Arrangement of the Elements in groups in the periodic Table

Arrangement of the Elements in groups in the Periodic Table - Inorganic Chemistry

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Periodic Table

The chemical properties of an element are largely governed by the number of electrons in the outer shell, and their arrangement. If the elements are arranged in groups which have the same outer electronic arrangement, then elements within a group should show · similarities in chemical ·· and physical properties. One great advantage of this is that initially, it is only necessary to learn the properties of each group rather than the properties of each individual element.
Elements with one electron in their outer shell are called Group I (the alkali metals) and elements with two s electrons in their outer shell are called Group II (the alkaline earth metals). These two groups are known as the s-block elements because their properties result from the presence of s electrons.
Elements with three electrons in their outer shell (two s electrons and one p electron) are called Group III, and similarly, Group IV elements have four outer electrons, Group V elements have five outer electrons, Group VI elements have six outer electrons and Group VII elements have seven outer electrons; Group 0 elements have a full outer shell of electrons so that the next shell is empty; hence the group name. Groups III, IV, V, VI, VII, and 0 all have p orbitals filled and because their properties are dependent on the · presence of p electrons, they are called jointly the p-block elements.
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In a similar way, elements, where d orbitals are being filled, are called the d-block, or transition elements. In these, d electrons are being added to the penultimate shell. · · Finally, elements, where f orbitals are filling, are called the /-block, and here the f electrons are entering the antepenultimate (or second from the outside) shell. In the periodic tab!~ (Table 1.4), the elements are arranged in order of increasing atomic number, which is in order of increased nuclear charge, or increased number of orbital electrons. Thus each element contains one more orbital electron than the preceding element. Instead of listing the 103 elements as one long list, the periodic table arranges them into several horizontal rows or periods, in such a way that each row begins with an alkali metal and ends with a noble gas. The sequence in which the various energy levels are filled determines the number of elements in each period, and the periodic table can be divided into four main regions according to whether the s, p, d, or f levels are bein~ filled.

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The alkali metals appear in a vertical column labeled Group I, in which all elements have one electron in their outer shell, and hence have similar properties. Thus w~en one element in a group reacts with a reagent, the other elements in the group will probably react similarly, forming compounds that have similar formulae. Thus reactions of new compounds and their formulae may be predicted by analogy with known compounds. Similarly, the noble gases all appear in a vertical column labeled Group 0, and all have a complete outer shell of electrons. This is called the long form of the periodic table. It has many advantages, the most important being that it emphasizes the similarity of properties within a group and the relation between the group and the electron structure. The d-block elements are referred to as the transition elements as they are situated between the s- and p-blocks.
Hydrogen and helium differ from the rest of the elements because there are no p orbitals in the first shell. Helium obviously belongs to Group 0, the noble gases, which are chemically inactive because their outer shell of electrons is full. Hydrogen is more difficult to place in a group. It could be included in Group I because it has one electron in its outer shell, is univalent, and commonly forms univalent positive ions .. However, hydrogen is not a metal and is a gas whilst Li, Na, K, Rb, and Cs are metals and are solids. Similarly, hydrogen could be included in Group VII because it is one electron short of a complete shell, or in Group IV because its outer shell is half full. Hydrogen does not resemble the alkali metals, the halogens, or Group IV very closely. Hydrogen atoms are extremely small and have many unique properties. Thus there is a case for placing hydrogen in a group on its own.

Pauli Exclusion Principle and Hund's Rule 

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Pauli Exclusion Principle

Three quantum: numbers n, I, and m are needed to define an orbital. Each orbital may hold up to two electrons, provided they have opposite spihs. An extra ·quantum number is required to define the spin of an electron in an orbital. Thus four quantum numbers are needed to define the energy of an electron in an atom. The Pauli exclusion principle states that no two electrons in one atom can have all four quantum numbers the ~ame. Ry permutating the quantum numbers, the maximum number of electrons that can be contained in each main energy level can be calculated (see Figure 1.12).
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Hund's Rule

When atoms are in their ground state, the electrons occupy the lowest possible energy levels.
The simplest element, hydrogen, has one electron, which occupies the ls level; this level has the principal quantum number n = 1, and the subsidiary quantum number I = 0.
Helium has two electrons. The second electron also occupies the level . This is possible because the two electrons have opposite spins. This level is now full.
The next atom lithium has three electrons. The third electron occupies the next lowest level. This is the 2s level, which has the principal quantum number n = 2 and subsidiary quantum number I = 0.

