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Chapter 18: Problem 13

Consider the following galvanic cell: What happens to \(\mathscr{E}\) 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
When the concentration of Zn\( ^{2+} \) ions increases, the cell potential (\( \mathscr{E} \)) decreases, while the standard cell potential (\( \mathscr{E}^{\circ} \)) remains unchanged. On the other hand, when the concentration of Ag\( ^{+} \) ions increases, the cell potential (\( \mathscr{E} \)) increases, while the standard cell potential (\( \mathscr{E}^{\circ} \)) remains unchanged. This is based on the Nernst equation, which relates cell potential to ion concentrations, temperature, and the number of electrons transferred in the redox reaction.

Step by step solution

01

Write the complete cell reaction

To write the complete cell reaction, add the oxidation and reduction half-reactions, making sure the number of electrons in each half-reaction is equal: Zn(s) + 2 Ag+(aq) -> Zn^2+(aq) + 2 Ag(s) Step 2: Write the Nernst equation
02

Write the Nernst equation

The Nernst equation helps us to quantify the relationship between the cell potential (E), standard cell potential (\( \mathscr{E}^{\circ} \)), concentration of reactants and products, temperature, and the number of electrons transferred in a redox reaction. The Nernst equation is written as: \( \mathscr{E} = \mathscr{E}^{\circ} - \dfrac{0.0592}{n} \log Q \) Where: - \( \mathscr{E} \) is the cell potential at non-standard conditions - \( \mathscr{E}^{\circ} \) is the standard cell potential - n is the number of electrons transferred in the redox reaction - Q is the reaction quotient, which represents the ratio of concentrations of products to reactants Step 3: Calculate the effect of increasing Zn^2+ concentration
03

Calculate the effect of increasing Zn^2+ concentration

We will first determine the effect of increasing the concentration of Zn^2+ ions on the cell potential (E). From the Nernst equation: \( \mathscr{E} = \mathscr{E}^{\circ} - \dfrac{0.0592}{2} \log \dfrac{[Zn^{2+}]}{[Ag^+]^2} \) As the concentration of Zn^2+ increases, the value of Q increases. This then results in a decrease in the value of E because of the negative sign in front of the second term. The standard cell potential (\( \mathscr{E}^{\circ} \)) remains unchanged as it is a constant value that does not depend on the concentration of ions. Step 4: Calculate the effect of increasing Ag+ concentration
04

Calculate the effect of increasing Ag+ concentration

Now we will determine the effect of increasing the concentration of Ag+ ions on the cell potential (E): \( \mathscr{E} = \mathscr{E}^{\circ} - \dfrac{0.0592}{2} \log \dfrac{[Zn^{2+}]}{[Ag^+]^2} \) As the concentration of Ag+ ions increases, the value of Q decreases. This then results in an increase in the value of E since we have a negative sign in front of the second term. The standard cell potential (\( \mathscr{E}^{\circ} \)) again remains unchanged, as it is a constant value that does not depend on the concentration of ions. Conclusions: - As the concentration of Zn^2+ ions increases, E decreases, and \( \mathscr{E}^{\circ} \) remains unchanged. - As the concentration of Ag+ ions increases, E increases, and \( \mathscr{E}^{\circ} \) remains unchanged.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nernst Equation
The Nernst Equation is a fundamental relationship used in electrochemistry to calculate the cell potential under non-standard conditions. It connects the cell potential (\( \mathscr{E} \)) to the standard cell potential (\( \mathscr{E}^{\circ} \)), as well as the concentrations of the reactants and products in the redox reaction. The equation has the form: \[ \mathscr{E} = \mathscr{E}^{\circ} - \dfrac{0.0592}{n} \log Q \]where:
  • \( \mathscr{E} \) is the cell potential under non-standard conditions.
  • \( \mathscr{E}^{\circ} \) is the standard cell potential, a constant.
  • \( n \) is the number of moles of electrons transferred in the redox reaction.
  • \( Q \) is the reaction quotient, which compares the concentrations of products to reactants.
Understanding the Nernst Equation allows us to predict how the cell potential changes with varying concentrations of the reactants and products. A rise in the product concentrations increases the value of Q, thus decreasing \( \mathscr{E} \), while an increase in reactant concentration lowers Q, which increases \( \mathscr{E} \). This equation is particularly useful in fields like biochemistry and electrochemical engineering.
By using the Nernst Equation, scientists can determine the efficiency of electrochemical cells and optimize their performance.
Redox Reactions
Redox reactions, short for reduction-oxidation reactions, form the backbone of electrochemical cells. During a redox reaction, electrons are transferred from one chemical species to another. This transfer causes changes in the oxidation states of the involved elements.
  • Oxidation involves the loss of electrons, leading to an increase in oxidation state.
  • Reduction involves the gain of electrons, resulting in a decrease in oxidation state.
Redox reactions are split into two half-reactions: oxidation (where electrons are lost) and reduction (where electrons are gained). Both half-reactions must occur simultaneously for charge balance. In electrochemical cells, these reactions convert chemical energy into electrical energy, allowing us to generate electricity through chemical processes.

Understanding redox reactions is essential not only in chemistry but also in broader applications, such as energy storage and conversion technologies, including batteries and fuel cells.
Electrochemical Cells
Electrochemical cells are devices that capture the energy from redox reactions to either produce electricity or drive chemical reactions. There are two main types of electrochemical cells: galvanic (or voltaic) cells and electrolytic cells.
  • Galvanic cells generate electrical energy from spontaneous redox reactions. In these cells, the reaction provides power to external circuits, like batteries.
  • Electrolytic cells use electrical energy to drive non-spontaneous chemical reactions. These are often employed in electroplating and metal purification.
Each cell comprises two electrodes: the anode (where oxidation occurs) and the cathode (where reduction occurs). The electrode reactions are separated physically by an electrolyte, which facilitates ion movement and sustains charge balance.

