A set of theoretical atomic radii corresponding to the principal maximum in the radial distribution function, 4πr^{2}_{R}^{2} for the outermost orbital has been calculated for the ground state of 103 elements of the periodic table using Slater orbitals. The set of theoretical radii are found to reproduce the periodic law and the Lother Meyer’s atomic volume curve and reproduce the expected vertical and horizontal trend of variation in atomic size in the periodic table. The d-block and f-block contractions are distinct in the calculated sizes. The computed sizes qualitatively correlate with the absolute size dependent properties like ionization potentials and electronegativity of elements. The radii are used to calculate a number of size dependent periodic physical properties of isolated atoms viz., the diamagnetic part of the atomic susceptibility, atomic polarizability and the chemical hardness. The calculated global hardness and atomic polarizability of a number of atoms are found to be close to the available experimental values and the profiles of the physical properties computed in terms of the theoretical atomic radii exhibit their inherent periodicity. A simple method of computing the absolute size of atoms has been explored and a large body of known material has been brought together to reveal how many different properties correlate with atomic size.

The concept of atomic and ionic radii has been found to be very useful in understanding, explaining, correlating and even predicting many physico-chemical properties of atoms, ions and molecules. The periodicity of chemical and physical properties of the elements has been recognized from the date of early history of chemistry. The atomic volume curve of Lother Meyer was a striking example of periodicity of physical properties of elements. The atomic size has been also an important periodic property of the elements. The atomic and ionic radii values are important input in many calculations of size dependent physico–chemical properties of isolated atoms. The crystal chemists have tried for many years to treat atoms and ions as hard spheres and published series of atomic and ionic radii with the only significance that such atomic and ionic radii, when added, reproduce the minimum distance of separation between atoms and ions respectively in crystal lattices. Approximate additivity of atomic and ionic radii were noted by early investigators like Goldschmidt et al [

According to quantum mechanical view, the atoms and the ions do not have any rigid shape or size and hence the question of atomic and ionic radii simply does not arise in the true sense of the term. However, chemical experience suggests that the atoms and ions do have an effective size because, the atoms and ions cannot approach each other beyond certain limiting distance under the influence of forces encountered in chemical interactions. The determination of the empirical atomic and ionic radii, on the basis of hard sphere approximation model, has a history stretching back to the work of Bragg^{4} in 1920’s to the work of Slater [

The inherent approximation of the SCF method suggests that each electron have its own one-electron function or orbital in many–electron situation. The fact that each electron in a many electron system should have one-electron function is a pre-quantum mechanical thinking [

Slater [_{nl}(r) is given by,
_{nl}(r) = (2ξ)^{n+1/2}[(2n)!]^{-1/2}r^{n-1}exp(−ξr)

Where,
^{*}

Where, Z= atomic number and S is the screening constant and n^{*} is the effective principal quantum number. The quantity, (Z–S) is further identified as effective nuclear charge, Z^{*}.

In terms of Slater orbitals the theoretical atomic or ionic radii,
_{max} = n / ξ

The eqn. (3) is the formula of computation of theoretical atomic and ionic radii in terms of Slater orbital. This shows that theoretical radii of atoms and ions can be easily computed only if ξ is known and one can know ξ, if the effective principal quantum number of the outermost shell of the atom (n^{*}) and the screening constant (S) are known. Although the SCF ξ is also available [^{*} to give good approximations to the best atomic orbitals of this type. The goal of this work has been to explore the possibility of developing a simple but effective method of computing the absolute radii atoms and ions relying upon Slater’s definition of absolute radii and using Slater’s orbitals. It is well known that orbital exponent, ξ obtained by Slater’s empirical rules are found useful in SCF calculation giving satisfactory results although use of SCF optimized ξ is simultaneously in use [

In the present venture we have computed the theoretical atomic radii of as many as 103 elements of the periodic table by semi-empirical method using Slater’s orbital and involving Slater’s empirical rule for the computation of ξ. Since there is no scope of comparing the atomic and ionic radii with the experimental values, we have tested the range of validity of calculated sizes in a number of ways. The periodicity exhibited by the computed radii is compared with the periodicity inherent in the periodic table. In order to explore the internal consistency between the calculated sizes and the size dependent properties of the atoms, a comparative study of size vis-a-vis the electronegativity and the experimental quantity like first ionization potential of the atoms is made. A comparative study of the relative magnitudes of the sizes of the atoms of present calculation and the radii published by other workers – both theoretical and experimental is also made. The computed theoretical radii are used to calculate a number of size dependent intrinsic physical properties of isolated atoms viz., (1) the diamagnetic part of magnetic susceptibility (χ_{dia}) of 54 elements, and (2) the atomic polarizability (α), and (3) the global hardness (η) of as many as 103 elements of the periodic table. We have compared such computed size dependent properties of the atoms with available results of such quantities.

