To determine which complex exists as a pair of enantiomers, we need to identify the chiral complex among the given options. Chirality in coordination complexes arises from the absence of symmetry elements like a plane of symmetry (σ) or a center of symmetry (i).

Chirality Analysis of Complexes

• Option 1: $trans-[Co(H_2NCH_2CH_2NH_2)_2Cl_2]^+$
  • This is an octahedral complex of the type [MA₂B₂], where A is the bidentate ligand ethylenediamine (en) and B is the monodentate ligand chloride (Cl).
  • The trans isomer, with the two Cl^- ligands opposite each other, possesses a plane of symmetry.
  • Therefore, this complex is achiral.
• Option 2: $cis-[Co(NH_3)_4Cl_2]^+$
  • This is an octahedral complex of the type [MA₄B₂], where A is ammonia (NH₃) and B is chloride (Cl).
  • The cis isomer, with the two Cl^- ligands adjacent, has a plane of symmetry passing through the metal and the two Cl ligands.
  • Therefore, this complex is achiral.
• Option 3: $[Pt(PPh_3)(Cl)(Br)(CH_3)]^-$
  • This is a square planar complex with four different monodentate ligands: PPh₃, Cl, Br, and CH₃.
  • A square planar complex with four distinct ligands is typically chiral.
• Option 4: $[Co(H_2NCH_2CH_2NH_2)_3]^{3+}$
  • This is an octahedral complex of the type [M(AA)₃]ⁿ⁺, where AA is the symmetrical bidentate ligand ethylenediamine (en).
  • Complexes of the type [M(AA)₃]ⁿ⁺, with symmetrical bidentate ligands like ethylenediamine, lack any plane or center of symmetry.
  • The arrangement of the three ethylenediamine ligands around the central cobalt ion creates a chiral structure.
  • Therefore, this complex exists as a pair of enantiomers.

Conclusion on Enantiomers

The complex that exhibits chirality and exists as a pair of enantiomers is the one lacking symmetry elements. Based on the analysis, the complex $[Co(H_2NCH_2CH_2NH_2)_3]^{3+}$ fits this criterion.