# A bottle of wine contains 12.5% ethanol (CH_3CH_2OH) by volume. The density of ethanol is 0.789 g/cm^3 and the density water is 1.000 g/cm^3. How do you calculate the concentration of ethanol in terms of mass percent of ethanol?

## You may assume wine i made entirely of water and ethanol.

Nov 18, 2016

The mass percent of ethanol is 10.1 %.

#### Explanation:

Percent of alcohol by volume (ABV) is the volume of ethanol in ${\text{100 cm}}^{3}$ of solution.

It is impossible to answer this question because we don’t know the volume or the density of the wine.

The problem arises because the volume obtained by mixing water and ethanol is less than the sum of the original volumes.

For example, the volume of ${\text{12.5 cm}}^{3}$ of ethanol plus ${\text{87.5 cm}}^{3}$ of water is only ${\text{97.8 cm}}^{3}$.

The density of 12.5 % ABV at 20 °C is ${\text{0.9784 g/cm}}^{3}$.

Now, we can calculate the mass percent of the ethanol.

$\text{Mass of ethanol" = 12.5 color(red)(cancel(color(black)("cm"^3color(white)(l) "ethanol"))) × "0.789 g ethanol"/(1 color(red)(cancel(color(black)("cm"^3color(white)(l) "ethanol")))) = "9.862 g ethanol}$

$\text{Mass of wine" = 100 color(red)(cancel(color(black)("cm"^3 color(white)(l)"wine"))) × "0.9784 g solution"/(1 color(red)(cancel(color(black)("cm"^3color(white)(l) "wine")))) = "97.84 g solution}$

"% by mass" = "mass of ethanol"/"mass of wine" × 100 % = (9.862 color(red)(cancel(color(black)("g"))))/(97.84 color(red)(cancel(color(black)("g")))) × 100 % = 10.1 %