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Chapter 18: Problem 11

Consider the following galvanic cell:What happens to \(\mathscr{C}\) as the concentration of \(\mathrm{Zn}^{2+}\) is increased? As the concentration of \(\mathrm{Ag}^{+}\) is increased? What happens to \(\mathscr{E}^{\circ}\) in these cases?

Short Answer

Expert verified
In summary, when the concentration of Zn²⁺ is increased in the galvanic cell, the cell potential (\(\mathscr{E}\)) decreases, while when the concentration of Ag⁺ is increased, the cell potential increases. However, the standard cell potential (\(\mathscr{E}^{\circ}\)) remains unchanged in both cases, as it is independent of the concentration of the species involved.

Step by step solution

01

Identify the half-cell reactions

The given galvanic cell has a Zn²⁺/Zn electrode and an Ag⁺/Ag electrode. The half-cell reactions are: Zn⁺⁺ (aq) + 2e⁻ → Zn (s) (Reduction half-cell reaction) Ag⁺ (aq) + e⁻ → Ag (s) (Reduction half-cell reaction)
02

Write the overall cell reaction

Balancing the half-cell reactions and adding them together, we obtain the overall cell reaction: Zn (s) → Zn²⁺ (aq) + 2e⁻ (Oxidation) 2Ag⁺ (aq) + 2e⁻ → 2Ag (s) (Reduction) Zn (s) + 2Ag⁺ (aq) → Zn²⁺ (aq) + 2Ag (s) (Overall cell reaction) Step 2: Apply the Nernst equation
03

Nernst equation

The Nernst equation is used to determine the cell potential under non-standard conditions. It is given by: \[\mathscr{E} = \mathscr{E}^{\circ} - \frac{RT}{nF} \ln Q\] Where: - \(\mathscr{E}\) is the cell potential - \(\mathscr{E}^{\circ}\) is the standard cell potential - R is the gas constant (8.314 J/mol·K) - T is the temperature in Kelvin (assuming 298 K) - n is the number of electrons transferred (in this case, 2) - F is the Faraday constant (96485 C/mol) - Q is the reaction quotient, defined as \(\frac{[Zn^{2+}]}{[Ag+]^2}\) for this reaction. Step 3: Analyze the effect of increasing Zn²⁺ and Ag⁺ concentrations on cell potential (C)
04

Increasing Zn²⁺ concentration

When the concentration of Zn²⁺ is increased, the value of Q increases. According to the Nernst equation, an increased Q value will result in a reduced cell potential (\(\mathscr{E}\)) as the value of \(ln (Q)\) will increase.
05

Increasing Ag⁺ concentration

When the concentration of Ag⁺ is increased, the value of Q decreases, since Ag⁺ appears in the denominator. According to the Nernst equation, a decreased Q value will result in an increased cell potential (\(\mathscr{E}\)) as the value of \(ln (Q)\) will become more negative. Step 4: Analyze the effect of increasing Zn²⁺ and Ag⁺ concentrations on standard cell potential (E°)
06

Effect on standard cell potential

The standard cell potential, \(\mathscr{E}^{\circ}\), depends only on the standard oxidation and reduction potentials of the half-cells and not on the concentration of the species involved. As a result, increasing the concentration of either Zn²⁺ or Ag⁺ ions does not affect the value of \(\mathscr{E}^{\circ}\). In summary, increasing the concentration of Zn²⁺ will decrease the cell potential (C), while increasing the concentration of Ag⁺ will increase the cell potential. The standard cell potential (E°) remains unchanged in both cases.

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Most popular questions from this chapter

Sketch the galvanic cells based on the following overall reactions. Show the direction of electron flow, and identify the cathode and anode. Give the overall balanced equation. Assume that all concentrations are \(1.0 \mathrm{M}\) and that all partial pressures are \(1.0 \mathrm{~atm}\). a. \(\mathrm{Cr}^{3+}(a q)+\mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}(a q)+\mathrm{Cl}^{-}(a q)\) b. \(\mathrm{Cu}^{2+}(a q)+\mathrm{Mg}(s) \rightleftharpoons \mathrm{Mg}^{2+}(a q)+\mathrm{Cu}(s)\)

It took \(2.30\) min using a current of \(2.00 \mathrm{~A}\) to plate out all the silver from \(0.250 \mathrm{~L}\) of a solution containing \(\mathrm{Ag}^{+}\). What was the original concentration of \(\mathrm{Ag}^{+}\) in the solution?

An aqueous solution of \(\mathrm{PdCl}_{2}\) is electrolyzed for \(48.6\) seconds, and during this time \(0.1064 \mathrm{~g}\) of \(\mathrm{Pd}\) is deposited on the cathode. What is the average current used in the electrolysis?

A galvanic cell is based on the following half-reactions: $$ \begin{aligned} \mathrm{Ag}^{+}+\mathrm{e}^{-} \longrightarrow \mathrm{Ag}(s) & \mathscr{E}^{\circ}=0.80 \mathrm{~V} \\ \mathrm{Cu}^{2+}+2 \mathrm{e}^{-} \longrightarrow \mathrm{Cu}(s) & \mathscr{E}^{\circ}=0.34 \mathrm{~V} \end{aligned} $$ In this cell, the silver compartment contains a silver electrode and excess \(\operatorname{AgCl}(s)\left(K_{\mathrm{sp}}=1.6 \times 10^{-10}\right)\), and the copper compartment contains a copper electrode and \(\left[\mathrm{Cu}^{2+}\right]=2.0 M\). a. Calculate the potential for this cell at \(25^{\circ} \mathrm{C}\). b. Assuming \(1.0 \mathrm{~L}\) of \(2.0 \mathrm{M} \mathrm{Cu}^{2+}\) in the copper compartment, calculate the moles of \(\mathrm{NH}_{3}\) that would have to be added to give a cell potential of \(0.52 \mathrm{~V}\) at \(25^{\circ} \mathrm{C}\) (assume no volume change on addition of \(\mathrm{NH}_{3}\) ). $$ \begin{aligned} \mathrm{Cu}^{2+}(a q)+4 \mathrm{NH}_{3}(a q) \rightleftharpoons & \\ \mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}{ }^{2+}(a q) & K=1.0 \times 10^{13} \end{aligned} $$

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?
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