Problem 119 - Chemistry: The Central Science

Question

The \(K_{\infty}\) value for \(\mathrm{PbS}(s)\) is \(8.0 \times 10^{-28} .\) By using this value together with an electrode potential from Appendix \(\mathrm{E}\), determine the value of the standard reduction potential for the reaction $$ \mathrm{PbS}(s)+2 \mathrm{e}^{-}-\cdots \mathrm{Pb}(s)+\mathrm{S}^{2-}(a q) $$

Step-by-Step Solution

Answer

Using the Nernst equation and combining the related reaction from Appendix E, we can find the standard reduction potential for the given reaction: $$ E^0 = E_1^0 - \frac{RT}{2F} \ln K_{\infty} $$ Plug in the values for \(R, T, F, K_{\infty}\), and \(E_1^0\) (from Appendix E) and solve for \(E^0\) to find the standard reduction potential.

Step 1: Write down the Nernst equation

The Nernst equation relates the reduction potential \(E\) to the standard reduction potential \(E^0\), the number of electrons involved in the reaction \(n\), and the reaction quotient \(Q\): $$ E = E^0 - \frac{RT}{nF} \ln Q $$ where \(R\) is the gas constant, \(T\) is the temperature in Kelvin, and \(F\) is the Faraday constant.

Step 2: Write the solubility equilibrium expression for \(\mathrm{PbS}\)

The solubility equilibrium can be represented by: $$ \mathrm{PbS}(s) \rightleftharpoons \mathrm{Pb}^{2+}(aq) + \mathrm{S}^{2-}(aq) $$ with the solubility product given as \(K_{\infty} = [\mathrm{Pb}^{2+}][\mathrm{S}^{2-}] = 8.0 \times 10^{-28}.\)

Step 3: Determine a related electrode potential

From Appendix E, we can find an electrode potential related to the given reaction such as the reaction: $$ \mathrm{Pb}^{2+}(aq) + 2 e^- \rightarrow \mathrm{Pb}(s), \quad E_1^0 $$

Step 4: Combine the reactions

We need to combine the reaction in Step 3 with the given reaction: $$ \mathrm{PbS}(s)+2 \mathrm{e}^{-}\rightleftharpoons \mathrm{Pb}(s)+\mathrm{S}^{2-}(aq) $$ Subtracting the given reaction from the reaction in Step 3, we get: $$ \mathrm{PbS}(s) \rightleftharpoons \mathrm{Pb}^{2+}(aq) + \mathrm{S}^{2-}(aq), \quad E - E_1^0 $$ The standard reduction potential for the given reaction will be \(E^0 = E_1^0\). To find \(E^0\), we will use the Nernst equation in Step 1.

Step 5: Calculate the standard reduction potential

Substituting the relevant values into the Nernst equation: $$ E_1^0 - E^0 = \frac{RT}{2F} \ln K_{\infty} $$ $$ E^0 = E_1^0 - \frac{RT}{2F} \ln K_{\infty} $$ Plug in the values for \(R, T, F, K_{\infty}\), and \(E_1^0\) and solve for \(E^0\) to find the standard reduction potential for the given reaction. Please make sure to look up the electrode potential (\(E_1^0\)) value from Appendix E or a similar resource for the accurate answer.

Key Concepts

Standard reduction potential

Electrochemistry involves many reactions that require knowledge of the standard reduction potential. This potential, symbolized as \( E^0 \), is like a chemical battery's voltage when the reaction is at equilibrium under standard conditions. The standard conditions include 1 M concentrations, 1 atm pressure, and a temperature of 25°C (298K).

In the given exercise, the calculation of the standard reduction potential \( E^0 \) for the reaction involving \( \text{PbS}(s) \) is necessary. To do this, we use the Nernst equation (explained later) to relate it to other known values. The electrode potential of a related reaction, found in Appendix E, serves as a point of reference.

• It helps to think of \( E^0 \) as the potential difference achieved when electrons flow as a result of the oxidation-reduction reaction under specified conditions.
• Once assessed, \( E^0 \) provides insight into how readily a species gains electrons compared to a reference hydrogen electrode.

Remember, the more positive \( E^0 \), the greater the species' propensity to gain electrons or be reduced. In multi-step calculations like this, finding \( E^0 \) guides understanding of the reaction's direction and feasibility.

Nernst equation

The Nernst equation is a crucial relationship in electrochemistry used for calculating the reduction potential of an electrochemical cell under non-standard conditions. It links the measured electrode potential \( E \) to factors like standard reduction potential \( E^0 \), temperature \( T \), the number of moles of electrons transferred \( n \), and the reaction quotient \( Q \):

\[ E = E^0 - \frac{RT}{nF} \ln Q \]

- \( R \) is the universal gas constant. - \( T \) is the absolute temperature in Kelvin (K). - \( n \) indicates the amount of electrons involved in the redox reaction. - \( F \) is the Faraday constant, representing the charge per mole of electrons.

In solving problems, such as the one about \( \text{PbS}(s) \), the Nernst equation helps calculate the cell potential by adjusting \( E^0 \) based on the actual conditions. You evaluate \( Q \), which here relates to the solubility product \( K_{\infty} \). Using \( K_{\infty} \) allows you to describe the non-equilibrium state and proceed to find \( E^0 \). Understanding the exact role each factor plays in the equation provides deep insights into electrochemical processes and reaction behaviors.

Solubility product (Ksp)

In the realm of electrochemistry and equilibrium, the solubility product \( K_{\infty} \) is an important constant that signifies a sparingly soluble salt's solubility under standard conditions. It is expressed as the product of the molar concentrations of an ionic compound's constituent ions, raised to the power of their stoichiometric coefficients.

For \( \text{PbS}(s) \), a slighty soluble compound, the dissociation in water can be represented by:

\[ \text{PbS}(s) \rightleftharpoons \text{Pb}^{2+}(aq) + \text{S}^{2-}(aq) \]

The solubility product expression \( K_{\infty} \) then becomes \([\text{Pb}^{2+}][\text{S}^{2-}]\).

• A low \( K_{\infty} \) value indicates very limited solubility in water, as seen with \( \text{PbS}(s) \).
• In electrochemical calculations, \( K_{\infty} \) can be used to explain how slight ion presence in aqueous solution can affect electrical potential readings.

In connection to the exercise, this leads to understanding how \( K_{\infty} \) affects the equilibrium constant \( Q \) inside the Nernst equation, guiding toward accurate computation of the standard reduction potential, \( E^0 \). Recognizing the intricacies of \( K_{\infty} \) offers essential insights into the interactions between solubility and electrochemical properties.