The radial charge density distribution function is defined [^{2} R^{2} or simply r^{2}R^{2} where R is the radial function. According to Slater [

Radial charge density distribution function ρ(r) is given by
^{2}R^{2}
^{2}(2ξ)^{2n+1}[(2n)!]^{-1}r^{2n-2}exp(-2ξr)^{2n}(2ξ)^{2n+1}[(2n)!]^{-1}exp(-2ξr)

Differentiating the left-hand side with respect to r, and equating the result with zero, we get the maximum of the radial charge density distribution function, the theoretical atomic radii,
^{2n+1}[(2n)!]^{-1}exp(-2ξr)][2nr^{2n-1}-2ξr^{2n}]
_{max}, we obtain,
_{max}^{2n-1} – ξr_{max}^{2n}) = 0
_{max} ^{–1} = ξ

From this relation we obtain the formula for computing the theoretical atomic or ionic radii already noted in eqn (3) above, i.e. the theoretical atomic or ionic radii r = r_{max}

The screening constant (S) may be calculated from Slater’s rule and are profusely available in any standard textbook of physical and inorganic chemistry. The values of n^{*} for principal quantum number n up to 6 and Z^{*} for about 26 elements are published by Pople [^{*} are calculated by simple extrapolation. and we have got the extrapolated value of n^{*} = 4.3 for principal quantum number 7. The Z^{*} and ξ for required atoms are calculated and for that matter we have relied upon the ground state electron–configuration published by Shriver and Atkins [

Plot of atomic radii (angstrom) as a function of atomic number.

Plot of atomic radii (angstrom) as a function of atomic number.

Calculated orbital exponents (ξ) and radii of atoms.

Atoms | Orbital Exponent | Atomic Radii (Å) | Atoms | Orbital Exponent | Atomic Radii (Å) | Atoms | Orbital Exponent | Atomic Radii (Å) |
---|---|---|---|---|---|---|---|---|

