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 \(\mathrm{PCB}\) products. The mixture is analyzed by decomposing the PCBs and then precipitating the resulting \(\mathrm{Cl}^{-}\) as 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

#tag_title#Short Answer#tag_content# The general equation relating the average value of n to the mass of a given mixture of PCBs and the mass of AgCl produced is: $$ n_{avg} = \frac{m_{PCB} \times M_{Cl}}{m_{AgCl}} \times M_{AgCl} \times \frac{1}{M_{Cl}} $$ Using the given values of 0.1947 g PCB sample and 0.4791 g AgCl produced, we can find the average value of n for this sample: $$ n_{avg} = \frac{0.1947 \times 35.45}{0.4791} \times 143.32 \times \frac{1}{35.45} $$ $$ n_{avg} \approx 4.79 $$

Step-by-step solution

Develop the general equation

To find a general equation that relates the average value of n, mass of PCB mixture (m_PCB), and mass of AgCl produced (m_AgCl), we should first look at the chemical reaction. We will label the PCB species as C12H(10-n)Cl_n, and will define the molar masses of all species involved: M_PCB, M_Cl2, M_AgCl, and M_HCl. $$ \mathrm{C}_{12} \mathrm{H}_{10}+n \mathrm{Cl}_{2} \rightarrow \mathrm{C}_{12} \mathrm{H}_{10-n} \mathrm{Cl}_{n}+n \mathrm{HCl} $$ Notice that for every mole of Cl_n produced in the reaction, n moles of Cl2 are used and n moles of HCl are produced. The mass of the Chlorine in PCB mixture (m_Cl) is also released as AgCl, so we can write: $$ m_{Cl} = m_{AgCl} \times \frac{M_{Cl}}{M_{AgCl}} = m_{PCB} \times \frac{M_{Cl.n}}{M_{PCB}} $$ Where M_Cl.n is the molar mass of Chlorine atoms in one mole of PCB in the given mixture. Now we can rewrite this equation to find the M_Cl.n in terms of m_PCB, and m_AgCl: $$ M_{Cl.n} = \frac{m_{PCB} \times M_{Cl}}{m_{AgCl}} \times M_{AgCl} $$ Now, the average value of n can be found by dividing M_Cl.n by the molar mass of Chlorine: $$ n_{avg} = \frac{M_{Cl.n}}{M_{Cl}} $$ Combining the equations: $$ n_{avg} = \frac{m_{PCB} \times M_{Cl}}{m_{AgCl}} \times M_{AgCl} \times \frac{1}{M_{Cl}} $$

Key concepts

Chemical Reaction Analysis

Analyzing chemical reactions is all about understanding how substances interact and transform into new substances. In the case of Polychlorinated Biphenyls (PCBs), their formation involves a combination of molecules participating in a controlled reaction.
The reaction for producing PCBs is given by:
- \(\text{C}_{12} \text{H}_{10} + n \text{Cl}_{2} \rightarrow \text{C}_{12} \text{H}_{10-n} \text{Cl}_{n} + n \text{HCl}\)
Here, patterns within the reaction show that for each mole of new PCB formed, \(n\) moles of \(\text{Cl}_2\) are incorporated, and \(n\) moles of \(\text{HCl}\) are generated as a byproduct.
This means analyzing the chemical reaction will involve:

• Identifying the number of chlorine molecules needed.
• Determining how many hydrogen chloride molecules are released.

Understanding these interactions helps in predicting the outcome and determining the composition of the PCB mixture.

Environmental Chemistry

Environmental Chemistry focuses on how chemical substances interact within the environment, especially concerning pollutants like PCBs. PCBs, due to their extensive use as dielectric materials, have polluted air, water, and soil ecosystems.
Key concerns include:

• Persistence: PCBs don't easily break down in the environment, leading to accumulation over time.
• Bioaccumulation: PCBs accumulate in the fatty tissues of living organisms, which can lead to long-term exposure and harm.
• Toxicity: Exposure to PCBs is linked to a range of negative health impacts in humans and wildlife.

Environmental Chemistry helps develop methods to detect PCBs and determine their impact, allowing scientists to assess and minimize their harmful effects through remediation and regulation.

Molar Mass Calculation

Calculating molar mass is crucial for understanding chemical reactions and the quantities of substances involved. For any compound, the molar mass is the sum of the atomic masses of all atoms present. In the case of PCBs:

• The molar mass of the original compound \(\text{C}_{12} \text{H}_{10}\) needs to be considered.
• The molar mass of \(\text{Cl}\), given as a multiple of \(n\), to account for the chlorinated forms.
• For AgCl, which is produced during the analysis, its molar mass assists in determining how many chlorine atoms in PCB have been precipitated.

By using the formula \( n_{avg} = \frac{m_{PCB} \times M_{Cl}}{m_{AgCl}} \times M_{AgCl} \times \frac{1}{M_{Cl}} \), you determine the average number of chlorine atoms in PCB, hence understanding the extent of chlorination in a given PCB sample.

Dielectric Materials

Dielectric materials, like PCBs, are essential in electrical applications due to their ability to store and manage energy efficiently without conducting electricity.

• PCBs were historically valued for their high dielectric constant, thermally stable properties, and chemical resilience.
• These features made them ideal for use in transformers and capacitors, where preventing electrical discharges is critical.
• However, the environmental and health drawbacks of using PCBs have led to the search for safer alternatives.

Understanding the role of dielectric materials helps in appreciating their widespread use and the ongoing transition towards more sustainable options. The case of PCBs highlights the balance between functional utility and environmental responsibility.