Wikipedia

Atomic radius

The atomic radius of a chemical element is a measure of the size of its atom, usually the mean or typical distance from the center of the nucleus to the outermost isolated electron. Since the boundary is not a well-defined physical entity, there are various non-equivalent definitions of atomic radius. Four widely used definitions of atomic radius are: Van der Waals radius, ionic radius, metallic radius and covalent radius. Typically, because of the difficulty to isolate atoms in order to measure their radii separately, atomic radius is measured in a chemically bonded state; however theoretical calculations are simpler when considering atoms in isolation. The dependencies on environment, probe, and state lead to a multiplicity of definitions.

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

Depending on the definition, the term may apply to atoms in condensed matter, covalently bonding in molecules, or in ionized and excited states; and its value may be obtained through experimental measurements, or computed from theoretical models. The value of the radius may depend on the atom's state and context.[1]

Electrons do not have definite orbits nor 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; these are referred to as atomic orbitals or electron clouds. 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 ångströms. 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]

History

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The concept of atomic radius was preceded in the 19th century by the concept of atomic volume, a relative measure of how much space would on average an atom occupy in a given solid or liquid material.[3] By the end of the century this term was also used in an absolute sense, as a molar volume divided by Avogadro constant.[4] Such a volume is different for different crystalline forms even of the same compound,[5] but physicists used it for rough, order-of-magnitude estimates of the atomic size, getting 10-8–10-7 cm for copper.[6]

The earliest estimates of the atomic size was made by opticians in the 1830s, particularly Cauchy,[7][8] who developed models of light dispersion assuming a lattice of connected "molecules".[9] In 1857 Clausius developed a gas-kinetic model which included the equation for mean free path. In the 1870s it was used to estimate gas molecule sizes, as well as an aforementioned comparison with visible light wavelength and an estimate from the thickness of soap bubble film at which its contractile force rapidly diminishes.[10] By 1900, various estimates of mercury atom diameter averaged around 275±20 pm[7] (modern estimates give 300±10 pm, see below).

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.[11] 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.[12]

Definitions

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Widely used definitions of atomic radius include:

  • 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.[13]
  • 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.[13]
  • 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).[15][16] 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, because it is equivalent to the quantum-mechanical most probable distance of the electron from the nucleus.

Empirically measured atomic radius

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The following table shows empirically measured covalent radii for the elements, as published by J. C. Slater in 1964.[17] 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 		*
  		Lu
175 		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 		**
  		Lr
  		Rf
  		Db
  		Sg
  		Bh
  		Hs
  		Mt
  		Ds
  		Rg
  		Cn
  		Nh
  		Fl
  		Mc
  		Lv
  		Ts
  		Og
 
*
  		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
**
  		Ac
195 		Th
180 		Pa
180 		U
175 		Np
175 		Pu
175 		Am
175 		Cm
  		Bk
  		Cf
  		Es
  		Fm
  		Md
  		No
 
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A graph comparing the atomic radius of elements with atomic numbers 1–100. Accuracy of ±5 pm.

Electrons in atoms fill electron shells from the lowest available energy level. As a consequence of the Aufbau principle, each new period begins with the first two elements filling the next unoccupied s-orbital. Because an atom's s-orbital electrons are typically farthest from the nucleus, this results in a significant increase in atomic radius with the first elements of each period.

The atomic radius of each element generally decreases across each period due to an increasing number of protons, since an increase in the number of protons increases the attractive force acting on the atom's electrons. The greater attraction draws the electrons closer to the protons, decreasing the size of the atom. Down each group, the atomic radius of each element typically increases because there are more occupied electron energy levels and therefore a greater distance between protons and electrons.

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 despite an increase in attractive force from the nucleus. Electron shielding causes the attraction of an atom's nucleus on its electrons to decrease, so electrons occupying higher energy states farther from the nucleus experience reduced attractive force, increasing the size of the atom. However, elements in the 5d-block (lutetium to mercury) are much smaller than this trend predicts due to the weak shielding of the 4f-subshell. This phenomenon is known as the lanthanide contraction. A similar phenomenon exists for actinides; however, the general instability of transuranic elements makes measurements for the remainder of the 5f-block difficult and for transactinides nearly impossible. Finally, for sufficiently heavy elements, the atomic radius may be decreased by relativistic effects.[18] This is a consequence of electrons near the strongly charged nucleus traveling at a sufficient fraction of the speed of light to gain a nontrivial amount of mass.