H | 1 | 0.5292 | Kr | 2.2297 | 0.9493 | Lu | 3 | 1.0583 |

He | 1.7 | 0.3113 | Rb | 0.55 | 4.8106 | Hf | 3.15 | 1.0079 |

Li | 0.65 | 1.6282 | Sr | 0.7125 | 3.7135 | Ta | 3.3095 | 0.9594 |

Be | 0.975 | 1.0855 | Y | 0.75 | 3.5278 | W | 3.4643 | 0.9165 |

B | 1.3 | 0.8141 | Zr | 0.7875 | 3.3598 | Re | 3.619 | 0.8773 |

C | 1.625 | 0.6513 | Nb | 0.825 | 3.2071 | Os | 3.7738 | 0.8413 |

N | 1.95 | 0.5427 | Mo | 0.8625 | 3.0677 | Ir | 3.9286 | 0.8182 |

O | 2.275 | 0.4652 | Tc | 0.9 | 2.9398 | Pt | 4.0833 | 0.7776 |

F | 2.6 | 0.4071 | Ru | 0.9375 | 2.8222 | Au | 4.2381 | 0.7492 |

Ne | 2.925 | 0.3618 | Rh | 0.975 | 2.7137 | Hg | 4.3929 | 3.0636 |

Na | 0.7333 | 2.1649 | Pd | 1.0125 | 2.6132 | Tl | 1.1905 | 2.667 |

Mg | 0.95 | 1.6711 | Ag | 1.05 | 2.5199 | Pb | 1.3452 | 2.3603 |

Al | 1.1667 | 1.3607 | Cd | 1.0875 | 2.433 | Bi | 1.5 | 2.1167 |

Si | 1.3833 | 1.1476 | In | 1.25 | 2.1167 | Po | 1.6548 | 1.9187 |

P | 1.6 | 0.9922 | Sn | 1.4125 | 1.8732 | At | 1.8095 | 1.7546 |

S | 1.8167 | 0.8738 | Sb | 1.575 | 1.6799 | Rn | 1.9643 | 1.6164 |

Cl | 2.0333 | 0.7807 | Te | 1.7375 | 1.5228 | Fr | 0.5116 | 7.2404 |

Ar | 2.25 | 0.7056 | I | 1.9 | 1.3926 | Ra | 0.6628 | 5.5887 |

K | 0.5946 | 3.5598 | Xe | 2.0625 | 1.2828 | Ac | 0.6977 | 5.3091 |

Ca | 0.7703 | 2.7479 | Cs | 0.5238 | 6.0615 | Th | 0.7326 | 5.0569 |

Sc | 0.8108 | 2.6106 | Ba | 0.6786 | 4.6788 | Pa | 1 | 3.7042 |

Ti | 0.8514 | 2.4861 | La | 0.8333 | 3.8102 | U | 1.1512 | 3.2177 |

V | 0.8919 | 2.3732 | Ce | 0.9881 | 3.2133 | Np | 1.3023 | 2.8443 |

Cr | 0.9324 | 2.2701 | Pr | 1.1429 | 2.778 | Pu | 1.5698 | 2.3596 |

Mn | 0.973 | 2.1754 | Nd | 1.2976 | 2.4468 | Am | 1.7209 | 2.1525 |

Fe | 1.0135 | 2.0885 | Pm | 1.4524 | 2.1861 | Cm | 1.7558 | 2.1097 |

Co | 1.0541 | 2.008 | Sm | 1.6071 | 1.9756 | Bk | 2.0233 | 1.8308 |

Ni | 1.0946 | 1.9337 | Eu | 1.7619 | 1.802 | Cf | 2.1744 | 1.7035 |

Cu | 1.1351 | 1.8648 | Gd | 1.9167 | 1.6565 | En | 2.3256 | 1.5928 |

Zn | 1.1757 | 1.8004 | Tb | 2.0714 | 1.5328 | Fm | 2.4767 | 1.4956 |

Ga | 1.3514 | 1.5663 | Dy | 2.2262 | 1.4262 | Md | 2.6279 | 1.4096 |

Ge | 1.527 | 1.3862 | Ho | 2.381 | 1.3335 | No | 2.7791 | 1.3329 |

As | 1.7027 | 1.2431 | Er | 2.5357 | 1.2521 | Lr | 2.814 | 1.3164 |

Se | 1.8784 | 1.1269 | Tm | 2.6905 | 1.1801 | |||

Br | 2.0541 | 1.0305 | Yb | 2.8452 | 1.1159 |

_{dia}), (B) atomic polarizabilities (α), and (C) the global hardness (η) are stated below.

Plot of atomic radii of first transition sereis (3-d block) elements in angstrom unit as a function of atomic number.

Plot of atomic radii of second transition series (4-d block) elements in angstrom unit as a function of atomic number.

Plot of atomic radii of lanthanides in angstrom unit as function of atomic number.

Plot of atomic radii of actinides in angstrom unit as a function of Atomic number.

_{dia}

The theoretical determination of atomic and ionic radii provides with a scope of a theoretical calculation of the diamagnetic part of the atomic susceptibility, which occurs even when an atom already has a permanent moment. We have computed the diamagnetic susceptibility of as many as 54 elements starting from hydrogen. Since atoms after element with atomic number 54 are strongly paramagnetic due to L-S coupling, the computation of diamagnetic susceptibility of atoms of such elements is not taken up.

The induced diamagnetic moment of an atom is opposite to the applied field and is proportional to the square of the orbit radius. The classical equation of diamagnetic susceptibility of atom [_{dia} = – 2.84 × 10^{-10} Σ_{n} <r^{2}>_{av}

Where, <r^{2}>_{av} is the mean square of the actual orbital radius or the average of the square of all the orbit radii and Σ implies the summation of <r^{2}>_{av} for all n electrons in atom as the total moment is given by the sum over all the electrons in the atom.

There can not be any experimental value for Σ_{n} <r^{2}>_{av} of an atom as such and quantum mechanics can compute it and it is interpreted as the average of the square of the distance from the center for the probability distribution [_{dia} = – 1.888 × 10^{–10} Σ_{n} <r^{2}>_{av}

We have calculated the <r^{2}>_{av} of each atom by calculating the radii of each orbital. We have computed the <r^{2}>_{av} for all the 54 elements. With these values of <r^{2}>_{av}, the molar diamagnetic susceptibility is computed through eqn.(9). The calculated <r^{2}>_{av} and χ_{dia} are shown in ^{2}>_{av} and χ_{dia} are plotted as a function of atomic number in

Plot of mean square radii and diamagnetic part of magnetic susceptibility of atoms as a function of atomic number.

