY RADIATION (the energy emitted by hot objects in thermal equilibrium with their environment) and proposed that matter could not absorb or emit an arbitrary quantity of energy. Instead, he said that energy is only transferred in WHOLE NUMBER multiples of a basic quantity of energy. This basic quantity of energy is a small energy packet called a “quantum” (plural is “quanta”) and the formula that governs the change in energy, ∆E, is
∆E = nh / (The Planck equation)where h = Planck's constant = 6.626 075 5 x 10–34 J•s
= the frequency of the radiation (in Hz)
and n = the number of quanta transferred
The above equation is normally used in the following form.
E = hDefinition:A PHOTON is a quantum of light energy.
(A photon is sometimes called a “particle” of light energy, but this is somewhat misleading since light has no mass and is not a “particle” in the conventional sense.)
The energy of the quantum or photon depends on the value of ; the greater the frequency, the greater the energy of the photon.
EXAMPLE:Calculate the energy of green light having a wavelength of 515 nm.
First, find the frequency of the light.
= = x = 5.825 x 1014 s–1
Next, find the energy corresponding to this frequency.
E = h = (6.63 x 10–34 J•s) x (5.825 x 1014 s–1) = 3.86 x 10–19 J (for a single photon)
The generally accepted wavelengths of pure colours in the visible region are given in the following table.
Colour / Wavelength (nm) / Colour / Wavelength (nm) / Colour / Wavelength (nm)violet / 390 – 455 / green / 492 – 577 / orange / 597 – 622
blue / 455 – 492 / yellow / 577 – 597 / red / 622 – 770
More excited an electron = higher frequency = smaller wavelength
Infrared (IR):770 nm – 3000 nm (near IR), 3000 nm – 30 m (middle IR), 30 m – 1 mm (far IR)
Ultraviolet (UV):390 nm – 200 nm (near UV), 200 nm – 40 nm (far UV)
EXERCISES:
5.Calculate the energy of a radio wave having a wavelength of 225 m.
6.Calculate the wavelength of a photon having an energy of 4.25 x 10–21 J.
7.Calculate the energy of an X-ray having a frequency of 1.38 x 1012 MHz.
8.Calculate the wavelength and frequency of a photon in the gamma ray region with an energy of
1.64 x 10–13J.
THE PHOTOELECTRIC EFFECT
———————————————————————————————————————————————
One consequence of Planck's equation was that the energy of radiation was only proportional to its frequency and NOT its intensity; that is, not on the total amount of radiation. This immediately helped Albert Einstein explain a puzzling phenomena called the PHOTOELECTRIC EFFECT.
If an alkali metal is bombarded with light energy, electrons are ejected from the metal atoms. However, no electrons are emitted if the radiation’s frequency is below a certain THRESHOLD FREQUENCY. The kinetic energy of ejected electrons depends only on the frequency of the incident radiation, not on the INTENSITY of the radiation.
Einstein explained the photoelectric effect as follows. Energy is transferred in packets of definite energy: a PHOTON. If the photon is of sufficient energy, it supplies the energy required to remove an electron.
Cs + h Cs+ + e–
The KE of an ejected electron is the energy in excess of the minimum energy required to eject an electron from an atom:
KE = h – wwhere w = the electron work function; that is, the energy required to eject an electron,
h = the energy of the photon.
Obviously, for an electron to be emitted with any kinetic energy requires h > w and the threshold (“minimum”) energy required to eject an electron is the electron work function for the electron.
EXERCISES:
9.The value of the electron work function for cesium (Cs) is 2.14 eV, where 1 eV (electron volt) has an energy of 1.602 177 x 10–19 J. If a photon of blue light with a wavelength of 462 nm strikes an atom of cesium, what is the kinetic energy of the electron ejected from the cesium?
10.Does green light with a wavelength of 560 nm possess sufficient energy to eject an electron with a work function value of 2.30 eV from an atom of potassium? Would green light with a wavelength of 510 nm eject such an electron?
ATOMIC SPECTRA (LINE SPECTRA)
———————————————————————————————————————————————
Light is absorbed or emitted by gaseous atoms only at distinct and specific frequencies.
The most noticeable thing observed about the line spectrum of HYDROGEN is that the lines appearing on the photographic plates are bunched together in a special way: the spacing between the lines gets smaller and smaller, as shown below.
