Atomic radius

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Diagram of a helium atom, showing the electron probability density as shades of gray.

The atomic radius of a chemical element is a measure of the size of its atoms, usually the mean or typical distance from the center of 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. Three widely used definitions of atomic radius are Van der Waals radius, ionic radius, and covalent 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.[1]

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.

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),[2] and less than 1/1000 of the wavelength of visible light (400–700 nm).

The approximate shape of a molecule of ethanol, CH3CH2OH. Each atom is modeled by a sphere with the element's Van der Waals radius.

For many purposes, atoms can be modeled as spheres. This 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.[citation needed]

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 radii decrease across the Periodic Table because as the atomic number increases, the number of protons increases across the period, but the extra electrons are only added to the same quantum shell. Therefore, the effective nuclear charge towards the outermost electrons increases, drawing the outermost electrons closer. As a result, the electron cloud contracts and the atomic radii decreases.

History[edit]

In 1920, shortly after it had become possible to determine the sizes of atoms using X-ray crystallography, it was suggested that all atoms of the same element have the same radii.[3] However, in 1923, when more crystal data had become available, it was found that the approximation of an atom as a sphere does not necessarily hold when comparing the same atom in different crystal structures.[4]

Definitions[edit]

Widely used definitions of atomic radius include:

  • 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.[5]
  • Ionic radius: the nominal radius of the ions of an element in a specific ionization state, deduced from the spacing of atomic nuclei in crystalline salts 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.[5]
  • Covalent radius: the nominal radius of the atoms of an element when covalently bound to other atoms, as deduced from 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.[5]
  • Metallic radius: the nominal radius of atoms of an element when joined to other atoms by metallic bonds.[citation needed]
  • Bohr radius: the radius of the lowest-energy electron orbit predicted by Bohr model of the atom (1913).[6][7] 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.

Empirically measured atomic radii[edit]

The following table shows empirically measured covalent radii for the elements, as published by J. C. Slater in 1964.[8] The values are in picometers (pm or 1×10−12 m,), with an accuracy of about 5 pm. The shade of the box ranges from red to yellow as the radius increases; gray indicates lack of data.

Group
(column)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Period
(row)
1 H
25
He
 
2 Li
145
Be
105
B
85
C
70
N
65
O
60
F
50
Ne
 
3 Na
180
Mg
150
Al
125
Si
110
P
100
S
100
Cl
100
Ar
 
4 K
220
Ca
180
Sc
160
Ti
140
V
135
Cr
140
Mn
140
Fe
140
Co
135
Ni
135
Cu
135
Zn
135
Ga
130
Ge
125
As
115
Se
115
Br
115
Kr
 
5 Rb
235
Sr
200
Y
180
Zr
155
Nb
145
Mo
145
Tc
135
Ru
130
Rh
135
Pd
140
Ag
160
Cd
155
In
155
Sn
145
Sb
145
Te
140
I
140
Xe
 
6 Cs
260
Ba
215
*
 
Hf
155
Ta
145
W
135
Re
135
Os
130
Ir
135
Pt
135
Au
135
Hg
150
Tl
190
Pb
180
Bi
160
Po
190
At
 
Rn
 
7 Fr
 
Ra
215
**
 
Rf
 
Db
 
Sg
 
Bh
 
Hs
 
Mt
 
Ds
 
Rg
 
Cn
 
Uut
 
Fl
 
Uup
 
Lv
 
Uus
 
Uuo
 
Lanthanides *
 
La
195
Ce
185
Pr
185
Nd
185
Pm
185
Sm
185
Eu
185
Gd
180
Tb
175
Dy
175
Ho
175
Er
175
Tm
175
Yb
175
Lu
175
Actinides **
 
Ac
195
Th
180
Pa
180
U
175
Np
175
Pu
175
Am
175
Cm
 
Bk
 
Cf
 
Es
 
Fm
 
Md
 
No
 
Lr
 

Explanation of the general trends[edit]

A graph comparing the calculated atomic radius of elements with atomic numbers 1–100. Accuracy of ±5 pm.

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 alkali 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 is one notable exception, known as the lanthanide contraction: the 5d block of elements are much smaller than one would expect, due to the shielding caused by the 4f electrons.

