ATOMIC STRUCTURE

 

Information collected from

http://chemed.chem.purdue.edu/genchem/topicreview/

http://wine1.sb.fsu.edu/chm1045/notes/Atoms/AtomStr1/

http://www.chemistry.ohio-state.edu/~grandinetti/teaching/Chem121/lectures/

http://library.thinkquest.org/19662/low/eng/exp-rutherford.html

http://library.thinkquest.org/19662/low/eng/model-bohr.html?tqskip1=1

http://www.chemtopics.com/lectures/unit04/lecture1/l1u4.htm

http://www-theory.chem.washington.edu/~trstedl/quantum/quantum.html

http://science.howstuffworks.com/atom5.htm

 

Fundamental subatomic particles

Particle

Symbol

Charge

Mass

electron

e-

-1

0.0005486 amu

proton

p+

+1

1.007276 amu

neutron

no

0

1.008665 amu

The first evidence for sub-atomic particles came from experiments with the conduction of electricity through gases in sealed glass tubes at low pressures. In the late 19th century, chemists and physicists were studying the relationship between electricity and matter. They were placing high voltage electric currents through glass tubes filled with low-pressure gas (mercury, neon, xenon) much like neon lights. Electric current was carried from one electrode (cathode) through the gas to the other electrode (anode) by a beam called cathode rays.

In 1897, a British physicist, J. J. Thomson did a series of experiments with the following results:

Thomson concluded the following:

Later, an American Physicist named Robert Milikan measured the electrical charge of an electron. With these two numbers (charge, charge to mass ratio), physicists calculated the mass of the electron as 9.10 x 10-28 grams. For comparison, a U.S. penny has a mass of 2.5 grams; so, 2.7 x 1027 or 2.7 billion billion billion electrons would weigh as much as a penny!

Two other conclusions came from the discovery of the electron:

From these results, Thomson proposed a model of the atom that was like a watermelon. The red part was the positive charge and the seeds were the electrons. In 1909 Robert Millikan used the classic oil drop experiment to determine the charge on these particles.

J.J. Thomson demonstrated in 1897 that the rays consist of a stream of negatively charged particles which he called electrons. He was able to measure the charge/mass ratio of these particles and found this to be the same regardless of what gas was in the tube or what metal the electrodes were made from.

Using the smallest charge obtained and Thomson's charge/mass ratio the electron mass is roughly 1/2000 the mass of the lightest atom. Thus there are obviously particles smaller than atoms.

Thomson's dilemma: how could matter containing electrons be neutral and where was all the mass?

Other experiments with discharge tubes suggested the existence of a positive particle with much greater mass (the proton).

Based on this evidence Thomson proposed the first atomic model with sub-atomic particles.

Neutrons (Chadwick (1932) discovered neutron-neutral charge particles in the nucleus)

Be + a = C + n + Energy

B + n = Li + a + Energy

 

Rutherford´s model of the atom

The alpha particles propagated on the atomic nucleusesIn the years 1909-1911 Ernest Rutherford and his students - Hans Geiger (1882-1945) and Ernest Marsden conducted some experiments to search the problem of alpha particles scattering by the thin gold-leaf. Rutherford knew that the particles contain the 2e charge. The experiment caused the creation of the new model of atom  ‘ the planetary model’.

Radioactivity

Wilhelm Roentgen (1895) discovered that when cathode rays struck certain materials (copper for example) a different type of ray was emitted. This new type of ray, called the "x" ray had the following properties:

Henri Becquerel (1896) was studying materials which would emit light after being exposed to sunlight (i.e. phosphorescent materials). The discovery by Roentgen made Becquerel wonder if the phosphorescent materials might also emit x- rays. He discovered that uranium containing minerals produced x-ray radiation (i.e. high energy photons).

Marie and Pierre Curie set about to isolate the radioactive components in the uranium mineral.

Ernest Rutherford studied alpha rays, beta rays and gamma rays, emitted by certain radioactive substances. He noticed that each behaved differently in response to an electric field:

The a and b "rays" were composed of (charged) particles and the g-"ray" was high energy radiation (photons) similar to x-rays

Rutherford model of the atom (1910)

Rutherford (1919) discovers protons - positively charged particles in the nucleus

Chadwick (1932) discovers neutron - neutral charge particles in the nucleus

The number of protons, neutrons, and electrons in an atom can be determined from a set of simple rules.

