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Step-by-Step Solution
Step 1: Identify the complexes and their coordination geometry
• The first complex is triamminetrinitrocobalt(III), written as [Co(NO_{2})_{3}(NH_{3})_{3}] .
• The second complex is trioxalatochromate(III), written as [Cr(C_{2}O_{4})_{3}]^{3-} .
Step 2: Determine the number of geometrical isomers for [Co(NO_{2})_{3}(NH_{3})_{3}]
• Each ligand set ( NO_{2} and NH_{3} ) occupies three positions around the cobalt center.
• The overall geometry for a six-coordinate complex is typically octahedral.
• We check how the three NO_{2} ligands can be arranged relative to the three NH_{3} ligands to see if distinct geometrical isomers can exist (i.e., different spatial arrangements that cannot be superimposed).
• In an octahedral arrangement with three identical ligands of one type and three identical ligands of another type, there are two possible geometrical isomers commonly referred to as the “fac” (facial) and “mer” (meridional) isomers:
fac-isomer: All three identical ligands occupy adjacent positions at the corners of one triangular face of the octahedron.
mer-isomer: The three identical ligands occupy a plane that includes the metal center, forming a meridian around the octahedron.
• Therefore, X = 2 for [Co(NO_{2})_{3}(NH_{3})_{3}] .
Step 3: Determine the number of geometrical isomers for [Cr(C_{2}O_{4})_{3}]^{3-}
• Each oxalate ( C_{2}O_{4}^{2-} ) is a bidentate ligand, binding through two oxygen atoms. Three oxalate ligands around a chromium(III) center in an octahedral environment is a symmetrical arrangement.
• All oxalate ligands are identical, and in this complex, any rearrangement simply gives the same spatial arrangement because each ligand spans two coordination sites in a symmetrical fashion.
• Hence, there are no distinct geometrical isomers for this complex.
• Therefore, Y = 0 for [Cr(C_{2}O_{4})_{3}]^{3-} .
Step 4: Calculate X + Y
• We have X = 2 and Y = 0 .
• Thus, X + Y = 2 + 0 = 2 .
Step 5: Final Answer
The value of X + Y is 2.
