Chapter 18, Problem 18.76QP

Consider a galvanic cell composed of the SHE and a half-cell using the reaction Ag⁺(aq) + e⁻ → Ag(s). (a) Calculate the standard cell potential. (b) What is the spontaneous cell reaction under standard-state conditions? (c) Calculate the cell potential when [H⁺] in the hydrogen electrode is changed to (i) 1.0 × 10⁻² M and (ii) 1.0 × 10⁻⁵ M, all other reagents being held at standard-state conditions. (d) Based on this cell arrangement, suggest a design for a pH meter.

Interpretation Introduction

Interpretation:

The standard electrode potential of the given cell and the spontaneous chemical reaction in the cell has to be found.  The cell potential of the given cell has to be found with the different concentrations of the hydrogen ion and a design for the ${\text{p}}^{\text{H}}$ meter has to be predicted.

Concept Introduction:

Galvanic cell is an electrochemical cell which converts the chemical energy of a reaction into electrical energy.

Standard hydrogen electrode (SHE) is a reference electrode whose potential is considered to be zero volts.  The potential of any other electrode is found by comparing with the SHE.

The standard electrode potential of a cell ${\text{(E}}^{°}{}_{\text{cell}})$ is the difference in electrode potential of the cathode and anode.

${\text{E}}^{\text{°}}{}_{\text{cell}}={\text{E}}^{\text{°}}{}_{\text{cathode}}-{\text{E}}^{\text{°}}{}_{\text{anode}}$

Nernst equation is one of the important equations in electrochemistry.  In Nernst equation the electrode potential of a cell reaction is related to the standard electrode potential, concentration or activities of the species that is involved in the chemical reaction and temperature.

${\text{E}}_{\text{cell}}{\text{=E}}^{\text{°}}{}_{\text{cell}}\text{-}\frac{\text{RT}}{\text{2}\text{.303nF}}\text{log}\frac{\left[\text{Red}\right]}{\left[\text{Oxd}\right]}$

Where,

${\text{E}}_{\text{cell}}$ is the potential of the cell at a given temperature

${\text{E}}^{\text{°}}{}_{\text{cell}}$ is the standard electrode potential

$\text{R}$ is the universal gas constant $\text{(R=8}{\text{.314JK}}^{\text{-1}}{\text{mol}}^{\text{-1}})$

$\text{T}$ is the temperature

$\text{n}$ is the number of electrons involved in a reaction

$\text{F}$ is the Faraday constant $\text{(F=9}\text{.64853399}\times {\text{10}}^4{\text{Cmol}}^{\text{-1}})$

$\left[\text{Red}\right]$ is the concentration of the reduced species

$\left[\text{Oxd}\right]$ is the concentration of the oxidised species

At room temperature $({25}^{°}C)$, after substituting the values of all the constants the equation can be written as

${\text{E}}_{\text{cell}}{\text{=E}}^{\text{°}}{}_{\text{cell}}\text{-}\frac{0.0591}{\text{n}}\text{log}\frac{\left[\text{Red}\right]}{\left[\text{Oxd}\right]}$

Answer to Problem 18.76QP

The standard electrode potential of the cell is found to be $\text{0}\text{.80V}$

Explanation of Solution

To calculate the standard electrode potential of the cell ${\text{(E}}^{\text{°}}{}_{\text{cell}})$

The standard electrode potential of the cell is the difference in standard electrode potential of the cathode and anode.

${\text{E}}^{\text{°}}{}_{\text{cell}}={\text{E}}^{\text{°}}{}_{\text{cathode}}-{\text{E}}^{\text{°}}{}_{\text{anode}}$

In order to determine the standard electrode potential we need to find out the half cell reactions in the cathode and anode of the given electrode.

