Symmetry Operations in Coordination Complexes
This solution calculates the difference in the total number of symmetry operations between two coordination complexes: $[PdCl_6]^{2-}$ and trans-$[PdBr_2Cl_4]^{2-}$.
Step 1: Analyze $[PdCl_6]^{2-}$
The complex $[PdCl_6]^{2-}$ possesses an octahedral geometry.
The point group corresponding to octahedral symmetry is $O_h$.
The total number of symmetry operations (order, $h$) for the $O_h$ point group is $48$.
Therefore, $x = 48$.
Step 2: Analyze trans-$[PdBr_2Cl_4]^{2-}$
Palladium(II) complexes like $[PdBr_2Cl_4]^{2-}$ typically exhibit a square planar geometry.
The 'trans' designation indicates that the two Bromine (Br) ligands are positioned opposite each other in the square plane.
This specific arrangement belongs to the $D_{4h}$ point group.
The total number of symmetry operations (order, $h$) for the $D_{4h}$ point group is $16$.
Therefore, $y = 16$.
Step 3: Calculate the Difference ($x - y$)
The question requires the calculation of $x - y$.
Using the values determined in the previous steps:
$ x - y = 48 - 16 $The result of the subtraction is:
$ x - y = 32 $The value of $x - y$ is 32.
