Q21.146CP - Chemistry: Molecular Nature Of Matter And Change

Question

Black-and-white photographic film is coated with silver halides. Because silver is expensive, the manufacturer monitors the ${\mathbf{Ag}}^{\mathbf{+}}$content of the waste stream, $\mathbf{[}\mathbf{A}{\mathbf{g}}^{\mathbf{+}}{\mathbf{]}}_{\mathbf{waste}}$, from the plant with an ${\mathbf{Ag}}^{\mathbf{+}}$selective electrode at ${\mathbf{25}}^{\mathbf{o}}\mathbf{C}$. A stream of known ${\mathbf{Ag}}^{\mathbf{+}}$ concentration, $\mathbf{[}\mathbf{A}{\mathbf{g}}^{\mathbf{+}}{\mathbf{]}}_{\mathbf{standard}}$, is passed over the electrode in turn with the waste stream and the data recorded by a computer. 

(a) Write the equations relating the nonstandard cell potential to the standard cell potential and [${\mathbf{Ag}}^{\mathbf{+}}$] for each solution. (b) Combine these into a single equation to find $\mathbf{[}\mathbf{A}{\mathbf{g}}^{\mathbf{+}}{\mathbf{]}}_{\mathbf{waste}}$. (c) Rewrite the equation from part (b) to find $\mathbf{[}\mathbf{A}{\mathbf{g}}^{\mathbf{+}}{\mathbf{]}}_{\mathbf{waste}}$ in ng/L.  (d) If ${\mathbf{E}}_{\mathbf{waste}}$ is  $0.003$V higher than , and the standard solution contains $\mathbf{1000}$ng/L, what is $\mathbf{[}\mathbf{A}{\mathbf{g}}^{\mathbf{+}}{\mathbf{]}}_{\mathbf{waste}}$? (e) Rewrite the equation in part (b) to find $\mathbf{[}\mathbf{A}{\mathbf{g}}^{\mathbf{+}}{\mathbf{]}}_{\mathbf{waste}}$ for a system in which T changes and ${\mathbf{T}}_{\mathbf{waste}}$ and  ${\mathbf{T}}_{\mathbf{s}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{d}\mathbf{a}\mathbf{r}\mathbf{d}}$ may be different.

Step-by-Step Solution

Answer

(a) The equations rewritten in standard cell potential and is: 

$\begin{array}{l}{\mathrm{E}}_{\mathrm{waste}}= {\mathrm{E}}_{\mathrm{cell}}^{\mathrm{o}}-\frac{\mathrm{RT}}{\mathrm{nF}}\times \ln {\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}................\left(1\right)\\ {\mathrm{E}}_{\mathrm{standard}}= {\mathrm{E}}_{\mathrm{cell}}^{\mathrm{o}}-\frac{\mathrm{RT}}{\mathrm{nF}}\times \ln {\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{standard}}..........\left(2\right)\end{array}$

(b) The equations combined in single equation is: 

${\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}= {\mathrm{e}}^{\frac{-\left({\mathrm{E}}_{\mathrm{waste}}-{\mathrm{E}}_{\mathrm{standard}}\right)\times \mathrm{nF}}{\mathrm{RT}}}\times {\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{standard}}..............\left(3\right)$

(c) The equation of ${\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}$ when converted in ng/L is:

${\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}={\mathrm{e}}^{\frac{-\left({\mathrm{E}}_{\mathrm{waste}}-{\mathrm{E}}_{\mathrm{standard}}\right)\times \mathrm{nF}}{\mathrm{RT}}}\times {\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{standard}}............\left(4\right)$

(d) The value ${\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}$ of is obtained as: $\mathrm{A}{{\mathrm{g}}^{+}}_{\mathrm{waste}}=889.8\frac{\mathrm{ng}}{\mathrm{L}}$ .

Step 1: Define chemical reaction

Chemical synthesis or, alternatively, chemical breakdown into two or more separate chemicals occurs when one component interacts with another to generate a new material. These processes are known as chemical reactions, and they are generally irreversible until followed by other chemical reactions.

Step 2: Rewrite the equation

(a) Cell potentials may be linked to ion concentrations using the Nernst equation.

$\begin{array}{c}{\mathrm{E}}_{\mathrm{waste}}{=\mathrm{E}}_{\mathrm{cell}}^{\mathrm{o}}-\frac{\mathrm{RT}}{\mathrm{nF}}\times \ln {\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}................\left(1\right)\\ {\mathrm{E}}_{\mathrm{standard}}{=\mathrm{E}}_{\mathrm{cell}}^{\mathrm{o}}-\frac{\mathrm{RT}}{\mathrm{nF}}\times \ln {\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{standard}}..........\left(2\right)\end{array}$

Therefore, the equations are:

