Chapter 21, Problem 21.143P

(a)

Interpretation Introduction

Interpretation:

The equations that relates the non-standard to standard cell potential and $\left[{\text{Ag}}^{\text{+}}\right]$ for each given solution has to be identified.

Concept Introduction:

An electrochemical cell is a device in which a redox reaction is used to convert chemical energy into electrical energy. Such device is also known as the galvanic or voltaic cell.

Galvanic cell consists of two half-cells. The redox reaction occurs in these half-cells. The half-cell in which the reduction reaction occurs is known as the reduction half-cell, whereas the half-cell in which the oxidation reaction occurs is known as the oxidation half-cell.

Anode: The electrode where the oxidation occurs is called as an anode. It is a negatively charged electrode.

Cathode: The electrode where reduction occurs is called as a cathode. It is a positively charged electrode.

Oxidation: The gain of oxygen or the loss of hydrogen or the loss of an electron in a species during a redox reaction is called as oxidation.

Reduction: The loss of oxygen or the gain of hydrogen or the gain of an electron in a species during a redox reaction is called as reduction.

Explanation of Solution

Using the Nernst equation the relationship is depicted as follows,

    $\begin{array}{l}A{g}^{+}\left(aq\right)\to Ag\left(s\right)+1e^{-}\\ \\ E_{cell}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{n}\log Q\\ \\ E_{waste}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{waste}\Rightarrow Non\text{-standard}\text{cell}\\ \\ E_{\text{standard}}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{\text{standard}}\Rightarrow S\text{tandard}\text{cell}\end{array}$

(b)

Interpretation Introduction

Interpretation:

Using the obtained relation equation ${\left[{\text{Ag}}^{\text{+}}\right]}_{\text{waste}}$ has to be calculated.

Concept Introduction:

An electrochemical cell is a device in which a redox reaction is used to convert chemical energy into electrical energy. Such device is also known as the galvanic or voltaic cell.

Galvanic cell consists of two half-cells. The redox reaction occurs in these half-cells. The half-cell in which the reduction reaction occurs is known as the reduction half-cell, whereas the half-cell in which the oxidation reaction occurs is known as the oxidation half-cell.

Anode: The electrode where the oxidation occurs is called as an anode. It is a negatively charged electrode.

Cathode: The electrode where reduction occurs is called as a cathode. It is a positively charged electrode.

Oxidation: The gain of oxygen or the loss of hydrogen or the loss of an electron in a species during a redox reaction is called as oxidation.

Reduction: The loss of oxygen or the gain of hydrogen or the gain of an electron in a species during a redox reaction is called as reduction.

Explanation of Solution

Using the Nernst equation the relationship is depicted as follows,

    $\begin{array}{l}A{g}^{+}\left(aq\right)\to Ag\left(s\right)+1e^{-}\\ \\ E_{cell}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{n}\log Q\\ \\ E_{waste}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{waste}\Rightarrow Non\text{-standard}\text{cell}\\ \\ E_{\text{standard}}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{\text{standard}}\Rightarrow S\text{tandard}\text{cell}\end{array}$

Combining the above equations the ${\left[{\text{Ag}}^{\text{+}}\right]}_{\text{waste}}$ is determined as follows,

    $\begin{array}{l}{E}_{waste}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{waste}\\ E^{\circ }{}_{cell}=E_{waste}\text{+}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{waste}\\ \\ E_{\text{standard}}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{\text{standard}}\\ E^{\circ }{}_{cell}=E_{\text{standard}}+\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{\text{standard}}\end{array}$

    $\begin{array}{c}{E}_{waste}-E_{\text{standard}}=\text{-}\frac{0.0592}{1e^{-}}\log \frac{{\left[A{g}^{+}\right]}_{waste}}{{\left[A{g}^{+}\right]}_{\text{standard}}}\\ \frac{E_{waste}-E_{\text{standard}}}{0.0592}=\log {\left[A{g}^{+}\right]}_{waste}-\log {\left[A{g}^{+}\right]}_{\text{standard}}\\ \log {\left[A{g}^{+}\right]}_{waste}=\frac{E_{\text{standard}}-E_{waste}}{0.0592V}+\log {\left[A{g}^{+}\right]}_{\text{standard}}\\ {\left[A{g}^{+}\right]}_{waste}=\left[\text{antilog}\left(\frac{E_{\text{standard}}-E_{waste}}{0.0592V}\right)\right]\left({\left[A{g}^{+}\right]}_{\text{standard}}\right)\end{array}$

(c)

Interpretation Introduction

Interpretation:

Using the obtained relation from subpart (b) the equation ${\left[{\text{Ag}}^{\text{+}}\right]}_{\text{waste}}$ in $\text{ng/L}$ has to be determined.

