Question

Mole fraction of the solute in a $1.00$ molal aqueous solution is:
A. $1.7700$
B. $0.1770$
C. $0.0177$
D. $0.0344$

Answer 1

We can calculate the mole fraction of the solute using the moles of the solute and moles of water. First, we have to calculate the moles of solute and then we have to calculate the moles of water using the molar mass of the water. From the moles of solute and moles of water, we have to calculate the total number of moles. From this total moles and moles of solute, we can calculate the mole fraction of solute.

Formula used: We can calculate the mole fraction using the formula,
${\chi _A} = \dfrac{{{\text{Moles of A}}\left( {{\text{in mol}}} \right)}}{{{\text{Moles of A}}\left( {{\text{in mol}}} \right) + {\text{Moles of B}}\left( {{\text{in mol}}} \right) + {\text{Moles of C}}\left( {{\text{in mol}}} \right)}}$
${\chi _A} = \dfrac{{{\text{Moles of A}}\left( {{\text{in mol}}} \right)}}{{{\text{Total moles}}}}$

Complete step by step answer:
We can characterize Mole fraction as the proportion of moles of one substance in a combination to the absolute number of moles of all substances. For a solution containing two substances A and B, we can figure the mole fraction utilizing the equation,
${\chi _A} = \dfrac{{{\text{Moles of A}}\left( {{\text{in mol}}} \right)}}{{{\text{Moles of A}}\left( {{\text{in mol}}} \right) + {\text{Moles of B}}\left( {{\text{in mol}}} \right) + {\text{Moles of C}}\left( {{\text{in mol}}} \right)}}$
${\chi _A} = \dfrac{{{\text{Moles of A}}\left( {{\text{in mol}}} \right)}}{{{\text{Total moles}}}}$
Now, coming back to the question, we have to determine the mole fraction of solute. We can calculate the mole fraction of solute using the moles of solute and total number of moles.
Given data contains,
One mole of solute dissolved in one kilogram of water is one molal aqueous solution.
So, the number of moles in one molal aqueous solution is one mole.
Let us now calculate the number of moles of water. The molar mass of water is $18g/mol$.
One kg of water is equal to thousand grams of water.
The number of moles of water is calculated as,
Number of moles of water = $\dfrac{{1000}}{{18g/mol}}$
Number of moles of water = $55.55moles$
The number of moles of water is $55.55moles$.
The total number of moles of solute is the sum of moles of solute and moles of water.
Total number of moles of solute = ${\text{Moles of solute}} + {\text{moles of water}}$
Total number of moles of solute = $1\,mol + 55.55mol$
Total number of moles of solute = $56.55mol$
The total number of moles is $56.55mol$.
We can calculate the mole fraction of solute as,
${\chi _{solute}} = \dfrac{{{\text{Moles of solute}}}}{{{\text{Total moles}}}}$
Let us now substitute the moles of solute and total moles in the above expression.
${\chi _{solute}} = \dfrac{{{\text{Moles of solute}}}}{{{\text{Total moles}}}}$
Now we can substitute the known values we get,
$\Rightarrow {\chi _{solute}} = \dfrac{{1mol}}{{56.55mol}}$
On substitution we get,
${\chi _{solute}} = 0.0177$
The mole fraction of the solute is $0.0177$.

Hence, the correct answer is C.

Note: We can ascertain mole fraction from mass percent and molarity. A portion of the properties of mole fraction are,

1. It doesn't rely upon temperature.
2. A combination of mole parts could be set up by weighting the amounts of the substances.
   We can likewise figure the mole fraction from Raoult's law. The equation we use to figure the mole fraction from Raoult's law is,
   ${P_{solv}} = {\chi _{solv}}{P^o}_{solv}$
   The vapor pressure of the solvent $\left( {{P_{solv}}} \right)$ above a dilute solution is equivalent to the mole fraction of the solvent $({\chi _{solv}})$ times the vapor pressure of the pure solvent $\left( {{P_{solv}}^o} \right)$.