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 \(200 .\) g. a. Calculate the cell potential when this battery is first connected. b. Calculate the cell potential after 10.0 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 A before it goes dead?

Short answer

a. The initial cell potential is approximately 1.07 V. b. After 10 hours, new cell potential can be found using the new concentrations of \(Cu^{2+}\) and \(Zn^{2+}\) ions in the solution. c. The mass of each electrode after 10 hours can be calculated using Faraday's Law: \[m_{Cu} = (n_{Cu})(63.546 \, g/mol)\] \[m_{Zn} = (n_{Zn})(65.38 \, g/mol)\] d. To find how long the battery can deliver a 10 A current, determine the moles of the limiting reagent, then use Faraday's Law to find the required time.

Step-by-step solution

Calculate the initial cell potential

To calculate the initial cell potential, we need to use the Nernst equation: \[E_{cell} = E^{\circ}_{cell} - \frac{RT}{nF} \ln Q\] Where: - \(E^{\circ}_{cell}\) is the standard cell potential. We can find it in tables or as given below: \[E^{\circ}_{cell} = E^{\circ}_{Cu^{2+}/Cu} - E^{\circ}_{Zn^{2+}/Zn}\] - R is the universal gas constant (\(8.314 \frac{J}{mol\cdot K}\)) - T is the temperature in Kelvin (we need to convert \(25^{\circ}\) C to Kelvin which is 298.15 K) - n is the number of electrons transferred - F is Faraday's constant (96485 C/mol) - Q is the reaction quotient, which is given by: \[Q = \frac{[Cu^{2+}]}{[Zn^{2+}]}\] First, we need to find the standard cell potential (\(E^{\circ}_{cell}\)). For zinc and copper, we have: \[E^{\circ}_{Cu^{2+}/Cu} = +0.34 \, V\] \[E^{\circ}_{Zn^{2+}/Zn} = -0.76 \, V\] Subtracting these values gives the standard cell potential: \[E^{\circ}_{cell} = 0.34 - (-0.76) = 1.10\, V\] Now, we can plug the values into the Nernst equation and find the initial cell potential: \[E_{cell} = 1.10 - \frac{8.314 \times 298.15}{(2)(96485)} \ln \frac{2.50}{0.10}\] \[E_{cell} \approx 1.07\, V\] So the initial cell potential is approximately 1.07 V.

Calculate the cell potential after 10 hours

To calculate the cell potential after 10 hours, we first need to find the new concentrations of \(Cu^{2+}\) and \(Zn^{2+}\) ions in the solution. We will use Faraday's Law to find the moles of electrons transferred during this time: \[q = n \cdot F\] \[q = (10.0 \, A) \times (10.0 \, h \times 3600 \, s/h)\] Now, we need to find the change in the molarity (ΔM) of \(Cu^{2+}\) and \(Zn^{2+}\): \[\Delta M_{Cu^{2+}} = M_{Cu^{2+}} - 2.50\, M\] \[\Delta M_{Zn^{2+}} = M_{Zn^{2+}} - 0.10\, M\] Replace these values in the Nernst equation and solve for the new cell potential.

Calculate the mass of each electrode after 10 hours

Using Faraday's Law, we can calculate the mass of each electrode after 10 hours of use: \[q = n_{Cu}F = n_{Zn}F\] From this, we can solve for \(n_{Cu}\) and \(n_{Zn}\): \[n_{Cu} = \frac{q}{F}\] \[n_{Zn} = \frac{q}{F}\] Now, we can find the mass of each electrode after 10 hours: \[m_{Cu} = (n_{Cu})(molar \, mass \, of \, Cu_{(s)}) = (n_{Cu})(63.546 \, g/mol)\] \[m_{Zn} = (n_{Zn})(molar \, mass \, of \, Zn_{(s)}) = (n_{Zn})(65.38 \, g/mol)\]

Determine how long the battery can deliver 10 A current

To know how long the battery can deliver a 10 A current, we need to find the moles of the limiting reagent according to the reaction: \[Zn_{(s)} + Cu^{2+}_{(aq)} \rightarrow Cu_{(s)} + Zn^{2+}_{(aq)}\] The limiting reagent is that one with the lowest ratio of moles to stoichiometric coefficients. Find this reagent, and then use Faraday's Law to determine how long the battery remains active while providing a current of 10 A.