 

We use the following symbol to describe the atom:

 

A= Z + N, where N is the number of neutrons.

If you add or subtract a proton from the nucleus, you create a new element.

If you add or subtract a neutron from the nucleus, you create a new isotope of the same element you started with.

In the planetary model of atom, the electron should emit energy and spirally fall on the nucleus.In a neutral atom, the number of positively charged protons in the nucleus is equal to the number of orbiting electrons.

But, the model created by Rutherford had still some serious discordance. According to the classic science, electron moving around the nucleus should emit an electromagnetic wave. That kind of emission is connected with the escape of some energy from the electron-ion circuit. Electron should than move not by the circle but helical and finally collide with the nucleus. But atom is stable. Other discordance regarded the radiation - it were to be constant (because the time of electron's cycle in accordance with the lost of energy should change constantly) and spectral lines shouldn't occur.

The model of atom created by Rutherford couldn't be the conclusive model of matter's constitution.

 

Electromagnetic Theory of Radiation

Much of what is known about the structure of the electrons in an atom has been obtained by studying the interaction between matter and different forms of electromagnetic radiation. Electromagnetic radiation has some of the properties of both a particle and a wave.

Particles have a definite mass and they occupy space. Waves have no mass and yet they carry energy as they travel through space. In addition to their ability to carry energy, waves have four other characteristic properties: speed, frequency, wavelength, and amplitude. The frequency (v) is the number of waves (or cycles) per unit of time. The frequency of a wave is reported in units of cycles per second (s-1) or hertz (Hz).

The idealized drawing of a wave in the figure below illustrates the definitions of amplitude and wavelength. The wavelength (l) is the smallest distance between repeating points on the wave. The amplitude of the wave is the distance between the highest (or lowest) point on the wave and the center of gravity of the wave.

If we measure the frequency (v) of a wave in cycles per second and the wavelength (l) in meters, the product of these two numbers has the units of meters per second. The product of the frequency (v) times the wavelength (l) of a wave is therefore the speed (s) at which the wave travels through space. l l = s

 

Light and Other Forms of Electromagnetic Radiation

Light is a wave with both electric and magnetic components. It is therefore a form of electromagnetic radiation.

Visible light contains the narrow band of frequencies and wavelengths in the portion of the electro-magnetic spectrum that our eyes can detect. It includes radiation with wavelengths between about 400 nm (violet) and 700 nm (red). Because it is a wave, light is bent when it enters a glass prism. When white light is focused on a prism, the light rays of different wavelengths are bent by differing amounts and the light is transformed into a spectrum of colors. Starting from the side of the spectrum where the light is bent by the smallest angle, the colors are red, orange, yellow, green, blue, and violet.

As we can see from the following diagram, the energy carried by light increases as we go from red to blue across the visible spectrum.

Because the wavelength of electromagnetic radiation can be as long as 40 m or as short as 10-5 nm, the visible spectrum is only a small portion of the total range of electromagnetic radiation.

The electromagnetic spectrum includes radio and TV waves, microwaves, infrared, visible light, ultraviolet, x-rays, g-rays, and cosmic rays, as shown in the figure above. These different forms of radiation all travel at the speed of light (c). They differ, however, in their frequencies and wavelengths. The product of the frequency times the wavelength of electromagnetic radiation is always equal to the speed of light.                                    vl = c

As a result, electromagnetic radiation that has a long wavelength has a low frequency, and radiation with a high frequency has a short wavelength.

Emission Spectrum of Hydrogen

When an electric current is passed through a glass tube that contains hydrogen gas at low pressure the tube gives off blue light. When this light is passed through a prism (as shown in the figure below), four narrow bands of bright light are observed against a black background.

These narrow bands have the characteristic wavelengths and colors shown in the table below.

Wavelength

 

 

Color

 

 

656.2

 

 

red

 

 

486.1

 

 

blue-green

 

 

434.0

 

 

blue-violet

 

 

410.1

 

 

violet

 

 

Four more series of lines were discovered in the emission spectrum of hydrogen by searching the infrared spectrum at longer wave-lengths and the ultraviolet spectrum at shorter wavelengths. Each of these lines fits the same general equation, where n1 and n2 are integers and RH is 1.09678 x 10-2 nm-1.

Equation

The system of the lines was called the Balmer spectral series. For hydrogen there are also some other series:

    The spectral lines of the other elements which are heavier than hydrogen are more complicated.

