Calculation of Wavelength at Maximum Absorption (λmax)

The wavelength at the maximum absorption (λmax) can be calculated using the crystal field stabilization energy (CFSE) and the spectrochemical series.

Spectrochemical Series

The spectrochemical series is a list of ligands arranged in order of increasing ability to cause ligand field splitting. The ligands at the top of the series cause the greatest degree of splitting and result in the largest CFSE values, while the ligands at the bottom of the series cause the least amount of splitting and result in the smallest CFSE values.

The spectrochemical series is as follows:

I- < br-="" />< s2-="" />< scn-="" />< cl-="" />< no3-="" />< n3-="" />< f-="" />< oh-="" />< c2o42-="" />< h2o="" />< ncs-="" />< ch3cn="" />< nh3="" />< en="" />< bipy="" />< phen="" />< no2-="" />< pph3="" />< cn-="" />

Calculation of CFSE

The CFSE of the [Ti(H2O)6]3+ complex is given as 7680 cm−1.

Identification of Ligand

Since the complex has octahedral geometry, the ligands surrounding the central Ti3+ ion are likely to be weak field ligands.

Application of Spectrochemical Series

Using the spectrochemical series, we can see that H2O is a weak field ligand, which is consistent with the octahedral geometry of the complex.

Calculation of λmax

The wavelength at the maximum absorption (λmax) is related to the CFSE by the following equation:

λmax = hc/CFSE

where h is Planck's constant, c is the speed of light, and λmax is in units of nm.

Substituting the given values, we get:

λmax = (6.626 x 10^-34 J s x 2.998 x 10^8 m/s)/(7680 cm^-1 x 100 cm/m)

λmax = 258 nm

Therefore, the wavelength at the maximum absorption (λmax) of the [Ti(H2O)6]3+ complex is 258 nm.

Explanation

The CFSE of a complex depends on the ligand field splitting, which in turn is determined by the nature of the ligands surrounding the central metal ion. In this case, the CFSE of the [Ti(H2O)6]3+ complex is relatively large because the ligands are weak field ligands. The wavelength at the maximum absorption (λmax) is related to the CFSE by an inverse relationship, such that a larger CFSE corresponds to a shorter λmax. Thus, the relatively large CFSE of the [Ti(H2O)6]3+ complex results in a relatively short wavelength at the maximum absorption (λmax).