Question

Calculate \(K_{\mathrm{sp}}\) for iron(II) sulfide given the following data: $$\begin{aligned}\mathrm{FeS}(s)+2 \mathrm{e}^{-} \longrightarrow \mathrm{Fe}(s)+\mathrm{S}^{2-}(a q) & & \mathscr{b}^{\circ} &=-1.01 \mathrm{~V} \\\\\mathrm{Fe}^{2+}(a q)+2 \mathrm{e}^{-} \longrightarrow \mathrm{Fe}(s) & & \mathscr{b}^{\circ} &=-0.44 \mathrm{~V} \end{aligned}$$

Short answer

The balanced equation for the dissolution of FeS is \(\mathrm{FeS}(s) \longrightarrow \mathrm{Fe}^{2+}(a q) + \mathrm{S}^{2-}(a q)\). The overall cell potential, \(E_{cell}^0\), is \(-0.57\,\mathrm{V}\). Using the Nernst equation at equilibrium and solving for solubility 's', we get \(s = \sqrt{e^{\frac{0.57 \times 4 \times 96485}{8.314 \times 298}}}\). Finally, the solubility product constant, \(K_{\mathrm{sp}}\), for iron(II) sulfide is calculated as \(K_{\mathrm{sp}} = s^2\).

Step-by-step solution

Write the balanced equation for the dissolution of FeS

We are given the following half-reactions: $$\mathrm{FeS}(s)+2 \mathrm{e}^{-} \longrightarrow \mathrm{Fe}(s)+\mathrm{S}^{2-}(a q)$$ $$\mathrm{Fe}^{2+}(a q)+2 \mathrm{e}^{-} \longrightarrow \mathrm{Fe}(s)$$ To get the balanced equation for the dissolution of FeS, reverse the second equation and add it to the first one: $$\mathrm{FeS}(s) \longrightarrow \mathrm{Fe}^{2+}(a q) + \mathrm{S}^{2-}(a q)$$

Calculate the overall cell potential, E°

For the two given half-reactions, we have the standard reduction potentials: $$E_1^0=-1.01\,\mathrm{V}$$ $$E_2^0=-0.44\,\mathrm{V}$$ Since we reversed the second half-reaction, the standard potential for the reversed reaction will be the negative of its value: $$E_2^0(reversed)=+0.44\,\mathrm{V}$$ Now add the standard potentials of both half-reactions to get the overall cell potential, E°: $$E_{cell}^0 = E_1^0 + E_2^0(reversed) = -1.01\,\mathrm{V} + 0.44\,\mathrm{V} = -0.57\,\mathrm{V}$$

Use the Nernst equation to solve for Ksp

The Nernst equation relates the cell potential, standard cell potential, and reaction quotient (Q) as follows: $$E_{cell} = E_{cell}^0 - \frac{RT}{nF} \ln Q$$ For the dissolution of FeS, the balanced equation is: $$\mathrm{FeS}(s) \longrightarrow \mathrm{Fe}^{2+}(a q) + \mathrm{S}^{2-}(a q)$$ Let the solubility of FeS be represented by 's'. The reaction quotient, Q, can then be written as: $$Q = [\mathrm{Fe}^{2+}][\mathrm{S}^{2-}] = (s)(s) = s^2$$ At equilibrium, we can replace E° by -0.57 V (from Step 2) and E_cell by 0 V, since at equilibrium, the cell potential is zero: $$0 = -0.57\,\mathrm{V} - \frac{RT}{4F} \ln s^2$$ Here, R is the gas constant (\(8.314\,\mathrm{J/(mol\cdot K)}\)), T is the temperature in Kelvin (assuming 298 K), n is the number of electrons transferred (4), and F is the Faraday constant (\(96485\,\mathrm{C/mol}\)). Now we can solve for solubility, s, and the solubility product, Ksp: $$\frac{0.57 \times 4F}{RT} = \ln s^2$$ $$\frac{0.57 \times 4 \times 96485}{8.314 \times 298}=\ln s^2$$ Calculate s by taking the square root of a = \(e^{0.57 \times 4 \times 96485 / (8.314 \times 298)}\). Now, calculate the solubility product, Ksp: $$K_{\mathrm{sp}} = [\mathrm{Fe}^{2+}][\mathrm{S}^{2-}] = s^2$$ In the end, plug in the value of 's' to get the solubility product, \(K_{\mathrm{sp}}\), for iron(II) sulfide.

Key concepts

Solubility Product Constant (Ksp)

The solubility product constant, or Ksp, is a measure of the extent to which a compound can dissolve in water. It is a special type of equilibrium constant that applies to the dissolution of sparingly soluble salts. When a salt like iron(II) sulfide (FeS) dissolves in water, it dissociates into its constituent ions, in this case, iron(II) ions (Fe²⁺) and sulfide ions (S^2-).

The Ksp expression for this dissolution can be written as follows:
$$K_{sp} = [\mathrm{Fe}^{2+}][\mathrm{S}^{2-}]$$
Here, the square brackets denote the concentrations of the ions at equilibrium. In the case of FeS, determining its solubility involves calculating the Ksp from known potentials, using electrochemical principles.

Standard Reduction Potential

The standard reduction potential (E°) is the voltage associated with a reduction reaction at an electrode when all solutes are at an effective concentration of 1 molar, and all gases are at 1 atmosphere pressure, with a standard state temperature (usually 298.15 K or 25°C). It is a measure of the tendency of a chemical species to acquire electrons and thereby be reduced. Each half-cell has its own standard reduction potential, and these values are used to determine the spontaneity of a reaction and to calculate cell potentials.

For iron(II) sulfide, the standard reduction potentials for the two half-reactions were given in the exercise. By convention, the more positive reduction potential tends to indicate a greater likelihood of the species being reduced in the electrochemical series.

Nernst Equation

The Nernst equation is a fundamental equation in electrochemistry that relates the electric potential of a chemical reaction to the concentration of the reactants. It is given by:
$$E_{cell} = E_{cell}^0 - \frac{RT}{nF} \ln Q$$
Where:

• \(E_{cell}\) is the cell potential.
• \(E_{cell}^0\) is the standard cell potential.
• \(R\) is the gas constant.
• \(T\) is the temperature in kelvin.
• \(n\) is the number of moles of electrons transferred in the reaction.
• \(F\) is Faraday's constant.
• \(Q\) is the reaction quotient, analogous to the equilibrium expression but for concentrations that are not at equilibrium.

With the help of the Nernst equation and the standard reduction potentials, we can calculate the Ksp for a given temperature.

Electrochemistry

Electrochemistry is the branch of chemistry that deals with the relations between electrical and chemical phenomena. It is concerned with the interconversion of chemical and electrical energy, as occurs in batteries, fuel cells, and electrolysis. Understanding electrochemistry is essential when dealing with chemical reactions that involve electron transfer, such as the dissolution of iron(II) sulfide.

By applying the electrochemical concepts of standard reduction potentials along with the Nernst equation, we can calculate the Ksp for FeS, which demonstrates the intrinsic relationship between the solubility of a substance, its standard reduction potential, and the principles of electrochemistry. This multiplicative nature of chemistry often unfolds as we find the connections between seemingly different concepts, providing a comprehensive understanding of chemical processes.