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Chapter 18: Problem 47

The organic anion is found in most detergents. Assume that the anion undergoes aerobic decomposition in the following manner: $$ \begin{array}{r} 2 \mathrm{C}_{18} \mathrm{H}_{29} \mathrm{SO}_{3}^{-}(a q)+51 \mathrm{O}_{2}(a q) \longrightarrow \\ 36 \mathrm{CO}_{2}(a q)+28 \mathrm{H}_{2} \mathrm{O}(l)+2 \mathrm{H}^{+}(a q)+2 \mathrm{SO}_{4}^{2-}(a q) \end{array} $$ What is the total mass of \(\mathrm{O}_{2}\) required to biodegrade \(10.0 \mathrm{~g}\) of this substance?

Short Answer

Expert verified
The total mass of \(\mathrm{O}_2\) required to biodegrade \(10.0 \space \mathrm{g}\) of the organic anion is \(22.7 \space \mathrm{g}\).

Step by step solution

01

Calculate the molar mass of the organic anion and \(\mathrm{O}_2\)

First, we need to find the molar mass of the given organic anion \(\mathrm{C}_{18}\mathrm{H}_{29}\mathrm{SO}_{3}^{-}\). To do this, we find the molar mass of each element in the anion and sum them up. The molar mass of C is 12.01 g/mol, H is 1.008 g/mol, S is 32.07 g/mol, and O is 16.00 g/mol. Molar mass of \(\mathrm{C}_{18}\mathrm{H}_{29}\mathrm{SO}_{3}^{-}\): \((18\times 12.01 \space g/mol) + (29 \times 1.008 \space g/mol) + (1\times 32.07 \space g/mol) + (3\times 16.00 \space g/mol) = 361.37 \space g/mol\) Molar mass of \(\mathrm{O}_2\): \((2\times 16.00 \space g/mol) = 32.00 \space g/mol\)
02

Convert the mass of the organic anion to moles

Now, we convert the mass of the organic anion (\(10.0 \space g\)) to moles using the calculated molar mass: Moles of \(\mathrm{C}_{18}\mathrm{H}_{29}\mathrm{SO}_{3}^{-}\) = \(\frac{10.0 \space g}{361.37 \space g/mol} = 0.0277 \space mol\)
03

Use stoichiometry to find the moles of \(\mathrm{O}_2\) needed

Given the balanced chemical equation, we can see that for every 2 moles of the organic anion, we require 51 moles of \(\mathrm{O}_2\). Using the ratio, we can calculate the moles of \(\mathrm{O}_2\) needed for the decomposition: Moles of \(\mathrm{O}_2\) = \(\frac{51 \,"\,\text{moles of } \mathrm{O}_2"}{2 \, "\,moles of }\mathrm{C}_{18}\mathrm{H}_{29}\mathrm{SO}_{3}^{-}\space "}\) \(\times 0.0277 \,"\,moles of }\mathrm{C}_{18}\mathrm{H}_{29}\mathrm{SO}_{3}^{-}\space "= 0.7094 \space mol\)
04

Convert the moles of \(\mathrm{O}_2\) to mass

Finally, we multiply the moles of \(\mathrm{O}_2\) by the molar mass of \(\mathrm{O}_2\) to find the mass required for the biodegradation process: Mass of \(\mathrm{O}_2\) = \(0.7094 \space mol \times 32.00 \space g/mol = 22.7 \space g\) Therefore, the total mass of \(\mathrm{O}_2\) required to biodegrade \(10.0 \space \mathrm{g}\) of the organic anion is \(22.7 \space \mathrm{g}\).

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