Question

A student wanted to measure the copper(II) concentration in aqueous solution. For the cathode half-cell she used a silver electrode with a 1.00 -M solution of \(\mathrm{AgNO}_{3}\). For the anode half-cell she used a copper electrode dipped into the aqueous sample. The cell gave \(E_{\text {cell }}=\) \(0.62 \mathrm{~V}\) at \(25^{\circ} \mathrm{C}\). Calculate the copper(II) ion concentration of the solution.

Short answer

The copper(II) ion concentration is approximately \(3.7 \times 10^{-6}\) M.

Step-by-step solution

Write the Cell Reaction

In this electrochemical cell, the reaction is as follows:\[ \text{Ag}^+ (aq) + e^- \rightarrow \text{Ag} (s) \quad \text{(Cathode, Reduction)} \]\[ \text{Cu} (s) \rightarrow \text{Cu}^{2+} (aq) + 2e^- \quad \text{(Anode, Oxidation)} \]Combining these half-reactions, the full cell reaction is:\[ 2\text{Ag}^+ (aq) + \text{Cu} (s) \rightarrow 2\text{Ag} (s) + \text{Cu}^{2+} (aq) \]

Determine the Standard Electrode Potentials

From standard electrode potential tables, we have:- Reduction potential for silver: \( E^\circ_{\text{Ag}^+/\text{Ag}} = +0.80 \text{ V} \)- Reduction potential for copper: \( E^\circ_{\text{Cu}^{2+}/\text{Cu}} = +0.34 \text{ V} \)

Calculate the Standard Cell Potential

The standard cell potential \( E^\circ_{\text{cell}} \) can be found using the formula:\[ E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}} \]\[ E^\circ_{\text{cell}} = 0.80 \text{ V} - 0.34 \text{ V} = 0.46 \text{ V} \]

Use Nernst Equation

The Nernst equation for this cell at 25°C (298 K) is:\[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF} \ln(Q) \]Where \( R = 8.314 \text{ J/mol K} \), \( T = 298 \text{ K} \), \( n = 2 \text{ mol e}^- \), and \( F = 96485 \text{ C/mol e}^- \).\( Q \) is the reaction quotient and in this case is \( \frac{[\text{Cu}^{2+}]}{[\text{Ag}^+]^2} \). Since \([\text{Ag}^+]\) is 1.00 M, \( Q = [\text{Cu}^{2+}] \).

Substitute Values into Nernst Equation

Given \( E_{\text{cell}} = 0.62 \text{ V} \), substitute all known values into the Nernst equation:\[ 0.62 = 0.46 - \frac{(8.314)(298)}{(2)(96485)} \ln([\text{Cu}^{2+}]) \]\[ 0.62 = 0.46 - 0.0128 \ln([\text{Cu}^{2+}]) \]

Solve for \( \ln([\text{Cu}^{2+}]) \)

Rearrange and solve for \( \ln([\text{Cu}^{2+}]) \):\[ 0.62 - 0.46 = -0.0128 \ln([\text{Cu}^{2+}]) \]\[ 0.16 = -0.0128 \ln([\text{Cu}^{2+}]) \]\[ \ln([\text{Cu}^{2+}]) = -12.5 \]

Calculate \([\text{Cu}^{2+}]\) Concentration

To find the concentration of \( \text{Cu}^{2+} \), exponentiate both sides:\[ [\text{Cu}^{2+}] = e^{-12.5} \]\[ [\text{Cu}^{2+}] \approx 3.7 \times 10^{-6} \text{ M} \]

Key concepts

Nernst Equation

The Nernst Equation is a crucial formula in electrochemistry that allows us to calculate the potential of an electrochemical cell under non-standard conditions. It is expressed as:\[E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF} \ln(Q)\]In this equation:

• \( E_{\text{cell}} \) is the cell potential under the given conditions.
• \( E^\circ_{\text{cell}} \) signifies the standard cell potential, derived from the standard reduction potentials of the involved half-reactions.
• \( R \) is the universal gas constant with a value of \( 8.314 \text{ J/mol K} \).
• \( T \) is the temperature in Kelvin.
• \( n \) indicates the number of electrons transferred in the cell reaction.
• \( F \) is the Faraday constant, \( 96485 \text{ C/mol e}^- \).
• \( Q \) represents the reaction quotient, detailing the concentration or pressures of reactants and products.

By inserting known values into this equation, it's possible to solve for unknowns like the ion concentration, as demonstrated in the exercise to find \([\text{Cu}^{2+}]\).
Through the Nernst Equation, practitioners can understand how various factors influence cell potential outside of standard conditions.

Standard Electrode Potential

Standard Electrode Potential, denoted by \( E^\circ \), is the voltage achieved when a half-cell is connected to a standard hydrogen electrode under standard conditions (1 M concentration, 1 atm pressure, and 25°C).
Each half-reaction's standard electrode potential is determined and used to predict the cell’s overall standard potential.For instance, in the given exercise, the standard electrode potential for silver, \( E^\circ_{\text{Ag}^+/\text{Ag}} \), is \( +0.80 \text{ V} \).
Meanwhile, copper’s standard electrode potential, \( E^\circ_{\text{Cu}^{2+}/\text{Cu}} \), is \( +0.34 \text{ V} \). These values are derived from experimental data and compiled in reference tables.
To find the standard cell potential \( E^\circ_{\text{cell}} \), the formula utilized is:\[E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}}\]This simplifies understanding the tendency of a redox reaction to proceed and its electrical potential difference under ideal conditions.
Grasping standard electrode potentials is vital, as they serve as building blocks to analyze more complex electrochemical systems.

Reaction Quotient

The Reaction Quotient, represented as \( Q \), plays a central role in thermodynamics and electrochemistry. It is used to gauge the direction a reaction must go to reach equilibrium.
For a reaction:\[aA + bB \rightleftharpoons cC + dD\]The reaction quotient \( Q \) is calculated as:\[Q = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]Here, \([A]\), \([B]\), \([C]\), and \([D]\) are the molar concentrations of the reactants and products. In scenarios where the reaction involves gases, partial pressures are used.
In the context of the provided exercise, the reaction quotient was simplified to \( Q = [\text{Cu}^{2+}] \) because the concentration of silver (\([\text{Ag}^+]\)) was given as 1 M.
As the Nernst Equation is dependent on \( Q \), understanding its value helps determine shifts from expected standard conditions, revealing the concentration or pressure changes needed for equilibrium.
Reaction Quotient analysis is quintessential in predicting and interpreting the behavior of chemical reactions in dynamic systems.