One more important size dependent physical property of the atoms is the atomic polarizability, α. By polarizability, Pearson [

Computed mean square radii, < r^{2} > and diamagnetic part of atomic susceptibility, (χ_{dia}) of atoms.

Atom | < r^{2} > X 10^{–16} sq.cm |
χ_{dia x 10}^{–6} c.c |
Atoms | < r^{2} > X 10^{–16} sq.cm |
χ_{dia x 10}^{–6} c.c |
---|---|---|---|---|---|

H | 0.2801 | -0.5287 | Ni | 1.5669 | -82.8339 |

He | 0.0969 | -0.3659 | Cu | 1.5013 | -82.2003 |

Li | 1.3447 | -7.6165 | Zn | 1.4422 | -81.6834 |

Be | 0.5994 | -4.5267 | Ga | 0.9694 | -56.7355 |

B | 0.3377 | -3.1882 | Ge | 0.7087 | -42.8176 |

C | 0.2164 | -2.4513 | As | 0.5452 | -33.912 |

N | 0.1504 | -1.9872 | Se | 0.4345 | -27.8915 |

O | 0.1106 | -1.6698 | Br | 0.3553 | -23.4787 |

F | 0.0847 | -1.4395 | Kr | 0.2965 | -20.1491 |

Ne | 0.0669 | -1.264 | Rb | 4.9346 | -344.7138 |

Na | 1.5972 | -33.1713 | Sr | 2.9131 | -208.9958 |

Mg | 0.9558 | -21.6546 | Y | 3.8844 | -286.0142 |

Al | 0.6368 | -15.6297 | Zr | 3.6483 | -275.519 |

Si | 0.4549 | -12.0238 | Nb | 3.444 | -266.593 |

P | 0.3413 | -9.6652 | Mo | 3.266 | -258.9836 |

S | 0.2655 | -8.0205 | Tc | 3.1099 | -252.4745 |

Cl | 0.2125 | -6.8214 | Ru | 2.9723 | -246.9151 |

Ar | 0.1741 | -5.915 | Rh | 2.8505 | -242.1745 |

K | 3.2677 | -117.2196 | Pd | 2.7419 | -238.1305 |

Ca | 1.9664 | -74.2518 | Ag | 2.6448 | -234.6907 |

Sc | 2.338 | -92.6958 | Cd | 2.5577 | -231.7906 |

Ti | 2.1789 | -90.5027 | In | 1.6726 | -154.7359 |

V | 2.0413 | -88.6433 | Sn | 1.2015 | -113.4178 |

Cr | 1.9214 | -87.0608 | Sb | 0.9035 | -87.9572 |

Mn | 1.8158 | -85.7077 | Te | 0.7218 | -70.8638 |

Fe | 1.723 | -84.5766 | I | 0.5866 | -58.7022 |

Co | 1.6403 | -83.6167 | Xe | 0.4872 | -49.6685 |

The induced dipole moment in an atom or molecule is proportional to the applied electric field [

It has been shown [^{3}.
^{3}
^{3}
^{3}

where, K is the proportionality constant.

For atoms, the value of ‘K’ was determined by Dimitrieva and Plindov [_{0}^{3}
_{0} is the Bohr radius.

Comparing eqns. (14) and (15) we see that K = 4.5

Relying more upon the quantum mechanical derivation [

Plot of atomic polarizability (c.c) as a function of atomic number.

Computed atomic polarizability ( α) and global hardness (η) of atoms.

Atoms | α x 10^{–24} c.c |
η (ev) | Atoms | α x 10^{–24} c.c |
η (ev) | Atoms | α x 10^{–24} c.c |
η (ev) |
---|---|---|---|---|---|---|---|---|