$\begin{array}{c}{\mathrm{E}}_{\mathrm{waste}}{=\mathrm{E}}_{\mathrm{cell}}^{\mathrm{o}}-\frac{\mathrm{RT}}{\mathrm{nF}}\times \ln {\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}................\left(1\right)\\ {\mathrm{E}}_{\mathrm{standard}}{=\mathrm{E}}_{\mathrm{cell}}^{\mathrm{o}}-\frac{\mathrm{RT}}{\mathrm{nF}}\times \ln {\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{standard}}..........\left(2\right)\end{array}$

Step 3: Combine to form single equation

(b) Equation ($2$) should be removed from equation ($1$) in part a in order to extract ${\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}$

$\begin{array}{c}{\mathrm{E}}_{\mathrm{waste}}-{\mathrm{E}}_{\mathrm{standard}}= {\mathrm{E}}_{\mathrm{cell}}^{\mathrm{o}}{-\mathrm{E}}_{\mathrm{cell}}^{\mathrm{o}}-\frac{\mathrm{RT}}{\mathrm{nF}}\times \ln {\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}-\frac{\mathrm{RT}}{\mathrm{nF}}\times \ln {\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{standard}}\\ {\mathrm{E}}_{\mathrm{waste}}-{\mathrm{E}}_{\mathrm{standard}}= -\frac{\mathrm{RT}}{\mathrm{nF}}\times \ln \frac{{\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}}{{\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{standard}}}\\ \ln \frac{{\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}}{{\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{standard}}}= \frac{-\left({\mathrm{E}}_{\mathrm{waste}}-{\mathrm{E}}_{\mathrm{standard}}\right)\times \mathrm{nF}}{\mathrm{RT}}\\ \frac{{\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}}{{\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{standard}}}= {\mathrm{e}}^{\frac{-\left({\mathrm{E}}_{\mathrm{waste}}-{\mathrm{E}}_{\mathrm{standard}}\right)\times \mathrm{nF}}{\mathrm{RT}}}\\ {\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}= {\mathrm{e}}^{\frac{-\left({\mathrm{E}}_{\mathrm{waste}}-{\mathrm{E}}_{\mathrm{standard}}\right)\times \mathrm{nF}}{\mathrm{RT}}}\times {\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{standard}}\end{array}$

Therefore, the equation is:

${\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}= {\mathrm{e}}^{\frac{-\left({\mathrm{E}}_{\mathrm{waste}}-{\mathrm{E}}_{\mathrm{standard}}\right)\times \mathrm{nF}}{\mathrm{RT}}}\times {\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{standard}}$

Step 4: Evaluate [ A g + ] waste in ng/L

The molarity of ${\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}$ and ${\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{standard}}$ must be multiplied by the molar mass of Ag in ng/mol to adapt the equation to use concentrations in ng/L.

The conversion factor will simply cancel out given the term $\ln \frac{{\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}}{{\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{standard}}}$. If alternative concentration units are used, the equation remains the same. It still stands:

${\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}={\mathrm{e}}^{\frac{-\left({\mathrm{E}}_{\mathrm{waste}}-{\mathrm{E}}_{\mathrm{standard}}\right)\times \mathrm{nF}}{\mathrm{RT}}}\times {\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{standard}}$

Therefore, the equation is:

${\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}={\mathrm{e}}^{\frac{-\left({\mathrm{E}}_{\mathrm{waste}}-{\mathrm{E}}_{\mathrm{standard}}\right)\times \mathrm{nF}}{\mathrm{RT}}}\times {\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{standard}}$

Step 5: Evaluate the value of [ A g + ] waste

Using the equation in with ${\mathrm{E}}_{\mathrm{waste}}-{\mathrm{E}}_{\mathrm{standard}}=0.003\mathrm{V}$ and ${\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{standard}}=1000\mathrm{ng}/\mathrm{L}$ , the concentration of ${\left[{\mathrm{Ag}}^{+}\right]}_{\mathrm{waste}}$ in ng/L may be calculated (c). At 298.15 K, the reduction of silver involves only one electron:

$\begin{array}{c}\mathrm{A}{{\mathrm{g}}^{+}}_{\mathrm{waste}}= {\mathrm{e}}^{\frac{-\left({\mathrm{E}}_{\mathrm{waste}}-{\mathrm{E}}_{\mathrm{standard}}\right)\times \mathrm{nF}}{\mathrm{RT}}}\times \mathrm{A}{{\mathrm{g}}^{+}}_{\mathrm{standard}}\\ \mathrm{A}{{\mathrm{g}}^{+}}_{\mathrm{waste}}\left(\frac{\mathrm{ng}}{\mathrm{L}}\right)= {\mathrm{e}}^{\frac{-0.003\times 1\times 96,500}{298.15\times 8.314}}\times 1000\left(\frac{\mathrm{ng}}{\mathrm{L}}\right)\\ \mathrm{A}{{\mathrm{g}}^{+}}_{\mathrm{waste}}= 889.8\frac{\mathrm{ng}}{\mathrm{L}}\end{array}$

Therefore, the value is: $\mathrm{A}{{\mathrm{g}}^{+}}_{\mathrm{waste}}=889.8\frac{\mathrm{ng}}{\mathrm{L}}$.