Niels Bohr showed that the energies of the lines can be fitted to a mathematical expression of the form
E = = RHwhere:m = the mass of the electron
e = the charge on the electron
n1 = an integer (see the n–values on the diagram on the next page)
n2 = an integer (that is larger than n1)
andRH = the Rydberg constant = 2.179 874 1 x 10–18 J.
The fact that the light emitted by a hydrogen atom can be analyzed completely in terms of a simple mathematical formula was the key to unravelling the problem of what goes on inside the atom. By 1913 NielsBohr proposed a model to explain why the line spectrum of hydrogen looks like it does and why only certain energies (frequencies) are possible in the spectrum. Bohr stated that the electron in a hydrogen atom only exists in specific (“discrete”) energy states. These energy states are associated with specific circular orbits that the electron occupies around the atom. When an electron absorbs or emits a specific, exact amount of energy it instantaneously moves from one orbit to another; the farther the orbit from the nucleus of the atom, the greater the energy of the orbit. (See the diagram on the next page.)
The pattern of lines in the spectrum reflects the pattern existing in the energy levels, and results from energy level differences. (The numbers to the right of the SPECTRUM represent the energy levels involved in the “transition”.)
A major problem existed with Bohr’s initial version of his theory: electrons moving through a magnetic field lose energy to the field. Since the nucleus possesses a large magnetic field, the electrons had to violate some classical laws of physics to allow them to move in their orbits without quickly losing all their energy. Later work by Bohr and others suggested that an electron simply exists at a certain level of energy and the concept of a moving electron in an orbit within the atom was abandoned.
The following table gives several energy levels of hydrogen.
n value / Energy (eV) / n value / Energy (eV) / n value / Energy (eV) / n value / Energy (eV)1 / 0 / 3 / 12.093 954 / 5 / 13.061 470 / 7 / 13.328 031
2 / 10.204 274 / 4 / 12.755 342 / 6 / 13.227 762 / 8 / 13.393 109
A useful conversion factor: 1 ev (electron volt) = 1.602 177 x 10–19 J.
Definitions:An ENERGY LEVEL is the energy possessed by an electron within an atom.
The GROUND STATE is the lowest energy level an electron can possess within an atom; that is, the n = 1 level for hydrogen.
An EXCITED STATE is any state above the ground state.
A QUANTUM is the energy absorbed or emitted by an electron undergoing a transition between two energy levels.
The QUANTIZATION OF ENERGY associated with an atomic spectrum is the natural result of taking energy differences between discrete energy levels. Since only specific energy levels exist, only certain energy differences are possible.
The model of the atom that Bohr put forward had tremendous explaining power but also had a significant “catch”: the theory would not work for any atom other than hydrogen (or ions having only one electron). A great deal of brilliant research produced the modern quantum theory, which is based on probabilities. Nevertheless, much of Bohr's work still remains valid; later refinements to the picture are based on the terminology he adopted.
EXAMPLE:Find the colour of the light emitted when an electron goes from the n=3 to the n=2 level of hydrogen.
E = RH = 2.18 x 10–18 J x = 3.028 x 10–19 J
Converting this to a frequency: = = = 4.569 x 1014 s–1
and converting to a wavelength: = = = 6.56 x 10–7 m x
= 657 nm.
This wavelength is in the RED region of the visible spectrum.
EXERCISES:
11.Find the colour of light emitted when an electron goes from the n=7 to n=2 level of hydrogen.
12.In what region of the energy spectrum are following electronic transitions for hydrogen found?
(a)n=5 to n=1(b)n=6 to n=2(c)n=4 to n=3
13.Each series of lines (Lyman, etc.) in the hydrogen spectrum has a highest wavelength called the “series limit”, corresponding to n1= lowest value in series and n2= an infinitely large number. Calculate the series limit for (a) the Lyman series, and (b) the Balmer series.
B. THE ENERGY LEVEL DIAGRAM FOR HYDROGEN (A REVIEW)
We now jump to a simplified form of the modern quantum theory. Some of the key features of the theory state that
• electrons within an atom exist in “orbitals”, each of which has a distinct energy.
• there are various types of orbitals, each of which has a distinct shape.
Extensive experiments indicate that the complete set of energy levels for hydrogen is arranged as shown below. Each dash represents the energy possessed by a particular “orbital” in the atom. The discussion that follows is concerned with 4 distinct “types” of orbitals: s, p, d and f (there are more but we will ignore them).