Electrochemical cells are fundamental to diverse technological applications ranging from batteries and fuel cells to sensors and metal recovery.
Cell Potential
Cell potential (\( \mathscr{E} \)) refers to the voltage or electric potential difference between two electrodes of an electrochemical cell. It measures the tendency of a redox reaction to occur and represents the ability of the cell to drive an electric current through an external circuit.The cell potential is influenced by several factors, including:
  • The inherent electrochemical properties of the substances involved, noted as standard potentials.
  • The concentration of reactants and products in the electrolyte solution.
The cell potential decreases when the concentration of products increases, and it increases when the concentration of reactants rises. This relationship is quantitatively described by the Nernst Equation, allowing for precise adjustments in chemical processes or battery operations.

Understanding cell potential is crucial in optimizing the operation of devices like batteries and in assessing the feasibility of electrochemical reactions in industrial applications.
Standard Cell Potential
The standard cell potential (\( \mathscr{E}^{\circ} \)) is the cell potential measured under standard conditions, which include 1 molar concentrations for all reactants and products, a pressure of 1 atmosphere for any gases involved, and a temperature of 25°C (298 K). This potential indicates the voltage that a galvanic cell can produce under these ideal conditions.Standard cell potential relies on the individual standard reduction potentials of the half-reactions occurring at the cathode and anode. It is calculated as: \[ \mathscr{E}^{\circ} = \mathscr{E}^{\circ}_{\mathrm{cathode}} - \mathscr{E}^{\circ}_{\mathrm{anode}} \]This standard measure allows scientists to compare the strengths of different reducing agents, helping in the design of batteries and other applications where redox reactions are pivotal.

The standard cell potential is a critical parameter for determining the feasibility of reactions and predicting which direction a redox reaction might proceed under specified conditions. Although it remains constant regardless of concentration changes, variations in the actual cell potential can be sustained by altering the concentration or pressure of substances using the Nernst Equation.

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Most popular questions from this chapter

Look up the reduction potential for \(\mathrm{Fe}^{3+}\) to \(\mathrm{Fe}^{2+} .\) Look up the reduction potential for \(\mathrm{Fe}^{2+}\) to Fe. Finally, look up the reduction potential for \(\mathrm{Fe}^{3+}\) to Fe. You should notice that adding the reduction potentials for the first two does not give the potential for the third. Why not? Show how you can use the first two potentials to calculate the third potential.

Consider the following reduction potentials: $$\begin{array}{ll}{\mathrm{Co}^{3+}+3 \mathrm{e}^{-} \longrightarrow \mathrm{Co}} & {\mathscr{E}^{\circ}=1.26 \mathrm{V}} \\ {\mathrm{Co}^{2+}+2 \mathrm{e}^{-} \longrightarrow \mathrm{Co}} & {\mathscr{E}^{\circ}=-0.28 \mathrm{V}}\end{array}$$ a. When cobalt metal dissolves in 1.0\(M\) nitric acid, will \(\mathrm{Co}^{3+}\) or \(\mathrm{Co}^{2+}\) be the primary product (assuming standard conditions)? b. Is it possible to change the concentration of \(\mathrm{HNO}_{3}\) to get a different result in part a? Concentrated \(\mathrm{HNO}_{3}\) is about 16 \(M\) .

How can one construct a galvanic cell from two substances, each having a negative standard reduction potential?

An electrochemical cell consists of a standard hydrogen electrode and a copper metal electrode. a. What is the potential of the cell at \(25^{\circ} \mathrm{C}\) if the copper electrode is placed in a solution in which \(\left[\mathrm{Cu}^{2+}\right]=\) \(2.5 \times 10^{-4} \mathrm{M} ?\) b. The copper electrode is placed in a solution of unknown \(\left[\mathrm{Cu}^{2+}\right] .\) The measured potential at \(25^{\circ} \mathrm{C}\) is 0.195 \(\mathrm{V}\) . What is \(\left[\mathrm{Cu}^{2+}\right] ?\) (Assume \(\mathrm{Cu}^{2+}\) is reduced.)

An electrochemical cell consists of a silver metal electrode immersed in a solution with \(\left[\mathrm{Ag}^{+}\right]=1.00 M\) separated by a porous disk from a compartment with a copper metal electrode immersed in a solution of 10.00\(M \mathrm{NH}_{3}\) that also contains \(2.4 \times 10^{-3} \mathrm{M} \mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}^{2+} .\) The equilibrium between \(\mathrm{Cu}^{2+}\) and \(\mathrm{NH}_{3}\) is: $$\mathrm{Cu}^{2+}(\mathrm{aq})+4 \mathrm{NH}_{3}(\mathrm{aq}) \rightleftharpoons \mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}^{2+}(\mathrm{aq}) \qquad K=1.0 \times 10^{13}$$ and the two cell half-reactions are: $$\begin{array}{rl}{\mathrm{Ag}^{+}+\mathrm{e}^{-} \longrightarrow \mathrm{Ag}} & {\mathscr{E}^{\circ}=0.80 \mathrm{V}} \\ {\mathrm{Cu}^{2+}+2 \mathrm{e}^{-} \longrightarrow \mathrm{Cu}} & {\mathscr{E}^{\circ}=0.34 \mathrm{V}}\end{array}$$ Assuming \(\mathrm{Ag}^{+}\) is reduced, what is the cell potential at \(25^{\circ} \mathrm{C} ?\)
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