Problem 90 - Chemistry: The Central Science

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 plated out?

Step-by-Step Solution

Answer

The maximum electrical work that the voltaic cell can accomplish under standard conditions when 50 grams of copper is plated out is 167,101 Joules.

Step 1: Determine the moles of copper

Given that 50 grams of copper is plated out, we need to find the number of moles of copper involved. The molar mass of copper (Cu) is 63.55 g/mol. We can use this to calculate the moles of Cu: Moles of Cu = (Mass of Cu) / (Molar mass of Cu) Moles of Cu = \( \frac{50}{63.55} \) = 0.787 moles

Step 2: Determine the number of moles of electrons

In the given cell reaction, 1 mole of copper (Cu) is produced by 1 mole of Cu2+ by gaining 2 moles of electrons. Since there are 0.787 moles of Cu being plated out, we can find the number of moles of electrons transferred: Moles of electrons = 0.787 moles of Cu × 2 moles of electrons/mole of Cu = 1.574 moles of electrons

Step 3: Calculate cell potential

The cell reaction given is: $$ \mathrm{Zn}(s)+\mathrm{Cu}^{2+}(a q) \longrightarrow \mathrm{Zn}^{2+}(a q)+\mathrm{Cu}(s) $$ We can use the standard reduction potentials of each half-cell to determine the overall cell potential. The standard reduction potentials are: Cu2+ + 2e- -> Cu (s) : Eº = +0.34 V Zn2+ + 2e- -> Zn (s) : Eº = -0.76 V The overall cell potential can be found by subtracting the reduction potential of the anode (Zn) from that of the cathode (Cu): Cell potential (Eºcell) = Eº (Cu2+/Cu) - Eº (Zn2+/Zn) = 0.34 - (-0.76) = 1.10 V

Step 4: Calculate the maximum electrical work

Now that we have the cell potential (Eºcell) and the number of moles of electrons involved, we can calculate the maximum work (in Joules) done by the voltaic cell. We can use Faraday's constant (F), which is 96,485 C/mol, to convert the number of moles of electrons to Coulombs of charge: Charge (Q) = moles of electrons × F = 1.574 moles × 96,485 C/mol = 151,910 C The maximum electrical work (W) is given by the product of the cell potential and the charge: W = Eºcell × Q = 1.10 V × 151,910 C = 167,101 Joules Therefore, the maximum electrical work that the voltaic cell can accomplish under standard conditions when 50 grams of copper is plated out is 167,101 Joules.

Key Concepts

Cell Potential

Cell potential is a key concept when discussing voltaic cells. It reflects the ability of a voltaic cell to produce electrical energy. In simplistic terms, it's like the cell's measure of oomph or vigor, which drives the electricity through the circuit. This potential difference is measured in volts (V). To determine the overall cell potential, one must focus on the two half-reactions taking place in a voltaic cell: the reduction at the cathode and the oxidation at the anode. These reactions each have a standard reduction potential, denoted usually as . The cell potential (Eºcell) is calculated by finding the difference between these potentials, specifically subtracting the anode potential from the cathode potential. For the copper-zinc cell in our problem, this is given by:

• Cathode (Cu²⁺/Cu) Reaction: Eº = +0.34 V
• Anode (Zn²⁺/Zn) Reaction: Eº = -0.76 V

Thus, the Eºcell = 0.34 V - (-0.76 V) = 1.10 V. This value suggests the maximum potential of the voltaic cell to do work or move electrons through the circuit. Remember, a positive cell potential indicates that the process is spontaneous, meaning it can proceed without additional energy input.

Faraday's Constant

Faraday's constant is a fundamental value in electrochemistry. Symbolized as F, it represents the charge of one mole of electrons. Its value is approximately 96,485 coulombs per mole of electrons (C/mol). This constant is integral when determining how much charge is passed through or utilized in an electrochemical reaction.
Within the context of a voltaic cell or any such electrochemical calculation, Faraday's constant provides a bridge between the moles of electrons involved in a reaction and the total electrical charge transferred. By calculating the number of moles of electrons exchanged in a given reaction and multiplying it by Faraday's constant, one can determine the specific charge involved.
This is crucial for calculating electrical work or energy output, enabling us to compute how much electrical energy is harnessed or used in practical terms. Additionally, Faraday's constant links microscopic molecular processes to macroscopic energy systems, facilitating an understanding of extensive systems based on atomic interactions.

Standard Reduction Potentials

Standard reduction potentials are intrinsic values that dictate the likelihood of a reduction reaction occurring under standard conditions. These potentials denote the tendency of a species to gain electrons and thereby be "reduced". Each half-cell in a voltaic cell possesses its specific standard reduction potential, often measured against the standard hydrogen electrode (SHE), which is set arbitrarily at 0.00 volts.
A positive reduction potential indicates a species strongly lends itself to reduction, while a negative value suggests less likelihood or an inclination towards oxidation in the right context. The standard reduction potentials are pivotal for understanding and calculating the overall cell potential, as mentioned in the cell potential section above.
In the copper-zinc voltaic cell:

• Zinc's reduction potential indicates a less favorable reduction, thus its role as the anode (site of oxidation).
• Copper's higher positive reduction potential favors its reduction, positioning it as the cathode.

Using these potentials helps predict the electromotive force and spontaneity of the cell reaction, indicating how effectively the reaction can produce electrical work.

Electrical Work

Electrical work in the realm of electrochemistry is the amount of work (energy) a voltaic cell can perform as it drives electrons through an external circuit. Calculating this work is essential to understand how much usable energy can be derived from an electrochemical reaction. The work (W) done by the cell involves converting chemical energy into electrical energy and is quantified by the formula: \[ W = Eºcell \times Q \] where \(Eºcell\) is the cell potential (explained earlier), and \(Q\) represents the charge (in coulombs) moved by the electrons.Here, \(Q\) is linked to the moles of electrons via Faraday's constant. For instance, if we have 1.574 moles of electrons (as calculated in the problem), then the charge \(Q\) is \(1.574 \times 96,485\,\text{C/mol}\) = 151,910 C.
This charge, multiplied by the cell potential, yields the maximum theoretical work, such as 167,101 Joules in the problem context. This estimation helps assess the cell’s efficiency and energy output under specified conditions, making it vital for applications ranging from batteries to electroplating.