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The fourth electron in beryllium also occupies the 2s level. Boron must have its fifth electron in the 2p level as the 2s level is full. The sixth electron in carbon is also in the 2p level. Hund's rule states . that the number of unpaired electrons in a given energy level is a maximum. Thus in the ground state the two p electrons in carbon are unpaired. They occupy separate p orbitals and have parallel spins. Similarly in nitrogen the three p electrons are unpaired and have parallel spins.
To show the positions of the electrons in an atom, the symbols Is, 2s, 2p, etc. are used to denote the main energy level and sub-level. A superscript indicates the number of electrons in each set of orbitals. Thus for 1 hydrogen, the Is orbital contains one electron, and this is shown as Is • For 2 helium the ls orbital contains two electrotls, denoted ls . The electronic structures of the first few 'atoms in the periodic table tnay be written:

An alternative way of showing the electroniC structure of an atom is .to draw boxes for orbitals, and arrows for the electrons.

Sequence of energy levels

It is important to know the sequence in which the energy levels are filled. Figure 1.13 is a useful aid. From this it can be seen that the order of filling of energy levels is: ls, 2s, 2p, 3s, 3p, 4s, 3d, 4p, Ss, 4d, Sp, 6s, 4/, Sd, 6p, 7s, etc.
After the ls, 2s, '{.p, 3s and 3p levels have been filled at argon, the next two electrons go into the 4s level. This gives the elements potassium and calcium. Once the 4s level is full the 3d level is the next lowest in energy, not the 3p level. Thus tl:ie 3d starts to fill at scandium. The elements from scandium to copper have two electrons in the 4s level and an incomplete 3d level, and all behave in a similar manner chemically. Such a series of atoms is known as a tninsition series.
A second transition series starts after the 5s orbital has been tilled, at strontium , because in the next element, yttrium, the 4d level begins to fill up. A third transition series starts at- lanthanum where the electrons start to fill the Sd level after the 6d level has been filled with two electrons.
A further complication arises here because after lanthanum, which has one electron in the 5d level, the 4/ level begins to fill, giving the elements from cerium to lutetium with from one to 14/ electrons. These are sometimes called the inner transition elements, but are usually known as the lanthanides or rare earth metals.

Radial And Angular Functions

Radial And Angular Functions - Atomic structure and the periodic table

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The Schrodinger equation can be solved completely for the hydrogen atom, and for related ions which have only one electron such as He+ and u2+. For other atoms only approx: create solutions can be obtained. For most calculations, it is simpler to solve the wave equation if the cartesian coordinates x, y, and z are converted into polar coordinates r, e, and. The coordinates of the point A measured from the origin are x, y, a.rid z in cartesian coordinates, and r, e, and in polar coordinates. It c·an be seen that the two sets of coordinates are related by the following expressions:

R(r) is a function that depends on the distance from the nucleus, which in turn depends on the quantum numbers n and I

Θ(θ) is a function of 0; which depends on the quantum numbers I and m
Φ(Φ) is a function of cp, which depends only on the quantum number m
Equation (1.6) may be rewritten

This splits the wave function into two. pa~ts which can be solved separately: 1. R(r) the radial function, which depends on the· quantum numbers n and/.
2. A_ml the total angular wave function, which depends on the quantum numbers m and I.
The radial function R has no physical meaning, but R² gives the probability of finding the electron in a small volume DV near the point at which R is measured. For a given value of r the number of small volumes is 4πr², so the probability of the electron being at a distance. r from ·the nucleus is 4πr²R²• This is called the radial distribution function. Graphs of the

the radial distribution function for hydrogen plotted against r is shown in Figure 1.8.