Explanation of the Emission Spectrum

Max Planck presented a theoretical explanation of the spectrum of radiation emitted by an object that glows when heated. He argued that the walls of a glowing solid could be imagined to contain a series of resonators that oscillated at different frequencies. These resonators gain energy in the form of heat from the walls of the object and lose energy in the form of electromagnetic radiation. The energy of these resonators at any moment is proportional to the frequency with which they oscillate.

To fit the observed spectrum, Planck had to assume that the energy of these oscillators could take on only a limited number of values. In other words, the spectrum of energies for these oscillators was no longer continuous. Because the number of values of the energy of these oscillators is limited, they are theoretically "countable." The energy of the oscillators in this system is therefore said to be quantized. Planck introduced the notion of quantization to explain how light was emitted.

Albert Einstein extended Planck's work to the light that had been emitted. At a time when everyone agreed that light was a wave (and therefore continuous), Einstein suggested that it behaved as if it was a stream of small bundles, or packets, of energy. In other words, light was also quantized. Einstein's model was based on two assumptions. First, he assumed that light was composed of photons, which are small, discrete bundles of energy. Second, he assumed that the energy of a photon is proportional to its frequency.                                           E = hv

In this equation, h is a constant known as Planck's constant, which is equal to 6.626 x 10-34 J-s.

Example: Let's calculate the energy of a single photon of red light with a wavelength of 700.0 nm and the energy of a mole of these photons.

Red light with a wavelength of 700.0 nm has a frequency of 4.283 x 1014 s-1. Substituting this frequency into the Planck-Einstein equation gives the following result.

Equation

A single photon of red light carries an insignificant amount of energy. But a mole of these photons carries about 171,000 joules of energy, or 171 kJ/mol.

Equation

Absorption of a mole of photons of red light would therefore provide enough energy to raise the temperature of a liter of water by more than 40oC.

The fact that hydrogen atoms emit or absorb radiation at a limited number of frequencies implies that these atoms can only absorb radiation with certain energies. This suggests that there are only a limited number of energy levels within the hydrogen atom. These energy levels are countable. The energy levels of the hydrogen atom are quantized.

The Bohr Model of the Atom

Niels Bohr proposed a model for the hydrogen atom that explained the spectrum of the hydrogen atom. The Bohr model was based on the following assumptions.

Some of the key elements of this hypothesis are illustrated in the figure below.

Diagram

Three points deserve particular attention. First, Bohr recognized that his first assumption violates the principles of classical mechanics. But he knew that it was impossible to explain the spectrum of the hydrogen atom within the limits of classical physics. He was therefore willing to assume that one or more of the principles from classical physics might not be valid on the atomic scale.

Second, he assumed there are only a limited number of orbits in which the electron can reside. He based this assumption on the fact that there are only a limited number of lines in the spectrum of the hydrogen atom and his belief that these lines were the result of light being emitted or absorbed as an electron moved from one orbit to another in the atom.

Finally, Bohr restricted the number of orbits on the hydrogen atom by limiting the allowed values of the angular momentum of the electron. Any object moving along a straight line has a momentum equal to the product of its mass (m) times the velocity (v) with which it moves. An object moving in a circular orbit has an angular momentum equal to its mass (m) times the velocity (v) times the radius of the orbit (r). Bohr assumed that the angular momentum of the electron can take on only certain values, equal to an integer times Planck's constant divided by 2p.

Equation

Bohr then used classical physics to show that the energy of an electron in any one of these orbits is inversely proportional to the square of the integer n.

The difference between the energies of any two orbits is therefore given by the following equation.

Equation

In this equation, n1 and n2 are both integers and RH is the proportionality constant known as the Rydberg constant.

Planck's equation states that the energy of a photon is proportional to its frequency.

E = hv

Substituting the relationship between the frequency, wavelength, and the speed of light into this equation suggests that the energy of a photon is inversely proportional to its wavelength. The inverse of the wavelength of electromagnetic radiation is therefore directly proportional to the energy of this radiation.

By properly defining the units of the constant, RH, Bohr was able to show that the wavelengths of the light given off or absorbed by a hydrogen atom should be given by the following equation.

Equation

Thus, once he introduced his basic assumptions, Bohr was able to derive an equation that matched the relationship obtained from the analysis of the spectrum of the hydrogen atom.