H | 0.6669 | 13.588 | Kr | 3.8497 | 7.5748 | Lu | 5.3338 | 6.7947 |

He | 0.1358 | 22.383 | Rb | 500.9683 | 1.4948 | Hf | 4.6075 | 7.1344 |

Li | 19.4239 | 4.4164 | Sr | 230.4462 | 1.9364 | Ta | 3.9739 | 7.4951 |

Be | 5.7558 | 6.6244 | Y | 197.5715 | 2.0383 | W | 3.4643 | 7.8459 |

B | 2.428 | 8.8328 | Zr | 170.6683 | 2.1402 | Re | 3.0385 | 8.1965 |

C | 1.2432 | 11.0407 | Nb | 148.4397 | 2.2422 | Os | 2.6796 | 8.5472 |

N | 0.7193 | 13.25 | Mo | 129.9126 | 2.344 | Ir | 2.3756 | 8.8973 |

O | 0.453 | 15.4574 | Tc | 114.3315 | 2.446 | Pt | 2.1175 | 9.2474 |

F | 0.3034 | 17.6634 | Ru | 101.1523 | 2.5479 | Au | 1.8924 | 9.598 |

Ne | 0.2131 | 19.875 | Rh | 89.9286 | 2.6498 | Hg | 129.646 | 2.3456 |

Na | 45.659 | 3.3215 | Pd | 80.3028 | 2.7517 | Tl | 85.3653 | 2.6962 |

Mg | 21 | 4.303 | Ag | 72.005 | 2.8536 | Pb | 59.1717 | 3.0466 |

Al | 11.337 | 5.2846 | Cd | 64.8095 | 2.9555 | Bi | 42.6767 | 3.3972 |

Si | 6.8011 | 6.2659 | In | 42.6767 | 3.3972 | Po | 31.7858 | 3.7477 |

P | 4.3955 | 7.2473 | Sn | 29.5777 | 3.8388 | At | 24.3079 | 4.0983 |

S | 3.0023 | 8.2293 | Sb | 21.3335 | 4.2805 | Rn | 19.0046 | 4.4487 |

Cl | 2.1412 | 9.2107 | Te | 15.8906 | 4.7221 | Fr | 1705 | 0.9913 |

Ar | 1.5808 | 10.191 | I | 12.1532 | 5.0636 | Ra | 785.4977 | 1.2867 |

K | 202.9969 | 2.02 | Xe | 9.4993 | 5.6056 | Ac | 673.4033 | 1.3544 |

Ca | 93.3717 | 2.6162 | Cs | 1002.1964 | 1.1863 | Th | 581.6815 | 1.4222 |

Sc | 80.0633 | 2.7545 | Ba | 460.9098 | 1.5369 | Pa | 228.7156 | 1.9413 |

Ti | 69.1712 | 2.8924 | La | 248.9177 | 1.8873 | U | 149.9164 | 2.2348 |

V | 60.1472 | 3.03 | Ce | 149.2883 | 2.2378 | Np | 103.5473 | 2.5281 |

Cr | 52.6438 | 3.1676 | Pr | 96.4738 | 2.5885 | Pu | 59.1266 | 3.0473 |

Mn | 46.3265 | 3.3055 | Nd | 65.9186 | 2.9389 | Am | 44.8789 | 3.3407 |

Fe | 40.9939 | 3.443 | Pm | 47.0135 | 3.2893 | Cm | 42.2547 | 3.4084 |

Co | 36.4555 | 3.5811 | Sm | 34.6984 | 3.6398 | Bk | 27.5918 | 3.9277 |

Ni | 32.5372 | 3.7187 | Eu | 26.3316 | 3.9905 | Cf | 22.2453 | 4.2212 |

Cu | 29.1769 | 3.8561 | Gd | 20.4544 | 4.341 | En | 18.1843 | 4.5146 |

Zn | 26.2615 | 3.994 | Tb | 16.2057 | 4.6913 | Fm | 15.0542 | 4.808 |

Ga | 17.2917 | 4.5909 | Dy | 13.0543 | 5.0419 | Md | 12.6038 | 5.1013 |

Ge | 11.9864 | 5.1874 | Ho | 10.6707 | 5.3924 | No | 10.6563 | 5.3949 |

As | 8.6443 | 5.7846 | Er | 8.8334 | 5.743 | Lr | 10.2631 | 5.4629 |

Se | 6.4397 | 6.381 | Tm | 7.3955 | 6.0934 | |||

Br | 4.9244 | 6.978 | Yb | 6.253 | 6.4439 |

The term hardness,η, as applied to atoms and molecules has no reference to its mechanical physical hardness [

The operational definition of hardness, η

The energy, E (q) of charging a conducting sphere of radius R with charge q is classically given by [^{2}/(4πε_{0}) 2R
^{2}/2R

The implication of the eqn (18) is that the E(q) is the energy in ergs, q is the charge in electrostatic unit and R should be measured in cm. Now increasing q by one unit and decreasing q by one unit, we get the changes in energy which are I and A, the ionization potential and electron affinity of atom respectively.