Concept Introduction:

An electrochemical cell is a device in which a redox reaction is used to convert chemical energy into electrical energy. Such device is also known as the galvanic or voltaic cell.

Galvanic cell consists of two half-cells. The redox reaction occurs in these half-cells. The half-cell in which the reduction reaction occurs is known as the reduction half-cell, whereas the half-cell in which the oxidation reaction occurs is known as the oxidation half-cell.

Anode: The electrode where the oxidation occurs is called as an anode. It is a negatively charged electrode.

Cathode: The electrode where reduction occurs is called as a cathode. It is a positively charged electrode.

Oxidation: The gain of oxygen or the loss of hydrogen or the loss of an electron in a species during a redox reaction is called as oxidation.

Reduction: The loss of oxygen or the gain of hydrogen or the gain of an electron in a species during a redox reaction is called as reduction.

Explanation of Solution

Using the Nernst equation the relationship is depicted as follows,

    $\begin{array}{l}A{g}^{+}\left(aq\right)\to Ag\left(s\right)+1e^{-}\\ \\ E_{cell}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{n}\log Q\\ \\ E_{waste}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{waste}\Rightarrow Non\text{-standard}\text{cell}\\ \\ E_{\text{standard}}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{\text{standard}}\Rightarrow S\text{tandard}\text{cell}\end{array}$

Combining the above equations the ${\left[{\text{Ag}}^{\text{+}}\right]}_{\text{waste}}$ is determined as follows,

    $\begin{array}{l}{E}_{waste}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{waste}\\ E^{\circ }{}_{cell}=E_{waste}\text{+}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{waste}\\ \\ E_{\text{standard}}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{\text{standard}}\\ E^{\circ }{}_{cell}=E_{\text{standard}}+\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{\text{standard}}\end{array}$

    $E_{waste}-E_{\text{standard}}=\text{-}\frac{0.0592}{1e^{-}}\log \frac{{\left[A{g}^{+}\right]}_{waste}}{{\left[A{g}^{+}\right]}_{\text{standard}}}$

The conversions gets cancelled since the silver ion concentration are in same units and it is indicated as follows,

    $\begin{array}{l}{E}_{waste}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{waste}\\ E^{\circ }{}_{cell}=E_{waste}\text{+}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{waste}\\ \\ {\left[A{g}^{+}\right]}_{waste}=\left[\text{antilog }\frac{E_{\text{standard}}-E_{waste}}{0.0592}\right]conc.{\left(A{g}^{+}\right)}_{\text{standard}}\end{array}$

(d)

Interpretation Introduction

Interpretation:

The value of ${\left[{\text{Ag}}^{\text{+}}\right]}_{\text{waste}}$ in $\text{ng/L}$ has to be determined using given values.

Concept Introduction:

An electrochemical cell is a device in which a redox reaction is used to convert chemical energy into electrical energy. Such device is also known as the galvanic or voltaic cell.

Galvanic cell consists of two half-cells. The redox reaction occurs in these half-cells. The half-cell in which the reduction reaction occurs is known as the reduction half-cell, whereas the half-cell in which the oxidation reaction occurs is known as the oxidation half-cell.

Anode: The electrode where the oxidation occurs is called as an anode. It is a negatively charged electrode.

Cathode: The electrode where reduction occurs is called as a cathode. It is a positively charged electrode.

Oxidation: The gain of oxygen or the loss of hydrogen or the loss of an electron in a species during a redox reaction is called as oxidation.

Reduction: The loss of oxygen or the gain of hydrogen or the gain of an electron in a species during a redox reaction is called as reduction.

Explanation of Solution

From sub part (c) the equation for ${\left[{\text{Ag}}^{\text{+}}\right]}_{\text{waste}}$ in $\text{ng/L}$ is determined. Plugging the given values into that equation will result to give the concentration values for ${\left[{\text{Ag}}^{\text{+}}\right]}_{\text{waste}}$ in $\text{ng/L}$.

    $\begin{array}{c}{E}_{waste}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{waste}\\ E^{\circ }{}_{cell}=E_{waste}\text{+}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{waste}\\ \\ {\left[A{g}^{+}\right]}_{waste}=\left[\text{antilog }\frac{-0.003}{0.0592}\right]\left(\text{1000}\text{.}\text{ng/L}\right)\\ =\text{ 900 ng/L}\end{array}$

(e)

Interpretation Introduction

Interpretation:

The equation of ${\left[{\text{Ag}}^{\text{+}}\right]}_{\text{waste}}$ in $\text{ng/L}$ for system which T changes and with different ${\text{T}}_{\text{waste}}\&T_{\text{standard}}$ has to be determined.

