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Applied Chemistery-I
Session 2009-10 , UET Lahore
Lecture # 3
Atomic Radius
To a first approximation, atoms generally behave like minute spherical objects. The atomic radius of a
chemical element is a measure of the size of its atoms, usually the distance from the nucleus to the
boundary of the surrounding cloud of electrons. Since the boundary is not a well-defined physical
entity, there are various non-equivalent definitions of atomic radius.Depending on the definition, the
term may apply only to isolated atoms, or also to atoms in condensed matter, covalently bound in
molecules, or in ionized and excited states; and its value may be obtained through experimental
measurements, or computed from theoretical models. Under some definitions, the value of the radius
may depend on the atom's state and context.
The concept is difficult to define because the electrons do not have definite orbits, or sharply defined
ranges. Rather, their positions must be described as probability distributions that taper off gradually as
one moves away from the nucleus, without a sharp cutoff. Moreover, in condensed matter and
molecules, the electron clouds of the atoms usually overlap to some extent, and some of the electrons
may roam over a large region encompassing two or more atoms.Despite these conceptual difficulties,
under most definitions the radii of isolated neutral atoms range between 30 and 300 pm (trillionths of a
meter), or between 0.3 and 3 angstroms. Therefore, the radius of an atom is more than 10,000 times the
radius of its nucleus (1–10 fm). and less than 1/1000 of the wavelength of visible light (400–700
nm).The modeling of atoms by spheres is only a crude approximation, but it can provide quantitative
explanations and predictions for many phenomena, such as the density of liquids and solids, the
diffusion of fluids through molecular sieves, the arrangement of atoms and ions in crystals, and the size
and shape of molecules
Atomic radii vary in a predictable and explicable manner across the periodic table. For instance, the
radii generally decrease along each period (row) of the table, from the alkali metals to the noble gases;
and increase down each group (column). The radius increases sharply between the noble gas at the end
of each period and the alkali metal at the beginning of the next period. These trends of the atomic radii
(and of various other chemical and physical properties of the elements) can be explained by the
electron shell theory of the atom; they provided important evidence for the development and
confirmation of quantum theory.The atomic radius is also called the width of the atom
Widely used definitions of atomic radius include:
1

Van der Waals radius: in principle, half the minimum distance between the nuclei of two
atoms of the element that are not bound to the same molecule.

Ionic radius: the nominal radius of the ions of an element in a specific ionization state,
deduced from the spacing of atomic nuclei in cyrstalline salts of that include that ion. In
principle, the spacing between two adjacent oppositely charged ions (the length of the ionic
bond between them) should equal the sum of their ionic radii

Covalent radius: the nominal radius of the atoms of an element when covalently bound to
other atoms, as deduced the separation between the atomic nuclei in molecules. In principle,
the distance between two atoms that are bound to each other in a molecule (the length of that
covalent bond) should equal the sum of their covalent radii

Metallic radius: the nominal radius of atoms of an element when joined to other atoms by
metallic bonds.

