Question

Zinc is an amphoteric metal; that is, it reacts with both acids and bases. The standard reduction potential is $-1.36\mathrm{V}$ for the reaction: $$\mathrm{Zn}(\mathrm{OH}{)}_4^{2-}(aq)+2e^{-}⟶\mathrm{Zn}(s)+4{\mathrm{OH}}^{-}(aq)$$ Calculate the formation constant $\left(K_{\mathrm{f}}\right)$ for the reaction: $${\mathrm{Zn}}^{2+}(aq)+4{\mathrm{OH}}^{-}(aq)\rightleftharpoons \mathrm{Zn}(\mathrm{OH}{)}_4^{2-}(aq)$$

Short answer

The formation constant $K_f$ is approximately $1.2\times {10}^{46}$.

Step-by-step solution

Understanding the Goal

We need to determine the formation constant $K_f$ for the reaction ${\mathrm{Zn}}^{2+}+4{\mathrm{OH}}^{-}\rightleftharpoons {\mathrm{Zn}(\mathrm{OH})}_4^{2-}$. Using the given standard reduction potential, we will calculate $K_f$.

Electrochemical Cell Setup

The given standard reduction potential is for the half-reaction ${\mathrm{Zn}(\mathrm{OH})}_4^{2-}+2e^{-}\to \mathrm{Zn}+4{\mathrm{OH}}^{-}$ with ${\text{E}}^{\circ }=-1.36\text{ V}$. This is a part of the electrochemical process involving zinc.

Relating Potential to Free Energy Change

The relation between the free energy change $\mathrm{\Delta }{G}^{\circ }$ and the standard reduction potential $E^{\circ }$ is given by $$\mathrm{\Delta }{G}^{\circ }=-nF{E}^{\circ }$$ where $n$ is the number of moles of electrons (which is 2), and $F$ is Faraday's constant $96485\text{ C/mol}$.

Calculate $\mathrm{\Delta }{G}^{\circ }$ for the Half-Reaction

Substitute the known values into the equation: $$\mathrm{\Delta }{G}^{\circ }=-2\times 96485\times (-1.36)=262,082\text{ J/mol}$$

Relate $\mathrm{\Delta }{G}^{\circ }$ to the Formation Constant

The relationship between $\mathrm{\Delta }{G}^{\circ }$ and the equilibrium constant $K$ is: $$\mathrm{\Delta }{G}^{\circ }=-RT\ln K$$ where $R=8.314\text{ J/mol K}$ is the ideal gas constant, and $T=298\text{ K}$ is the temperature.

Solve for $K_f$

Rearrange the expression for $K$: $$\ln K_f=\frac{-\mathrm{\Delta }{G}^{\circ }}{RT}$$Substitute the values: $$\ln K_f=\frac{-262082}{8.314\times 298}$$ Calculate $K_f$:$$\ln K_f\approx 105.65$$ $$K_f\approx e^{105.65}$$

Final Calculation of $K_f$

We perform the exponential calculation: $$K_f\approx e^{105.65}\approx 1.2\times {10}^{46}$$ Therefore, the formation constant $K_f$ for the reaction is approximately $1.2\times {10}^{46}$.

Key concepts

Amphoteric Metals

Amphoteric metals are quite fascinating as they possess unique chemical properties. These metals can react not only with acids but also with bases. This duality allows them to participate in a wider range of chemical reactions compared to metals that solely react with one type of reagent. A classic example is zinc, which demonstrates these properties vividly.
For instance, when zinc reacts with hydrochloric acid (an acid), it releases hydrogen gas and forms zinc chloride. Conversely, when it reacts with sodium hydroxide (a base), it forms a complex ion, namely, zincate $(ext{Zn(OH)}_4^{2-})$. This property of reacting with both acids and bases is fundamentally due to the amphoteric nature of zinc. Being amphoteric is advantageous in applications like metallurgy and materials science, where such properties can be harnessed to prepare various compounds.

Formation Constant

Formation constants, also known as stability constants, are crucial in understanding how strongly a complex ion is formed in a solution. The formation constant, denoted by $K_f$, indicates the equilibrium constant for the formation of a complex ion from its constituent ions. A high value of $K_f$ suggests that the complex ion is very stable, while a low $K_f$ indicates less stability.
In the context of zinc chemistry, the formation constant is associated with the equilibrium between zinc ions and hydroxide ions forming zincate, $${\text{Zn}}^{2+}+4{\text{OH}}^{-}\rightleftharpoons {\text{Zn(OH)}}_4^{2-}$$
Calculating the formation constant involves considering the change in free energy $\mathrm{\Delta }{G}^{\circ }$, which in turn is related to the standard reduction potential. The relationship can be mathematically defined through the equation:$$\mathrm{\Delta }{G}^{\circ }=-RT\ln K$$
This formula allows us to determine $K_f$ once $\mathrm{\Delta }{G}^{\circ }$ is known. In practical terms, a calculated $K_f$ of $1.2\times {10}^{46}$ for this zinc reaction suggests an extremely stable zincate formation.

Standard Reduction Potential

Understanding standard reduction potentials is key in predicting the direction and feasibility of redox reactions. The standard reduction potential $E^{\circ }$ is a measure of the tendency of a chemical species to acquire electrons and be reduced. It is measured in volts and compared against a standard hydrogen electrode.
In the zinc electrochemical cell, with a given $E^{\circ }=-1.36\text{ V}$ for the reduction$${\text{Zn(OH)}}_4^{2-}+2e^{-}\to \text{Zn}+4{\text{OH}}^{-}$$
We see a negative potential indicating that this is not spontaneous under standard conditions. This negative sign implies that energy must be applied for the reaction to proceed in the forward direction, making it an essential point when calculating the free energy change and subsequently the formation constant. Therefore, standard reduction potentials play a pivotal role in understanding the electrochemical characteristics of metals like zinc.

Electrochemical Reactions

Electrochemical reactions encompass processes where redox reactions generate electricity or are driven by an electrical current. In zinc electrochemistry, these reactions are paramount as they enable reactions such as the reduction of zinc ate to solid zinc.
Electrochemical cells make use of separate half-reactions occurring at respective electrodes. For zinc, the half-reaction of interest is often the reduction part:$${\text{Zn(OH)}}_4^{2-}+2e^{-}\to \text{Zn}+4{\text{OH}}^{-}$$
The interplay between these redox reactions and standard reduction potentials allows us to calculate thermodynamic quantities such as $\mathrm{\Delta }{G}^{\circ }$, which is directly linked to the feasibility and directionality of the overall reaction.
Thus, electrochemical reactions provide invaluable insights into the practical applications of zinc, particularly in the field of batteries and galvanization where zinc's reactive tendencies are harnessed for energy storage and protective coatings.