The Bohr Model vs. Reality

At first glance, the Bohr model looks like a two-dimensional model of the atom because it restricts the motion of the electron to a circular orbit in a two-dimensional plane. In reality the Bohr model is a one-dimensional model, because a circle can be defined by specifying only one dimension: its radius, r. As a result, only one coordinate (n) is needed to describe the orbits in the Bohr model.

Unfortunately, electrons aren't particles that can be restricted to a one-dimensional circular orbit. They act to some extent as waves and therefore exist in three-dimensional space. The Bohr model works for one-electron atoms or ions only because certain factors present in more complex atoms are not present in these atoms or ions. To construct a model that describes the distribution of electrons in atoms that contain more than one electron we have to allow the electrons to occupy three-dimensional space. We therefore need a model that uses three coordinates to describe the distribution of electrons in these atoms.

 

 

 

 

Quantum mechanics

Wave-Particle Duality

The theory of wave-particle duality developed by Louis-Victor de Broglie eventually explained why the Bohr model was successful with atoms or ions that contained one electron. It also provided a basis for understanding why this model failed for more complex systems. Light acts as both a particle and a wave. In many ways light acts as a wave, with a characteristic frequency, wavelength, and amplitude. Light carries energy as if it contains discrete photons or packets of energy.

When an object behaves as a particle in motion, it has an energy proportional to its mass (m) and the speed with which it moves through space (s).

E = ms2

When it behaves as a wave, however, it has an energy that is proportional to its frequency:

Equation

By simultaneously assuming that an object can be both a particle and a wave, de Broglie set up the following equation.

Equation

By rearranging this equation, he derived a relationship between one of the wave-like properties of matter and one of its properties as a particle.

As noted in the previous section, the product of the mass of an object times the speed with which it moves is the momentum (p) of the particle. Thus, the de Broglie equation suggests that the wavelength (l) of any object in motion is inversely proportional to its momentum.

Equation

De Broglie concluded that most particles are too heavy to observe their wave properties. When the mass of an object is very small, however, the wave properties can be detected experimentally. De Broglie predicted that the mass of an electron was small enough to exhibit the properties of both particles and waves. In 1927 this prediction was confirmed when the diffraction of electrons was observed experimentally by C. J. Davisson.

De Broglie applied his theory of wave-particle duality to the Bohr model to explain why only certain orbits are allowed for the electron. He argued that only certain orbits allow the electron to satisfy both its particle and wave properties at the same time because only certain orbits have a circumference that is an integral multiple of the wavelength of the electron, as shown in the figure below.

 

Wave Functions and Orbitals

We still talk about the Bohr model of the atom even if the only thing this model can do is explain the spectrum of the hydrogen atom because it was the last model of the atom for which a simple physical picture can be constructed. It is easy to imagine an atom that consists of solid electrons revolving around the nucleus in circular orbits.

Erwin Schrödinger combined the equations for the behaviour of waves with the de Broglie equation to generate a mathematical model for the distribution of electrons in an atom. The advantage of this model is that it consists of mathematical equations known as wave functions that satisfy the requirements placed on the behaviour of electrons. The disadvantage is that it is difficult to imagine a physical model of electrons as waves.

The Schrödinger model assumes that the electron is a wave and tries to describe the regions in space, or orbitals, where electrons are most likely to be found. Instead of trying to tell us where the electron is at any time, the Schrödinger model describes the probability that an electron can be found in a given region of space at a given time. This model no longer tells us where the electron is; it only tells us where it might be.

 

 

The Heisemberg uncertainty principle

People are familiar with measuring things in the macroscopic world around them. Someone pulls out a tape measure and determines the length of a table. A state trooper aims his radar gun at a car and knows what direction the car is traveling, as well as how fast. They get the information they want and don't worry whether the measurement itself has changed what they were measuring. After all, what would be the sense in determining that a table is 80 cm long if the very act of measuring it changed its length!

At the atomic scale of quantum mechanics, however, measurement becomes a very delicate process. Let's say you want to find out where an electron is and where it is going (that trooper has a feeling that any electron he catches will be going faster than the local speed limit). How would you do it? Get a super high powered magnifier and look for it? The very act of looking depends upon light, which is made of photons, and these photons could have enough momentum that once they hit the electron they would change its course! It's like rolling the cue ball across a billiard table and trying to discover where it is going by bouncing the 8-ball off of it; by making the measurement with the 8-ball you have certainly altered the course of the cue ball. You may have discovered where the cue ball was, but now have no idea of where it is going (because you were measuring with the 8-ball instead of actually looking at the table).