Bohr radius: the radius of the lowest-energy electron orbit predicted by Bohr model of the
atom (1913). It is only applicable to atoms and ions with a single electron, such as hydrogen,
singly-ionized helium, and positronium. Although the model itself is now obsolete, the Bohr
radius for the hydrogen atom is still regarded as an important physical constant.
Hasan Qayyum Chohan , Reg. No. 2009-CH-204
Chemical Engineering
UET Lahore
Applied Chemistery-I
Session 2009-10 , UET Lahore
Explanation of the general trends
The way the atomic radius varies with increasing atomic number can be explained by the arrangement
of electrons in shells of fixed capacity. The shells are generally filled in order of increasing radius,
since the negatively charged electrons are attracted by the positively charged protons in the nucleus. As
the atomic number increases along each row of the periodic table, the additional electrons go into the
same outermost shell; whose radius gradually contracts, due to the increasing nuclear charge. In a noble
gas, the outermost shell is completely filled; therefore, the additional electron of next alkaly metal will
go into the next outer shell, accounting for the sudden increase in the atomic radius.The increasing
nuclear charge is partly counterbalanced by the increasing number of electrons, a phenomenon that is
known as shielding; which explains why the size of atoms usually increases down each column.
However, there are two occasions where shielding is less effective: in these cases, the atoms are smaller
than would otherwise be expected
The following table summarizes the main pheonomena that influence the atomic radius of an element:
factor
principle
increase with...
tend to
effect
electron
shells
quantum mechanics
principal and azimuthal
quantum numbers
atomic radius
increase
increase down each
column
nuclear
charge
attractive force acting on electrons by
protons in nucleus
atomic number
atomic radius
decrease
decrease along each
period
shielding
repulsive force acting on outermost shell
electrons by inner electrons
number of electron shells
atomic radius
increase
reduce the effect of
the 2nd factor
Lanthanide contraction
The electrons in the 4f-subshell, which is progressively filled from cerium (Z = 58) to lutetium (Z =
71), are not particularly effective at shielding the increasing nuclear charge from the sub-shells further
out. The elements immediately following the lanthanides have atomic radii which are smaller than
would be expected and which are almost identical to the atomic radii of the elements immediately
above them.[7] Hence hafnium has virtually the same atomic radius (and chemistry) as zirconium, and
tantalum has an atomic radius similar to niobium, and so forth. The effect of the lanthanide contraction
is noticeable up to platinum (Z = 78), after which it is masked by a relativistic effect known as the inert
pair effect.
Due to lanthanide contraction, the 5 following observations can be drawn:
1.
2.
3.
4.
5.
The size of Ln3+ ions regularly decreases with atomic number. According to Fajans' rules,
decrease in size of Ln3+ ions increases the covalent character and decreases the basic character
between Ln3+ and OH- ions in Ln(OH)3. Hence the order of size of Ln3+ is given:
La3+ > Ce3+ > ... , ... > Lu3+.
There is a regular decrease in their ionic radii.
There is a regular decrease in their tendency to act as a reducing agent, with increase in atomic
number.
The second and third rows of d-block transition elements are quite close in properties.
Consequently, these elements occur together in natural minerals and are difficult to separate.
d-Block contraction
Main article: d-block contraction
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Hasan Qayyum Chohan , Reg. No. 2009-CH-204
Chemical Engineering
UET Lahore
Applied Chemistery-I
Session 2009-10 , UET Lahore
The d-block contraction is less pronounced than the lanthanide contraction but arises
from a similar cause. In this case, it is the poor shielding capacity of the 3d-electrons
which affects the atomic radii and chemistries of the elements immediately following the first row of
the transition metals, from gallium (Z = 31) to bromine (Z = 35).
Calculated atomic radii
The following table shows atomic radii computed from theoretical models, as published by E. Clementi
and others in 1967. The values are in picometres (pm).
Group
1
(vertical)
Period
(horizontal
)
H
1
53
Li
16
2
7
3
4
5
6
7
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18
He
31
Be
11
2
M
Na
g
19
14
0
5
M
K Ca Sc Ti V Cr
Fe
n
24 19 18 17 17 16
15
16
3 4 4 6 1 6
6
1
M
Rb Sr Y Zr Nb
Tc Ru
o
26 21 21 20 19
18 17
19
5 9 2 6 8
3 8
0
Cs Ba
Hf Ta W Re Os
*
29 25
20 20 19 18 18
8 3
8 0 3 8 5
B C N O F Ne
87 67 56 48 42 38
Al
Si P S Cl Ar
11
111 98 88 79 71
8
Co Ni Cu
Ga
Zn
Ge As Se Br Kr
15 14 14
13
142
125 114 103 94 88
2 9 5
6
Rh Pd Ag
In
I
Cd
Sn Sb Te
Xe
17 16 16
15
11
161
145 133 123
108
3 9 5
6
5
Ir Pt Au
Tl
Hg
18 17 17
15
171
0 7 4
6
Uu Uu
Fr Ra ** Rf Db Sg Bh Hs Mt Ds Rg
b t
Pr Nd Pm Sm Eu Gd Tb
Lanthanid * La Ce
24 20 20 23 23 23 22
es
7 6 5 8 1 3 5
A C
** Ac Th Pa U Np Pu
Bk
Actinides
m m
3
Pb Bi Po At Rn
154 143 135
120
Uu Uu Uu Uu Uu
q p h s o
Dy
Er
Ho
Tm Yb Lu
22
22
222 222 217
8
6
Cf Es Fm Md No Lr
Hasan Qayyum Chohan , Reg. No. 2009-CH-204
Chemical Engineering
UET Lahore
Applied Chemistery-I
Session 2009-10 , UET Lahore
Ionic radius
Ionic Radius, rion, is a measure of the size of an ion in a crystal lattice. It is measured in either
picometres (pm) or Angstrom (Å), with 1 Å = 100 pm. Typical values range from 30 pm (0.3 Å) to