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These diagrams show that the probability is zero at the nucleus (as r = 0), and by examining the plots for ls, 2s and 3s that the most probable distance increases markedly as the principal quantum number increases. Furthermore, by comparing the plots for 2s and 2p, or 3s, 3p, and 3d it can be seen that the most probable radius decreases slightly as the subsidi_ary quantum number increases. All the~· orbitals except the first one (ls) have a shell-like structure, rather like an onion or a hailstone, consisting of concentric layers of electron density. Similarly, all bun he first p orbitals (2p) and the first d orbitals (3d) have a shell structure.
The angular function A depends only on the direction and is independent of the distance from the nucleus (r). Thus A² is the probability of

finding an electron at a given direction θ, Φ  at any distance from the nucleus to infinity. The angular functions Aare plotted as polar diagrams in Figure 1.9. It must be emphasized that these polar diagrams do not represent the total wave function u, but only the angular part of the wave function. (The total wave function is made up of contributions from both the radial and the angular functions.)

Thus the probability of finding an electron simultaneously at a distance r and in a given direction θ, Φ is Ψ²_rθ, Φ

Polar diagrams, that are drawings of the angular part of the wave function, are commonly used to illustrate the overlap of orbitals giving bonding between atoms. Polar diagrams are quite good for this purpose, as they show the signs + and - relating to the symmetry of the angular function. For bonding like signs must overlap. These shapes are slightly different from the shapes of the total wave function. There are several points about such diagrams:
l. It is difficult to picture an angular wave function as a mathematical equation. It is much easier to visualize a boundary surface, which is a solid shape. which for example contains 90% of the electron density. To emphasize that Ψ is continuous. function, the boundary surfaces have been extended up to the nucleus in Figure 1.9. For p orbitals, the electron density is zero at the nucleus, and stone texts show a p orbital a' two spheres which do not touch.
2. These drawings show the symmetry for the ls, 2p, 3d orbitals. However, in the others, 2s, 3s, 4s . .. , 3p, 4p, Sp . .. , 4d, Sd . .. the sign (symmetry) changes inside the boundary surface Of the orbital. this is readily seen as nodes in the graphs Of the radial functions (Figure 1.8).

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3. The probability of finding an electron at a direction θ, Φ is the wave function squared, A² , or more precisely Ψ²_θ, Ψ²_ΦThe diagrams in Figure 1. 9 are of the angular part of the wave function A, not A² Squaring does not change the shape of an s orbital, but it elongates the lobes of p orbitals (Figure 1.10). Some books use elongated p orbitals, but strictly these should not have signs, as squaring removes· any sign from the symmetry. Despite this, many authors draw shapes approximating to the probabilities, i.e. squared wave functions, and put the signs of the wave function on the lobes, and refer to both the shapes and the wave functions as orbitals.
4. A full representation of the probability of finding an electron requires the total wave function squared and includes both the radial and angular probabilities squared. It really needs a three-dimensional model to display this probability and show the shapes of the orbitals. It is difficult to do this adequately on a two-dimensional piece of paper, but a representation is shown in Figure 1.11. The orbitals are not drawn to scale. Note that the p orbitals are not simply two spheres, but are ellipsoids of revolution. Thus the 2p_x  orbital is spherically symmetrical about the x axis, but is not spherical in the other direction. Similarly, the P_y orbital is spherically symmetrical about the y axis, and both the Pz and the 3d_z2 are spherically symmetrical about the z axis.

The Schrodinger Wave Equation definition

The Schrodinger Wave Equation - Atomic structure and the periodic table

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For a standing wave (such as a vibrating string) of wavelength λ, the whose amplitude at any point along x may be described by a function f (x). it can be shown that:

If an electron is considered as a wave which moves in only one dimension then:

An electron may move in three directions x. y and z so this becomes

Using the · symbol ∇ instead of the three partial differentials, this is shortened to

The de Broglie relationship states that

(where h is Planck's constant, m is the mass of an electron and v its velocity); hence:

or

However, the total energy of the system Eis made up of the kinetic energy K plus the potential energy V

Substituting for v² in equation (1.5) gives the well-known form of the Schrodinger equation

Acceptable solutions to the wave equation, that are solutions which are physically possible, must have certain properties:
1. ♆ the must be continuous.
2. ♆ the must be fi.nite.
3. ♆ the must be single-valued.
4. The probability of finding the electron over all the space from plus infinity to minus infinity must be equal to one.
The probability of finding an electron at a point x, y, z is ♆², so

