Solve this mole fraction calculation with incomplete information

Question

I am currently trying to solve a problem that seems quite simple at the beginning but the complexity reveals itself soon thereafter. The problem is the following:

You mix a known mass of water with a known mass of ethanol. Then you fill up the flask to the 50 mL mark with water, but cannot weigh the contents when you're done. What is the molar fraction of ethanol in this mixture?

My attempt at solving this problem is to basically calculate the initial molar fraction, and then stepwise add small virtual amounts of water. Then the new molar fraction is calculated, and somehow the excess volume (which is negative for the water-ethanol mixture) would also be included somewhere here. This is repeated until the final volume is reached.

An other attempt would be to simply neglect the excess volume. While this might be fine at the edges of the molar fractions, it would introduce a grave error towards the middle. Or is the error still negligible?

Are these reasonable attempts or am I completely missing something here?

Answer 1

I don't think you can solve that unless you know additional information like mass of water added or volume of water added, or the density of the final solution.

You could approximate by dividing mass of ethanol by density of ethanol to get volume of ethanol, and saying volume of water added is 50mL-(volume ethanol), but this is only an approximation because volumes are not additive when the two solutions are different.

The error could be estimated from this graph:

source: http://en.wikipedia.org/wiki/Partial_molar_property

or the data from the graph could be used to improve the accuracy of the answer

or use this density table: http://www.handymath.com/cgi-bin/ethanolwater3.cgi?submit=Entry