Question

An electrochemical cell consists of a nickel metal electrode immersed in a solution with \(\left[\mathrm{Ni}^{2+}\right]=1.0 \mathrm{M}\) separated by a porous disk from an aluminum metal electrode immersed in a solution with \(\left[\mathrm{Al}^{3+}\right]=1.0 \mathrm{M} .\) Sodium hydroxide is added to the aluminum compartment, causing \(\mathrm{Al}(\mathrm{OH})_{3}(s)\) to precipitate. After precipitation of Al(OH) \(_{3}\) has ceased, the concentration of \(\mathrm{OH}^{-}\) is \(1.0 \times 10^{-4} \mathrm{M}\) and the measured cell potential is \(1.82 \mathrm{V}\). Calculate the \(K_{\mathrm{sp}}\) value for \(\mathrm{Al}(\mathrm{OH})_{3}\) $$ \mathrm{Al}(\mathrm{OH})_{3}(s) \rightleftharpoons \mathrm{Al}^{3+}(a q)+3 \mathrm{OH}^{-}(a q) \quad K_{\mathrm{sp}}=? $$

Short answer

The solubility product constant (Ksp) for Al(OH)₃ can be calculated using the Nernst equation and the provided information about the electrochemical cell. First, we find the standard cell potential (E°cell) to be 1.41 V. Then, we calculate the reaction quotient (Q) as 1 and use the Nernst equation to find the concentration of Al³⁺. Finally, we use this Al³⁺ concentration, along with the given OH⁻ concentration, to calculate Ksp for Al(OH)₃: \[ K_\text{sp} = [\mathrm{Al^{3+}}([\mathrm{OH^-}])^3 = \mathrm{Al}^{3+} (1.0 \times 10^{-4})^3 \]

Step-by-step solution

Understanding the Nernst equation

The Nernst equation relates the cell potential (Ecell) to the reduction potential (E°) and the reaction quotient (Q) of a redox reaction. By determining the Ecell from the provided information, we can use the Nernst equation to calculate the Al³⁺ concentration. The Nernst equation is given by: \[ E_\text{cell} = E°_\text{cell} - \frac{RT}{nF} \ln Q \] Where: - Ecell is the cell potential - E°cell is the standard reduction potential - R is the ideal gas constant (8.314 J/mol·K) - T is the temperature (in Kelvin) - n is the number of moles of electrons transferred in the redox reaction (urea) - F is the Faraday's constant (96485 C/mol) - Q is the reaction quotient

Calculate the standard reduction potential

The standard reduction potential for the given electrochemical cell (nickel and aluminum electrodes) can be found in a table of standard reduction potentials. The values are as follows: \[ E°_\text{Ni²⁺/Ni} = -0.25\,\text{V} \] \[ E°_\text{Al³⁺/Al} = -1.66\,\text{V} \] To find the standard reduction potential of the cell, we have to subtract the values: \[ E°_\text{cell} = E°_\text{Ni²⁺/Ni} - E°_\text{Al³⁺/Al} = (-0.25) - (-1.66) = 1.41\,\text{V} \]

Determine the reaction quotient (Q)

The reaction quotient (Q) can be calculated using the given concentrations of the reacting species: \[ Q = \frac{\text{[Al³⁺]}}{\text{[Ni²⁺]}} = \frac{1}{1.0} = 1\]

Calculate the concentration of Al³⁺

We know the cell potential (1.82 V), the standard cell potential (1.41 V), the number of moles of electrons transferred (n=2), and the temperature (assume room temperature = 298 K). We can now use the Nernst equation to calculate the concentration of Al³⁺: \[ 1.82 = 1.41 - \frac{8.314(298)}{2(96485)}\ln1\] \[ \mathrm{Al}^{3+}(aq) \]

Calculate the Ksp for Al(OH)₃

Using the concentration of Al³⁺ and the concentration of OH⁻ (1.0 x 10⁻⁴ M), we can now calculate the Ksp for Al(OH)₃: \[ \mathrm{Al(OH)_3 (s) \rightleftharpoons Al^{3+}(aq) + 3OH^-(aq)} \] \[ K_\text{sp} = [\mathrm{Al^{3+}}([\mathrm{OH^-}])^3 = \mathrm{Al}^{3+} (1.0 \times 10^{-4})^3 \] Plug in the previously calculated concentration of Al³⁺ and evaluate the expression to find the Ksp value for Al(OH)₃.

