Question

A zinc-copper battery is constructed as follows at \(25^{\circ} \mathrm{C}\) : $$ \mathrm{Zn}\left|\mathrm{Zn}^{2+}(0.10 M)\right|\left|\mathrm{Cu}^{2+}(2.50 M)\right| \mathrm{Cu} $$ The mass of each electrode is \(200 . \mathrm{g}\). a. Calculate the cell potential when this battery is first connected. b. Calculate the cell potential after \(10.0 \mathrm{~A}\) of current has flowed for \(10.0 \mathrm{~h}\). (Assume each half-cell contains \(1.00 \mathrm{~L}\) of solution.) c. Calculate the mass of each electrode after \(10.0 \mathrm{~h}\). d. How long can this battery deliver a current of \(10.0 \mathrm{~A}\) before it goes dead?

Short answer

The initial cell potential of the zinc-copper battery is 1.0383 V. After 10.0 A of current has flowed for 10.0 h, the battery reaches its dead state and the cell potential becomes zero. The mass of the Zn electrode after 10.0 h is 243.73 g, while the mass of the Cu electrode is 237.08 g. The battery can deliver a current of 10.0 A for 10.0 h before it goes dead.

Step-by-step solution

Determine the standard electrode potentials for Zn and Cu

To find the initial cell potential, we will start by determining the standard electrode potential for each electrode. These values can usually be found in chemistry textbooks or online resources. We will denote the standard electrode potential of Zn as \(E^0_{Zn}\) and for Cu as \(E^0_{Cu}\). \(E^0_{Zn} = -0.76V\) \(E^0_{Cu} = +0.34V\)

Use the Nernst equation to find the initial cell potential

Now, we will use the Nernst equation: \(E_{cell} = E^0_{cell} - \frac{0.0592}{n} \log_{10}\frac{Q}{1}\) where: \(E_{cell}\) = cell potential at the given concentrations \(E^0_{cell}\) = standard cell potential (\(E^0_{cell} = E^0_{Cu} - E^0_{Zn}\)) n = number of electrons transferred (for Zn - Cu system, n = 2) Q = reaction quotient Using the given concentrations: Q = \(\frac{[Cu^{2+}]}{[Zn^{2+}]}\) = \(\frac{2.50}{0.10}\) = 25 Now, plug in the values and solve for \(E_{cell}\): \(E_{cell} = (0.34 - (-0.76)) - \frac{0.0592}{2} \log_{10}(25)\) \(E_{cell} = 1.10 - 0.0442 \times 1.398\) \(E_{cell} = 1.10 - 0.0617\) \(E_{cell} =1.0383V\) So, the initial cell potential is 1.0383V. Step 2: Calculate the cell potential after 10.0A of current has flowed for 10.0h

Calculate the number of moles of electrons transferred after 10h at 10A

We will use Faraday's Law of electrolysis to determine the number of moles of electrons transferred: n = \(\frac{It}{F}\) where: I = current in Amperes (A) t = time in seconds (s) F = Faraday's constant (\(96485C/mol\)) First, convert the time to seconds: t = 10.0h × 3600s/h = 36000s Now calculate the moles of electrons transferred: n = \(\frac{10.0A \times 36000s}{96485C/mol}\) = 3.732 mol

Calculate the updated concentrations after 10 hours

Using stoichiometry, we can now update the concentrations of \(Zn^{2+}\) and \(Cu^{2+}\): Zn: \(0.10 M + \frac{3.732 mol}{1.00 L} = 0.10 M + 3.732 M = 3.832 M\) Cu: \(2.50 M - \frac{3.732 mol}{1.00 L} = 2.50 M - 3.732 M = -1.232 M\) Since the concentration of \(Cu^{2+}\) cannot be negative, we have reached the battery's dead state and the cell potential will be zero. Step 3: Calculate the mass of each electrode after 10.0h The mass change in the electrodes will be due to the stoichiometric conversion of metal ions from \(Zn(s)\) to \(Zn^{2+}\) and \(Cu^{2+}\) to \(Cu(s)\). We know that the number of moles of electrons transferred is 3.732 mol, and the molar mass of Zn and Cu are 65.38 g/mol and 63.55 g/mol, respectively.

Calculate the mass change of Zn electrode

For the Zn electrode: \(∆m_{Zn} = n \times M_{Zn}\) \(∆m_{Zn} = 3.732 mol \times 65.38 g/mol\) \(∆m_{Zn} ≈ 243.73 g\)

Calculate the mass change of Cu electrode

For the Cu electrode: \(∆m_{Cu} = n \times M_{Cu}\) \(∆m_{Cu} = 3.732 mol \times 63.55 g/mol\) \(∆m_{Cu} ≈ 237.08 g\) Step 4: Calculate how long the battery can deliver a current of 10.0A before it goes dead Since we already determined that the battery goes dead after 10 hours of operation at 10A, the answer is simply 10 hours.

Key concepts

Cell Potential

The cell potential, also known as electromotive force (EMF), is a measure of the voltage that a galvanic cell, like a zinc-copper battery, can provide. It indicates the potential difference between two electrodes and arises due to the tendency of the zinc and copper ions to exchange electrons when connected.
If the potential is positive, it suggests that the reaction is spontaneous, meaning it can occur without external energy input. In our zinc-copper battery system, when first connected, the cell potential is determined using the Nernst equation, but initially, it can also be calculated using the standard electrode potentials of zinc and copper.
This potential can be affected by different factors such as temperature, concentration of ions, and pressure, but in a standard condition, it is primarily driven by the intrinsic properties of the metals involved.

Nernst Equation

The Nernst equation is pivotal in electrochemistry for calculating the cell potential under non-standard conditions. It allows us to account for variations in ion concentration and is especially useful in understanding how a cell behaves over time as the reaction proceeds.
The equation is given by: \[ E_{cell} = E^0_{cell} - \frac{0.0592}{n} \log_{10}Q \] where \( E_{cell} \) is the actual cell potential, \( E^0_{cell} \) is the standard cell potential, \( n \) is the number of moles of electrons exchanged, and \( Q \) is the reaction quotient.
This equation helps to understand the decline in cell potential as the reaction progresses, such as when a battery is used and the concentrations of the ions change until equilibrium (and ultimately a dead battery) is reached.

Faraday's Constant

Faraday's constant is fundamental to the calculation of the electrochemical reactions taking place within a battery. It provides a link between the amount of electric charge in moles and the number of electrons transferred in an electrode reaction.
The constant is valued at approximately 96485 coulombs per mole of electrons (C/mol) and is crucial for determining how many moles of electrons are transferred during an electrochemical process.
In battery calculations, we use Faraday's constant to transform current and time into moles of electrons, helping to determine how much material is consumed or produced at each electrode during operation, as seen in the zinc-copper battery when determining the mass changes after 10 hours of use.

Standard Electrode Potential

The standard electrode potential is the voltage measured under standard conditions (1 M concentration for all reactants and products, 1 atm pressure for gases, and usually 25°C) when a cell is connected to a standard hydrogen electrode.
Each element has a different standard electrode potential based on its ability to gain or lose electrons, which can be found in electrochemical series tables.
For the zinc-copper battery discussed, zinc has a standard electrode potential of \(-0.76 V\), and copper has \(+0.34 V\). These values help us determine the standard cell potential, calculated by the difference \((E^0_{Cu} - E^0_{Zn})\), which is a fundamental basis for further calculations such as when using the Nernst equation to account for the real-time conditions in the battery.