The half cell reactions are

$\begin{array}{l}{\text{2H}}^{\text{+}}{\text{+2e}}^{\text{-}}\to {\text{H}}_{\text{2}}{}_{\text{(g)}}{\text{E}}^{\text{°}}\text{anode = 0}\text{.00V}\\ {\text{Ag}}^{\text{+}}{}_{\text{(aq)}}{\text{+e}}^{\text{-}}\to {\text{Ag}}_{\text{(s)}}{\text{ E}}^{\text{°}}\text{cathode = 0}\text{.80V}\end{array}$

The standard electrode potential is calculated as given below

$\begin{array}{l}{\text{E}}^{\text{°}}{}_{\text{cell}}={\text{E}}^{\text{°}}{}_{\text{cathode}}-{\text{E}}^{\text{°}}{}_{\text{anode}}\\ \text{=}\text{0}\text{.80V-0}\text{.00V}\\ \text{=0}\text{.80V}\end{array}$

Interpretation Introduction

Interpretation:

The standard electrode potential of the given cell and the spontaneous chemical reaction in the cell has to be found.  The cell potential of the given cell has to be found with the different concentrations of the hydrogen ion and a design for the ${\text{p}}^{\text{H}}$ meter has to be predicted.

Concept Introduction:

Galvanic cell is an electrochemical cell which converts the chemical energy of a reaction into electrical energy.

Standard hydrogen electrode (SHE) is a reference electrode whose potential is considered to be zero volts.  The potential of any other electrode is found by comparing with the SHE.

The standard electrode potential of a cell ${\text{(E}}^{°}{}_{\text{cell}})$ is the difference in electrode potential of the cathode and anode.

${\text{E}}^{\text{°}}{}_{\text{cell}}={\text{E}}^{\text{°}}{}_{\text{cathode}}-{\text{E}}^{\text{°}}{}_{\text{anode}}$

Nernst equation is one of the important equations in electrochemistry.  In Nernst equation the electrode potential of a cell reaction is related to the standard electrode potential, concentration or activities of the species that is involved in the chemical reaction and temperature.

${\text{E}}_{\text{cell}}{\text{=E}}^{\text{°}}{}_{\text{cell}}\text{-}\frac{\text{RT}}{\text{2}\text{.303nF}}\text{log}\frac{\left[\text{Red}\right]}{\left[\text{Oxd}\right]}$

Where,

${\text{E}}_{\text{cell}}$ is the potential of the cell at a given temperature

${\text{E}}^{\text{°}}{}_{\text{cell}}$ is the standard electrode potential

$\text{R}$ is the universal gas constant $\text{(R=8}{\text{.314JK}}^{\text{-1}}{\text{mol}}^{\text{-1}})$

$\text{T}$ is the temperature

$\text{n}$ is the number of electrons involved in a reaction

$\text{F}$ is the Faraday constant $\text{(F=9}\text{.64853399}\times {\text{10}}^4{\text{Cmol}}^{\text{-1}})$

$\left[\text{Red}\right]$ is the concentration of the reduced species

$\left[\text{Oxd}\right]$ is the concentration of the oxidised species

At room temperature $({25}^{°}C)$, after substituting the values of all the constants the equation can be written as

${\text{E}}_{\text{cell}}{\text{=E}}^{\text{°}}{}_{\text{cell}}\text{-}\frac{0.0591}{\text{n}}\text{log}\frac{\left[\text{Red}\right]}{\left[\text{Oxd}\right]}$

Answer to Problem 18.76QP

The spontaneous reaction taking place in the cell is the reduction of silver ion and oxidation of hydrogen gas.

${\text{2Ag}}^{\text{+}}{}_{\text{(aq)}}{\text{+2H}}^{+}\to {\text{Ag}}_{\text{(s)}}{\text{+2H}}^{+}{}_{\text{(aq)}}$

Explanation of Solution

To write the spontaneous cell reaction under the given standard conditions

In the given cell composed of standard hydrogen electrode and silver electrode,  The silver ions in the solution will be reduced into solid silver and the hydrogen molecules will be oxidised into hydrogen ions.

${\text{2Ag}}^{\text{+}}{}_{\text{(aq)}}{\text{+2H}}^{+}\to {\text{Ag}}_{\text{(s)}}{\text{+2H}}^{+}{}_{\text{(aq)}}$

Interpretation Introduction

Interpretation:

The standard electrode potential of the given cell and the spontaneous chemical reaction in the cell has to be found.  The cell potential of the given cell has to be found with the different concentrations of the hydrogen ion and a design for the ${\text{p}}^{\text{H}}$ meter has to be predicted.

Concept Introduction:

Galvanic cell is an electrochemical cell which converts the chemical energy of a reaction into electrical energy.

Standard hydrogen electrode (SHE) is a reference electrode whose potential is considered to be zero volts.  The potential of any other electrode is found by comparing with the SHE.