Werner Heisenberg was the first to realize that certain pairs of measurements have an intrinsic uncertainty associated with them. For instance, if you have a very good idea of where something is located, then, to a certain degree, you must have a poor idea of how fast it is moving or in what direction. We don't notice this in everyday life because any inherent uncertainty from Heisenberg's principle is well within the acceptable accuracy we desire. For example, you may see a parked car and think you know exactly where it is and exactly how fast it is moving. But would you really know those things exactly? If you were to measure the position of the car to an accuracy of a billionth of a billionth of a centimeter, you would be trying to measure the positions of the individual atoms which make up the car, and those atoms would be jiggling around just because the temperature of the car was above absolute zero!

Heisenberg's uncertainty principle completely flies in the face of classical physics. After all, the very foundation of science is the ability to measure things accurately, and now quantum mechanics is saying that it's impossible to get those measurements exact! But the Heisenberg uncertainty principle is a fact of nature, and it would be impossible to build a measuring device which could get around it.

What is the Schrödinger equation?

Every quantum particle is characterized by a wave function. In 1925 Erwin Schrödinger developed the differential equation which describes the evolution of those wave functions. By using Schrödinger's equation scientists can find the wave function which solves a particular problem in quantum mechanics. Unfortunately, it is usually impossible to find an exact solution to the equation, so certain assumptions are used in order to obtain an approximate answer for the particular problem.

Schrodinger equation {5 kB}

 

Quantum Numbers and Electron configurations

Quantum Numbers

Four numbers used to describe the electrons in an atom.

The Bohr model was a one-dimensional model that used one quantum number to describe the electrons in the atom. Only the size of the orbit was important, which was described by the n quantum number. Schrödinger described an atomic model with electrons in three dimensions. This model required three coordinates, or three quantum numbers, to describe where electrons could be found.

The three coordinates from Schrödinger's wave equations are the principal (n), angular (l), and magnetic (m) quantum numbers. These quantum numbers describe the size, shape, and orientation in space of the orbitals on an atom.

1. Principal (shell) quantum number - n

The principal quantum number (n) describes the size of the orbital. Orbitals for which n = 2 are larger than those for which n = 1, for example. Because they have opposite electrical charges, electrons are attracted to the nucleus of the atom. Energy must therefore be absorbed to excite an electron from an orbital in which the electron is close to the nucleus (n = 1) into an orbital in which it is further from the nucleus (n = 2). The principal quantum number therefore indirectly describes the energy of an orbital.

2. Momentum (subshell) quantum number - l

The orbital (angular) quantum number (l) describes the shape of the orbital. Orbitals have shapes that are best described as spherical (l = 0), polar (l = 1), or cloverleaf (l = 2). They can even take on more complex shapes as the value of the angular quantum number becomes larger.

3. Magnetic quantum number - m

This diagram shows the three possible orientations of a p orbital - px, py, pz.

There is only one way in which a sphere (l = 0) can be oriented in space. Orbitals that have polar (l = 1) or cloverleaf (l = 2) shapes, however, can point in different directions. We therefore need a third quantum number, known as the magnetic quantum number (m), to describe the orientation in space of a particular orbital. (It is called the magnetic quantum number because the effect of different orientations of orbitals was first observed in the presence of a magnetic field.) For example, the p orbital can line up with the x axis, y axis, or z axis. Numerically, the options are expressed as –1, 0, and +1. The "middle" orientation is always expressed as zero, and the others are +/- integers.

4. Spin quantum number - s

To distinguish between the two electrons in an orbital, we need a fourth quantum number. This is called the spin quantum number (s) because electrons behave as if they were spinning in either a clockwise or counterclockwise fashion. One of the electrons in an orbital is arbitrarily assigned an s quantum number of +1/2, the other is assigned an s quantum number of -1/2. Thus, it takes three quantum numbers to define an orbital but four quantum numbers to identify one of the electrons that can occupy the orbital.