Definitions:A SHELL is the set of all orbitals with the same n–value.
Example: The 3rd shell consists of the 3s, 3p and 3d orbitals.
A SUBSHELL is a set of orbitals of the same type.
Example:The set of five 3d–orbitals is a subshell.
A set of orbitals with the SAME energy is said to be a DEGENERATESET, and they are said to have DEGENERATEENERGIES.
Example:At the n = 3 level there are a total of 9 degenerate orbitals (one s–type, three p–type and 5 d–type)
C. THE SHAPES OF ELECTRON ORBITALS
Definition:The ELECTRON DENSITY in an orbital is the probability of finding an electron at a given point in the orbital.
s-Orbitals
The shape of each type of orbital differs from the shapes of other types. Plotting the radial behaviour (outward from the origin) versus angular behaviour (3–dimensional rotation about the origin) of the 1s electron densities gives the orbital shape shown below.
A cross–section in the yz–plane shows the following electron density behaviour.
Similarly, a cross–section through the yz–plane of the 2s orbital shows the following electron density. (It looks similar to the above diagram because the electron density is the same in every direction.)
Because an orbital is described by a mathematical probability function having no definite boundaries (it tapers off rapidly, but never becomes zero), an orbital is frequently represented by a SURFACE enclosing 90% of the total electron probability.
This boundary–surface convention is used to show the p–orbitals and d–orbitals that follow.
2p-Orbitals
Each of the “carrot” shapes is called a “LOBE”. The “x” in “2px” indicates that the orbital lobes lie along the x–axis.
3p-Orbitals
3d-Orbitals
D. THE ENERGY LEVEL DIAGRAM FOR POLYELECTRONIC ATOMS (A REVIEW)
As mentioned previously, the initial Bohr model of the electrons in an atom only worked for hydrogen (or ions having a single electron). The energy level diagram is modified for atoms having more than one electron. Fortunately, the modified diagram below can be used for ALL polyelectronic atoms.
The principal difference between the energy level arrangements for a single–electron system and a many– electron system is a general “up–sweep” of the p, d and f–orbitals to higher energies in many–electron systems. This causes the 3d energies, for example, to get mixed between the 4s and 4p energies, the 4d energies to get mixed between the 4s and 5p energies, and the 4f and 5d energies to go between the 6s and 6p energies.
Note:In order to simplify the diagram below, only one of each of the p, d and f–orbitals is shown and the energy gaps between levels is not shown to scale.
SIMPLIFIED ENERGY LEVEL DIAGRAMS
HYDROGEN ALL OTHER ATOMS
E. ELECTRON CONFIGURATIONS (A REVIEW)
The addition of electrons to atomic orbitals follows three simple rules.
(a) The Aufbau ("Building Up") Principle: As the atomic number increases, electrons are added to the available orbitals. To ensure the LOWEST POSSIBLE ENERGY for the atom, electrons are added to the lowest–energy available orbitals.
(b) The Pauli Exclusion Principal: Each orbital can have a maximum of 2 electrons.
(c) Hund's Rule: Electrons placed in degenerate orbitals tend to remain unpaired.
Writing Electron Configurations for Neutral Atoms
———————————————————————————————————————————————
An ELECTRON CONFIGURATION lists the orbitals that contain electrons in an atom and how many electrons are present in each orbital. In order to show where the electrons exist, the above three rules are used for placing electrons in an atom. Refer to the energy level diagram on p. 9 to see how the “electron–filling” proceeds.
Hydrogen has 1 electron, which goes in the lowest (1s) energy level [Aufbau Principle]. Therefore, H has the ELECTRON CONFIGURATION
H (1s1) — pronounced “Hydrogen, one s one”.
Helium has 2 electrons. Since each orbital can contain up to 2 electrons [Pauli Exclusion Principle], both of helium’s electrons go into the 1s orbital, and helium has the electron configuration
He (1s2) — pronounced “Helium, one s two”.
Two electrons fill the 1st electron shell completely. The 1st electron shell, having n = 1, consists of only one orbital, the 1s.
Lithium has 3 electrons. The first 2 electrons fill the 1s orbital and the 3rd electron goes into the orbital with the next–higher energy [according to the Aufbau Principle]: the 2s. The electron configuration of Li is
Li (1s22s1).
