Problem 49 - Chemistry The Central Science In Si Units

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: $$\begin{array}{l} \mathrm{C}_{18} \mathrm{H}_{35} \mathrm{O}_{2}^{-}(a q)+26 \mathrm{O}_{2}(a q) \longrightarrow \\ 17 \mathrm{CO}_{2}(a q)+17 \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{HCO}_{3}^{-}(a q) \end{array}$$ What is the total mass of \(\mathrm{O}_{2}\) required to biodegrade \(3.0 \mathrm{~g}\) of this substance?

Step-by-Step Solution

Answer

8.144 grams of \(O_2\) are required.

Step 1: Calculate the molar mass of sodium stearate

First, find the molar mass of sodium stearate \(C_{18}H_{35}O_2Na\). You can add the atomic masses of each element:- Carbon (C): \(18 \times 12.01 = 216.18 \text{ g/mol}\).- Hydrogen (H): \(35 \times 1.01 = 35.35 \text{ g/mol}\).- Oxygen (O): \(2 \times 16.00 = 32.00 \text{ g/mol}\).- Sodium (Na): \(22.99 \text{ g/mol}\).Sum these values to get the molar mass: \(216.18 + 35.35 + 32.00 + 22.99 = 306.52 \text{ g/mol}\).

Step 2: Find the moles of sodium stearate

To find the number of moles in \(3.0\, .0.0.0/\text{He} \) of sodium stearate, use the formula:\[\text{Moles of } C_{18}H_{35}O_2Na = \frac{\text{mass}}{\text{molar mass}} = \frac{3.0\, \text{g}}{306.52\, \text{g/mol}} \approx 0.00979\, \text{mol}\]

Step 3: Determine the moles of \(O_2\) required

From the balanced equation, \(1\, \text{mol}\) of sodium stearate requires \(26\, \text{mol}\) of \(O_2\). Therefore, the moles of \(O_2\) needed are:\[0.00979\, \text{mol}\, C_{18}H_{35}O_2Na \times 26\, \text{mol}\, O_2/\text{mol}\, C_{18}H_{35}O_2Na = 0.2545\, \text{mol}\, O_2\]

Step 4: Calculate the mass of \(O_2\) required

The molar mass of \(O_2\) is \(32.00\, \text{g/mol}\). Therefore, the mass of \(O_2\) required is:\[\text{Mass of } O_2 = 0.2545\, \text{mol} \times 32.00\, \text{g/mol} = 8.144\, \text{g}\]

Key Concepts

Understanding Molar Mass Calculation

Molar mass is a key concept in chemistry that enables us to relate the number of moles of a substance to its mass in grams. It represents the mass of one mole of a given substance and is typically expressed in grams per mole (g/mol). To calculate the molar mass of a compound like sodium stearate, you essentially add up the atomic masses of all the atoms that make up the compound.For sodium stearate, which has the chemical formula \( \mathrm{C}_{18}\mathrm{H}_{35}\mathrm{O}_{2}\mathrm{Na} \), you sum the atomic masses of carbon, hydrogen, oxygen, and sodium:

• Carbon: There are 18 carbon atoms, each weighing approximately 12.01 g/mol. The total mass from carbon is \( 18 \times 12.01 = 216.18 \text{ g/mol} \).
• Hydrogen: With 35 hydrogen atoms at roughly 1.01 g/mol each, this gives \( 35 \times 1.01 = 35.35 \text{ g/mol} \).
• Oxygen: Two oxygen atoms at 16.00 g/mol each contribute \( 2 \times 16.00 = 32.00 \text{ g/mol} \).
• Sodium: One sodium atom adds another 22.99 g/mol.

Adding these components together provides the molar mass of sodium stearate: \( 216.18 + 35.35 + 32.00 + 22.99 = 306.52 \text{ g/mol} \). This value is used for further calculations in stoichiometry and chemical reactions.

Stoichiometry: Balancing Chemical Equations

Stoichiometry involves using balanced chemical equations to determine the relationships between reactants and products in a reaction. This section will guide you through how stoichiometry is applied in calculating the required mass of oxygen for a reaction.In the reaction where sodium stearate undergoes aerobic decomposition, the balanced chemical equation is given as:\[ \mathrm{C}_{18} \mathrm{H}_{35} \mathrm{O}_{2}^{-} (aq) + 26 \mathrm{O}_{2} (aq) \rightarrow 17 \mathrm{CO}_{2} (aq) + 17 \mathrm{H}_{2} \mathrm{O} (l) + \mathrm{HCO}_{3}^{-} (aq) \]The stoichiometric coefficients in the equation indicate how many moles of each substance are involved in the reaction. One mole of sodium stearate reacts with 26 moles of oxygen \( \mathrm{O}_2 \). This ratio is crucial for determining how much oxygen is necessary to completely react with a given amount of sodium stearate.To find out how much \( \mathrm{O}_2 \) is needed for 3.0 grams of sodium stearate, first calculate the number of moles of sodium stearate using its molar mass. Then use the stoichiometric ratio from the balanced equation to find the equivalent amount of oxygen. This method allows for precise calculations when planning reactions or analyzing reaction yields in a laboratory setting.

Exploring the Biodegradation Process

Biodegradation refers to the breakdown of organic compounds by the action of living organisms, typically microbes. In the context of sodium stearate, biodegradation is the process through which this compound is decomposed by microorganisms in the presence of oxygen. This specific reaction is an example of aerobic decomposition, where oxygen is consumed as microbes break down organic molecules. During the biodegradation of sodium stearate, it reacts with oxygen to eventually form carbon dioxide, water, and a bicarbonate ion. This transformation plays a crucial role in the environment as it converts potentially harmful substances into simpler, non-toxic forms. Understanding the biodegradation process is important for:

• Environmental Protection: Efficient biodegradation reduces pollution and mitigates the impact of chemicals like soaps and detergents on ecosystems.
• Sustainability: Encourages the development of biodegradable products that lessen the environmental footprint compared to non-biodegradable counterparts.
• Scientific Research: Provides insights into the metabolic pathways of microorganisms, which can be exploited for bioremediation techniques.

By studying these processes, we can better design chemical compounds and industrial processes that align with sustainability goals.