Now in e.s.u, the one unit of charge, which is associated with the process of ionization I, and the electron affinity A, should be e, the electronic charge. Hence,
^{2}/2R – q^{2}/2R
^{2}/2R – (q – e) ^{2}/2R

Or,
^{2}/ 2R

But the formula derived by Pearson [

We have computed the global hardness of atoms, η, through eqn (19) and ultimately it is converted to electron volt (eV). The theoretically computed values of global hardness of as many as 103 atoms are tabulated in

Plot of Global hardness (eV) of atoms as a function of atomic number.

The periodicity of elements, along with Darwin’s theory of evolution and Planck’s quantum theory, ranks as one of the greatest generalizations in Science [

Atomic and ionic size decreases along the row.

Atomic and ionic size increases down the main group.

There is d–block contraction.

There is f–block contraction.

The periodicity of atomic size should be isomorphic with the periodicity of ionization potential.

The periodicity of size should be matched with the periodicity of electronegativity

The size relationship as a function of vertically downward or horizontally rightward movement will be straightforward if the sizes of the atoms are extrapolated as a function of atomic number. We have drawn the sizes of atoms as a function of atomic numbers in two different fashions in

Thereafter, in order to explore a correlation between the computed atomic size with the absolute periodic properties of atoms we now examine the nature of the curves of the first ionization potential and the electronegativities drawn as a function of atomic number in ^{35} method utilized the regularities of the curve found by plotting radii vs. ionization potential for the determination of atomic radii of elements from graphical extrapolation.

Plot of atomic radii and first ionization potential of atoms as a function of atomic number.

A close look at the

Plot of atomic radii and electronegativity of atoms as a function of atomic number.

To make a comparative study of the theoretical radii vis-à-vis the so called experimental radii, we have chosen the experimental values of Pauling [

The following general observations transpire from a comparative study of the sizes of atoms.

(i) From

Comparative study of theoretical radii and Pauling’s experimental metallic radii of atoms.

Atoms | Theoretical atomic radii (Å) | Single bond metallic radii (Å) | Atoms | Theoretical atomic radii (Å) | Single bond metallic radii (Å) |
---|---|---|---|---|---|

Li | 1.6282 | 1.225 | Mo | 3.0677 | 1.296 |

Be | 1.0855 | 0.889 | Tc | 2.9398 | 1.271 |

Na | 2.1649 | 1.572 | Ru | 2.8222 | 1.246 |

Mg | 1.6711 | 1.364 | Rh | 2.7137 | 1.252 |

Al | 1.3607 | 1.248 | Pd | 2.6132 | 1.283 |

K | 3.5598 | 2.025 | Ag | 2.5199 | 1.339 |

Ca | 2.7479 | 1.736 | Cd | 2.433 | 1.413 |

Sc | 2.6106 | 1.439 | In | 2.1167 | 1.497 |

Ti | 2.4861 | 1.324 | Sn | 1.8732 | 1.399 |

V | 2.3732 | 1.224 | Cs | 6.0615 | 2.35 |

Cr | 2.2701 | 1.176 | Ba | 4.6788 | 1.981 |

Mn | 2.1754 | 1.171 | La | 3.8102 | 1.69 |

Fe | 2.0885 | 1.165 | Hf | 1.0079 | 1.442 |

Co | 2.008 | 1.162 | Ta | 0.9594 | 1.343 |

NI | 1.9337 | 1.154 | W | 0.9165 | 1.304 |

Cu | 1.8648 | 1.173 | Re | 0.8773 | 1.283 |

Zn | 1.8004 | 1.249 | Os | 0.8413 | 1.26 |

Ga | 1.5663 | 1.245 | Ir | 0.8082 | 1.265 |

Ge | 1.3862 | 1.223 | Pt | 0.7776 | 1.295 |

Rb | 4.8106 | 2.16 | Au | 0.7492 | 1.336 |

Sr | 3.7135 | 1.914 | Hg | 3.0656 | 1.44 |

Y | 3.5278 | 1.616 | Tl | 2.667 | 1.549 |

Zr | 3.3598 | 1.454 | Pb | 2.3603 | 1.539 |

Nb | 3.2071 | 1.342 |

Comparative study of theoretical radii and Pauling’s experimental covalent radii of atoms.

