A chemist designs an ion-specific probe for measuring \(\left[\mathrm{Ag}^{+}\right]\) in an \(\mathrm{NaCl}\) solution saturated with \(\mathrm{AgCl}\). One half-cell has an Ag wire electrode immersed in the unknown AgCl-saturated \(\mathrm{NaCl}\) solution. It is connected through a salt bridge to the other half-cell, which has a calomel reference electrode [a platinum wire immersed in a paste of mercury and calomel (Hg \(_{2} \mathrm{Cl}_{2}\) ) ] in a saturated KCl solution. The measured \(E_{\text {cell }}\) is \(0.060 \mathrm{~V}\). (a) Given the following standard half-reactions, calculate \(\left[\mathrm{Ag}^{+}\right]\). Calomel: \(\mathrm{Hg}_{2} \mathrm{Cl}_{2}(s)+2 \mathrm{e}^{-} \longrightarrow 2 \mathrm{Hg}(I)+2 \mathrm{Cl}^{-}(a q) \quad E^{\circ}=0.24 \mathrm{~V}\) Silver: \(\mathrm{Ag}^{+}(a q)+\mathrm{e}^{-} \longrightarrow \operatorname{Ag}(s)\) $$E^{\circ}=0.80 \mathrm{~V}$$ (Hint: Assume that [Cl \(^{-}\) ] is so high that it is essentially constant.) (b) A mining engineer wants an ore sample analyzed with the \(\mathrm{Ag}^{+}-\) selective probe. After pretreating the ore sample, the chemist measures the cell voltage as \(0.53 \mathrm{~V}\). What is \(\left[\mathrm{Ag}^{+}\right] ?\)

Step by step solution

01

- Write the Nernst Equation

The Nernst equation for the cell potential is: \[ E_{\text{cell}} = E^\text{cathode} - E^\text{anode} - \frac{RT}{nF} \text{ln}(Q) \]Here, \( Q \) is the reaction quotient, \( R \) is the gas constant (8.314 J/(mol·K)), \( T \) is the temperature in Kelvin (usually 298 K), \( n \) is the number of moles of electrons transferred, and \( F \) is the Faraday constant (96485 C/mol).

02

- Calculate E°cell

Calculate the standard cell potential, \( E^\text{cell}_0 \), using the standard reduction potentials given:\[ E^\text{cell}_0 = E^\text{cathode}_0 - E^\text{anode}_0 \]Given: \[ E^\text{cathode}_0 = 0.80 \text{ V (Silver)} \]\[ E^\text{anode}_0 = 0.24 \text{ V (Calomel)} \]So,\[ E^\text{cell}_0 = 0.80 \text{ V} - 0.24 \text{ V} = 0.56 \text{ V} \]

03

- Formulate Nernst Equation for the Silver Half-Cell

The Nernst equation for the silver half-cell is: \[ E_{\text{Ag}} = E^\text{Ag}_0 - \frac{RT}{nF} \text{ln}([\text{Ag}^+]) \]Using the given cell potential (\( E_{\text{cell}} = 0.060 \text{ V} \)) and assuming a constant \( [\text{Cl}^-] \), we have: \[ 0.060 = 0.56 - \frac{RT}{F} \text{ln} (\frac{[\text{Ag}^+]}{[\text{Cl}^-]^2}) \]Since \( [\text{Cl}^-] \) is constant, this simplifies to \[ 0.060 = 0.56 - \frac{RT}{F} \text{ln} ([\text{Ag}^+]) \]

04

- Isolate [Ag+]