Rules Governing the Allowed Combinations of Quantum Numbers

 

Shells and Subshells of Orbitals

Orbitals that have the same value of the principal quantum number form a shell. Orbitals within a shell are divided into subshells that have the same value of the angular quantum number. Chemists describe the shell and subshell in which an orbital belongs with a two-character code such as 2p or 4f. The first character indicates the shell (n = 2 or n = 4). The second character identifies the subshell. By convention, the following lowercase letters are used to indicate different subshells.

s:

 

l = 0

p:

 

l = 1

d:

 

l = 2

f:

 

l = 3

Although there is no pattern in the first four letters (s, p, d, f), the letters progress alphabetically from that point (g, h, and so on). Some of the allowed combinations of the n and l quantum numbers are shown in the figure below.

 

The third rule limiting allowed combinations of the n, l, and m quantum numbers has an important consequence. It forces the number of subshells in a shell to be equal to the principal quantum number for the shell. The n = 3 shell, for example, contains three subshells: the 3s, 3p, and 3d orbitals.

 

 

 

Possible Combinations of Quantum Numbers

There is only one orbital in the n = 1 shell because there is only one way in which a sphere can be oriented in space. The only allowed combination of quantum numbers for which n = 1 is the following.

n

 

l

 

m

 

 

1

 

0

 

0

 

1s

There are four orbitals in the n = 2 shell.

n

 

l

 

m

 

 

2

 

0

 

0

 

2s

 

2

 

1

 

-1

bracket

 

2

 

1

 

0

2p

2

 

1

 

1

 

There is only one orbital in the 2s subshell. But, there are three orbitals in the 2p subshell because there are three directions in which a p orbital can point. One of these orbitals is oriented along the X axis, another along the Y axis, and the third along the Z axis of a coordinate system, as shown in the figure below. These orbitals are therefore known as the 2px, 2py, and 2pz orbitals.

There are nine orbitals in the n = 3 shell.

n

 

l

 

m

 

 

3

 

0

 

0

 

3s

 

 

 

 

 

 

 

3

 

1

 

-1

Bracket

 

3

 

1

 

0

3p

3

 

1

 

1

 

 

 

 

 

 

 

 

3

 

2

 

-2

Bracket

 

3

 

2

 

-1

3d

3

 

2

 

0

3

 

2

 

1

3

 

2

 

2

 

There is one orbital in the 3s subshell and three orbitals in the 3p subshell. The n = 3 shell, however, also includes 3d orbitals.

The five different orientations of orbitals in the 3d subshell are shown in the figure below. One of these orbitals lies in the XY plane of an XYZ coordinate system and is called the 3dxy orbital. The 3dxz and 3dyz orbitals have the same shape, but they lie between the axes of the coordinate system in the XZ and YZ planes. The fourth orbital in this subshell lies along the X and Y axes and is called the 3dx2-y2 orbital. Most of the space occupied by the fifth orbital lies along the Z axis and this orbital is called the 3dz2 orbital.

The number of orbitals in a shell is the square of the principal quantum number: 12 = 1, 22 = 4, 32 = 9. There is one orbital in an s subshell (l = 0), three orbitals in a p subshell (l = 1), and five orbitals in a d subshell (l = 2). The number of orbitals in a subshell is therefore 2(l) + 1.

Before we can use these orbitals we need to know the number of electrons that can occupy an orbital and how they can be distinguished from one another. Experimental evidence suggests that an orbital can hold no more than two electrons.

To distinguish between the two electrons in an orbital, we need a fourth quantum number. This is called the spin quantum number (s) because electrons behave as if they were spinning in either a clockwise or counterclockwise fashion. One of the electrons in an orbital is arbitrarily assigned an s quantum number of +1/2, the other is assigned an s quantum number of -1/2. Thus, it takes three quantum numbers to define an orbital but four quantum numbers to identify one of the electrons that can occupy the orbital.

The allowed combinations of n, l, and m quantum numbers for the first four shells are given in the table below. For each of these orbitals, there are two allowed values of the spin quantum number, s.

