Question

Consider a Daniell cell with \(1.0 \mathrm{M}\) ion concentrations: Does the cell voltage increase, decrease, or remain the same when each of the following changes is made? Explain. (a) \(5.0 \mathrm{M} \mathrm{CuSO}_{4}\) is added to the cathode compartment. (b) \(5.0 \mathrm{M} \mathrm{H}_{2} \mathrm{SO}_{4}\) is added to the cathode compartment. (c) \(5.0 \mathrm{M} \mathrm{Zn}\left(\mathrm{NO}_{3}\right)_{2}\) is added to the anode compartment. (d) \(1.0 \mathrm{M} \mathrm{Zn}\left(\mathrm{NO}_{3}\right)_{2}\) is added to the anode compartment.

Short answer

(a) Increases, (b) Remains the same, (c) Decreases, (d) Remains the same.

Step-by-step solution

Understanding the Daniell Cell Components

A Daniell cell consists of a zinc electrode in a zinc sulfate solution as the anode and a copper electrode in a copper sulfate solution as the cathode. Changes made to these compartments can affect the cell potential based on the Nernst equation.

Applying the Nernst Equation

The Nernst equation explains how the cell potential is influenced by ion concentration: \(E = E^{\circ} - \frac{RT}{nF} \ln Q\), where \(Q\) is the reaction quotient representing the ratio of product ions to reactant ions.

Evaluating the effect of CuSO4 at the Cathode - (a)

Adding \(5.0 \ M \ CuSO_4\) increases the concentration of \(Cu^{2+}\) ions in the cathode. According to the Nernst equation, an increase in the concentration of reactants (\( Cu^{2+} \) ions) will decrease \( Q\), increasing the cell potential.

Considering H2SO4 effect at the Cathode - (b)

\(H_2SO_4\) dissociates into ions, but since it doesn't provide additional \(Cu^{2+}\) or change the reaction quotient directly, its main impact is modifying the ionic strength rather than cell voltage. Therefore, the cell potential remains unchanged.

Impact of Adding Zn(NO3)2 at the Anode - (c)

Adding \(5.0 \ M \ Zn(NO_3)_2\) increases the \(Zn^{2+}\) ion concentration, which increases \(Q\) since the anode reaction involves \(Zn^{2+}\) as a product. This increases the value of \(Q\) and thus decreases the cell potential.

Evaluating the Addition of 1.0 M Zn(NO3)2 at the Anode - (d)

Since the concentration of \(Zn^{2+}\) ions doesn't change from the initial set-up (as it remains at \(1.0 \ M\)), there is no change to \(Q\). Therefore, the cell potential remains unchanged.

Key concepts

Daniell cell

The Daniell cell is a type of electrochemical cell created from a zinc electrode immersed in a zinc sulfate solution and a copper electrode immersed in a copper sulfate solution. This setup involves two half-cells connected by a salt bridge or porous barrier that allows ions to flow between them without the solutions mixing directly.
The core function of a Daniell cell is to convert chemical energy into electrical energy through spontaneous redox reactions.
In this process:

• Zinc acts as the anode, where oxidation occurs, releasing electrons and Zn²⁺ ions.
• Copper acts as the cathode, where reduction occurs, as Cu²⁺ ions in the solution gain electrons to become copper metal.

This flow of electrons from zinc to copper generates an electric current, which can be harnessed to do work. The unique composition of the Daniell cell makes it a perfect example to study cell potentials and the effects of various ion concentration changes.

Nernst equation

The Nernst equation is a fundamental formula in electrochemistry, used to determine the cell potential of an electrochemical cell under non-standard conditions. The equation is expressed as:\[ E = E^\circ - \frac{RT}{nF} \ln Q \]Here, \( E \) represents the cell potential at specific conditions, \( E^\circ \) is the standard cell potential, \( R \) is the universal gas constant (8.314 J mol^-1 K^-1), \( T \) denotes temperature in Kelvin, \( n \) stands for the number of moles of electrons exchanged in the reaction, \( F \) is the Faraday constant (96485 C mol^-1), and \( Q \) is the reaction quotient.
The Nernst equation helps us understand how variations in ion concentrations can affect the cell's potential.

• If the concentration of reactants increases, \( Q \) decreases, thus increasing the cell potential.
• Conversely, an increase in product concentration raises \( Q \), which results in a decrease in cell potential.

This equation beautifully links chemical kinetics and thermodynamics, providing insight into how real-world conditions impact electrochemical reactions.

Cell potential

Cell potential, also known as electromotive force (EMF), is a measure of the voltage, or electric potential difference, produced by an electrochemical cell. It represents the driving force behind the flow of electrons from one electrode to the other.
In terms of measurement, standard cell potential (\( E^\circ \)) is evaluated under standard conditions:

• temperature of 298 K (25°C)
• pressure of 1 atm for gases
• ion concentrations of 1.0 M

The cell potential can change with various factors, including ion concentration and temperature. By applying the Nernst equation, these changes can be precisely calculated, providing insight into how a reaction might proceed under different environmental conditions.
This is especially relevant in a Daniell cell, where adjustments in the concentrations of ions like \( Cu^{2+} \) or \( Zn^{2+} \) can significantly influence the cell potential, either enhancing or reducing the cell's ability to do work.

Reaction quotient

The reaction quotient, represented by \( Q \), is a dimensionless number that describes the relative amounts of products and reactants present during a reaction at any given time. Similar in structure to the equilibrium constant \( K \), but applicable to non-equilibrium conditions, it is calculated using the formula:\[ Q = \frac{\text{products (concentration raised to their stoichiometric coefficients)}}{\text{reactants (concentration raised to their stoichiometric coefficients)}} \]In electrochemistry, \( Q \) reflects the instantaneous state of a reaction, revealing how far the system is from equilibrium. This influences the cell potential according to the Nernst equation.
For example:

• An increase in reactant concentration decreases \( Q \), often increasing cell potential.
• Increased product concentration raises \( Q \), thus decreasing the cell potential.

Understanding \( Q \) provides insight into reaction progress, measures how changes in concentration affect a cell's efficiency, and predicts how these changes can shift the balance of the electrochemical reaction in cells like the Daniell cell.