Question

Bacterial digestion is an economical method of sewage treatment. The reaction \(5 \mathrm{CO}_{2}(g)+55 \mathrm{NH}_{4}^{+}(a q)+76 \mathrm{O}_{2}(g) \stackrel{\text { hacteria }}{\longrightarrow}\) \(\mathrm{C}_{3} \mathrm{H}_{7} \mathrm{O}_{2} \mathrm{~N}(s)+54 \mathrm{NO}_{2}^{-}(a q)+52 \mathrm{H}_{2} \mathrm{O}(t)+109 \mathrm{H}^{+}(a q)\) bacterial tissue is an intermediate step in the conversion of the nitrogen in organic compounds into nitrate ions. What mass of bacterial tissue is produced in a treatment plant for every \(1.0 \times 10^{4} \mathrm{~kg}\) of wastewater containing \(3.0 \% \mathrm{NH}_{4}{ }^{+}\) ions by mass? Assume that \(95 \%\) of the ammonium ions are consumed by the bacteria.

Short answer

First, find the mass of NH₄⁺ ions in the wastewater: \( Mass_{NH_4^+} = (3.0 / 100) \times 1.0 \times 10^7 \,g\). Next, calculate the mass of NH₄⁺ ions consumed: \(Mass_{NH_4^+ \, consumed} = (95 / 100) \times Mass_{NH_4^+}\). Convert the mass of consumed NH₄⁺ ions to moles: \(n_{NH_4^+} = Mass_{NH_4^+ \, consumed} / 18.05\, g/mol\). Use stoichiometry to find moles of bacterial tissue produced: \(n_{C_3H_7O_2N} = (1\,mol\,C_3H_7O_2N / 55\,mol\,NH_4^+) \times n_{NH_4^+}\). Finally, calculate the mass of bacterial tissue produced: \(Mass_{C_3H_7O_2N} = n_{C_3H_7O_2N} \times 60.07\, g/mol\,C_3H_7O_2N\).

Step-by-step solution

Find the mass of NH₄⁺ ions in the wastewater.

Since the wastewater contains 3.0% NH₄⁺ ions by mass, we can find the mass of ammonium ions as follows: Total mass of wastewater = 1.0 x 10^4 kg = 1.0 x 10^7 g Mass of NH₄⁺ ions = (3.0 / 100) × 1.0 x 10^7 g

Calculate the mass of NH₄⁺ ions consumed by the bacteria.

As we know that 95% of the total NH₄⁺ ions are consumed by the bacteria, the mass of NH₄⁺ ions consumed can be calculated as: Mass of NH₄⁺ ions consumed = (95 / 100) × Mass of NH₄⁺ ions

Determine the number of moles of NH₄⁺ ions consumed.

Now, we need to find the number of moles of NH₄⁺ ions consumed, using the following formula: Number of moles = mass / molar mass The molar mass of NH₄⁺ ions is equal to the sum of the atomic masses of 1 nitrogen (N) and 4 hydrogen (H) atoms: Molar mass of NH₄⁺ ions = 14.01 g/mol (N) + 4 × 1.01 g/mol (H) = 18.05 g/mol. Number of moles of NH₄⁺ ions consumed = Mass of NH₄⁺ ions consumed / 18.05 g/mol

Use stoichiometry to find the number of moles of bacterial tissue produced.

From the balanced equation, we can see that 55 moles of NH₄⁺ ions produce 1 mole of bacterial tissue (C₃H₇O₂N). Using this stoichiometric ratio, we can find the number of moles of bacterial tissue produced: Number of moles of bacterial tissue = (1 mol C₃H₇O₂N / 55 mol NH₄⁺ ions) × Number of moles of NH₄⁺ ions consumed

Calculate the mass of bacterial tissue produced using the molar mass of C₃H₇O₂N.

The molar mass of bacterial tissue (C₃H₇O₂N) is equal to the sum of atomic masses of 3 carbon (C), 7 hydrogen (H), 2 oxygen (O), and 1 nitrogen (N) atoms: Molar mass of C₃H₇O₂N = 3 × 12.01 g/mol (C) + 7 × 1.01 g/mol (H) + 2 × 16.00 g/mol (O) + 14.01 g/mol (N) = 60.07 g/mol. Now, using the number of moles of bacterial tissue and molar mass, we can find the mass of bacterial tissue produced: Mass of bacterial tissue = Number of moles of bacterial tissue × 60.07 g/mol C₃H₇O₂N By following these steps, we can find the mass of bacterial tissue produced in a treatment plant for every 1.0 x 10^4 kg of wastewater containing 3.0% NH₄⁺ ions by mass, provided that 95% of the ammonium ions are consumed by the bacteria.

Key concepts

Stoichiometric Calculations

Stoichiometry is the heart of chemical reactions, essentially serving as a recipe that allows chemists—and environmental engineers working with sewage treatment—to predict the products and byproducts of a chemical reaction. In the case of bacterial digestion, we deduce how much bacterial biomass, or tissue, can be generated from a given amount of ammonium ions in wastewater.

In our example, the stoichiometric calculation leans on a balanced chemical equation where the coefficients indicate the proportional relationship between reactants and products. This relationship helps us determine how moles of one substance (ammonium ions) convert to moles of another (bacterial tissue). With sewage treatment, such precision ensures that the treatment plant operates efficiently, managing resources and predicting the extent of waste treatment.

Molar Mass

Understanding molar mass is crucial in stoichiometric calculations. It is the weight of one mole of a substance, typically measured in grams per mole (g/mol), and can be calculated by summing the atomic masses of all the atoms in a molecule. For instance, the molar mass of the ammonium ion, NH₄⁺, is tallied by adding the atomic mass of nitrogen (14.01 g/mol) with four times the atomic mass of hydrogen (4 x 1.01 g/mol), which totals 18.05 g/mol.

Knowing this value is instrumental in converting between mass and moles, a frequent task in stoichiometry, allowing us to pivot from the physical weight of a sampled wastewater to the more abstract but chemically relevant quantity of moles. This step is essential for stoichiometric calculations that lead to figuring out how much bacterial tissue can be produced from ammonium ions.

Bacterial Digestion

Bacterial digestion is a biotechnological marvel wherein bacteria metabolize waste, such as ammonium ions in sewage, transforming pollutants into harmless or even useful substances. These microorganisms act as natural recyclers, turning potential environmental hazards into stable compounds that can be safely reintroduced into the ecosystem.

In sewage treatment, the bacteria's role is pivotal, as they enable the biochemical conversion of wastewater contaminants through metabolic processes into less harmful substances like nitrates, which can then be dealt with safely. This not only purifies the water but also results in the formation of bacterial biomass, which can be studied quantitatively through stoichiometric calculations.

Ammonium Ion Consumption

In wastewater treatment, monitoring the consumption of ammonium ions by bacteria is key to assessing the efficiency of the treatment process. In our scenario, it is estimated that 95% of the ammonium ions in the wastewater are consumed by bacteria, which reflects the efficacy of the bacterial digestion in transforming these ions.

The consumption rate is critical because it directly influences the amount of bacterial tissue produced. With high consumption rates, we can anticipate more biomass generation—a variable that can be precisely measured using stoichiometry. Therefore, understanding the nuances of ammonium ion consumption can inform the operational parameters of wastewater treatment facilities, leading to optimized processes that better support the environment and public health.