Similarly, the addition of a 4th electron gives
Be (1s22s2).
A problem arises in filling up the 2p orbitals. The three 2p orbitals are designated as 2px, 2py and 2pz and can be filled in any order. Since all three p–orbitals have the same energy (they are DEGENERATE), sequential addition of electrons DOES NOT completely fill one orbital, say 2px, before going on to 2py and 2pz. Hund's Rule for degenerate energy levels requires that one electron be placed in each of the 2p orbitals before pairing up the electrons in the 2p orbitals.
or simplyB (1s22s22p1)
or simply C (1s22s22p2)
or simplyN (1s22s22p3)
or simplyO (1s22s22p4)
or simplyF (1s22s22p5)
or simplyNe (1s22s22p6)
The constant reference to 2px, 2py and 2pz is quite cumbersome, so that the simpler method (shown above, to the right) is used in which the TOTAL number of electrons in a given subshell is shown, rather than specifying which individual orbitals in the subshell actually contain electrons.
At this point the second shell is completely filled (all orbitals having n = 2 are completely filled)and the second shell is said to be CLOSED (as in “closed to further filling”).
The diagram below shows the manner in which the electron energy levels dictate the structure of the periodic table.
EXAMPLE:Arsenic, As, has 33 electrons and is situated halfway along the “4p” block of orbitals. Getting to As on the table requires passing through all the intervening blocks of orbitals and filling them from left to right in a given horizontal line before going down to the next line and continuing.
As (1s22s22p63s23p64s23d104p3)
Technetium, Tc, has 43 electrons and is situated in the “4d” block of orbitals. Again, all the intervening orbitals are filled in order to get to the 4d orbitals.
Tc (1s22s22p63s23p64s23d104p65s24d5)
EXERCISE:
14.Predict the electron configuration of the following.
(a) P (e) Sr (i) Ca (m) Ga
(b) Ti (f) Nb (j) Kr (n) Y
(c) Co (g) Ge (k) Cs (o) Xe
(d) Br (h) Cd (l) Ar (p) Rh
Core Notation
———————————————————————————————————————————————
An atom’s electrons are divided into two subsets: CORE electrons and OUTER electrons.
Definition:The CORE of an atom is the set of electrons having the configuration of the nearest inert gas with an atomic number LESS than that of the atom being considering.
The OUTER electrons consist of all electrons outside the core.
Since core electrons normally do not take part in chemical reactions, core electrons are frequently not explicitly included when writing the electron configuration of an atom.
CORE NOTATION is a way of writing the electron configuration in terms of the core and outer electrons. The rules for writing an electron configuration using core notation are straight–forward.
EXAMPLE:Write the core notation for aluminum atom.
• Locate the atom and note the noble gas at the end of the row above the element
Ne is the noble gas at the end of the row above Al.
• Start to write the electron configuration as usual, but replace the part of the electron configuration corresponding to the configuration of the noble gas with the symbol for the noble gas in square brackets: [...]. Follow the core symbol with the configuration of the remaining electrons in the row containing the element.
Al (1s22s22p63s23p1) becomes Al ( [Ne] 3s23p1)
EXAMPLES: / Full notation / Core notationS (1s22s22p63s23p4) / S ( [Ne] 3s23p4)
Rb (1s22s22p63s23p64s23d104p65s1) / Rb ( [Kr] 5s1)
Kr (1s22s22p63s23p64s23d104p6) / Kr ( [Ar] 4s23d104p6)
Note:If you are given the electron configuration of an element in the 3rd or greater row, and are asked to re–write the configuration in core notation, look backward from the end of the given electron configuration until you find a "p6": this marks off the end of the core electrons.
EXERCISE:
15.Re–write the electron configurations in exercise 14 using core notation.
Electron Configuration Exceptions
———————————————————————————————————————————————
There are two exceptions to the expected configurations of the elements up to Kr. The configurations expected for Cr and Cu are
Cr ( [Ar] 4s23d4) ;d4 is one electron short of a half–filled subshell
Cu ( [Ar] 4s23d9) ;d9 is one electron short of a filled subshell
but experiments show that the configurations are
Cr ( [Ar] 4s13d5) ;4s1 and 3d5 are two half–filled subshells
Cu ( [Ar] 4s13d10) ;4s1 is a half–filled subshell, and 3d10 is a filled subshell