An applet showing atomic and molecular orbitals from


Summary of Allowed Combinations of Quantum Numbers

n

 

 

l

 

 

 

m

Subshell Notation

Number of Orbitals in the Subshell

Number of Electrons Needed to Fill Subshell

Total Number of Electrons in Subshell

 

1

 

 

0

 

 

 

0

1s

1

2

2

 

2

 

 

0

 

 

 

0

2s

1

2

 

2

 

 

1

 

 

 

1,0,-1

2p

3

6

8

 

3

 

 

0

 

 

 

0

3s

1

2

 

3

 

 

1

 

 

 

1,0,-1

3p

3

6

 

3

 

 

2

 

 

 

2,1,0,-1,-2

3d

5

10

18

 

4

 

 

0

 

 

 

0

4s

1

2

 

4

 

 

1

 

 

 

1,0,-1

4p

3

6

 

4

 

 

2

 

 

 

2,1,0,-1,-2

4d

5

10

 

4

 

 

3

 

 

 

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

4f

7

14

32

The Relative Energies of Atomic Orbitals

Because of the force of attraction between objects of opposite charge, the most important factor influencing the energy of an orbital is its size and therefore the value of the principal quantum number, n. For an atom that contains only one electron, there is no difference between the energies of the different subshells within a shell. The 3s, 3p, and 3d orbitals, for example, have the same energy in a hydrogen atom. The Bohr model, which specified the energies of orbits in terms of nothing more than the distance between the electron and the nucleus, therefore works for this atom.

The hydrogen atom is unusual, however. As soon as an atom contains more than one electron, the different subshells no longer have the same energy. Within a given shell, the s orbitals always have the lowest energy. The energy of the subshells gradually becomes larger as the value of the angular quantum number becomes larger.

Relative energies: s < p < d < f

As a result, two factors control the energy of an orbital for most atoms: the size of the orbital and its shape, as shown in the figure below.

 

A very simple device can be constructed to estimate the relative energies of atomic orbitals. The allowed combinations of the n and l quantum numbers are organized in a table, as shown in the figure below and arrows are drawn at 45 degree angles pointing toward the bottom left corner of the table.

 

The order of increasing energy of the orbitals is then read off by following these arrows, starting at the top of the first line and then proceeding on to the second, third, fourth lines, and so on. This diagram predicts the following order of increasing energy for atomic orbitals.

1s < 2s < 2p < 3s < 3p <4s < 3d <4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f < 6d < 7p < 8s ...

 

 

 

 

 

Electron Configurations, the Aufbau Principle, Degenerate Orbitals, and Hund's Rule

The electron configuration of an atom describes the orbitals occupied by electrons on the atom. The basis of this prediction is a rule known as the aufbau principle, which assumes that electrons are added to an atom, one at a time, starting with the lowest energy orbital, until all of the electrons have been placed in an appropriate orbital.

A hydrogen atom (Z = 1) has only one electron, which goes into the lowest energy orbital, the 1s orbital. This is indicated by writing a superscript "1" after the symbol for the orbital.

H (Z = 1): 1s1

The next element has two electrons and the second electron fills the 1s orbital because there are only two possible values for the spin quantum number used to distinguish between the electrons in an orbital.

He (Z = 2): 1s2

The third electron goes into the next orbital in the energy diagram, the 2s orbital.

Li (Z = 3): 1s2 2s1

The fourth electron fills this orbital.

Be (Z = 4): 1s2 2s2

After the 1s and 2s orbitals have been filled, the next lowest energy orbitals are the three 2p orbitals. The fifth electron therefore goes into one of these orbitals.

B (Z = 5): 1s2 2s2 2p1

When the time comes to add a sixth electron, the electron configuration is obvious.

C (Z = 6): 1s2 2s2 2p2

However, there are three orbitals in the 2p subshell. Does the second electron go into the same orbital as the first, or does it go into one of the other orbitals in this subshell?

To answer this, we need to understand the concept of degenerate orbitals. By definition, orbitals are degenerate when they have the same energy. The energy of an orbital depends on both its size and its shape because the electron spends more of its time further from the nucleus of the atom as the orbital becomes larger or the shape becomes more complex. In an isolated atom, however, the energy of an orbital doesn't depend on the direction in which it points in space. Orbitals that differ only in their orientation in space, such as the 2px, 2py, and 2pz orbitals, are therefore degenerate.

Electrons fill degenerate orbitals according to rules first stated by Friedrich Hund. Hund's rules can be summarized as follows.

When the time comes to place two electrons into the 2p subshell we put one electron into each of two of these orbitals. (The choice between the 2px, 2py, and 2pz orbitals is purely arbitrary.)

C (Z = 6): 1s2 2s2 2px1 2py1

The fact that both of the electrons in the 2p subshell have the same spin quantum number can be shown by representing an electron for which s = +1/2 with an

arrow pointing up and an electron for which s = -1/2 with an arrow pointing down.

The electrons in the 2p orbitals on carbon can therefore be represented as follows.