Rearrange the Nernst equation to solve for \( [\text{Ag}^+] \) : \[ \frac{RT}{F} \text{ln} ([\text{Ag}^+]) = 0.56 - 0.060 \]\[ \text{ln} ([\text{Ag}^+]) = \frac{0.56 - 0.060}{\frac{RT}{F}} \]Insert \( R \) (8.314 J/(mol·K)), \( T \) (298 K), and \( F \) (96485 C/mol): \[ \text{ln} ([\text{Ag}^+]) = \frac{0.50 \text{ V}}{\frac{(8.314 × 298)}{96485}} \]Calculate the value: \[ \text{ln} ([\text{Ag}^+]) = \frac{0.50}{0.0256} = 19.53 \]Therefore,\[ [\text{Ag}^+] = e^{19.53} \]

05

- Calculate [Ag+] in Part (b)

Given the voltage \( E_{\text{cell}} = 0.53 \text{ V} \) in part (b), use the same steps: Rearrange the Nernst equation: \[ E_{\text{cell}} = E^\text{cell}_0 - \frac{RT}{nF} \text{ln}([\text{Ag}^+]) \]\[ \frac{RT}{F} \text{ln} ([\text{Ag}^+]) = 0.56 - 0.53 \]\[ \text{ln} ([\text{Ag}^+]) = \frac{0.03}{0.0256} = 1.17 \]Finally,\[ [\text{Ag}^+] = e^{1.17} \]

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Key Concepts

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

Electrochemistry is the foundation

Electrochemistry studies the relationship between electricity and chemical reactions. It involves reactions where electrons are transferred between substances. These reactions are crucial in devices like batteries and electrochemical cells, including the setup involving a silver electrode and a calomel reference electrode as described in the problem. Understanding electrochemistry requires grasping concepts like oxidation-reduction reactions (redox), where one substance loses electrons (oxidation) and another gains electrons (reduction). The half-reactions provided in the problem are examples of such redox processes.

Cell Potential Explained

Cell potential, denoted as E_cell, measures the voltage generated by an electrochemical cell. It can be thought of as the 'driving force' of the electrochemical reaction. In the exercise, the cell potential is used to determine ion concentration. It depends on the nature of the electrodes and the concentration of ions in the solutions. To find the cell potential, we use the Nernst equation, which accounts for these factors and allows us to calculate the ion concentration in the solution based on the measured voltage.

Reaction Quotient and Its Role

The reaction quotient, denoted as Q, indicates the ratio of product concentrations to reactant concentrations at any point during a reaction. For the Nernst equation, it's particularly crucial because it influences the cell potential. In simple terms, Q shows us how far a reaction has proceeded. For the silver electrode reaction given, Q involves the concentration of Ag^+ ions. Assuming the Cl^- concentration is constant, the Nernst equation simplifies, making it easier to solve for the unknown ion concentration.

Understanding Standard Reduction Potential

Standard reduction potential (E°) is the inherent potential (voltage) of a half-reaction when all components are in their standard states (typically 1 M concentration, 1 atm pressure, and 25°C). It shows how readily a substance gains electrons (reduces). The exercise uses standard reduction potentials to calculate the cell potential. For instance, the standard reduction potential for Ag^+ to Ag is 0.80 V. This value is critical in determining the overall cell potential and thereby understanding the cell's capacity to drive the reaction.

The Importance of Faraday Constant

The Faraday constant (F) represents the charge of one mole of electrons, approximately 96485 coulombs per mole (C/mol). In electrochemistry, it’s a vital factor because it connects the amount of substance with the electric charge. In the Nernst equation, it serves to scale the potential difference to the electrochemical scale. Due to this constant, we can convert between electrical terms (voltage, charge) and chemical terms (moles of substance), facilitating the calculation of ion concentrations from cell potentials.

Gas Constant and Its Usage

The gas constant (R) is 8.314 J/(mol·K) and it's crucial in many physical chemistry equations, including the Nernst equation. It relates the energy scale of gas behavior to temperature and moles of gas. In electrochemistry, R helps account for temperature effects on cell potential. For standard temperature calculations, we typically use 298 K (25°C). Combining R with other constants and variables in the Nernst equation allows us to predict how changes in conditions affect the cell potential and thus the ion concentrations.