Question

The mole fraction of a solvent in aqueous solution of a solute is 0.8. the molality(in $mol\text{ }k{{g}^{-1}}$) of the aqueous solution is:
(a) $13.88\times {{10}^{-1}}$
(b) $13.88\times {{10}^{-2}}$
(c) $13.88$
(d) $13.88\times {{10}^{-3}}$

Answer 1

. In this first we have to find the no of moles of the solute by using the given mole fraction and then we will find the mass of the solvent (water) by using the formula as Moles of ${{H}_{2}}O$ =$\dfrac{given\text{ }mass}{molecular\text{ }mass}$ and then, we can easily find the molality of the solution by using the formula as; Molality =$\dfrac{\text{no of moles of the solute}}{\text{ total mass of the solvent in kg}}$, Now solve it.

Complete step by step answer:
First of all, let’s discuss mole fraction and molality. By the term mole fraction we mean the ratio of the number of a particular component to the total number of moles of the solution (i.e. solute + solvent). If the substance A is dissolved in solvent B and ${{n}_{A}}$ and ${{n}_{B}}$ are the mole fractions of the solute A and solvent B , then;
Mole fraction of solute A=$\dfrac{{{n}_{A}}}{{{n}_{A}}+{{n}_{B}}}$
Mole fraction of solvent B= $\dfrac{{{n}_{B}}}{{{n}_{A}}+{{n}_{B}}}$

And by the term molality we mean the no of moles of the solutes to the total mass of the solvent in kilograms. i.e.
Molality =$\dfrac{\text{no of moles of the solute}}{\text{ total mass of the solvent in kg}}$ -----------(A)
Now, considering the numerical;
The mole fraction of the solvent=0.8 (given)
Mole fraction of solute = 1- 0.8
=0.2

Now, calculating the molality of the solution using the equation (A) as;
Molality =$\dfrac{\text{no of moles of the solute}}{\text{ total mass of the solvent in kg}}$
We know, the no of moles solute=0.2
Since, the solution is aqueous ,so the solvent used is water.
Mass of the solvent (water)can be calculated as:
As we know that,
Moles of ${{H}_{2}}O$ =$\dfrac{given\text{ }mass}{molecular\text{ }mass}$
$moles\text{ }of\text{ }{{H}_{2}}O\times molecular\text{ }mass=given\text{ }mass$ ---------(1)
Moles of water=0.8 (given)

Mass of water=18 g
= $\dfrac{18}{1000}$ kg ( 1kg=1000g)
Put these values in equation (1), we can calculate the mass of solvent i.e. water
$0.8\times \dfrac{18}{1000}=given\text{ }mass$

Now, put all these values in equation(A) , we get;
Molality =$\dfrac{0.2\times 1000}{0.8\times 18}$
=$\dfrac{200}{14.4}$
= 13.88

So, the correct answer is “Option C”.

Note: The sum of the mole fraction of the solute and solvent is always equal to one and can never be greater than one but can be less than one and molality of the solution is independent of the temperature and depends only on the mass of the solvent.