Nanotechnology has become an important field, with applications ranging from high-density data storage to the design of “nano machines.” One common building block of nano structured architectures is manganese oxide nano particles. The particles can be formed from manganese oxalate nano rods, the formation of which can be described as follows: $$\mathrm{Mn}^{2+}(a q)+\mathrm{C}_{2} \mathrm{O}_{4}^{2-}(a q) \rightleftharpoons \mathrm{MnC}, \mathrm{O}_{4}(a q) \quad K_{1}=7.9 \times 10^{3}$$ $$\mathrm{MnC}_{2} \mathrm{O}_{4}(a q)+\mathrm{C}_{2} \mathrm{O}_{4}^{2-}(a q) \rightleftharpoons \mathrm{Mn}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{2}^{2-}(a q) \quad K_{2}=7.9 \times 10^{1}$$ Calculate the value for the overall formation constant for \(\mathrm{Mn}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{2}^{2-} :\) $$K=\frac{\left[\mathrm{Mn}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{2}^{2-}\right]}{\left[\mathrm{Mn}^{2+}\right]\left[\mathrm{C}_{2} \mathrm{O}_{4}^{2-}\right]^{2}}$$

Step by step solution

01

Write down the first equilibrium reaction and its constant

Mn²⁺(aq) + C₂O₄²⁻(aq) ⟷ MnC₂O₄(aq), K₁ = 7.9 × 10³ Step 2: Reaction 2

02

Write down the second equilibrium reaction and its constant

MnC₂O₄(aq) + C₂O₄²⁻(aq) ⟷ Mn(C₂O₄)₂²⁻(aq), K₂ = 7.9 × 10¹ Step 3: Overall Formation Reaction

03

Determine the overall reaction for the formation of Mn(C₂O₄)₂²⁻

To obtain the overall formation reaction, we add Reaction 1 and Reaction 2: Mn²⁺(aq) + 2 C₂O₄²⁻(aq) ⟷ Mn(C₂O₄)₂²⁻(aq) Step 4: Calculate the Overall Formation Constant

04

Use the given equilibrium constants to find the overall formation constant, K

The overall formation constant K is the product of K₁ and K₂: K = K₁ × K₂ = (7.9 × 10³) × (7.9 × 10¹) K = 62.41 × 10⁴ K = 6.24 × 10⁵ The overall formation constant for the species Mn(C₂O₄)₂²⁻ is approximately 6.24 × 10⁵.

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