over 200 pm (2 Å).
The concept of ionic radius was developed independently by Goldschmidt and Pauling in the 1920s to
summarize the data being generated by the (at the time) new technique of X-ray crystallography: it is
Pauling's approach which proved to be the more influential. X-ray crystallography can readily give the
length of the side of the unit cell of a crystal, but it is much more difficult (in most cases impossible,
even with more modern techniques) to distinguish a boundary between two ions. For example, it can be
readily determined that each side of the unit cell of sodium chloride is 564.02 pm in length, and that
this length is twice the distance between the centre of a sodium ion and the centre of a chloride ion:
2[rion(Na+) + rion(Cl−)] = 564.02 pm
However, it is not apparent what proportion of this distance is due to the size of the sodium ion and
what proportion is due to the size of the chloride ion. By comparing many different compounds, and
with a certain amount of chemical intuition, Pauling decided to assign a radius of 140 pm to the oxide
ion O2−, at which point he was able to calculate the radii of the other ions by subtraction. [1]
A major review of crystallographic data led to the publication of a revised set of ionic radii in 1976, [2]
and these are preferred to Pauling's original values. Some sources have retained Pauling's reference of
rion(O2−) = 140 pm, while other sources prefer to list "effective" ionic radii based on rion(O2−) = 126 pm.
The latter values are thought to be a more accurate approximation to the "true" relative sizes of anions
and cations in ionic crystals.
The ionic radius is not a fixed property of a given ion, but varies with coordination number, spin state
and other parameters. Nevertheless, ionic radius values are sufficiently transferable to allow periodic
trends to be recognized. As with other types of atomic radius, ionic radii increase on descending a
group. Ionic size (for the same ion) also increases with increasing coordination number, and an ion in a
high-spin state will be larger than the same ion in a low-spin state. Anions (negatively charged) are
almost invariable larger than cations (positively charged), although the fluorides of some alkali metals
are rare exceptions. In general, ionic radius decreases with increasing positive charge and increases
with increasing negative charge.
An "anomalous" ionic radius in a crystal is often a sign of
X−
NaX
AgX
significant covalent character in the bonding. No bond is
F
464
492
completely ionic, and some supposedly "ionic" compounds,
especially of the transition metals, are particularly covalent in
Cl
564
555
character. This is illustrated by the unit cell parameters for
Br
598
577
sodium and silver halides in the table. On the basis of the
+
+
Unit cell parameters (in pm, equal to two M–
fluorides, one would say that Ag is larger than Na , but on the
X bond lengths) for sodium and silver
basis of the chlorides and bromides the opposite appears to be
halides. All compounds crystallize in the
[3]
true. This is because the greater covalent character of the bonds NaCl structure.
in AgCl and AgBr reduces the bond length and hence the
apparent ionic radius of Ag+, an effect which is not present in the
halides of the more electropositive sodium, nor in silver fluoride in which the fluoride ion is relatively
unpolarizable.
Covalent radius
The covalent radius, rcov, is a measure of the size of atom that forms part of a covalent bond. It is
measured either in picometres (pm) or ångströms (Å), with 1 Å = 100 pm.
In principle, the sum of the two covalent radii should equal the covalent bond length between two
atoms. This relationship does not hold exactly because the size of an atom is not constant but depends
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Hasan Qayyum Chohan , Reg. No. 2009-CH-204
Chemical Engineering
UET Lahore
Applied Chemistery-I
Session 2009-10 , UET Lahore
on its chemical environment. In particular, polar covalent bonds tend to be shorter than
would be expected on the basis of the sum of covalent radii. Tabulated values of
covalent radii are either average or idealized values, which nevertheless show a certain transferability
between different situations.
Covalent radii are measured by X-ray diffraction (more rarely, neutron diffraction on molecular
crystals). Rotational spectroscopy can also give extremely accurate values of bond lengths. One method
takes the covalent radius to be half the single bond length in the element, e.g. d(H–H, in H2) =
74.14 pm so rcov(H) = 37.07 pm: in practice, it is usual to obtain an average value from a variety of
covalent compounds, although the difference is usually small. Sanderson has published a recent set of
non-polar covalent radii for the main-group elements,[1] but the availability of large collections of bond
lengths, which are more transferable, from the Cambridge Crystallographic Database[2] has rendered
covalent radii obsolete in many situations.
Van der Waals radius
The van der Waals radius, rw, of an atom is the radius of an imaginary hard sphere which can be used
to model the atom for many purposes. It is named after Johannes Diderik van der Waals, winner of the
1910 Nobel Prize in Physics, as he was the first to recognise that atoms had a finite size (ie, that atoms
were not simply points) and to demonstrate the physical consequences of their size through the van der
Waals equation of state.
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Hasan Qayyum Chohan , Reg. No. 2009-CH-204
Chemical Engineering
UET Lahore