Several wave functions called ♆₁. ♆₂ , ♆₃ . will satisfy these conditions to the wave equation, and each of these has a corresponding energy E₁, E₂ , E₃ . Each of these wave functions ♆₁ ♆₂ , etc. is called an orbital, by analogy with the orbits in the Bohr theory. In a hydrogen atom, the single electron normally occupies the lowest bf the energy levels E₁• This is called the ground state. The corresponding wave function ♆₁ describes the orbital, which is the volume in space where there is a high probability of finding the electron.
For a given type of atom, there are a number of solutions to the wave equation which are acceptable, and each orbital may be described uniquely by a set of three quantum numbers, n, I, and m. (These are t_he same quantum numbers - principal, subsidiary and magnetic - as were used in the Bohr theory).
The subsidiary quantum number I describe the shape of the orbital occupied by the electron. l may have values 0, 1, 2 or 3. When I= 0, the orbital is spherical and is called an s orbital; when I = l, the orbital is dumb-bell shaped and is called a p orbital; when I= 2, the orbital is double dumb-bell shaped and is called a d orbital; and when I = 3 a more complicated f orbital is formed (see Figure 1.6). the letters s, p, d and i come from the spectroscopic terms sharp, principal, diffuse and

.fundamental, which were used to describe the lines in the atomic spectra.

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Examination of a list of all the allowed solutions to the wave equation shows that the orbitals fall into groups. In the first group of solutions the value of the wave function ♆ and
hence the probability of finding the electron ♆² depends only on the distance r from the nucleus and is the same in all directions.

♆ = f(r)

This leads to a spherical orbital and occurs when the subsidiary quantum number I is zero. These are called s orbitals. When I = 0, the magnetic quantum number m = 0, so there is only one such orbital for each value of n.
In the second group of solutions to the wave equation, ♆ depends both on the distance from the nucleus, and on the direction in space (x, y or z). Orbitals of this kind occur when the subsidiary quantum number I = 1. These are called p orbitals and there are three possible values of the magnetic quantum number (m = -1, 0and+1). There are therefore three orbitals that are identical in energy, sh.ape, and size, which differ only in their direction in space. These three solutions to the wave equation may be written

Orbitals that are identical in energy are termed degenerate, and thus three degenerate p orbitals occur for each of the values of n = 2, 3, 4 ... The third group of solutions to the wave equation depends on the

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distance from the nucleus r and also on two directions in space, for example

This group of orbitals has I = 2, and these are called d orbitals. there are five solutions corresponding tom = -2, -1, 0, + 1 and +2, and these are all equal in energy. Thus five degenerated orbitals occur for each of the values of n = 3, 4, 5 .... A further set of solutions occurs when I = 3, and- these are called f orbitals. There are seven values of tn: -3, -2, -l, O", +l, +2 and +3, and seven degenerate f orbitals are formed when n = 4, 5, 6 ....

The Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle - Atomic structure and the periodic table

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Calculations on the Bohr model of an atom require precise information about the position of an electron an(f its velocity. It is difficult to measure both quantities accurately at the same time. An electron is too small to see and may only be observed if perturbed. For example, we could hit the electron with another particle such as a photon or an electron, or we could apply an electric or magnetic force to the electron. This will inevitably change the position of the electron, or its velocity and direction. Heisenberg stated that the more precisely we cart define the position of an electron. the less certain we are able to define its velocity. and vice versa. If l:!..x is the uncertainty in defining the position and !:!.. v the uncertainty in the velocity, the uncertainty principle may be expressed mathematically as:
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where h = Planck's constant = 6.6262 x 10- 3⁴ J s. This implies that it is impossible to know both the position and the velocity exactly. The concept of an electron following a definite orbit, where its position and velocity are known exactly, must therefore be replaced by the probability of finding an electron in a particular position, or in a particular volume of space. The Schrodinger wave equation provides a satisfactory description of an atom in these terms. Solutions to the wave equation are
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called wave functions and given the symbol Ψ. The probability of finding an electron at a point in space whose coordinates are x, y and z is Ψ² (x, y, z).

The Dual Nature of Electrons Particles or Waves

The Dual Nature of Electrons Particles or Waves - Atomic structure and the periodic table

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The planetary theory of atomic structure put forward by Rutherford and Bohr describes the atom as a central nucleus surrounded by electrons in certain orbits. The electron is thus considered as a particle. In the 1920s it was shown that moving particles such as electrons behaved in some ways as waves. This is an important concept in explaining the electronic structure of atoms .. For some time light has been considered as either particles or waves. Certain materials such as potassium emit electrons when irradiated with visible light, or in some cases with ultraviolet light.

