Number of Paramagnetic Complexes Among Given Coordination Compounds

To determine the total number of paramagnetic complexes among the given coordination compounds, we analyze the oxidation state of the central metal ion, its $d$-electron configuration, the strength of the ligands according to the spectrochemical series, and the number of unpaired electrons ($n$). A complex is paramagnetic if $n > 0$ and diamagnetic if $n = 0$.

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Detailed Analysis of Each Complex:

1. $[\text{MnBr}_4]^{2-}$

   • Central metal ion: $\text{Mn}^{2+}$ ($3d^5$)
   • Geometry: Tetrahedral (Weak field ligand $\text{Br}^-$)
   • Electronic configuration: $e^2 t_2^3$
   • Unpaired electrons ($n$): $5$
   • Nature: Paramagnetic

2. $[\text{NiCl}_4]^{2-}$

   • Central metal ion: $\text{Ni}^{2+}$ ($3d^8$)
   • Geometry: Tetrahedral (Weak field ligand $\text{Cl}^-$)
   • Electronic configuration: $e^4 t_2^4$
   • Unpaired electrons ($n$): $2$
   • Nature: Paramagnetic

3. $[\text{Ni(CN)}_4]^{2-}$

   • Central metal ion: $\text{Ni}^{2+}$ ($3d^8$)
   • Geometry: Square planar ($dsp^2$, Strong field ligand $\text{CN}^-$ causes pairing)
   • Electronic configuration: $(d_{xz}, d_{yz})^4 (d_{z^2})^2 (d_{xy})^2 (d_{x^2-y^2})^0$
   • Unpaired electrons ($n$): $0$
   • Nature: Diamagnetic

4. $[\text{Ni(CO)}_4]$

   • Central metal ion: $\text{Ni}^0$ ($3d^8 4s^2 \xrightarrow{\text{rearrangement}} 3d^{10}$)
   • Geometry: Tetrahedral ($sp^3$, Strong field ligand $\text{CO}$)
   • Unpaired electrons ($n$): $0$
   • Nature: Diamagnetic

5. $[\text{CoF}_6]^{3-}$

   • Central metal ion: $\text{Co}^{3+}$ ($3d^6$)
   • Geometry: Octahedral (Weak field ligand $\text{F}^- \implies \Delta_o < P$, high spin)
   • Electronic configuration: $t_{2g}^4 e_g^2$
   • Unpaired electrons ($n$): $4$
   • Nature: Paramagnetic

6. $[\text{Fe(CN)}_6]^{4-}$

   • Central metal ion: $\text{Fe}^{2+}$ ($3d^6$)
   • Geometry: Octahedral (Strong field ligand $\text{CN}^- \implies \Delta_o > P$, low spin)
   • Electronic configuration: $t_{2g}^6 e_g^0$
   • Unpaired electrons ($n$): $0$
   • Nature: Diamagnetic

7. $[\text{Mn(CN)}_6]^{3-}$

   • Central metal ion: $\text{Mn}^{3+}$ ($3d^4$)
   • Geometry: Octahedral (Strong field ligand $\text{CN}^- \implies \Delta_o > P$, low spin)
   • Electronic configuration: $t_{2g}^4 e_g^0$
   • Unpaired electrons ($n$): $2$
   • Nature: Paramagnetic

8. $[\text{Ti(CN)}_6]^{3-}$

   • Central metal ion: $\text{Ti}^{3+}$ ($3d^1$)
   • Geometry: Octahedral
   • Electronic configuration: $t_{2g}^1 e_g^0$
   • Unpaired electrons ($n$): $1$
   • Nature: Paramagnetic

9. $[\text{Cu(H}_2\text{O)}_6]^{2+}$

   • Central metal ion: $\text{Cu}^{2+}$ ($3d^9$)
   • Geometry: Octahedral
   • Electronic configuration: $t_{2g}^6 e_g^3$
   • Unpaired electrons ($n$): $1$
   • Nature: Paramagnetic

10. $[\text{Co(C}_2\text{O}_4)_3]^{3-}$

    • Central metal ion: $\text{Co}^{3+}$ ($3d^6$)
    • Geometry: Octahedral (Oxalate $\text{C}_2\text{O}_4^{2-}$ acts as a strong field ligand with $\text{Co}^{3+}$, causing pairing)
    • Electronic configuration: $t_{2g}^6 e_g^0$
    • Unpaired electrons ($n$): $0$
    • Nature: Diamagnetic

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Conclusion:

The paramagnetic complexes are:

1. $[\text{MnBr}_4]^{2-}$
2. $[\text{NiCl}_4]^{2-}$
3. $[\text{CoF}_6]^{3-}$
4. $[\text{Mn(CN)}_6]^{3-}$
5. $[\text{Ti(CN)}_6]^{3-}$
6. $[\text{Cu(H}_2\text{O)}_6]^{2+}$

Thus, the total number of paramagnetic complexes is 6.