Question

Polychlorinated biphenyls (PCBs) have been used extensively as dielectric materials in electrical transformers. Because PCBs have been shown to be potentially harmful, analysis for their presence in the environment has become very important. PCBs are manufactured according to the following generic reaction:

${\mathbf{C}}_{\mathbf{12}}{\mathbf{H}}_{\mathbf{10}}\mathbf{+}\mathbf{nC}{\mathbf{l}}_{\mathbf{2}}\underset{\mathbf{catalyst}}{\overset{\mathbf{Fe}}{\mathbf{\to }}}{\mathbf{C}}_{\mathbf{12}}{\mathbf{H}}_{\mathbf{10}\mathbf{-}\mathbf{n}}{\mathbf{Cl}}_{\mathbf{n}}\mathbf{+}\mathbf{nHCl}$

This reaction results in a mixture of PCB products. The mixture is analyzed by decomposing the PCBs and then precipitating the resulting ${\mathbf{Cl}}^{\mathbf{-}}$as role="math" localid="1657264500444" $\mathbf{AgCl}$.
a. Develop a general equation that relates the average value of $\mathbf{n}$ to the mass of a given mixture of PCBs and the mass of $\mathbf{AgCl}$ produced.
b. A role="math" localid="1657264553500" $\mathbf{0}\mathbf{.}\mathbf{1947}\mathbf{g}$ sample of a commercial PCB yielded $\mathbf{0}\mathbf{.}\mathbf{4971}\mathbf{g}$of $\mathbf{AgCl}$. What is the average value of $\mathbf{n}$ for this sample?

Short answer

In solution (a), the general equation is ${\mathrm{m}}_{\mathrm{AgCl}}[154+34.5\mathrm{n}]=\left[\mathrm{n}(143.4)\right]{\mathrm{m}}_{\mathrm{PCB}}$.

In solution (b), the average value is 7.108.

Step-by-step solution

Introduction

Mole is a SI unit for measuring the amount of a material. The mole is the amount of a material that contains approximately the same number of atoms as $\mathbf{12}\mathbf{.}\mathbf{00}\mathbf{g}$ of $\mathbf{Carbon}\mathbf{-}\mathbf{12}$.

The number of moles of a substance is defined as the ratio of the compound's given mass to its molar or molecular mass.

The mathematical formula is as follows:

$\mathrm{Number}\mathrm{of}\mathrm{moles}=\frac{\mathrm{Mass}\mathrm{of}\mathrm{the}\mathrm{compound}}{\mathrm{Molar}\mathrm{mass}\mathrm{of}\mathrm{the}\mathrm{compound}}$

Explanation for (a)

The given reaction is:

$\begin{array}{l}{\mathbf{C}}_{\mathbf{12}}{\mathbf{H}}_{\mathbf{10}}\mathbf{+}\mathbf{nC}{\mathbf{l}}_{\mathbf{2}}\underset{\mathbf{catalyst}}{\overset{\mathbf{Fe}}{\mathbf{\to }}}{\mathbf{C}}_{\mathbf{12}}{\mathbf{H}}_{\mathbf{10}\mathbf{-}\mathbf{n}}{\mathbf{Cl}}_{\mathbf{n}}\mathbf{+}\mathbf{nHCl}\\ {\mathbf{C}}_{\mathbf{12}}{\mathbf{H}}_{\mathbf{10}\mathbf{-}\mathbf{n}}{\mathbf{Cl}}_{\mathbf{n}}\mathbf{+}{\mathbf{nAg}}^{\mathbf{+}}\mathbf{\to }\mathbf{nAgCl}\end{array}$

The molar mass of Biphenyl $\left({\mathrm{C}}_{12}{\mathrm{H}}_{10}\right)=154.21\mathrm{g}/\mathrm{mole}$

The molar mass of chlorine$=35.5\mathrm{g}/\mathrm{mole}$

The molar mass of $\mathrm{AgCl}=143.4\mathrm{g}/\mathrm{mole}$

On calculating:

$\begin{array}{c}\mathrm{Mass}\mathrm{of}\mathrm{PCB}=\mathrm{moles}[(12\times \mathrm{molar}\mathrm{mass}\mathrm{of}\mathrm{carbon})+(10-\mathrm{n})\times \mathrm{molar}\mathrm{mass}\mathrm{of}\mathrm{hydrogen}+\mathrm{n}\left(\mathrm{molar}\mathrm{mass}\mathrm{of}\mathrm{chlorine}\right)]\\ =\mathrm{moles}[(12\times 12)+(10-\mathrm{n})\times \left(1\right)+\mathrm{n}(35.5)]\\ =\mathrm{moles}[\left(144\right)+(10-\mathrm{n})+\mathrm{n}(35.5)]\\ =\mathrm{moles}[154+34.5\mathrm{n}]\end{array}$

According to the reaction:

Explanation for (b)

The mass of PCB given is 0.1947 g.

The mass of AgCl is 0.4971 g.

The average value of “n” can be calculated as shown below:

$\begin{array}{c}{\mathrm{m}}_{\mathrm{AgCl}}[154+34.5\mathrm{n}]=\left[\mathrm{n}(143.4)\right]{\mathrm{m}}_{\mathrm{PCB}}\\ 0.4971\mathrm{g}[154+34.5\mathrm{n}]=\left[\mathrm{n}(143.4)\right]0.1947\mathrm{g}\\ 76.5534+17.14995\mathrm{n}=27.991998\mathrm{n}\\ \mathrm{On}\mathrm{simplification},\\ \mathrm{n}=\frac{76.5534}{10.77003}\\ \mathrm{n}=7.108\end{array}$

The average value of$\mathrm{n}$ for the given sample is 7.10.

Interpretation

Thus, the equation was developed using standard values, and the average value was derived based on theamount of moles.