Question

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 nanostructured architectures is manganese oxide nanoparticles. The particles can be formed from manganese oxalate nanorods, the formation of which can be described as follows: \(\mathrm{Mn}^{2+}(a q)+\mathrm{C}_{2} \mathrm{O}_{4}^{2-}(a q) \rightleftharpoons \mathrm{MnC}_{2} \mathrm{O}_{4}(a q) \quad K_{1}=7.9 \times 10^{3}\) \(\mathrm{MnC}_{2} \mathrm{O}_{4}(a q)+\mathrm{C}_{2} \mathrm{O}_{4}^{2-}(a q) \rightleftharpoons \mathrm{Mn}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{2}^{2-}(a q)\) $$ K_{2}=7.9 \times 10^{1} $$ Calculate the value for the overall formation constant for \(\mathrm{Mn}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{2}^{2-}\) $$ K=\frac{\left[\mathrm{Mn}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{2}^{2-}\right]}{\left[\mathrm{Mn}^{2+}\right]\left[\mathrm{C}_{2} \mathrm{O}_{4}^{2-}\right]^{2}} $$

Short answer

The overall formation constant (K) for \(\mathrm{Mn}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{2}^{2-}\) is calculated by multiplying the individual equilibrium constants: \(K = K_{1} \times K_{2} = (7.9 \times 10^{3}) \times (7.9 \times 10^{1}) = 6.24 \times 10^{5}\).

Step-by-step solution

Write down the reactions and equilibrium constants

Reaction 1: \(\mathrm{Mn}^{2+}(a q)+\mathrm{C}_{2} \mathrm{O}_{4}^{2-}(a q) \rightleftharpoons \mathrm{MnC}_{2} \mathrm{O}_{4}(a q)\) \(K_{1}=7.9 \times 10^{3}\) Reaction 2: \(\mathrm{MnC}_{2} \mathrm{O}_{4}(a q)+\mathrm{C}_{2} \mathrm{O}_{4}^{2-}(a q) \rightleftharpoons \mathrm{Mn}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{2}^{2-}(a q)\) \(K_{2}=7.9 \times 10^{1}\)

Add the reactions

Add the two reactions so that you are left only with the desired ions on each side of the equilibrium: \(\mathrm{Mn}^{2+}(a q) + 2\mathrm{C}_{2} \mathrm{O}_{4}^{2-}(a q) \rightleftharpoons \mathrm{Mn}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{2}^{2-}(a q)\)

Calculate the overall formation constant (K)

When combining two reactions, the overall equilibrium constant K is the product of the individual equilibrium constants: \(K = K_{1} \times K_{2} = (7.9 \times 10^{3}) \times (7.9 \times 10^{1})\) Calculate the product: \(K = 6.24 \times 10^{5}\) This overall formation constant represents the affinity of \(\mathrm{Mn}^{2+}\) for the oxalate ion in forming the complex ion \(\mathrm{Mn}\left(\mathrm{C}_{2}\mathrm{O}_{4}\right)_{2}^{2-}\).

Key concepts

Manganese Oxide Nanoparticles

Manganese oxide nanoparticles hold a pivotal role in the vast field of nanotechnology. These tiny particles have applications in various domains, due to their unique properties and small size. One significant attribute is their large surface area to volume ratio, which makes them highly reactive.

Key uses of manganese oxide nanoparticles include:

• Electronics: They are used in high-density data storage systems, improving the efficiency and storage capacity of modern devices.
• Catalysts: Due to their reactivity, they serve as catalysts in numerous chemical reactions, reducing the need for more cumbersome compounds.
• Energy Storage: Manganese oxide nanoparticles are utilized in batteries, particularly in enhancing lithium-ion battery performance.

These nanoparticles are often formed from manganese oxalate nanorods, a precursor that's easier to manipulate and control in a laboratory setting. The transition from nanorods to nanoparticles involves understanding complex chemical reactions and equilibriums, which are crucial in the creation of these nanostructures.

Formation Constant

The formation constant is a crucial concept in chemistry, especially when dealing with equilibrium reactions and complex ions. It's represented as \(K\), denoting the stability of a complex ion formed from its constituent ions. In simpler terms, it quantifies how readily reactants combine into the complex ion.

For instance, consider the overall formation constant for the complex \(\mathrm{Mn}(\mathrm{C}_{2}\mathrm{O}_{4})_{2}^{2-}\):

• This complex formation involves combining manganese ions \(\mathrm{Mn}^{2+}\) with oxalate ions \(\mathrm{C}_{2}\mathrm{O}_{4}^{2-}\).
• The reactions can be split into steps, each having its own equilibrium constant \(K_{1}\) and \(K_{2}\).
• The overall formation constant \(K\) is the product of these individual constants, illustrating the gradual formation process.

Mathematically, the overall formation constant is expressed as:\[K = \frac{[\mathrm{Mn}(\mathrm{C}_{2}\mathrm{O}_{4})_{2}^{2-}]}{[\mathrm{Mn}^{2+}]\cdot [\mathrm{C}_{2}\mathrm{O}_{4}^{2-}]^2}\]This equation accentuates the stoichiometry of the ions involved and their specific concentrations, making it a valuable tool for chemists in predicting reaction yields.

Equilibrium Reactions

Equilibrium reactions lie at the heart of many chemical processes, providing insight into how substances interact until reaching a balance. In the reversible process of forming manganese oxide nanoparticles from manganese oxalate, understanding equilibrium is essential.

Here’s why equilibrium reactions are important:

• Dynamic State: Even though the macroscopic properties remain constant, the forward and reverse reactions continue simultaneously at equilibrium.
• Predicting Concentrations: Using equilibrium constants, chemists can determine the concentrations of reactants and products at equilibrium. This helps in adjusting conditions to yield desired products.
• Reaction Extent: The value of the equilibrium constant \(K\) indicates whether reactants or products will be favored when equilibrium is reached. A larger \(K\) suggests a favor towards products, while a smaller \(K\) indicates predominant reactants.

In the example given, equilibrium concepts are applied in calculating how manganese and oxalate ions combine to form complex ions. By doing so, chemists can engineer reaction conditions that optimize nanoparticle formation, exploiting the intricate balance dictated by equilibrium principles.