Problem 28 - Chemistry and Chemical Reactivity

Question

One half-cell in a voltaic cell is constructed from a copper wire electrode in a \(4.8 \times 10^{-3} \mathrm{M}\) solution of \(\mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2} .\) The other half-cell consists of a zinc electrode in a \(0.40 \mathrm{M}\) solution of \(\mathrm{Zn}\left(\mathrm{NO}_{3}\right)_{2} .\) Calculate the cell potential.

Step-by-Step Solution

Answer

The cell potential is approximately 1.03 V.

Step 1: Identifying Standard Reduction Potentials

First, we need to find the standard reduction potentials for the half-reactions involved. For copper, the reduction half-reaction is: \( \mathrm{Cu}^{2+}(aq) + 2e^- \rightarrow \mathrm{Cu}(s) \) with a standard reduction potential \( E^\circ_{\mathrm{Cu}^{2+}/\mathrm{Cu}} = +0.34 \, \mathrm{V} \). For zinc, the reaction is: \( \mathrm{Zn}^{2+}(aq) + 2e^- \rightarrow \mathrm{Zn}(s) \) with \( E^\circ_{\mathrm{Zn}^{2+}/\mathrm{Zn}} = -0.76 \, \mathrm{V} \).

Step 2: Setting Up the Nernst Equation

The Nernst equation allows us to calculate the cell potential under non-standard conditions. The formula is: \[ E = E^\circ - \frac{RT}{nF} \ln Q \] where \( E \) is the cell potential, \( E^\circ \) is the standard cell potential, \( R \) is the gas constant (8.314 J/(mol·K)), \( T \) is the temperature in Kelvin, \( n \) is the number of moles of electrons transferred, \( F \) is the Faraday constant (96485 C/mol), and \( Q \) is the reaction quotient.

Step 3: Writing the Net Cell Reaction and Calculating Standard Cell Potential

The cell reaction combines the two half-reactions: \[ \mathrm{Zn}(s) + \mathrm{Cu}^{2+}(aq) \rightarrow \mathrm{Zn}^{2+}(aq) + \mathrm{Cu}(s) \]The standard cell potential \( E^\circ_{\mathrm{cell}} \) is given by: \[ E^\circ_{\mathrm{cell}} = E^\circ_{\mathrm{cathode}} - E^\circ_{\mathrm{anode}} \]Substituting the values: \[ (+0.34) - (-0.76) = +1.10 \, \mathrm{V} \]

Step 4: Calculating the Reaction Quotient (Q)

The reaction quotient \( Q \) is calculated from the concentrations of the ionic species involved: \[ Q = \frac{[\mathrm{Zn}^{2+}]}{[\mathrm{Cu}^{2+}]} \]Given that \([\mathrm{Zn}^{2+}] = 0.40 \, \mathrm{M}\) and \([\mathrm{Cu}^{2+}] = 4.8 \times 10^{-3} \, \mathrm{M}\), we find: \[ Q = \frac{0.40}{4.8 \times 10^{-3}} = 83.33 \]

Step 5: Applying the Nernst Equation

Assuming standard temperature of 298 K, and substituting known values into the Nernst equation:\[ E = 1.10 - \frac{(8.314)(298)}{(2)(96485)} \ln(83.33) \]Simplifying:\[ E = 1.10 - \frac{0.0257}{2} \ln(83.33) \approx 1.10 - 0.0396 \times 4.42 \approx 1.0275 \, \mathrm{V} \]

Step 6: Conclusion

The calculated cell potential under the given non-standard conditions is approximately 1.03 V.

Key Concepts

Nernst Equation

The Nernst equation is an essential tool for calculating cell potential under non-standard conditions. It links the concentration of reactants and products to the cell's potential. The formula is given by:\[ E = E^\circ - \frac{RT}{nF} \ln Q \]
Here, \( E \) is the cell potential, \( E^\circ \) is the standard cell potential, \( R \) is the gas constant, \( T \) is the temperature in Kelvin, \( n \) is the number of moles of electrons exchanged, \( F \) is the Faraday constant, and \( Q \) is the reaction quotient.
Using this equation, chemists can predict how changes in concentration affect the cell potential, making it incredibly useful in understanding electrochemical cells like the voltaic cell.

• It allows the calculation of a cell's electrical potential when concentrations deviate from standard state conditions.
• The equation demonstrates the effect of concentration changes on cell potential.
• The Nernst equation helps in determining equilibrium positions and predicting cell behavior.

Voltaic Cell

Voltaic cells, also known as galvanic cells, convert chemical energy into electrical energy through spontaneous redox reactions.
They consist of two different metals connected by a wire and an electrolyte in which ions can move freely.
Each metal acts as a half-cell, and a salt bridge maintains charge balance by allowing ions to flow between the two solutions. These cells are fundamental in powering electronic devices and are the basis for batteries.
Within a voltaic cell, one electrode undergoes oxidation (anode), releasing electrons that travel through the wire to the other electrode, where reduction occurs (cathode).
Some properties include:

• Spontaneous redox reaction creates voltage.
• The flow of electrons produces electricity.
• Commonly used in everyday electronic devices such as batteries.
• Consists of two half-cells connected by a conductive medium.

Standard Reduction Potential

Standard reduction potentials refer to the intrinsic tendency of a substance to gain electrons and be reduced, compared relative to the standard hydrogen electrode.
It is measured in volts (V) under standard conditions, which include all solutes at a concentration of 1 M, gases at a pressure of 1 atm, and a temperature of 298 K (25°C).
This value indicates how easily a substance will act as an oxidizing agent. The standard reduction potential is crucial in predicting the direction of electron flow in an electrochemical cell.
By comparing the potential of the cathode and anode, one can determine the overall cell potential and whether a reaction can occur spontaneously.
Key points include:

• Positive standard reduction potential means a strong tendency to be reduced.
• Help in identifying an electrochemical cell's spontaneous reaction.
• Values are tabulated for many common ions and compounds.
• The higher the reduction potential, the greater the substance's ability to gain electrons.

Reaction Quotient

The reaction quotient, denoted by \( Q \), measures the relative amounts of products and reactants present during a reaction at a given point in time.
It differs from the equilibrium constant, \( K \), as it calculates the potential at any moment before equilibrium is reached.Calculating \( Q \) requires substituting the molar concentrations of the products and reactants into a balanced chemical equation.
In a voltaic cell context, \( Q \) is used to calculate the Nernst equation to find the cell potential under non-standard conditions.

• Reflects changes in concentration from initial to equilibrium states.
• Helps in predicting the direction in which the reaction will proceed to reach equilibrium.
• Used extensively in conjunction with the Nernst equation for electrochemical calculations.
• If \( Q < K \), the reaction will proceed forward towards equilibrium.