The standard electrode potential of a cell ${\text{(E}}^{°}{}_{\text{cell}})$ is the difference in electrode potential of the cathode and anode.

${\text{E}}^{\text{°}}{}_{\text{cell}}={\text{E}}^{\text{°}}{}_{\text{cathode}}-{\text{E}}^{\text{°}}{}_{\text{anode}}$

Nernst equation is one of the important equations in electrochemistry.  In Nernst equation the electrode potential of a cell reaction is related to the standard electrode potential, concentration or activities of the species that is involved in the chemical reaction and temperature.

${\text{E}}_{\text{cell}}{\text{=E}}^{\text{°}}{}_{\text{cell}}\text{-}\frac{\text{RT}}{\text{2}\text{.303nF}}\text{log}\frac{\left[\text{Red}\right]}{\left[\text{Oxd}\right]}$

Where,

${\text{E}}_{\text{cell}}$ is the potential of the cell at a given temperature

${\text{E}}^{\text{°}}{}_{\text{cell}}$ is the standard electrode potential

$\text{R}$ is the universal gas constant $\text{(R=8}{\text{.314JK}}^{\text{-1}}{\text{mol}}^{\text{-1}})$

$\text{T}$ is the temperature

$\text{n}$ is the number of electrons involved in a reaction

$\text{F}$ is the Faraday constant $\text{(F=9}\text{.64853399}\times {\text{10}}^4{\text{Cmol}}^{\text{-1}})$

$\left[\text{Red}\right]$ is the concentration of the reduced species

$\left[\text{Oxd}\right]$ is the concentration of the oxidised species

At room temperature $({25}^{°}C)$, after substituting the values of all the constants the equation can be written as

${\text{E}}_{\text{cell}}{\text{=E}}^{\text{°}}{}_{\text{cell}}\text{-}\frac{0.0591}{\text{n}}\text{log}\frac{\left[\text{Red}\right]}{\left[\text{Oxd}\right]}$

Answer to Problem 18.76QP

(i) The electrode potential, when the concentration of hydrogen ion is $1.0\times {10}^{-2}\text{M}$ found to be $\text{0}\text{.92V}$

(ii) The electrode potential, when the concentration of hydrogen ion is $1.0\times {10}^{-5}\text{M}$ found to be $\text{1}\text{.10V}$

Explanation of Solution

(i)

To calculate the electrode potential when the concentration of hydrogen ion is $1.0\times {10}^{-2}\text{M}$

The electrode potential of the cell can be calculated using the Nernst equation.

$\begin{array}{l}{\text{E}}_{\text{cell}}{\text{=E}}^{\text{°}}{}_{\text{cell}}\text{-}\frac{0.0591}{\text{n}}\text{log}\frac{\left[\text{Red}\right]}{\left[\text{Oxd}\right]}\\ {\text{E}}_{\text{cell}}{\text{=E}}^{\text{°}}{}_{\text{cell}}\text{-}\frac{0.0591}{\text{2}}\text{log}\frac{{\left[{\text{H}}^{+}\right]}^2}{{\left[{\text{Ag}}^{+}\right]}^2{P}_{H_2}}\end{array}$

Where,$P_{H_2}$ is the partial pressure of hydrogen gas.

In the standard state all the species will have concentration equal to unity.  In this case only the concentration of hydrogen ion is changed.  On plugging in the concentration of the oxidised and reduced species to the given equation the electrode potential of the cell can be calculated.

$\begin{array}{l}{\text{E}}_{\text{cell}}\text{=0}\text{.80V-}\frac{0.0591}{\text{2}}\text{log}\frac{{\left[1.0\times {10}^{-2}\right]}^2}{{\left[1.0\right]}^2\left[1.0\right]}\\ =\text{0}\text{.92V}\end{array}$

(ii)

To calculate the electrode potential when the concentration of hydrogen ion is $1.0\times {10}^{-5}\text{M}$

The electrode potential of the cell can be calculated using the Nernst equation.