Key concepts

Nernst Equation

The Nernst equation provides a quantitative connection between the cell potential under any conditions and the standard cell potential. This valuable equation takes into account the effect of temperature and concentration on cell potential, making it an essential tool in analyzing electrochemical cells. For example, when given a standard reduction potential and the concentrations of reactants and products, the Nernst equation enables us to calculate the actual cell voltage.

The general form of the Nernst equation is:\[\[\begin{align*}E_{\text{cell}} &= E^{\circ}_{\text{cell}} - \frac{RT}{nF}\ln Q \&= E^{\circ}_{\text{cell}} - \frac{0.05916}{n}\log Q \text{ at 298 K}\end{align*}\]\]Where:

• E_{\text{cell}} is the actual cell potential,
• E^{\circ}_{\text{cell}} is the standard cell potential,
• R is the universal gas constant,
• T is the temperature in Kelvin,
• n is the number of moles of electrons transferred in the redox reaction,
• F is Faraday's constant,
• Q is the reaction quotient, which reflects the ratio of concentrations of products to reactants at any given moment in the cell's operation.

As concentrations and temperatures vary, the Nernst equation allows for calculations reflecting these changes, providing insights into the workings of the cell that the standard reduction potential alone cannot offer.

Reaction Quotient

The Reaction Quotient (Q) is a measure of the relative amounts of products and reactants present during a reaction at a given point in time. It has the same form as the equilibrium constant (K) expression but isn't limited to equilibrium conditions. For the reaction quotient, concentrations are used directly, whereas, for the equilibrium constant, one considers the concentrations at equilibrium.

The reaction quotient is crucial in the Nernst equation, as it indicates how far a reaction has proceeded towards equilibrium. It is calculated using:
\[\[\begin{align*}Q = \prod (\text{products})^{\text{stoichiometric coefficients}} / \prod (\text{reactants})^{\text{stoichiometric coefficients}}\end{align*}\]\]A Q value of less than the equilibrium constant (K) suggests that the reaction will proceed forward to create more products, while a value greater than K indicates that the reaction is likely to reverse and form more reactants. At equilibrium, Q equals K, indicating that the rates of the forward and reverse reactions are equal.

Solubility Product Constant (Ksp)

The Solubility Product Constant (Ksp) is an important equilibrium constant that is used to measure the solubility of a sparingly soluble compound. It is especially useful in predicting whether a precipitate will form in a solution when certain ions are present.

For a general precipitation reaction:\[\[\begin{align*}AB_{(s)} &\rightleftharpoons A^{+}_{(aq)} + B^{-}_{(aq)}\end{align*}\]\]Ksp is defined as:\[\[\begin{align*}Ksp &= [A^{+}]^m[B^{-}]^n\end{align*}\]\]where m and n are the stoichiometric coefficients of the ions A+ and B- in the balanced chemical equation, and the brackets represent the equilibrium concentrations of the ions. The Ksp value is specific to each compound and is affected by temperature. Substances with a high Ksp are considered more soluble in water. Calculating Ksp is essential in environmental science, chemical engineering, and whenever the solubility of compounds is important to the field of study.

Standard Reduction Potential

The Standard Reduction Potential (E°) is a measure of the tendency of a chemical species to acquire electrons and thereby be reduced. It is usually expressed in volts and is determined under standard conditions, which include a pressure of 1 bar, a concentration of 1 M for each ion participating in the reaction, and a temperature of 298 K. Each half-cell in an electrochemical cell has a standard reduction potential, and the cell potential can be calculated by taking the difference between the potentials of the cathode and anode.

In a standard table of reduction potentials, all values are given relative to the standard hydrogen electrode, which is arbitrarily assigned a potential of 0 volts. Thus:\[\[\begin{align*}E^{\circ}_{\text{cell}} &= E^{\circ}_{\text{cathode}} - E^{\circ}_{\text{anode}}\end{align*}\]\]The more positive the standard reduction potential, the greater the species' ability to gain electrons. Conversely, a more negative standard reduction potential indicates a stronger propensity to lose electrons. In summary, standard reduction potentials provide crucial information to predict the direction of electron flow and the feasibility of redox reactions.