Concept Introduction:

An electrochemical cell is a device in which a redox reaction is used to convert chemical energy into electrical energy. Such device is also known as the galvanic or voltaic cell.

Galvanic cell consists of two half-cells. The redox reaction occurs in these half-cells. The half-cell in which the reduction reaction occurs is known as the reduction half-cell, whereas the half-cell in which the oxidation reaction occurs is known as the oxidation half-cell.

Anode: The electrode where the oxidation occurs is called as an anode. It is a negatively charged electrode.

Cathode: The electrode where reduction occurs is called as a cathode. It is a positively charged electrode.

Oxidation: The gain of oxygen or the loss of hydrogen or the loss of an electron in a species during a redox reaction is called as oxidation.

Reduction: The loss of oxygen or the gain of hydrogen or the gain of an electron in a species during a redox reaction is called as reduction.

Explanation of Solution

From sub part (c) the equation for ${\left[{\text{Ag}}^{\text{+}}\right]}_{\text{waste}}$ in $\text{ng/L}$ is determined. Plugging the given values into that equation will result to give the concentration values for ${\left[{\text{Ag}}^{\text{+}}\right]}_{\text{waste}}$ in $\text{ng/L}$

  $\begin{array}{c}{E}_{waste}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{waste}\\ E^{\circ }{}_{cell}=E_{waste}\text{+}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{waste}\end{array}$

  $\begin{array}{l}{E}_{waste}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{waste}\\ E^{\circ }{}_{cell}=E_{waste}\text{+}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{waste}\\ \\ E_{\text{standard}}=E^{\circ }{}_{cell}\text{-}\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{\text{standard}}\\ E^{\circ }{}_{cell}=E_{\text{standard}}+\frac{0.0592}{1e^{-}}\log {\left[A{g}^{+}\right]}_{\text{standard}}\end{array}$    $\begin{array}{l}{\text{E}}_{\text{standard}}{\text{-E}}_{\text{waste}}\text{=}\left(\frac{\text{2}\text{.303R}}{\text{nF}}\right)\left({\text{T}}_{\text{waste}}\text{log}{\left[{\text{Ag}}^{\text{+}}\right]}_{\text{waste}}{\text{-T}}_{\text{standard}}\text{log}{\left[{\text{Ag}}^{\text{+}}\right]}_{\text{standard}}\right)\\ \left({\text{E}}_{\text{standard}}{\text{-E}}_{\text{waste}}\right)\left(\frac{\text{nF}}{\text{2}\text{.303R}}\right){\text{=T}}_{\text{waste}}\text{log}{\left[{\text{Ag}}^{\text{+}}\right]}_{\text{waste}}{\text{-T}}_{\text{standard}}\text{log}{\left[{\text{Ag}}^{\text{+}}\right]}_{\text{standard}}\\ \left({\text{E}}_{\text{standard}}{\text{-E}}_{\text{waste}}\right)\left(\frac{\text{nF}}{\text{2}\text{.303}\text{R}}\right){\text{+T}}_{\text{standard}}\text{log}{\left[{\text{Ag}}^{\text{+}}\right]}_{\text{standard}}{\text{=T}}_{\text{waste}}\text{log}{\left[{\text{Ag}}^{\text{+}}\right]}_{\text{waste}}\\ \text{log}{\left[{\text{Ag}}^{\text{+}}\right]}_{\text{waste}}\text{=}\left(\frac{\left({\text{E}}_{\text{standard}}{\text{-E}}_{\text{waste}}\right)\left(\text{nF/2}\text{.303}\text{R}\right){\text{+T}}_{\text{standard}}\text{log}{\left[{\text{Ag}}^{\text{+}}\right]}_{\text{standard}}}{{\text{T}}_{\text{waste}}}\right)\\ {\left[{\text{Ag}}^{\text{+}}\right]}_{\text{waste}}\text{=antilog}\left(\frac{\left({\text{E}}_{\text{standard}}{\text{-E}}_{\text{waste}}\right)\left(\text{nF/2}\text{.303}\text{R}\right){\text{+T}}_{\text{standard}}\text{log}{\left[{\text{Ag}}^{\text{+}}\right]}_{\text{standard}}}{{\text{T}}_{\text{waste}}}\right)\end{array}$