Diagram

When we get to N (Z = 7), we have to put one electron into each of the three degenerate 2p orbitals.

N (Z = 7):

 

1s2 2s2 2p3

 

Diagram

Because each orbital in this subshell now contains one electron, the next electron added to the subshell must have the opposite spin quantum number, thereby filling one of the 2p orbitals.

O (Z = 8):

 

1s2 2s2 2p4

 

diagram

The ninth electron fills a second orbital in this subshell.

F (Z = 9):

 

1s2 2s2 2p5

 

Diagram

The tenth electron completes the 2p subshell.

Ne (Z = 10):

 

1s2 2s2 2p6

 

Diagram

There is something unusually stable about atoms, such as He and Ne, that have electron configurations with filled shells of orbitals. By convention, we therefore write abbreviated electron configurations in terms of the number of electrons beyond the previous element with a filled-shell electron configuration. Electron configurations of the next two elements in the periodic table, for example, could be written as follows.

Na (Z = 11): [Ne] 3s1

Mg (Z = 12): [Ne] 3s2

The aufbau process can be used to predict the electron configuration for an element. The actual configuration used by the element has to be determined experimentally.

Exceptions to Predicted Electron Configurations

There are several patterns in the electron configurations listed in the table in the previous section. One of the most striking is the remarkable level of agreement between these configurations and the configurations we would predict. There are only two exceptions among the first 40 elements: chromium and copper.

Strict adherence to the rules of the aufbau process would predict the following electron configurations for chromium and copper.

predicted electron configurations:

 

Cr (Z = 24): [Ar] 4s2 3d4

 

 

Cu (Z = 29): [Ar] 4s2 3d9

The experimentally determined electron configurations for these elements are slightly different.

actual electron configurations:

 

Cr (Z = 24): [Ar] 4s1 3d5

 

 

Cu (Z = 29): [Ar] 4s1 3d10

In each case, one electron has been transferred from the 4s orbital to a 3d orbital, even though the 3d orbitals are supposed to be at a higher level than the 4s orbital.

Once we get beyond atomic number 40, the difference between the energies of adjacent orbitals is small enough that it becomes much easier to transfer an electron from one orbital to another. Most of the exceptions to the electron configuration predicted from the aufbau diagram shown earlier therefore occur among elements with atomic numbers larger than 40. Although it is tempting to focus attention on the handful of elements that have electron configurations that differ from those predicted with the aufbau diagram, the amazing thing is that this simple diagram works for so many elements.


Electron Configurations and the Periodic Table

When electron configuration data are arranged so that we can compare elements in one of the horizontal rows of the periodic table, we find that these rows typically correspond to the filling of a shell of orbitals. The second row, for example, contains elements in which the orbitals in the n = 2 shell are filled.

Li (Z = 3):

 

[He] 2s1

Be (Z = 4):

 

[He] 2s2

B (Z = 5):

 

[He] 2s2 2p1

C (Z = 6):

 

[He] 2s2 2p2

N (Z = 7):

 

[He] 2s2 2p3

O (Z = 8):

 

[He] 2s2 2p4

F (Z = 9):

 

[He] 2s2 2p5

Ne (Z = 10):

 

[He] 2s2 2p6

There is an obvious pattern within the vertical columns, or groups, of the periodic table as well. The elements in a group have similar configurations for their outermost electrons. This relationship can be seen by looking at the electron configurations of elements in columns on either side of the periodic table.

Group IA

 

 

 

Group VIIA

 

 

H

 

1s1

 

 

 

 

Li

 

[He] 2s1

 

F

 

[He] 2s2 2p5

Na

 

[Ne] 3s1

 

Cl

 

[Ne] 3s2 3p5

K

 

[Ar] 4s1

 

Br

 

[Ar] 4s2 3d10 4p5

Rb

 

[Kr] 5s1

 

I

 

[Kr] 5s2 4d10 5p5

Cs

 

[Xe] 6s1

 

At

 

[Xe] 6s2 4f14 5d10 6p5

The figure below shows the relationship between the periodic table and the orbitals being filled during the aufbau process. The two columns on the left side of the periodic table correspond to the filling of an s orbital. The next 10 columns include elements in which the five orbitals in a d subshell are filled. The six columns on the right represent the filling of the three orbitals in a p subshell. Finally, the 14 columns at the bottom of the table correspond to the filling of the seven orbitals in an f subshell.