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This is called the photoelectric effect. It is explained by light traveling as particles called photons. If a photon collides with an electron, it can transfer its energy to the electron. If the energy of the photon is sufficiently large it can remove the electron from the surface of the metal. However, the phenomena of diffraction and interference of light can only be explained by assuming that light behaves like waves. In 1924, de Broglie postulated that the same dual character existed with electrons - sometimes they are considered as particles, and at other times it is more convenient to consider them as waves.
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Experimental evidence for the wave nature of electrons was obtained when diffraction rings were observed photographically when a stream of elec;trons was passed through a thin metal foil. Electron diffraction has now become a useful tool in determining molecular structure, particularly of gases. Wave mechanics is a means of studying the build-up of electron shells in atoms, and the shape of orbitals occupied by the electrons.

Refinements of The Bohr Theory

Refinements of The Bohr Theory - Atomic structure and the periodic table

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It has been assumed that the nucleus remains stationary except for rotating on its own axis. This would be true if the mass of the nucleus were infinite, but the ratio of the mass of an electron to the mass of the hydrogen nucleus is 111836. The nucleus actually oscillates slightly about the center of gravity, and to allow for this the mass of the electron m is replaced by the. reduced mass µ in equation (1.4):

where M is the mass of the nucleus. The inclusion of the mass of the nucleus explains why different isotopes of an element produce lines in the spectrum at slightly different wavenumbers. The orbits are sometimes denoted by the letters K, L, M, N ... counting outwards from the nucleus, and they are also numbered 1, 2, 3, 4 ... This number is called the principal quantum number, which is given the symbol

Figure 1.4 Bohr orbits of hydrogen and the various series of spectral lines.

n. It is, therefore, possible t<> define which circular orbit is under consideration by specifying the principal quantum number. When an electron moves from one orbit to another it should give a single sharp line in the spectrum, corresponding precisely to the energy difference between the initial and final orbits. If the hydrogen spectrum is observed ~ with a high-resolution spectrometer it is found that some of the lines reveal 'fine structure'. This means that a line is really composed of several lines close together. Sommerfeld explained this splitting of lines by assuming that some of the orbits were elliptical; and that they precessed in space around the nucleus. For the orbit does to the nucleus, the principal quantum number n = 1, and there is a circular orbit. For the next orbit, #1e principal quantum number n = 2, and both circular and elliptical orbits ~re possible. To define an elliptical orbit, a second quantum number k is needed. The shape of the ellipse is defined by the ratio Of the lengths of the major arid minor axes. Thus

k is called the azimuthal or subsidiary quantum number and may have values from 1, 2 ... Ii. Thus for fl = 2, all may have the values 2/2 (circular orbit) and 2/1 (elliptical orbit). For the principal quantum number n = 3, nlk may have values 3/3 (circular),

3/2 (ellipse), and 3/1 (narrower ellipse). The presence of these extra orbits, which have slightly different energies k = 1 from each other, accounts for the extra lines in the spectrum revealed under high resolution. The original quantum number k has now been replaced by a new quantum number /, where I = k - 1. Thus for

This explained why some of the spectral lines are split into two, three, four, or more lines. In addition, some spectral lines are split still further into two lines (a double). This is explained by assuming that an electron spins on its axis in either a clockwise or an anticlockwise direction. Energy is quantized. and the value of the spin angular momentum was first considered to be ms· h/2n, where ms is the spin quantum number with values of ±~- (Quantum mechanics has since shown the exact expression to be Vs(s + I)· h/2n, where s is either the spin quantum number or the resultant of several spins.)
Zeeman showed that if atoms were placed in a strong magnetic field additional lines appeared on the spectrum. This is because elliptical orbits can only take up certain orientations with respect to the external field. rather than precessing freely. Each of these orientations is associated with a fourth quantum number m which can have values of l. (/ - I) .. . . 0 ... (-/ + I).-/. Thus a single line in the normal spectrum will appear as (2/ + 1) lines if a magnetic field is applied. Thus in order to explain the spectrum· of the hydrogen atom, four quantum numbers are needed, as shown in Table 1.2. The spectra of other atoms may be explained in a similar manner.