$\begin{array}{l}{\text{E}}_{\text{cell}}{\text{=E}}^{\text{°}}{}_{\text{cell}}\text{-}\frac{0.0591}{\text{n}}\text{log}\frac{\left[\text{Red}\right]}{\left[\text{Oxd}\right]}\\ {\text{E}}_{\text{cell}}{\text{=E}}^{\text{°}}{}_{\text{cell}}\text{-}\frac{0.0591}{\text{2}}\text{log}\frac{{\left[{\text{H}}^{+}\right]}^2}{{\left[{\text{Ag}}^{+}\right]}^2{P}_{H_2}}\end{array}$

 Where $P_{H_2}$ is the partial pressure of hydrogen gas.

In the standard state all the species will have concentration equal to unity.  In this case only the concentration of hydrogen ion is changed.  On plugging in the concentration of the oxidised and reduced species to the given equation the electrode potential of the cell can be calculated.

$\begin{array}{l}{\text{E}}_{\text{cell}}\text{=0}\text{.80V-}\frac{0.0591}{\text{2}}\text{log}\frac{{\left[1.0\times {10}^{-5}\right]}^2}{{\left[1.0\right]}^2\left[1.0\right]}\\ =\text{1}\text{.10V}\end{array}$

Interpretation Introduction

Interpretation:

The standard electrode potential of the given cell and the spontaneous chemical reaction in the cell has to be found.  The cell potential of the given cell has to be found with the different concentrations of the hydrogen ion and a design for the ${\text{p}}^{\text{H}}$ meter has to be predicted.

Concept Introduction:

Galvanic cell is an electrochemical cell which converts the chemical energy of a reaction into electrical energy.

Standard hydrogen electrode (SHE) is a reference electrode whose potential is considered to be zero volts.  The potential of any other electrode is found by comparing with the SHE.

The standard electrode potential of a cell ${\text{(E}}^{°}{}_{\text{cell}})$ is the difference in electrode potential of the cathode and anode.

${\text{E}}^{\text{°}}{}_{\text{cell}}={\text{E}}^{\text{°}}{}_{\text{cathode}}-{\text{E}}^{\text{°}}{}_{\text{anode}}$

Nernst equation is one of the important equations in electrochemistry.  In Nernst equation the electrode potential of a cell reaction is related to the standard electrode potential, concentration or activities of the species that is involved in the chemical reaction and temperature.

${\text{E}}_{\text{cell}}{\text{=E}}^{\text{°}}{}_{\text{cell}}\text{-}\frac{\text{RT}}{\text{2}\text{.303nF}}\text{log}\frac{\left[\text{Red}\right]}{\left[\text{Oxd}\right]}$

Where,

${\text{E}}_{\text{cell}}$ is the potential of the cell at a given temperature

${\text{E}}^{\text{°}}{}_{\text{cell}}$ is the standard electrode potential

$\text{R}$ is the universal gas constant $\text{(R=8}{\text{.314JK}}^{\text{-1}}{\text{mol}}^{\text{-1}})$

$\text{T}$ is the temperature

$\text{n}$ is the number of electrons involved in a reaction

$\text{F}$ is the Faraday constant $\text{(F=9}\text{.64853399}\times {\text{10}}^4{\text{Cmol}}^{\text{-1}})$

$\left[\text{Red}\right]$ is the concentration of the reduced species

$\left[\text{Oxd}\right]$ is the concentration of the oxidised species

At room temperature $({25}^{°}C)$, after substituting the values of all the constants the equation can be written as

${\text{E}}_{\text{cell}}{\text{=E}}^{\text{°}}{}_{\text{cell}}\text{-}\frac{0.0591}{\text{n}}\text{log}\frac{\left[\text{Red}\right]}{\left[\text{Oxd}\right]}$

Answer to Problem 18.76QP

The given cell is sensitive to the hydrogen in concentration.  Hence it can be used as the ${\text{p}}^{\text{H}}$ meter.  The ${\text{p}}^{\text{H}}$ meter can be represented as below.

${\text{Pt(s)|H}}_{\text{2}}{\text{(g)|HCl || Ag}}^{\text{+}}\text{|Ag(s)}$

Explanation of Solution

To suggest a design for a ${\text{p}}^{\text{H}}$ meter

From the results obtained in the question (c) it is clear that the given cell itself can be used as a ${\text{p}}^{\text{H}}$ meter.  The cell potential of the given cell is sensitive to the hydrogen ion concentration in the solution.

${\text{Pt(s)|H}}_{\text{2}}{\text{(g)|HCl || Ag}}^{\text{+}}\text{|Ag(s)}$