Question

The organic anion CCCCCC(C)c1ccc(S(=O)(=O)[O-])cc1 is found in most detergents. Assume that the anion undergoes aerobic decomposition in the following manner: $2 \mathrm{C}_{18} \mathrm{H}_{2} \mathrm{SO}_{3}^{-}(a q)+51 \mathrm{O}_{2}(a q) \longrightarrow$ $36 \mathrm{CO}_{2}(a q)+28 \mathrm{H}_{2} \mathrm{O}(l)+2 \mathrm{H}^{+}(a q)+2 \mathrm{SO}_{4}^{2-}(a q)$ What is the total mass of \(\mathrm{O}_{2}\) required to biodegrade $10.0 \mathrm{~g}$ of this substance?

Short answer

The total mass of O2 required to biodegrade 10.0 grams of the given anion is approximately 22.7 grams.

Step-by-step solution

Calculate the molar mass of the anion

To calculate the molar mass of the given anion (C18H29SO3-), we'll sum the molar masses of all the atoms in the formula. The anion's formula units are: - 18 Carbon atoms, each with a molar mass of 12.01 g/mol - 29 Hydrogen atoms, each with a molar mass of 1.008 g/mol - 1 Sulfur atom, with a molar mass of 32.07 g/mol - 3 Oxygen atoms, each with a molar mass of 16.00 g/mol Adding these together, we get the total molar mass of the anion: \(Molar \, mass \, (anion) = 18 \times 12.01 + 29 \times 1.008 + 32.07 + 3 \times 16.00 \approx 361.508\, g/mol\)

Determine the moles of the anion

Now, we can figure out the moles of the anion in 10.0 grams of the substance. Use the molar mass you calculated in Step 1: \(Moles \, (anion) = \cfrac{10.0 \, g}{361.508 \, g/mol} \approx 0.0277 \, moles\)

Use stoichiometric coefficients to determine moles of O2 required

The given balanced chemical equation is: \(2 C_{18}H_{29}SO_{3}^{-} + 51 O_{2} \longrightarrow 36 CO_{2} + 28 H_{2}O + 2 H^{+} + 2 SO_{4}^{2-}\) From the balanced equation, we can see that 2 moles of the anion react with 51 moles of O2. We'll set up a proportion to find the moles of O2 required for 0.0277 moles of the anion: \(\cfrac{0.0277 \, moles \, (anion)}{2 \, moles \, (anion)} = \cfrac{x \, moles \, (O_2)}{51 \, moles(O_2)}\) To solve for x, multiply both sides by 51 moles (O2): \(x \, moles \, (O_2) = 0.0277 \times \cfrac{51}{2} \approx 0.708 \, moles\)

Calculate the total mass of O2

Now we have the moles of O2 needed for the reaction, and we can find the mass of O2. The molar mass of O2 is: \(Molar\,mass\,(O_2) = 2 \times 16.00\, g/mol = 32.00\, g/mol\) We can now calculate the total mass of O2 required for the decomposition of 10 g of anion: \(Mass\,(O_2) = 0.708 \, moles \, (O_2) \times 32.00 \, g/mol \approx 22.7\,g\) Therefore, 22.7 grams of O2 are required to biodegrade 10.0 grams of the given anion.