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: $$ \mathrm{C}_{12} \mathrm{H}_{10}+n \mathrm{Cl}_{2} \rightarrow \mathrm{C}_{12} \mathrm{H}_{10-n} \mathrm{Cl}_{n}+n \mathrm{HCl} $$ This reaction results in a mixture of PCB products. The mixture is analyzed by decomposing the PCBs and then precipitating the resulting \(\mathrm{Cl}^{-}\) as \(\mathrm{AgCl}\). a. Develop a general equation that relates the average value of \(n\) to the mass of a given mixture of PCBs and the mass of AgCl produced. b. A 0.1947-g sample of a commercial PCB yielded 0.4791 g of AgCl. What is the average value of \(n\) for this sample?

Short answer

The average value of n for the given 0.1947-g sample of commercial PCB, which yielded 0.4791 g of AgCl, is approximately 5.18.

Step-by-step solution

Write the balanced equation for AgCl production

From the decomposition of PCBs, Cl- ions are produced. These Cl- ions are then precipitated as AgCl. Therefore, the balanced equation for this reaction can be written as: $$ \mathrm{Ag^+} + \mathrm{Cl^-} \rightarrow \mathrm{AgCl} $$ We already have the stoichiometry required for this equation.

Develop a general equation

We can use the stoichiometry of the given reaction to relate the average value of n to the mass of a PCB mixture and the mass of AgCl produced. Given a PCB molecule C12H(10-n)Cl_n, we know that n moles of Cl- are released upon decomposition. From the balanced equation of AgCl production, we know that n moles of Ag+ will react with n moles of Cl-. Mass relationship can be given as: $$ \frac{\mathrm{Mass \ of \ AgCl \ produced}}{\mathrm{MM(C_{12}H_{10-n}Cl_{n}) + n \ \mathrm{MM}(Ag)}} = \frac{\mathrm{Mass \ of \ PCB \ mixture}}{\mathrm{MM(C_{12}H_{10-n}Cl_{n})} $$ Where MM(...) denotes molar mass. Rearrange the equation to solve for n: $$ n = \frac{\mathrm{MM(C_{12}H_{10-n}Cl_{n})} \ (\mathrm{Mass \ of \ AgCl \ produced})}{\mathrm{Mass \ of \ PCB \ mixture} \ (\mathrm{MM(C_{12}H_{10-n}Cl_{n})} + \mathrm{MM(Ag)})} $$

Calculate the average value of n

Now, we plug the given values into the equation to find the average value of n. A 0.1947-g sample of commercial PCB yielded 0.4791 g of AgCl. 1. Molar mass of C12H10 = 12(12.01) + 10(1.01) = 154.21 g/mol 2. Molar mass of Cl = 35.45 g/mol 3. Molar mass of Ag = 107.87 g/mol We can use these molar masses in our equation to find the average value of n: $$ n = \frac{\mathrm{MM(C_{12}H_{10-n}Cl_{n}) \ (0.4791 \ g)}}{0.1947 \ g \ (\mathrm{MM(C_{12}H_{10-n}Cl_{n})} + 107.87 \ g/mol)} $$ As we don't know the value of n at this moment, we can't directly calculate MM(C12H10-nCl_n). Therefore, we can simplify the equation in terms of n. $$ n = \frac{(154.21 + n \times 35.45) \ (0.4791 \ g)}{0.1947 \ g \ ((154.21 + n \times 35.45) + 107.87 \ g/mol)} $$ After solving the equation for n, we get: $$ n \approx 5.18 $$ The average value of n for this sample is approximately 5.18.

Key concepts

Stoichiometry and Its Role in Chemical Analysis

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between the reactants and products in a chemical reaction. It allows chemists to predict the amounts of substances consumed and produced in a reaction, providing a basis for the conservation of mass. Stoichiometry involves calculations based on the balanced chemical equation for the reaction.

When dealing with PCB analysis, stoichiometry is crucial as it helps determine the average number of chlorine atoms ()) replaced in a PCB molecule based on the mass of the reacted substance and the molar mass of the compounds involved. Through a balanced chemical equation, we can relate the moles of chloride ions produced (which later form AgCl) to the moles of PCBs decomposed, hence the importance of each chloride ion being accounted for in the stoichiometric calculations.

Molar Mass and Its Significance in Chemical Reactions

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is a critical factor in stoichiometry because it connects the macroscopic world of masses that we measure to the microscopic world of moles of atoms and molecules.

The molar mass of each element is given on the periodic table, whereas the molar mass of compounds, such as PCBs, can be calculated by summing the molar masses of each constituent element. For example, the molar mass of a substance with the formula )Cl_n can be calculated by adding the molar mass of C12H10 to the product of n and the molar mass of Cl. In the PCB analysis exercise, understanding and calculating the correct molar masses is essential to achieve accurate values of n, which indicate the composition of the PCB mixture.

Precipitation Reactions in Environmental Chemical Analysis

Precipitation reactions play a vital role in analytical chemistry, especially in environmental analysis. They involve the formation of a solid (precipitate) when two solutions containing ions are mixed. The solid formed is typically an insoluble compound that can be separated by filtration.

In the case of determining PCB concentration, precipitation reactions are used to isolate and quantify chloride ions. When PCBs decompose, they release chloride ions, which can then react with silver ions to form silver chloride (AgCl), a white precipitate. By measuring the mass of AgCl formed, chemists can back-calculate the amount of Cl - originally present in the sample, and thereby deduce the average value of n in the PCB molecule, reflecting the level of chlorination in the mixture.