Question

Which of the following statements are correct?
1.Mole fraction of a solute $ + $ mole fraction of a solvent $ = 1$
2.If equal weights of helium and methane are present in a gaseous mixture, then the mole fraction of helium is $\dfrac{4}{5}$ .
3.The mole fraction of water in the aqueous solution of $NaOH$ is $0.8$ . The molality of solution is nearly $14moles/kg$ .
A. $1,2$
B. $2,3$
C. $1,3$
D. All are correct.

Answer 1

Hint: mole fraction is the number of molecules of each component present in a mixture divided by total number of moles. Molality is the number of solute in $1kg$ of solvent. It is independent of the temperature because volume is not involved in it.

Complete step by step answer:
1.Mole fraction of a solute $ + $ mole fraction of a solvent $ = 1$
Explanation: mole fraction is defined as the number of molecules of a particular component present in a mixture divided by the total number of moles present in a mixture.
For component $A$ ,
It is given by the formula: ${x_a} = \dfrac{{{n_a}}}{{{n_a} + {n_b}}}$
Where, $x = $ mole fraction of solvent
${n_a} = $ moles of solvent
${n_b} = $ moles of solute.
Similarly for component $B$ ,
It is given by the formula: ${x_b} = \dfrac{{{n_b}}}{{{n_a} + {n_b}}}$
Where, $x = $ mole fraction of solute
${n_a} = $ moles of solvent
${n_b} = $ moles of solute.
Therefore, ${x_a} + {x_b} = 1$
Thus, the sum of all the mole fraction of components is always equal to one.
Hence, the given statement is true.

2.If equal weights of helium and methane are present in a gaseous mixture, then the mole fraction of helium is $\dfrac{4}{5}$ .
Molecular weight of helium is $4$ .
Molecular weight of methane $\left( {C{H_4}} \right) = $ atomic mass of carbon $ + 4 \times $atomic mass of hydrogen
Molecular weight of methane $\left( {C{H_4}} \right) = 12 + 4 \times 1$
Molecular weight of methane $\left( {C{H_4}} \right) = 16$
We will use mole fraction formula to calculate the mole fraction of helium
${x_a} = \dfrac{{{n_a}}}{{{n_a} + {n_b}}}$
But, $n = \dfrac{W}{{MW}}$
$n = $ number of moles, $W = $ Weight , $MW = $ molecular weight.
Let $X$ be the weight of helium and methane.
Substituting the value of $n$ in mole fraction formula we get,
${x_a} = \dfrac{{\dfrac{{{W_1}}}{{M{W_1}}}}}{{\dfrac{{{W_1}}}{{M{W_1}}} + \dfrac{{{W_2}}}{{M{W_2}}}}}$
Substituting the values we get,
${x_a} = \dfrac{{\dfrac{X}{4}}}{{\dfrac{X}{4} + \dfrac{X}{{16}}}}$
${x_a} = \dfrac{{\dfrac{X}{4}}}{{\dfrac{{4X + X}}{{16}}}}$
${x_a} = \dfrac{{\dfrac{X}{4}}}{{\dfrac{{5X}}{{16}}}}$
${x_a} = \dfrac{{X \times 16}}{{5X \times 4}}$
${x_a} = \dfrac{4}{5}$
Thus the mole fraction of helium is $\dfrac{4}{5}$ .
Therefore, the above statement is true.

3.The mole fraction of water in the aqueous solution of $NaOH$ is $0.8$ . The molality of solution is nearly $14moles/kg$ .
Molality is defined as the number of moles of solute present in $1kg$of the solvent.
It is given by the formula as follows:
$m = \dfrac{{{W_a}}}{{M{W_a} \times {W_b}}} \times 1000$
Where, $m = $ molality
${W_a} = $ grams of solute
$M{W_a} = $ molecular mass
${W_b} = $ grams of solvent.
The relationship between molality and mole fraction is given by the formula as follows:
$m = \dfrac{{{X_{}} \times 1000}}{{\left( {1 - {X_{}}} \right)MW}}$
Where, $m = $ molality, $X = $ mole fraction, $MW = $ molecular weight
Using this relationship we will find the molality.
Given: mole fraction of water $ = 0.8$
Molecular weight of water $ = 18$
To find: molality $ = ?$
Formula to be used: $m = \dfrac{{{X_{}} \times 1000}}{{\left( {1 - {X_{}}} \right)MW}}$
$ \Rightarrow m = \dfrac{{{X_{}} \times 1000}}{{\left( {1 - {X_{}}} \right)MW}}$
Substituting the values we get,
$ \Rightarrow m = \dfrac{{0.8 \times 1000}}{{\left( {1 - 0.8} \right) \times 18}}$
$ \Rightarrow m = \dfrac{{800}}{{0.2 \times 18}}$
$ \Rightarrow m = \dfrac{{800}}{{3.6}}$
$ \Rightarrow m = \dfrac{{800}}{{3.6}}$
$ \Rightarrow m = 222.2$
The above statement given is incorrect.

So, the correct answer is Option A.

Note: mole fraction is the way of expressing the concentration of solution. Molality does not change with temperature. Only the mass changes with the temperature. Mole fraction is used when there are two or more components present in the solution.