Question

What are the mole fractions of hydrochloric acid (HCl) and water in a 20% w/w aqueous HCl solution?

Answer 1

As we know that mole fraction can be described as the number of molecules or moles of a particular component in a mixture divided by the total number of moles present in the given mixture. It is also a way of expressing the concentration of a solution.
Formula used:
We will require the following formula to calculate the mole fraction:-
${{x}_{A}}=\dfrac{\text{number of moles of A}}{\text{total moles present in the mixture}}$

Complete answer:
Let us begin with the definition of mole fraction as follows:-
-Mole fraction: It can be defined as the number of molecules or moles of a particular component in a mixture divided by the total number of moles present in the given mixture. It is denoted by ‘x’.
It can be calculated by the following formula:-
${{x}_{A}}=\dfrac{\text{number of moles of A}}{\text{total moles present in the mixture}}$
-As we have been given that 20% w/w aqueous HCl solution which means that out of 100 grams of solution, the mass of HCl present is 20 grams. So the mass of water present = (100 - 20) grams = 80 grams.
-Calculation of moles of HCl and water:-
Molar mass of H = 1g/mol
Molar mass of Cl = 35.5g/mol
Molar mass of O = 16g/mol
So the molar mass of HCl = (1+35.5)g/mol = 36.5g/mol
And molar mass of ${{H}_{2}}O$ = ( 2(1) + 16 )g/mol = 18g/mol
As we know that moles (n) = $\dfrac{\text{given mass}}{\text{molar mass}}$
Moles of HCl (${{n}_{HCl}}$) = $\dfrac{\text{20g}}{\text{36.5g/mol}}$= 0.547moles
Moles of ${{H}_{2}}O$(${{n}_{{{H}_{2}}O}}$) = $\dfrac{\text{80g}}{\text{18g/mol}}$= 4.444moles
-Calculation of mole fractions of hydrochloric acid (HCl) and water in a 20% w/w aqueous HCl solution:-
Mole fraction of HCl:-
$\begin{aligned} & \Rightarrow {{x}_{HCl}}=\dfrac{\text{number of moles of HCl}}{\text{total moles present in the mixture}} \\\ & \Rightarrow {{x}_{HCl}}=\dfrac{0.547moles}{0.547moles+4.444moles} \\\ & \Rightarrow {{x}_{HCl}}=\dfrac{0.547moles}{4.991moles} \\\ & \Rightarrow {{x}_{HCl}}=0.109\simeq 0.1(approximately) \\\ \end{aligned}$
Mole fraction of${{H}_{2}}O$:-
$\begin{aligned} & \Rightarrow {{x}_{{{H}_{2}}O}}=\dfrac{\text{number of moles of }{{H}_{2}}O}{\text{total moles present in the mixture}} \\\ & \Rightarrow {{x}_{{{H}_{2}}O}}=\dfrac{4.444moles}{0.547moles+4.444moles} \\\ & \Rightarrow {{x}_{{{H}_{2}}O}}=\dfrac{4.444moles}{4.991moles} \\\ & \Rightarrow {{x}_{{{H}_{2}}O}}=0.890\simeq 0.9(approximately) \\\ \end{aligned}$
Hence, the mole fraction of HCl is 0.1 and mole fraction of water is 0.9 in a 20% w/w aqueous HCl solution.

Note:
-Remember that the sum of mole fractions of all the components of mixture or a solution is always equal to 1. So the moles fraction of water can also be directly calculated as follows:-
Mole fraction of HCl + Mole fraction of water = 1
Mole fraction of water = 1 – 0.1 = 0.9