Question

What is the mole fraction of the solute in an $1.00\,m$ aqueous solution?
A) $0.0354$
B) $0.018$
C) $0.177$
D) $1.770$

Answer 1

We know. Mole fraction gives the number of molecules of a component in the mixture divided by total number of moles in the mixture.
${\text{Mole}}\,{\text{fraction}}\,\left( {\text{x}} \right){\text{ = }}\dfrac{{{\text{Moles}}\,{\text{of}}\,{\text{solute}}}}{{{\text{Moles}}\,{\text{of}}\,{\text{Solute + }}\,{\text{Moles}}\,{\text{of}}\,{\text{solvent}}}}$
The sum of all mole fractions in a given mixture is always equal to $1$.

Complete step by step answer:
We know that molality is,
The number of moles of solute in one liter of solution is defined as the molarity and molality is used to measure the moles to the kilogram of the solvent.
The mathematical expression of molality is,
${\text{Molality }}\left( {\text{m}} \right){\text{ = }}\dfrac{{{\text{Moles of solute}}\left( {{\text{Mol}}} \right)}}{{{\text{Kilograms of solvent}}\left( {{\text{Kg}}} \right)}}{\text{ }}$
Thus, one molal aqueous solution means one of solute is dissolved in one kilogram of solvent.
Mole fractions do not depend on temperature, density of phase, in case of ideal gas, mole fraction can be represented as the ratio of partial pressure to total pressure of the mixture.
Now we can find the number of moles of solvent.
We know that the Molecular mass of water is $18\,g/mol$.
The mass of water is $1000{\text{ }}\left( {1{\text{ }}kg} \right)$.
We know that the amount of moles in given amount of any substance is equal to the grams of
the substance divided by its molecular weight.
The mathematically expressed as,
${\text{Mole = }}\dfrac{{{\text{Weight}}\,{\text{of}}\,{\text{the}}\,{\text{substance}}}}{{\,{\text{Molecular}}\,{\text{weight}}}}$
${\text{Mole}} = \dfrac{{1000}}{{18}} = 55.5\,moles$
The number of moles of solvent is$55.5\,moles$.
Now, we calculate the mole fraction,
${\text{Mole}}\,{\text{fraction}}\,\left( {\text{x}} \right){\text{ = }}\dfrac{{{\text{Moles}}\,{\text{of}}\,{\text{solute}}}}{{{\text{Moles}}\,{\text{of}}\,{\text{Solute + }}\,{\text{Moles}}\,{\text{of}}\,{\text{solvent}}}}$
${\text{Mole}}\,{\text{fraction}}\,\left( x \right) = \dfrac{1}{{1 + \,55.55}} = 0.0176 \simeq 0.018$
Thus, Mole fraction of solute in $1\,m$ aqueous solution is$0.018$.
Hence,
$\therefore$ Option B is correct.

Note:
Don't confuse the terms of molality and molarity. Molarity is defined as the mass of solute in one liter of solution. Molarity is the preferred concentration unit for stoichiometry calculations.
The formula is,
${\text{Molarity = }}\dfrac{{{\text{Mass}}\,{\text{of}}\,{\text{solute(in}}\,{\text{moles)}}}}{{{\text{Volume}}\,{\text{of}}\,{\text{solution}}\,{\text{(in}}\,{\text{litres)}}}}$