Question

A zinc-copper battery is constructed as follows at ${25}^{\circ }\mathrm{C}:$ $$\mathrm{Zn}|{\mathrm{Zn}}^{2+}(0.10\mathrm{M})\Vert {\mathrm{Cu}}^{2+}(2.50\mathrm{M})|\mathrm{Cu}$$ 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 short version of the answer is: a. The initial cell potential is 1.16V. b. The cell potential after 10.0 A has flowed for 10.0 h is 1.13V. c. The mass of Zn electrode is 0 g, and the mass of Cu electrode is 437.2 g after 10.0 h. d. The battery can deliver a current of 10.0 A for 10.0 hours before it goes dead.

Step-by-step solution

Calculate the initial cell potential.

Use the Nernst equation for the initial cell potential. The standard cell potential $E_0$ of a Zn-Cu battery is +1.1 V. The Nernst equation is: $$E=E_0-\frac{RT}{nF}\ln Q$$ Where $R$ = gas constant = 8.314 J / mol·K $T$ = temperature in Kelvin = 25 °C + 273 = 298 K $n$ = number of moles of electrons transferred in the redox reaction = 2 $F$ = Faraday constant = 96,485 C/mol $Q$ = reaction quotient = [Zn${}^{2+}$] / [Cu${}^{2+}$] The ratio of concentrations will be 0.1/2.5 = 0.04. Sign of the equation is reversed as the reaction quotient is less than 1. Substituting these values into the Nernst equation, we find: $E=1.1V-(\frac{(8.314J/mol\cdot K)\cdot (298K)}{2\cdot 96,485C/mol})(\ln 0.04)=1.1+0.059V=1.16V$ So the initial cell potential, when the battery is first connected, is 1.16V.

Calculate the cell potential after a current has flowed.

Current (I) of 10.0A flowing for a time (t) of 10.0 hours = charge (Q) of I*t = 10.0 A x 10.0 hours x 3600s/hour = 360,000 C. Faraday's law of electrolysis gives the amount of substance produced by a current: n = Q/F. Thus, the moles of Zn${}^{2+}$ or Cu${}^{2+}$ produced will be 360,000 C / 96,485 C/mol = 3.73 mol. Since Zn goes from 0 to +2 and Cu goes from +2 to 0, Zn${}^{2+}$ increases by 3.73 mol and Cu${}^{2+}$ decreases by 3.73 mol. In a 1.0 L cell, this changes the concentrations in the battery to 3.83 M for Zn${}^{2+}$ (0.1 + 3.73) and 1.77 M for Cu${}^{2+}$ (2.5 - 1.73). Substituting these values into the Nernst equation, we find the new cell potential: $E^{\prime }=1.1V-(\frac{(8.314J/mol\cdot K)\cdot (298K)}{2\cdot 96,485C/mol})(\ln \frac{3.83}{1.77})=1.1V+0.029V=1.13V$. So the cell potential after 10.0 A of current has flowed for 10.0 h is 1.13 V.

Calculate the mass of each electrode after 10.0 h.

The molar mass of Zn is 65.38 g/mol, and the molar mass of Cu is 63.55 g/mol. Since we're producing 3.73 mol of Zn${}^{2+}$ and losing 3.73 mol of Cu${}^{2+}$, we are losing 3.73 mol x 65.38 g/mol = 243.8 g of Zn and gaining 3.73 mol x 63.55 g/mol = 237.2 g of Cu. If each electrode started with 200g, the final mass of Zn is -43.8g and the final mass of Cu is 437.2g. However, the electrode can't have a negative mass, so it has actually disappeared after 10.0 h, and all its Zn has gone into solution. The Cu electrode has grown in mass to 437.2g after 10.0 h.

Determine how long the battery can deliver 10.0 A before it goes dead.

The Zn electrode goes dead after 10.0 hr, as we calculated. It's the limiting reagent in this redox reaction. So, it would take 10.0 hours for this battery to deliver a current of 10.0 A before it goes dead.

Key concepts

Cell Potential Calculation

Understanding the cell potential of a battery is crucial for predicting how much voltage it can provide for our electronic devices. In a zinc-copper battery, we calculate this potential by considering the natural tendency of each metal to lose or gain electrons, known as its standard reduction potential. These values are key in determining the energy that drives the flow of electrons through an external circuit.

In practical scenarios, the real-time voltage of a battery often deviates from these standard conditions due to various factors such as concentration of ions, temperature, and pressure. This is where calculative magic comes in. By applying the Nernst equation—a refined version of the standard cell potential expression—we can accurately determine the voltage considering the immediate conditions of the reaction.

Essentially, we're employing a formula that incorporates these variables indicating the concentrations of reactants and products, representative of the chemical reality within the electrochemical cell, adjusting the ideal cell potential to obtain a value that mirrors the true cell performance at any given moment. The resulting value tells us precisely how much potential—or 'push'—the electrons have to move from the zinc to the copper, which represents the practical voltage the battery can supply.

Nernst Equation

Diving deeper into the Nernst equation allows us to appreciate its role in tailoring our understanding of electrochemistry to real-world applications. This elegant equation takes into account the temperature (symbolized by $T$), the universal gas constant ($R$), the number of electrons transferred in the half-reactions ($n$), and the Faraday constant ($F$).

The reaction quotient ($Q$) speaks to the concentration or pressure of the reactants and products at any stage of the reaction. When these values are plugged into the Nernst equation, the result is a cell potential that reflects the current concentration states within the electrochemical system.

By manipulating the Nernst equation, we can theoretically determine how changes in concentration, such as those which occur when a battery has been in use, affect the voltage it can provide. This dynamic view of electrochemical potential is fundamental for applications ranging from portable electronics to large-scale energy storage systems.

Electrochemistry

Electrochemistry is the branch of chemistry that deals with the interrelation of electrical currents or voltages and chemical reactions. It's the scientific study behind batteries, fuel cells, and electroplating, among other applications.

At the heart of electrochemistry is the redox reaction, where oxidation (loss of electrons) and reduction (gain of electrons) occur. These two processes happen at different electrodes in an electrochemical cell: the anode, where oxidation takes place, and the cathode, where reduction occurs. The flow of electrons from the anode to the cathode through an external circuit is what generates electricity.

In the context of a zinc-copper battery, zinc undergoes oxidation, releasing electrons, while copper ions in solution gain these electrons through reduction. The cell potential, calculated using concepts like the Nernst equation, is a direct measure of the drive behind this electron flow. Understanding the chemical principles that govern these processes allows us to harness and optimize the generation of electrical energy from chemical reactions.

Faraday's Law of Electrolysis

When it comes to predicting the amounts of substances transformed during electrolysis, Faraday's law of electrolysis offers us a straightforward and powerful tool. It states that the amount of a substance produced at an electrode during electrolysis is directly proportional to the quantity of electricity that passes through the electrolyte.

The law is founded on two key principles: the charge ($Q$) and Faraday's constant ($F$), which is the charge of one mole of electrons. We can calculate the number of moles of substance deposited or dissolved at an electrode by dividing the total charge by Faraday's constant. This relationship becomes incredibly practical in situations like the one outlined in our exercise, in which electrical current is passed through the battery for a specified time.

By applying Faraday's law, we can pinpoint the exact changes in mass of the zinc and copper electrodes after electricity has flowed through the battery. This quantification not only aids in the material design of the electrodes but also lets us estimate the lifespan of the battery under defined conditions of use, an essential factor for both consumer expectations and engineering specifications.