Question

Bacterial digestion is an economical method of sewage treatment. The reaction \(5 \mathrm{CO}_{2}(g)+55 \mathrm{NH}_{4}^{+}(a q)+76 \mathrm{O}_{2}(g)\) \(\mathrm{C}_{5} \mathrm{H}_{7} \mathrm{O}_{2} \mathrm{~N}(s)+54 \mathrm{NO}_{2}^{-}(a q)+52 \mathrm{H}_{2} \mathrm{O}(l)+109 \mathrm{H}^{+}(a q)\)

Short answer

In this bacterial digestion reaction, the balanced chemical equation given is: \(5 CO_2(g) + 55 NH_4^+(aq) + 76 O_2(g) \rightarrow C_5H_7O_2N(s)+ 54 NO_2^-(aq) + 52 H_2O(l) + 109 H^+(aq)\) To solve a related problem, follow these steps: 1. Identify known and unknown values in the problem. 2. Convert grams to moles or vice versa using molar masses. 3. Apply stoichiometry by using the stoichiometric coefficients in the balanced chemical equation to find the unknown values. 4. Convert the result back into grams if needed. 5. Check the answer for reasonableness and consistency with the given information and the balanced chemical equation.

Step-by-step solution

Understand the balanced chemical equation

In this bacterial digestion reaction, 5 molecules of carbon dioxide (\(CO_2\)), 55 ammonium ions (\(NH_4^+\)), and 76 molecules of oxygen (\(O_2\)) are consumed. On the other hand, one molecule of cellular material (\(C_5H_7O_2N\)), 54 nitrite ions (\(NO_2^-\)), 52 water molecules (\(H_2O\)), and 109 hydrogen ions (\(H^+\)) are produced.

Identify the known and unknown values

Suppose we are given a certain quantity of one or more reactants or products and asked to determine the quantity of another reactant or product in the reaction. First, identify the given information in the problem and determine what you need to find.

Convert moles to grams and grams to moles

When converting the given quantities from grams to moles or vice versa, use the molar mass of each involved substance. For example, if you are given the mass of \(O_2\) in grams, divide it by the molar mass of \(O_2\) (32 g/mol) to find the moles of \(O_2\). Similarly, if you have the moles of \(NH_4^+\) and need to find the mass in grams, multiply it by the molar mass of \(NH_4^+\) (18.04 g/mol).

Use stoichiometry to find the unknown values

Apply stoichiometry to determine the amount of other reactants or products based on the given information. For instance, if you know the moles of \(NH_4^+\), you can determine the moles of \(NO_2^-\) by using the stoichiometric coefficient: moles of \(NO_2^-\) = (moles of \(NH_4^+\) × 54) / 55 Once you have the moles of the unknown substance, you can convert it into grams by multiplying it by the molar mass of the substance.

Check your answer

Make sure your answer is reasonable in the context of the problem. Double-check the calculations and units to ensure they are consistent with the balanced chemical equation and with the given information.

Key concepts

Balanced Chemical Equations

Understanding balanced chemical equations is essential in the field of stoichiometry. A balanced chemical equation accurately represents the conservation of mass in a chemical reaction. It means that the number of atoms of each element in the reactants side is equal to the number of atoms of that element in the products side.

For the bacterial digestion equation, the balancing act looks like this: for every five molecules of carbon dioxide ((CO_2)), 55 ammonium ions ((NH_4^+)), and 76 molecules of oxygen ((O_2)) consumed, the bacteria produce one molecule of cellular material ((C_5H_7O_2N)), 54 nitrite ions ((NO_2^-)), 52 water molecules ((H_2O)), and 109 hydrogen ions ((H^+)). Balancing is critical because without it, stoichiometric calculations would not yield accurate and meaningful results.

Molar Mass Conversions

Molar mass conversions are a cornerstone of chemical calculations, allowing us to transition between the mass of a substance and the number of moles. The molar mass is the weight of one mole of a substance, typically expressed in grams per mole (g/mol).

In our exercise, to go from the mass of a reactant like oxygen to its molar equivalent, we divide the mass by oxygen's molar mass (32 g/mol). Conversely, if we've determined the number of moles of ammonium ions needed, we can find the mass required by multiplying by the molar mass of (NH_4^+), which is 18.04 g/mol. Mastering these conversions is key to performing accurate stoichiometric calculations.

Stoichiometric Calculations

Stoichiometric calculations are the mathematical methods we use to quantify the relationships in a balanced chemical equation. They enable us to predict the amount of reactants required to form a certain amount of product or vice versa.

For example, in a bacterial digestion scenario, if we know the amount of ammonium ions ((NH_4^+)) involved, we can calculate the corresponding amount of nitrite ions ((NO_2^-)) formed using the stoichiometric coefficients from the balanced equation. The coefficients tell us that for every 55 moles of (NH_4^+), 54 moles of (NO_2^-) are produced. Through these calculations, we can deduce the precise ratios and make practical predictions about the outcomes of the chemical reaction.

Chemical Reaction Quantification

Chemical reaction quantification involves determining the extent to which a chemical reaction has proceeded. This encompasses identifying the limiting reactant, calculating the theoretical yield, and possibly determining the reaction's percent yield.

In our exercise concerning bacterial digestion, if we are given a certain amount of (CO_2), (NH_4^+), and (O_2), we can use the balanced equation to ascertain how much (C_5H_7O_2N), (NO_2^-), and other products would be generated under ideal conditions. This is where the practical application of stoichiometry comes alive, enabling us to predict and measure the actual efficiency of bacterial digestion in treating sewage.