Atoms | Theoretical atomic radii (Å) | Experimental covalent radii (Å) | Atoms | Theoretical atomic radii (Å) | Experimental covalent radii (Å) |
---|---|---|---|---|---|

B | 0.8141 | 0.88 | Cl | 0.7807 | 0.99 |

C | 0.6531 | 0.77 | As | 1.2431 | 1.18 |

N | 0.5427 | 0.7 | Se | 1.1269 | 1.14 |

O | 0.4652 | 0.66 | Br | 1.0305 | 1.11 |

F | 0.4071 | 0.64 | Sb | 1.6799 | 1.36 |

Si | 1.1476 | 1.17 | Te | 1.5228 | 1.32 |

P | 0.9922 | 1.1 | I | 1.3926 | 1.28 |

S | 0.8738 | 1.04 |

Comparative study of absolute radii and covalent radii of Politzer-Parr-Murphy.

Atoms | Covalent radii (Å) | Absolute radii (Å) |
---|---|---|

Li | 1.357 | 1.6282 |

Na | 1.463 | 2.1649 |

Al | 1.487 | 1.3607 |

K | 1.802 | 3.5598 |

Cr | 1.494 | 2.2701 |

Ni | 1.3 | 1.9337 |

Cu | 1.166 | 1.8648 |

Rb | 1.924 | 4.8106 |

Ag | 1.28 | 2.5199 |

Sn | 1.492 | 1.8732 |

Te | 1.381 | 1.5228 |

B | 1.091 | 0.8141 |

C | 0.912 | 0.6513 |

N | 0.814 | 0.5427 |

O | 0.764 | 0.4652 |

F | 0.671 | 0.4071 |

Si | 1.296 | 1.1476 |

P | 1.185 | 0.9922 |

S | 1.12 | 0.8738 |

Cl | 0.999 | 0.7807 |

As | 1.258 | 1.2431 |

Se | 1.209 | 1.1269 |

Br | 1.116 | 1.0305 |

Sb | 1.433 | 1.6799 |

I | 1.299 | 1.3926 |

(ii) From

(iii) A comparative study is also made between theoretical covalent radii of atoms of Politzer, Parr and Murphy [

It is sporadically sprinkled impression in the chemical literature that the inert gas atoms are unusually big in size. The published data very often shows that radii of the inert gas atoms is the biggest in the row where it occurs and that too, in violation of periodic law. Looking at the

Comparative study of theoretical radii and Pauling’s experimental van der Waal’s radii of atoms.

Atoms | Theoretical atomic radii (Å) | Experimental van der Waal's radii (Å) | Atoms | Theoretical atomic radii (Å) | Experimental van der Waal's radii (Å) |
---|---|---|---|---|---|

N | 0.5427 | 1.5 | Se | 1.1269 | 2 |

P | 0.9922 | 1.9 | Te | 1.5228 | 2.2 |

As | 1.2431 | 2 | F | 0.4071 | 1.35 |

Sb | 1.6799 | 2.2 | Cl | 0.7807 | 1.8 |

H | 0.5292 | 1.2 | Br | 1.0305 | 1.95 |

O | 0.4652 | 1.4 | I | 1.3926 | 2.15 |

S | 0.8738 | 1.85 |

The available data regarding the size of the inert gas atoms in the chemical literature are all van der Waal radii. Since the van der Waal force of attraction is very weak, the gap between the atoms bonded by van der Waal bond is large and the van der Waal radii so computed must be very large. Thus, there is no reason of mysterious swelling of size of inert gas atoms as implied by their experimental radii. Justifiably, the absolute sizes of such atoms must be considerably smaller than their van der Waals radii. The radii of inert gas atoms in the present calculation are in conformity with the periodic law.

From the

We know that according to the suggestion of Seaborg [

Waber and Cromer [

Partington [^{2}>_{av} and χ_{dia} are shown in ^{2}>_{av} is quite periodic in nature and mimics pattern of the atomic size curve and it further transpires that the diamagnetic part of magnetic susceptibility of the atoms, χ_{dia,} basically a size property, varies periodically like other periodic properties of elements. The study of the nature of the profiles of the two curves in _{dia}, and the curve of <r^{2}>_{av} are homomorphic to each other. Now comparing the pattern of the profile of the diamagnetic susceptibility of atoms in _{dia} of the present calculation (_{dia}, of inert gas atoms as calculated by the present method and that published in the CRC Handbook [

Chattaraj and Maity [

From the eqn.14 we find that the atomic polarizability,α is directly proportional to the size of the atoms. Comparing the experimental [

Comparative study of theoretical and experimental atomic polarizability of some elements.