Key concepts

Nernst Equation

The Nernst equation is a fundamental formula in electrochemistry used to calculate the cell potential at non-standard conditions. It takes into account the effect of temperature and concentration on the cell voltage. To understand it better, let's break it down:

The equation is expressed as:
\[E_{cell} = E^{\circ}_{cell} - \frac{RT}{nF} \ln Q\]
Where:

• \(E_{cell}\) is the actual cell potential.
• \(E^{\circ}_{cell}\) represents the standard cell potential, which is the voltage of the cell under standard conditions.
• \(R\) is the universal gas constant in joules per mole-kelvin (\(8.314 J/mol\cdot K\)).
• \(T\) is the temperature in kelvin, which affects the kinetic energy and thus the potential of the cell.
• \(n\) is the number of moles of electrons exchanged in the redox reaction.
• \(F\) is Faraday's constant (\(96485 C/mol\)) and represents the charge of one mole of electrons.
• \(Q\) is the reaction quotient, measuring the ratio of products to reactants at any point in time.

In a zinc-copper battery, for example, the Nernst equation is used to adjust the cell potential from the standard state to realistic battery conditions by plugging in the concentrations of the zinc and copper ions.

Cell Potential

Cell potential, also known as electromotive force (EMF), is the driving force behind the electrical current in a galvanic cell. It's the difference in potential energy between two electrodes. When we calculate cell potential using the Nernst equation, we're able to predict how voltage output will change as a reaction progresses and concentrations shift. This potential is what drives electrons from the anode to the cathode, creating a useful electrical flow.

In our zinc-copper battery scenario, the initial cell potential is a measure of how much voltage the battery can provide when it's first connected. This potential decreases over time as the chemical reactants are consumed, which the Nernst equation can help us calculate, as was done in the textbook exercise.

Faraday's Law

Principle of Faraday's Law

Faraday's Law is essential for understanding how electric current relates to chemical changes in a galvanic cell. It states that the amount of chemical change is proportional to the amount of electricity that flows through the cell. This law can be expressed as:\[q = n \cdot F\]Where \(q\) is the total electric charge, \(n\) is the number of moles of electrons, and \(F\) is Faraday's constant. This becomes especially useful when calculating things like the change in concentration of ions in a solution after a certain amount of current has passed through it for a given time.

Application in Electrode Mass Calculation

By utilizing Faraday's Law, we can calculate how the mass of an electrode changes as the galvanic cell operates. The process involves determining the moles of electrons transferred as the current flows and then relating that to the mass of the electroactive material consumed or produced at the electrode. This way, we can predict changes in electrode mass after running the battery for a specific period, as outlined in the example problem.

Galvanic Cells

Galvanic cells, also known as voltaic cells, are devices that convert chemical energy into electrical energy through spontaneous redox reactions. They consist of two different metals connected by a salt bridge or a permeable membrane. These metals, referred to as electrodes, have differing tendencies to lose or gain electrons.

In a typical galvanic cell setup, the anode is the electrode where oxidation occurs (loss of electrons), while the cathode is the electrode where reduction happens (gain of electrons). The flow of electrons from the anode to the cathode through an external circuit generates electric current. The copper-zinc battery discussed in the exercise is an example of a galvanic cell, where the spontaneous reaction between the two different metals creates a flow of electrons that can be harnessed as electrical power.

Electrode Mass Calculation

Electrode mass calculation involves determining the mass of the reactive material at the electrodes before and after the galvanic cell operates. The change in mass reflects the extent of the reaction that has taken place. According to Faraday's Law, the amount of substance that reacts is directly proportional to the amount of electricity that passes through the cell.

To calculate the change in electrode mass:

• Determine the number of moles of electrons transferred using the total charge (current times time) and Faraday's constant.
• Relate the moles of electrons to the moles of metal involved in the electrode reaction using the mole ratio from the balanced equation for the redox reaction.
• Finally, calculate the mass by multiplying the moles of metal by its molar mass.

This calculation is crucial when predicting how long a galvanic cell can operate before the reactants are exhausted, as illustrated in the exercise, where we predicted the new mass of zinc and copper electrodes after the battery has been discharging for a specific time.