Question

Sodium stearate \(\left(\mathrm{C}_{18} \mathrm{H}_{35} \mathrm{O}_{2} \mathrm{Na}\right)\) is the most common soap. Assume that the stearate anion undergoes aerobic decomposition in the following manner: $$ \begin{aligned} \mathrm{C}_{18} \mathrm{H}_{35} \mathrm{O}_{2}^{-}(a q)+26 \mathrm{O}_{2}(a q) & \longrightarrow \\ & 17 \mathrm{CO}_{2}(a q)+17 \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{HCO}_{3}^{-}(a q) \end{aligned} $$ What is the total mass of \(\mathrm{O}_{2}\) required to biodegrade \(3.0 \mathrm{~g}\) of this substance?

Short answer

The total mass of O₂ required to biodegrade 3.0 g of sodium stearate is \(8.64 \: \text{g}\).

Step-by-step solution

Find the moles of sodium stearate

First, we need to find the molar mass of sodium stearate (C₁₈H₃₅O₂Na). To do this, we add the molar masses of each element in the compound: Molar mass of C₁₈H₃₅O₂Na = 18×(12.01 amu) + 35×(1.01 amu) + 2×(16.00 amu) + 1×(22.99 amu) = 287.46 g/mol Now, we can calculate the moles of sodium stearate: moles of C₁₈H₃₅O₂Na = (mass of sodium stearate)/(molar mass of sodium stearate) moles of C₁₈H₃₅O₂Na = (3.0 g)/(287.46 g/mol) = 0.0104 mol

Balance the given reaction

The given reaction is already balanced, so we can use the stoichiometric coefficients to find the moles of O₂ required to react with sodium stearate: C₁₈H₃₅O₂⁻ + 26 O₂ → 17 CO₂ + 17 H₂O + HCO₃⁻ According to the balanced equation, 26 moles of O₂ are needed for every 1 mole of sodium stearate.

Calculate the moles of O₂ required

Using the stoichiometric coefficients from Step 2, we can calculate the moles of O₂ needed: moles of O₂ = (moles of C₁₈H₃₅O₂Na) × (26 moles O₂ / 1 mole C₁₈H₃₅O₂Na) moles of O₂ = 0.0104 mol C₁₈H₃₅O₂Na × (26 moles O₂/1 mole C₁₈H₃₅O₂Na) = 0.270 mol O₂

Calculate the mass of O₂ required

Now, we can find the mass of O₂ required by multiplying the moles of O₂ by the molar mass of O₂: mass of O₂ = moles of O₂ × molar mass of O₂ mass of O₂ = 0.270 mol × 32.00 g/mol = 8.64 g So, 8.64 g of O₂ is required to biodegrade 3.0 g of sodium stearate.

Key concepts

Sodium Stearate

Sodium stearate is a common ingredient found in soap, and it serves as a key component due to its ability to emulsify oils and dirt, making them washable with water. Its chemical formula is represented as \(\text{C}_{18}\text{H}_{35}\text{O}_{2}\text{Na}\), where each element plays a critical role in its function. It consists of a stearate anion, \(\text{C}_{18}\text{H}_{35}\text{O}_{2}^{-}\), and a sodium cation, \(\text{Na}^{+}\).

The structure is derived from stearic acid, a long-chain fatty acid, reacting with sodium hydroxide to form the salt known as sodium stearate. This compound is crucial for the cleansing action because it forms micelles in water.

Micelles trap the oil-based dirt on surfaces using their hydrophobic tails while their hydrophilic heads interact with the water, thereby enabling the dirt to be washed away during rinsing.

Molar Mass

Molar mass is an important concept in stoichiometry and is used for converting grams of a substance into moles, a standard scientific unit for measuring the amount of substance. It acts like a "bridge" in calculations, allowing us to switch between the mass of a compound and the number of molecules or atoms it contains.

To find the molar mass of sodium stearate, \(\text{C}_{18}\text{H}_{35}\text{O}_{2}\text{Na}\), we sum up the atomic masses of its constituent atoms. Specifically, we calculate it as follows:

• Carbon (C) = 18 atoms × 12.01 g/mol
• Hydrogen (H) = 35 atoms × 1.01 g/mol
• Oxygen (O) = 2 atoms × 16.00 g/mol
• Sodium (Na) = 1 atom × 22.99 g/mol

Adding these values gives us the molar mass of 287.46 g/mol. This value is essential when calculating the amount of reactants and products during a chemical process.

Decomposition Reaction

Decomposition reactions involve a single compound breaking down into two or more products. These reactions are important in understanding various chemical processes, especially when analyzing the byproducts and end results of a reaction involving complex molecules.

In this context with sodium stearate, the decomposition refers to the breakdown of \(\text{C}_{18}\text{H}_{35}\text{O}_{2}^{-}\) (stearate anion) when exposed to oxygen, forming carbon dioxide \(\text{(CO}_2)\), water \(\text{(H}_2\text{O})\), and bicarbonate ion \(\text{(HCO}_{3}^{-})\).

The balance of the chemical equation reflects the stoichiometry:

• 1 mole of \(\text{C}_{18}\text{H}_{35}\text{O}_{2}^{-}\) breaks down with 26 moles of \(\text{O}_{2}\).
• Producing 17 moles of \(\text{CO}_{2}\), 17 moles of water \(\text{H}_2\text{O}\), and 1 mole of \(\text{HCO}_{3}^{-}\).

This decomposition allows us to determine the exact amounts of each product formed from the initial reactants.

Aerobic Decomposition

Aerobic decomposition is a type of decomposition reaction where oxygen is actively involved. This kind of reaction occurs in environments rich in oxygen and contrasts with anaerobic decomposition that occurs in the absence of oxygen.

In the reaction with sodium stearate, the term "aerobic" highlights the consumption of oxygen during the process. As seen in the equation provided, \(\text{O}_{2}\) is a crucial component that allows the stearate ion to break down efficiently. The presence of oxygen facilitates complete decomposition, creating carbon dioxide and water.

Understanding the mechanics behind aerobic processes is vital, especially in environmental contexts where waste decomposition needs to occur swiftly and effectively, contributing to reducing atmosphere pollutants. Aerobic decomposition is generally faster and more energy efficient, often resulting in fewer odors compared to its anaerobic counterpart due to the absence of partially decomposed products.