Question

Calculate the mole fraction of ethylene glycol $\left( {{{\rm{C}}_{\rm{2}}}{{\rm{H}}_{\rm{6}}}{{\rm{O}}_{\rm{2}}}} \right)$ in a solution containing 20% of ${{\rm{C}}_{\rm{2}}}{{\rm{H}}_{\rm{6}}}{{\rm{O}}_{\rm{2}}}$ by mass.

Answer 1

Hint: We know that mole fraction is the one way to express concentration of a solution. Here, first we have to calculate the moles of solvent and solute and then we have to use the formula of mole fraction of solute.

Complete step by step answer:
Let's first discuss mole fraction in detail. Mole fraction is the ratio of moles of one component to the total number of moles of all the components present in the solution. Let's understand mole fraction with the help of a formula.

We consider a binary solution comprising two components namely A and B. So, the formula of mole fraction of A is,

${X_A} = \dfrac{{{n_A}}}{{{n_A} + {n_B}}}$

Where, ${X_A}$ is a mole fraction of A, ${n_A}$ is moles of A and ${n_B}$ is moles of B.

Similarly, mole fraction of B is,

${X_B} = \dfrac{{{n_B}}}{{{n_A} + {n_B}}}$

Now, come to the question. The mass percentage of ethylene glycol $\left( {{{\rm{C}}_{\rm{2}}}{{\rm{H}}_{\rm{6}}}{{\rm{O}}_{\rm{2}}}} \right)$ in a solution is 20% that means mass of solute is (ethylene glycol) 20 g and the mass of solvent (water) is 80 g.

Now, we have to find the moles of solute and solvent.

The formula to calculate moles is,

Number of moles$ = \dfrac{{{\rm{Mass}}}}{{{\rm{Molar}}\,{\rm{mass}}}}$ …… (1)

For ethylene glycol,

Mass=20 g
Molar mass of ethylene glycol$\left( {{{\rm{C}}_{\rm{2}}}{{\rm{H}}_{\rm{6}}}{{\rm{O}}_{\rm{2}}}} \right) = 2 \times 12 + 6 \times 1 + 2 \times 16 = 24 + 6 + 32 = 62\,{\rm{g}}\,{\rm{mo}}{{\rm{l}}^{ - 1}}$

Now, we have to use the equation (1).

${\rm{Number\ of\ moles\ of\ ethylene\ glycol}} = \dfrac{{20}}{{62}} = 0.322$

Similarly, we have to calculate the moles of water.

Mass of water=80 g
Molar mass of water$\left( {{{\rm{H}}_{\rm{2}}}{\rm{O}}} \right) = 2 \times 1 + 16 = 18\,{\rm{g}}\,{\rm{mo}}{{\rm{l}}^{ - 1}}$

${\rm{Number\ of\ moles\ of\ water}} = \dfrac{{80}}{{18}} = 4.44\,{\rm{g}}\,{\rm{mo}}{{\rm{l}}^{ - 1}}$

So, now we have to calculate the mole fraction of ethylene glycol.

${X_{{\rm{ethylene}}\,{\rm{glycol}}}} = \dfrac{{{n_{{\rm{ethylene}}\,\,{\rm{glycol}}}}}}{{{n_{{\rm{ethylene}}\,\,{\rm{glycol}}}} + {n_{{\rm{water}}}}}}$

$ \Rightarrow {X_{{\rm{ethylene}}\,\,{\rm{glycol}}}} = \dfrac{{0.322}}{{0.322 + 4.44}} = 0.068$

Hence, mole fraction of solute ethylene glycol is 0.068.

Note: Always remember mole fraction is a unit less quantity. Also the addition of mole fraction of different components in a solution is equal to 1. For example, a binary solution comprising two components A and B, then ${X_A} + {X_B} = 1$. So, if one of the component’s mole fraction is known then we can easily calculate the mole fraction of another component.