The conditions and consequences that favour the \( t_{2g}^3 e_g^1 \) configuration in a metal complex are:
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The strength of the field ligand influences the distribution of electrons between \( t_{2g} \) and \( e_g \) orbitals. Weak field ligands lead to high spin configurations, while strong field ligands lead to low spin configurations.
The \( t_{2g}^3 e_g^1 \) configuration is typical for a weak field ligand. In such cases, the electrons tend to occupy the higher energy \( e_g \) orbitals, as opposed to pairing up in the lower energy \( t_{2g} \) orbitals. This leads to a low spin complex.
Final Answer: Weak field ligand, low spin complex.