Problem 60 - Chemistry The Central Science In Si Units

Question

Consider the voltaic cell illustrated in Figure \(20.5,\) which is based on the cell reaction $$ \mathrm{Zn}(s)+\mathrm{Cu}^{2+}(a q) \longrightarrow \mathrm{Zn}^{2+}(a q)+\mathrm{Cu}(s) $$ Under standard conditions, what is the maximum electrical work, in joules, that the cell can accomplish if \(50.0 \mathrm{~g}\) of copper is formed?

Step-by-Step Solution

Answer

The maximum electrical work is approximately \(-166899 \text{ J}\.}

Step 1: Calculate Molar Mass of Copper

First, find the molar mass of copper (Cu), which is approximately 63.55 g/mol. This value is needed to determine how many moles of copper are formed.

Step 2: Determine Moles of Copper Formed

Use the desired mass of copper (50.0 g) to calculate moles. Use the formula: \( \text{moles of Cu} = \frac{\text{mass of Cu}}{\text{molar mass of Cu}} = \frac{50.0 \text{ g}}{63.55 \text{ g/mol}} = 0.787 \text{ mol} \).

Step 3: Identify the Number of Electrons Transferred

The balanced reaction indicates that 2 electrons are transferred per mole of reaction. Therefore, for 0.787 mol of copper, twice the number of moles of electrons are transferred: \( 2 \times 0.787 = 1.574 \text{ mol of electrons} \).

Step 4: Calculate Maximum Work Using Cell Potential

Use the Nernst equation for electrochemical cells: \( W = -nFE \), where \( n = 1.574 \text{ mol} \), \( F = 96485 \text{ C/mol} \) (Faraday's constant), and \( E = 1.10 \text{ V} \) (standard cell potential for this redox reaction). Hence, the maximum work is \( W = -1.574 \times 96485 \times 1.10 \).

Step 5: Solve for Maximum Electrical Work

Calculate the work: \( W = -1.574 \times 96485 \times 1.10 = -166899.2 \text{ J} \). The maximum electrical work is approximately \(-166899 \text{ J}\).

Key Concepts

Voltaic Cell

A voltaic cell, also known as a galvanic cell, is a device that generates electrical energy through a spontaneous chemical reaction. It consists of two different metals connected by a salt bridge or a porous membrane. These metals are placed in separate containers filled with their own ionic solutions and are linked externally by a wire through which electrons can flow. This arrangement allows the conversion of chemical energy into electrical energy.
The essential parts of a voltaic cell include:

• Two half-cells: Each half-cell contains a metal electrode and a solution of ions.
• A salt bridge or porous membrane: This part connects the two half-cells, allowing ions to move and maintain charge balance.
• An external circuit: Electrons flow through this wire, powering any connected devices with the electric current.

In our example, zinc ( Zn ) and copper ( Cu ) electrodes are used. When zinc oxidizes and loses electrons, they travel through the wire to the copper electrode, where they reduce Cu^{2+} ions, creating an electric current.

Cell Potential

Cell potential, also known as electromotive force (EMF), is the measure of the potential difference between two electrodes in an electrochemical cell. It reflects the cell's ability to produce an electric current. The standard cell potential ( E^0_{cell} ) is measured under standard conditions, which include 1 M concentration, 1 atm pressure, and a temperature of 25°C (298 K).
The standard cell potential of a voltaic cell is calculated based on the reduction potentials of the electrodes. In our reaction involving zinc and copper, the standard reduction potential is given for copper ( Cu^{2+}/Cu i.e. +0.34 V) and zinc ( Zn^{2+}/Zn i.e. -0.76 V). The overall standard cell potential ( E^0_{cell} ) is the difference between these values, resulting in 1.10 V.

Electrical Work

The electrical work done by an electrochemical cell can be calculated by using the relationship between cell potential and the amount of charge transferred. The equation for this work is:\[ W = -nFE \]Where:

• W is the work in joules
• n is the number of moles of electrons transferred
• F is Faraday's constant (96485 C/mol)
• E is the cell potential in volts

In the context of the example given, the cell can accomplish a maximum work of -166899 J. This negative sign indicates that the work is done by the system (the voltaic cell) as it releases energy.

Nernst Equation

The Nernst equation is crucial for understanding how cell potential changes with varying conditions. It gives us a way to calculate the cell potential of an electrochemical cell under non-standard conditions and is expressed as:\[ E = E^0 - \frac{RT}{nF} \ln Q \]Where:

• E is the cell potential under non-standard conditions.
• E^0 is the standard cell potential.
• R is the universal gas constant (8.314 J/mol·K).
• T is the temperature in Kelvin.
• n is the number of moles of electrons exchanged.
• F is Faraday's constant.
• Q is the reaction quotient, a measure of the concentration of reactants and products.

This equation highlights the dependency of cell potential on concentration and temperature, enabling predictions of cell behavior in different scenarios. Understanding the Nernst equation allows us to delve deeper into electrochemical processes, extending beyond the standard state.