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

Factor Principle increase in... tend to effect on radius
electron shells quantum mechanics principal and azimuthal quantum numbers increase down each column increases the atomic radius
nuclear charge attractive force acting on electrons by protons in nucleus atomic number increase along each period (left to right) decreases the atomic radius
shielding repulsive force acting on outermost shell electrons by inner electrons number of electrons in inner shells reduce the effect of nuclear charge increases the atomic radius

Lanthanide contraction

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Main article: Lanthanide contraction

The electrons in the 4f-subshell, which is progressively filled from lanthanum (Z = 57) to ytterbium (Z = 70), 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.[19] Hence lutetium is in fact slightly smaller than yttrium, 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.[citation needed]

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, to the point that Yb(OH)3 and Lu(OH)3 can dissolve with difficulty in hot concentrated NaOH. 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 an 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

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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).[19]

Calculated atomic radius

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The following table shows atomic radii computed from theoretical models, as published by Enrico Clementi and others in 1967.[20] 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 		*
  		Lu
217 		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
127 		Rn
120
7 Fr
  		Ra
  		**
  		Lr
  		Rf
  		Db
  		Sg
  		Bh
  		Hs
  		Mt
  		Ds
  		Rg
  		Cn
  		Nh
  		Fl
  		Mc
  		Lv
  		Ts
  		Og
 
*
  		La
226 		Ce
210 		Pr
247 		Nd
206 		Pm
205 		Sm
238 		Eu
231 		Gd
233 		Tb
225 		Dy
228 		Ho
226 		Er
226 		Tm
222 		Yb
222
**
  		Ac
  		Th
  		Pa
  		U
  		Np
  		Pu
  		Am
  		Cm
  		Bk
  		Cf
  		Es
  		Fm
  		Md
  		No
 

See also

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References

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  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. ^ Knight, Charles (1859). The English Cyclopaedia: A New Dictionary of Universal Knowledge. Bradbury and Evans.
  4. ^ Fessenden, Reginald A. (1892-07-22). "The Laws and Nature of Cohesion". Science. ns-20 (494): 48–52. doi:10.1126/science.ns-20.494.48.b. ISSN 0036-8075.
  5. ^ Watts, Henry (1882). A Dictionary of chemistry and the allied branches of other sciences v. 3, 1882. Longmans, Green & Company.
  6. ^ Electrical World. McGraw-Hill. 1893.
  7. ^ a b Fessenden, Reginald Aubrey (February 1900). "A Determination of the Nature of the Electric and Magnetic Quantities and of the Density and Elasticity of the Ether, II". Physical Review (Series I). 10 (2): 83–115. doi:10.1103/PhysRevSeriesI.10.83. ISSN 1536-6065.
  8. ^ Thomson, W. (1870-07-01). "On the size of atoms". American Journal of Science. s2-50 (148): 38–44. doi:10.2475/ajs.s2-50.148.38.
  9. ^ Darrigol, Olivier (2012). A History of Optics from Greek Antiquity to the Nineteenth Century. OUP Oxford. ISBN 978-0-19-162745-3.
  10. ^ The American chemist: a monthly journal of theoretical, analytical and technical chemistry. C. F. & W. H. Chandler. 1877.
  11. ^ Bragg, W. L. (1920). "The arrangement of atoms in crystals". Philosophical Magazine. 6. 40 (236): 169–189. doi:10.1080/14786440808636111.
  12. ^ 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.
  13. ^ a b c Pauling, L. (1945). The Nature of the Chemical Bond (2nd ed.). Cornell University Press. LCCN 42034474.
  14. ^ Federov, Dmitry V.; Sadhukhan, Mainak; Stöhr, Martin; Tkatchenko, Alexandre (2018). "Quantum-Mechanical Relation between Atomic Dipole Polarizability and the van der Waals Radius". Physical Review Letters. 121 (18): 183401. arXiv:1803.11507. Bibcode:2018PhRvL.121r3401F. doi:10.1103/PhysRevLett.121.183401. PMID 30444421. S2CID 53564141. Retrieved 9 May 2021.
  15. ^ Bohr, N. (1913). "On the Constitution of Atoms and Molecules, Part I. – Binding of Electrons by Positive Nuclei" (PDF). Philosophical Magazine. 6. 26 (151): 1–24. Bibcode:1913PMag...26....1B. doi:10.1080/14786441308634955. Archived (PDF) from the original on 2011-09-02. Retrieved 8 June 2011.
  16. ^ Bohr, N. (1913). "On the Constitution of Atoms and Molecules, Part II. – Systems containing only a Single Nucleus" (PDF). Philosophical Magazine. 6. 26 (153): 476–502. Bibcode:1913PMag...26..476B. doi:10.1080/14786441308634993. Archived (PDF) from the original on 2008-12-09. Retrieved 8 June 2011.
  17. ^ 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.
  18. ^ Pitzer, Kenneth S. (1 August 1979). "Relativistic effects on chemical properties". Accounts of Chemical Research. 12 (8): 271–276. doi:10.1021/ar50140a001.
  19. ^ a b Jolly, W. L. (1991). Modern Inorganic Chemistry (2nd ed.). McGraw-Hill. p. 22. ISBN 978-0-07-112651-9.
  20. ^ 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.
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