Calculating ionic radius in Angstrom (Å) is not a direct calculation but rather relies on experimental data and theoretical models. Ionic radii are typically determined indirectly through methods like X-ray diffraction on crystalline ionic compounds and then assigned based on certain assumptions and conventions. Since 1 Å = 100 picometers (pm), the value is often initially calculated in pm and then converted to Å. Here's a breakdown of the process and considerations:
1. Experimental Determination of Interionic Distances:
- X-ray Diffraction: This is the primary method. When X-rays are diffracted by a crystal, the diffraction pattern can be used to determine the distances between ions in the crystal lattice. This provides the sum of the radii of the cation and anion ($d = r+ + r-$).
2. Assigning Individual Ionic Radii:
The challenge is separating the measured distance into the individual radii of the cation and anion. Several approaches have been used:
- Pauling's Method (Isoelectronic Series): This method assumes that for ions in an isoelectronic series (ions with the same electron configuration, like Na+, Mg2+, Al3+), the radius is inversely proportional to the effective nuclear charge ($Z{eff}$). $Z{eff}$ can be estimated using Slater's rules.
- $r{+} = \frac{C}{Z{eff}^{+}}$ and $r{-} = \frac{C}{Z{eff}^{-}}$, where C is a constant determined using the experimental interionic distance.
- Goldschmidt's Method: Goldschmidt established a set of "reference" ionic radii based on the assumption that the radius of $O^{2-}$ in many oxides is approximately 1.40 Å. He then used interionic distances in various oxide crystals to determine the radii of other cations.
- Lande's Method: Lande assumed that in crystals where the anion is much larger than the cation (e.g., LiI), there is anion-anion contact. This allows for the estimation of the anion radius, which can then be used to find the cation radius in other compounds.
- Shannon-Prewitt Radii: This is a more modern and comprehensive set of ionic radii, often considered the standard. These radii are based on a large database of crystal structures and incorporate corrections for coordination number. These are effective ionic radii, reflecting the dependence of ionic size on the coordination environment.
3. Converting from Picometers (pm) to Angstroms (Å):
Once the ionic radius is determined in picometers (pm), convert it to Angstroms (Å) using the following conversion factor:
- 1 Å = 100 pm
- Therefore, $r (Å) = \frac{r (pm)}{100}$
Example:
Let's say you've determined the radius of a sodium ion (Na+) to be 98 pm. To convert this to Angstroms:
$r (Na^+) = \frac{98 pm}{100} = 0.98 Å$
Important Considerations:
- Coordination Number: The ionic radius is influenced by the coordination number (the number of ions surrounding a central ion). Radii are often reported for a specific coordination number (typically 6). Shannon-Prewitt radii provide values for various coordination numbers.
- Charge: Ions with a higher positive charge (more protons relative to electrons) tend to be smaller, while ions with a higher negative charge tend to be larger.
- Ionic vs. Covalent Character: The presence of significant covalent character in a bond can affect the apparent ionic radii.
- Approximations and Assumptions: Remember that the determination of ionic radii involves approximations and assumptions. Different methods can yield slightly different values.
- Solvation: In solution, ions are solvated, meaning they are surrounded by solvent molecules. This solvation shell significantly affects the effective size of the ion in solution, and calculating this hydrated ionic radius is complex.
In summary, calculating ionic radius in Angstroms involves experimental measurement of interionic distances, assigning individual radii using theoretical models and assumptions (like Pauling's, Goldschmidt's or Shannon-Prewitt radii), and then converting from picometers to Angstroms using the conversion factor 1 Å = 100 pm. Remember that ionic radii are not fixed values and depend on factors such as coordination number and charge.
