Q144CP - Chemistry: Molecular Nature Of Matter And Change

Question

 Methanol $\mathbf{\left(}{\mathbf{\text{CH}}}_{\mathbf{\text{3}}}\mathbf{\text{OH}}\mathbf{\right)}$and ethanol $\mathbf{\left(}{\mathbf{\text{C}}}_{\mathbf{\text{2}}}{\mathbf{\text{H}}}_{\mathbf{\text{5}}}\mathbf{\text{OH}}\mathbf{\right)}$ are miscible because the major intermolecular force for each is hydrogen bonding. In some methanol-ethanol solutions, the mole fraction of methanol is higher, but the mass percent of ethanol is higher. What is the range of mole fraction of methanol for these solutions?

Step-by-Step Solution

Answer

The range of mole fraction of methanol is $50-59\%$.

Definitions:

The concentration of a chemical component is expressed using a concentration unit called a mole fraction. By dividing the total number of moles of all the components in a solution by the number of moles of one component in a solution, the mole fraction can be computed.

Number of moles:

To calculate the mole fraction of methanol, first, calculate the moles of ethanol and methanol. 

Assume you have $10\text{ g}$  of the total mass of ethanol and methanol. Therefore,

$\text{10 g of total mass}=\text{5 g ethanol}+\text{5 g methanol}$ 

 

The molar mass of methanol is $32\raisebox{1ex}{${\displaystyle \text{g}}$}\!\left/ \!\raisebox{-1ex}{$\text{mol}$}\right.$.

The molar mass of ethanol is $\text{46 }\raisebox{1ex}{${\displaystyle \text{g}}$}\!\left/ \!\raisebox{-1ex}{$\text{mol}$}\right.$.

 

Hence, you can calculate the moles by dividing mass by the molar mass.

$\begin{array}{rcl}{n}_{methanol}& =& \frac{\text{mass\hspace{0.17em}of\hspace{0.17em}methanol}}{\text{molar\hspace{0.17em}mass\hspace{0.17em}of\hspace{0.17em}methanol}}\\ & =& \frac{5\text{\hspace{0.17em}g}}{32{\displaystyle \raisebox{1ex}{$\text{g}$}\!\left/ \!\raisebox{-1ex}{$\text{mol}$}\right.}}\\ & =& 0.156\text{ mol}\end{array}$

$\begin{array}{rcl}{n}_{ethanol}& =& \frac{\text{mass\hspace{0.17em}of\hspace{0.17em}ethanol}}{\text{molar\hspace{0.17em}mass\hspace{0.17em}of\hspace{0.17em}ethanol}}\\ & =& \frac{5\text{\hspace{0.17em}g}}{46{\displaystyle \raisebox{1ex}{$\text{g}$}\!\left/ \!\raisebox{-1ex}{$\text{mol}$}\right.}}\\ & =& 0.108\text{\hspace{0.17em}mol}\end{array}$

 

Mole fraction of methanol:

The mole fraction of methanol can be calculated as follow.

$\begin{array}{rcl}{\chi }_{methanol}& =& \frac{n_{methanol}}{n_{methanol}+n_{ethanol}}\\ & =& \frac{0.156\text{ mol}}{0.156\text{ mol}+0.108\text{ mol}}\\ & =& 0.59\end{array}$

Hence, the mole fraction of methanol should be in range between $50-59\%$.