Black-and-white photographic film is coated with silver halides. Because silver is expensive, the manufacturer monitors the \(\mathrm{Ag}^{+}\) content of the waste stream, \(\left[\mathrm{Ag}^{+}\right]_{\text {waste }},\) from the plant with an \(\mathrm{Ag}^{+}\) -selective electrode at \(25^{\circ} \mathrm{C}\). A stream of known \(\mathrm{Ag}^{+}\) concentration, \(\left[\mathrm{Ag}^{+}\right]_{\text {standand }},\) is passed over the electrode in turn with the waste stream and the data recorded by a computer. (a) Write the equations relating the nonstandard cell potential to the standard cell potential and \(\left[\mathrm{Ag}^{+}\right]\) for each solution. (b) Combine these into a single equation to find [Ag \(^{+}\) ] waste (c) Rewrite the equation from part (b) to find [Ag \(^{+}\) ] \(_{\text {waste }}\) in \(\mathrm{ng} / \mathrm{L}\). (d) If \(E_{\text {waste }}\) is \(0.003 \mathrm{~V}\) higher than \(E_{\text {standard }},\) and the standard solution contains \(1000 . \mathrm{ng} / \mathrm{L},\) what is \(\left[\mathrm{Ag}^{+}\right]_{\text {wast }} ?\) (e) Rewrite the equation from part (b) to find [Ag \(^{+}\) ] waste for a system in which \(T\) changes and \(T_{\text {waste }}\) and \(T_{\text {standard }}\) may be different.

Step by step solution

01

Write the Nernst Equation for the Standard Solution

The Nernst equation for the standard solution can be given as: \[ E_{\text{standard}} = E^\theta - \frac{RT}{nF} \times \frac{1}{z} \times \text{ln}([\text{Ag}^+]_{\text{standard}}) \] where:- \( E^\theta \) is the standard cell potential.- \( R \) is the gas constant \( (8.314 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1}) \).- \( T \) is the temperature in Kelvin \( (298 \, \text{K}) \).- \( n \) is the number of moles of electrons exchanged \( (n = 1) \).- \( F \) is the Faraday constant \( (96485 \, \text{C} \, \text{mol}^{-1}) \).- \( z \) is the charge on the ion which is \(1\).

02

Write the Nernst Equation for the Waste Solution

Similarly, the Nernst equation for the waste solution can be given as: \[ E_{\text{waste}} = E^\theta - \frac{RT}{nF} \times \frac{1}{z} \times \text{ln}([\text{Ag}^+]_{\text{waste}}) \]

03

Combine the Equations

To find the concentration of \([\text{Ag}^+]_{\text{waste}}\), we can subtract the equation for the standard solution from the equation for the waste solution: \[ E_{\text{waste}} - E_{\text{standard}} = - \frac{RT}{nF} \times \frac{1}{z} \times \text{ln} \frac{[\text{Ag}^+]_{\text{waste}}}{[\text{Ag}^+]_{\text{standard}}} \]

04

Solve for [Ag⁺]waste

Rearrange the equation to solve for \([\text{Ag}^+]_{\text{waste}}\): \[ \text{ln} \frac{[\text{Ag}^+]_{\text{waste}}}{[\text{Ag}^+]_{\text{standard}}} = \frac{nF}{RT} \times (E_{\text{standard}} - E_{\text{waste}}) \]\[ \frac{[\text{Ag}^+]_{\text{waste}}}{[\text{Ag}^+]_{\text{standard}}} = e^{\frac{nF}{RT} \times (E_{\text{standard}} - E_{\text{waste}})} \]\[ [\text{Ag}^+]_{\text{waste}} = [\text{Ag}^+]_{\text{standard}} \times e^{\frac{nF}{RT} \times (E_{\text{standard}} - E_{\text{waste}})} \]

05

Convert Concentration to ng/L

Assuming the standard solution concentration \([\text{Ag}^+]_{\text{standard}}\) is given in ng/L, the output will directly be in the same units. If conversion is necessary, use atomic mass of Ag.

