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The mole fraction of water in $20%$ (wt./wt.) aqueous solution of ${{\text{H}}_{2}}{{\text{O}}_{2}}$is:\[\]
A. $\dfrac{77}{68}$
B. $\dfrac{68}{77}$
C. $\dfrac{20}{80}$
D. $\dfrac{80}{20}$

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Last updated date: 13th Jun 2024
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Answer
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Hint: Mole fraction can be defined as the number of molecules of a particular component present in a mixture divided by the total number of moles present in the given mixture. The mole fraction is represented by \[X\].

Formula used:
${{\text{x}}_{\text{A}}}=\dfrac{{{\text{n}}_{\text{A}}}}{{{\text{n}}_{\text{A}}}+{{\text{n}}_{\text{B}}}}$, \[\text{n}=\dfrac{\text{Given}\text{.wt}\text{.or}%}{\text{Molarmass}}\]
\[{{\text{x}}_{\text{A}}}\]= mole fraction of A
\[{{\text{n}}_{\text{A}}}\] = moles for A
\[{{\text{n}}_{\text{B}}}\] = moles for B

Complete step by step answer:
Moles for water be\[{{\text{n}}_{\text{A}}}\] let say.
So,\[{{\text{n}}_{\text{A}}}=\dfrac{80}{\text{Molar mass of water}}\] (If${{\text{H}}_{2}}{{\text{O}}_{2}}$is \[20%\]then water is \[100-20=80%\])
Molar mass of water \[\left( {{\text{H}}_{2}}\text{O} \right)\]= $2\times 1$ (for hydrogen)\[+16\] (for 0)
\[=2+16\]
\[=18\]
So, \[{{\text{n}}_{\text{A}}}=80/18\text{ }\]
 \[=40/9\]
Moles for \[{{\text{H}}_{2}}{{\text{O}}_{2}}\] be \[{{n}_{B}}=\dfrac{20}{\text{Molar mass of }{{\text{H}}_{2}}{{\text{O}}_{2}}}\]
Molar mass of \[{{\text{H}}_{2}}{{\text{O}}_{2}}\]= \[2\times 1\] (for hydrogen) \[+\text{ }2\times 16\] (for oxygen)
\[=2+32\]
\[=\text{ }34\]
So, \[{{\text{n}}_{\text{B}}}=\dfrac{20}{34}=\dfrac{10}{17}\]
So, Mole fraction of water
          ${{\text{x}}_{\text{A}}}=\dfrac{{{\text{n}}_{\text{A}}}}{{{\text{n}}_{\text{A}}}+{{\text{n}}_{\text{B}}}}\text{ }$
$=\dfrac{\dfrac{40}{9}}{\dfrac{40}{9}+\dfrac{10}{7}}$
$=\dfrac{\dfrac{40}{9}}{\dfrac{680+90}{153}}=\dfrac{\dfrac{40}{9}}{\dfrac{770}{153}}=\dfrac{40\times 153}{9\times 770}=\dfrac{68}{77}$
So, mole fraction of water in \[20%\] aqueous solution of \[{{\text{H}}_{2}}{{\text{O}}_{2}}\]is\[\dfrac{68}{77}\].
Here, \[20%\]aqueous \[{{\text{H}}_{2}}{{\text{O}}_{2}}\] solution means 20 gm \[{{\text{H}}_{2}}{{\text{O}}_{2}}\]in 100 gm solution and the rest 80gm water.

So, the correct answer is “Option B”.

Additional Information:
Hydrogen peroxide\[\left( {{\mathbf{H}}_{\mathbf{2}}}{{\mathbf{O}}_{\mathbf{2}}} \right)\]: It is a pale blue liquid which is slightly more viscous than water. It is used as an oxidizer, bleaching agent, and antiseptic. Its molar mass is\[34\cdot 0147\text{ g/mol}\]. Hydrogen peroxide is usually not dangerous. It can create a lot of foam though. Household hydrogen peroxide can be irritating to eyes and skin. Solutions containing more than about 8 percent hydrogen peroxide are corrosive to the skin. It decomposes into water and oxygen upon heating or in the presence of numerous substances, particularly salts of metals like iron, copper, manganese, nickel or chromium.

Note:
Learn how to find the moles and the mole fraction. One should know how to find the molar masses of various compounds which requires knowledge for the atomic number which can be gained through the periodic Table.
Mole fraction is defined as the unit of amount of constituents divided by the total amount of all constituents in a mixture.
It decomposes into water and oxygen upon heating or in the presence of numerous substances, particularly salts of metals like iron, copper, manganese, nickel or chromium.