Atoms | Theoretical Atomic Polarizability (x 10^{–24} cc.) |
Experimental Atomic Polarizability (x 10^{–24} cc.) |
---|---|---|

H | 0.6669 | 0.66 |

He | 0.1358 | 0.21 |

Li | 19.4239 | 12 |

Be | 5.7558 | 9.3 |

C | 2.4279 | 1.5 |

Ne | 1.2432 | 0.4 |

Na | 45.659 | 27 |

Ar | 1.5808 | 1.6 |

K | 202.9969 | 34 |

The hardness is an intrinsic periodic physical property of atoms [

The terms atomic and ionic radii are very popular in chemical literature and the concept has been largely employed in rationalizing various physico-chemical properties of atoms and molecules and is a useful parameter of electronic structure theory. We have explored a simple method of computing the absolute size of atoms and brought together a large body of known material to reveal how many different properties correlate with atomic size. Although a large body of scattered information on sizes of atoms and ions has appeared in the literature, a critical analysis of the status of a reported set of radii

Comparative study of theoretical and absolute global hardness (atoms having theoretical hardness close within 0.1–1.0 eV to the experimental are given asterisks).

Atoms | Theoretical Global hardness of atoms (eV) | Absolute Global hardness of atoms (eV) | Atoms | Theoretical Global hardness of atoms (eV) | Absolute Global hardness of atoms (eV) |
---|---|---|---|---|---|

H | 13.588 | 6.43 | Rb* | 1.4948 | 1.85 |

Li | 4.4164 | 2.39 | Sr | 1.9364 | 3.7 |

Be | 6.6244 | 4.5 | Yb | 2.0383 | 3.19 |

B | 8.8328 | 4.01 | Zr | 2.1402 | 3.21 |

C | 11.0407 | 5 | Nb* | 2.2422 | 3 |

N | 13.25 | 7.23 | Mo* | 2.344 | 3.1 |

O | 15.4574 | 6.08 | Ru* | 2.5479 | 3 |

F | 17.6634 | 7.01 | Rh* | 2.6498 | 3.16 |

Na* | 3.3215 | 2.3 | Pd | 2.7517 | 3.89 |

Mg* | 4.303 | 3.9 | Ag* | 2.8536 | 3.14 |

Al | 5.2846 | 2.77 | Cd | 2.9555 | 4.66 |

Si | 6.2659 | 3.38 | In* | 3.3972 | 2.8 |

P | 7.2473 | 4.88 | Sn* | 3.8388 | 3.05 |

S | 8.2293 | 4.14 | Sb* | 4.2805 | 3.8 |

Cl | 9.2107 | 4.68 | Te | 4.7221 | 3.52 |

K* | 2.02 | 1.92 | I | 5.1636 | 3.69 |

Ca | 2.6168 | 4 | Cs* | 1.1863 | 1.71 |

Sc* | 2.7545 | 3.2 | Ba | 1.5369 | 2.9 |

Ti* | 2.8924 | 3.37 | La* | 1.8873 | 2.6 |

V* | 3.03 | 3.1 | Hf | 7.1344 | 3 |

Cr* | 3.0676 | 3.06 | Ta | 7.4951 | 3.79 |

Mn* | 3.3055 | 3.72 | W | 7.8459 | 3.58 |

Fe* | 3.443 | 3.81 | Re | 8.1965 | 3.87 |

Co* | 3.5811 | 3.6 | Os | 8.5472 | 3.8 |

Ni* | 3.7187 | 3.25 | Ir | 8.8973 | 3.8 |

Cu | 3.8561 | 3.25 | Pt | 9.2474 | 3.5 |

Zn* | 3.994 | 4.94 | Au | 9.598 | 3.46 |

Ga | 4.5909 | 2.9 | Hg | 2.3456 | 5.54 |

Ge | 5.1874 | 3.4 | Tl* | 2.6962 | 2.9 |

As | 5.7846 | 4.5 | Pb* | 3.0466 | 3.53 |

Se | 6.381 | 3.87 | Bi* | 3.3972 | 3.74 |

Br | 6.978 | 4.22 |

One of the authors, R. B, is grateful to the University of Kalyani for financial assistance.