06

Calculate Based on Given Data

With provided values: - \( E_{\text{waste}} = E_{\text{standard}} + 0.003 \, V \), - \([\text{Ag}^+]_{\text{standard}} = 1000 \, \text{ng/L} \).Use the formula: \[ [\text{Ag}^+]_{\text{waste}} = [\text{Ag}^+]_{\text{standard}} \times e^{\frac{nF}{RT} \times (E_{\text{standard}} - E_{\text{waste}})} \] Since \( E_{\text{waste}} = E_{\text{standard}} + 0.003 \): \[ E_{\text{standard}} - E_{\text{waste}} = -0.003 \]Substitute into the equation: \[ [\text{Ag}^+]_{\text{waste}} = 1000 \times e^{\frac{1 \times 96485}{8.314 \times 298} \times (-0.003)} \ [\text{Ag}^+]_{\text{waste}} = 1000 \times e^{-0.116} \]\[ [\text{Ag}^+]_{\text{waste}} \ \approx 1000 \times 0.890 \approx 890 \, \text{ng/L} \]

07

Equation with Variable Temperature

If the temperature is variable: \[ E_{\text{waste}} - E_{\text{standard}} = - \frac{R}{nF} \times (\frac{1}{T_{\text{waste}}} - \frac{1}{T_{\text{standard}}}) \times \text{ln} \frac{[\text{Ag}^+]_{\text{waste}}}{[\text{Ag}^+]_{\text{standard}}} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nernst Equation

The Nernst Equation connects the cell potential with the concentrations of the ions involved in the reaction. It's a foundational concept for understanding electrochemical cells. Here's the general form:
\[ E = E^\theta - \frac{RT}{nF} \times \frac{1}{z} \times \text{ln}(Q) \]
where:

• \(E\) is the cell potential under non-standard conditions.
• \(E^\theta\) is the standard cell potential.
• \(R\) is the gas constant (8.314 J mol^-1 K^-1).
• \(T\) is the temperature in Kelvin (298 K).
• \(n\) is the number of moles of electrons transferred in the reaction (for silver, n = 1).
• \(F\) is the Faraday constant (96485 C mol^-1).
• \(z\) is the charge on the ion (for silver, z = 1).

This equation helps us quantify how the potential of an electrochemical cell changes with varying concentrations. It's vital for calculations involving electrochemical cells, such as the ones in waste stream analysis of silver ions.

Electrochemical Cells

Electrochemical cells are devices that convert chemical energy into electrical energy through redox reactions. There are two main types: Galvanic cells and Electrolytic cells. In the given problem, we're dealing with a form of concentration cell where the same substance (Ag⁺) is involved, but at different concentrations in the waste stream and the standard stream.
Each cell consists of two electrodes (anode and cathode) immersed in an electrolyte. The potential difference between the electrodes, measured in volts, is what drives the electrical current. This potential difference is influenced by the concentrations of ions, as described by the Nernst Equation. The goal here is to maintain a check on the concentration of \(Ag^+\) ions, as they are costly and environmental pollutants.

Waste Stream Analysis

Waste stream analysis is crucial for both environmental protection and cost efficiency, especially when dealing with valuable and potentially toxic materials like silver ions (\(Ag^+\)).
The process typically involves:

• Monitoring the concentration of \(Ag^+\) ions using a selective electrode.
• Comparing the results with a standard solution of known concentration.
• Using these measurements to calculate the \(Ag^+\) concentration in the waste stream employing the Nernst Equation.

In this problem, a consistent difference in potential (0.003 V) between the waste and standard solutions is observed. This potential difference is then used to calculate the concentration of silver in the waste stream. Monitoring such levels ensures that the plant minimizes wastage and complies with environmental regulations.

Standard Cell Potential

The standard cell potential (\(E^\theta\)) is a measure of the electromotive force (emf) of a cell under standard conditions (1 M concentration, 1 atm pressure, and 298 K temperature). For silver, the standard reduction potential is significant:

• It serves as a reference point in the Nernst Equation.
• Allows comparison between different electrochemical cells.

It's calculated using standard reduction potentials from electrochemical series tables. The value remains constant for a given reaction and is used to predict whether a reaction will proceed spontaneously.
In our context, knowing the standard cell potential is necessary to determine how deviations (due to changes in concentration) affect the cell's behavior. It's especially useful for getting accurate measurements and ensuring reliable waste stream analysis.