The following table summarizes the main phenomena that influence the atomic radius of an element:

factor principle increase with... tend to effect on radius
electron shells quantum mechanics principal and azimuthal quantum numbers increase atomic radius increases down each column
nuclear charge attractive force acting on electrons by protons in nucleus atomic number decrease atomic radius decreases along each period
shielding repulsive force acting on outermost shell electrons by inner electrons number of electrons in inner shells increase atomic radius reduces the effect of the 2nd factor

Lanthanide contraction[edit]

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.[9] 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. 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+.
  2. There is a regular decrease in their ionic radii.
  3. There is a regular decrease in their tendency to act as a reducing agent, with increase in atomic number.
  4. The second and third rows of d-block transition elements are quite close in properties.
  5. Consequently, these elements occur together in natural minerals and are difficult to separate.

d-Block contraction[edit]

Main article: d-block contraction

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).[9]

Calculated atomic radii[edit]

The following table shows atomic radii computed from theoretical models, as published by Enrico Clementi and others in 1967.[10] The values are in picometres (pm).

Group
(column)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Period
(row)
1 H
53
He
31
2 Li
167
Be
112
B
87
C
67
N
56
O
48
F
42
Ne
38
3 Na
190
Mg
145
Al
118
Si
111
P
98
S
88
Cl
79
Ar
71
4 K
243
Ca
194
Sc
184
Ti
176
V
171
Cr
166
Mn
161
Fe
156
Co
152
Ni
149
Cu
145
Zn
142
Ga
136
Ge
125
As
114
Se
103
Br
94
Kr
88
5 Rb
265
Sr
219
Y
212
Zr
206
Nb
198
Mo
190
Tc
183
Ru
178
Rh
173
Pd
169
Ag
165
Cd
161
In
156
Sn
145
Sb
133
Te
123
I
115
Xe
108
6 Cs
298
Ba
253
*
 
Hf
208
Ta
200
W
193
Re
188
Os
185
Ir
180
Pt
177
Au
174
Hg
171
Tl
156
Pb
154
Bi
143
Po
135
At
 
Rn
120
7 Fr
 
Ra
 
**
 
Rf
 
Db
 
Sg
 
Bh
 
Hs
 
Mt
 
Ds
 
Rg
 
Cn
 
Uut
 
Fl
 
Uup
 
Lv
 
Uus
 
Uuo
 
Lanthanides *
 
La
 
Ce
 
Pr
247
Nd
206
Pm
205
Sm
238
Eu
231
Gd
233
Tb
225
Dy
228
Ho
 
Er
226
Tm
222
Yb
222
Lu
217
Actinides **
 
Ac
 
Th
 
Pa
 
U
 
Np
 
Pu
 
Am
 
Cm
 
Bk
 
Cf
 
Es
 
Fm
 
Md
 
No
 
Lr
 

See also[edit]

References[edit]

  1. ^ Cotton, F. A.; Wilkinson, G. (1988). Advanced Inorganic Chemistry (5th ed.). Wiley. p. 1385. ISBN 978-0-471-84997-1. 
  2. ^ Basdevant, J.-L.; Rich, J.; Spiro, M. (2005). Fundamentals in Nuclear Physics. Springer. p. 13, fig 1.1. ISBN 978-0-387-01672-6. 
  3. ^ Bragg, W. L. (1920). "The arrangement of atoms in crystals". Philosophical Magazine. 6 40 (236): 169–189. doi:10.1080/14786440808636111. 
  4. ^ Wyckoff, R. W. G. (1923). "On the Hypothesis of Constant Atomic Radii". Proceedings of the National Academy of Sciences of the United States of America 9 (2): 33–38. Bibcode:1923PNAS....9...33W. doi:10.1073/pnas.9.2.33. PMC 1085234. PMID 16576657. 
  5. ^ a b c Pauling, L. (1945). The Nature of the Chemical Bond (2nd ed.). Cornell University Press. LCCN 42034474. 
  6. ^ Bohr, N. (1913). "On the Constitution of Atoms and Molecules, Part I. – Binding of Electrons by Positive Nuclei". Philosophical Magazine. 6 26 (151): 1–24. doi:10.1080/14786441308634955. Retrieved 8 June 2011. 
  7. ^ Bohr, N. (1913). "On the Constitution of Atoms and Molecules, Part II. – Systems containing only a Single Nucleus". Philosophical Magazine. 6 26 (153): 476–502. doi:10.1080/14786441308634993. Retrieved 8 June 2011. 
  8. ^ Slater, J. C. (1964). "Atomic Radii in Crystals". Journal of Chemical Physics 41 (10): 3199–3205. Bibcode:1964JChPh..41.3199S. doi:10.1063/1.1725697. 
  9. ^ a b Jolly, W. L. (1991). Modern Inorganic Chemistry (2nd ed.). McGraw-Hill. p. 22. ISBN 978-0-07-112651-9. 
  10. ^ Clementi, E.; Raimond, D. L.; Reinhardt, W. P. (1967). "Atomic Screening Constants from SCF Functions. II. Atoms with 37 to 86 Electrons". Journal of Chemical Physics 47 (4): 1300–1307. Bibcode:1967JChPh..47.1300C. doi:10.1063/1.1712084.