# Is the negative excess volume of ethanol/water the cause of my math error?

I have a binary mixture, say ethanol at $$90\%wt$$ ethanol, $$10\%wt$$ water. According to Perry's Chemical Engineers' Handbook 9th Ed., that mixture has a density of $$0.81797g/cc$$ (at $$20^{\circ}C$$). So my $$1L$$ of mixture should have a mass of $$817.97g$$.

Now, I want to create this mixture. I know the total mass and the mass %'s, so I just multiply them out: we find that we need $$736.17g$$ ethanol and $$81.80g$$ water. Ok, so far so good. However, say all of my balances are broken, so I only have graduated cylinders to work with. Easy, just convert those masses back to volumes using their respective pure substance densities: $$0.78934g/cc$$ for ethanol and $$0.99823g/cc$$ for water.

However, doing this tells me that I should have $$932.6mL$$ of ethanol and $$81.9mL$$ of water, which adds up to a total volume of $$1014.6mL$$ of binary mixture, so somehow I magicked $$14.6mL$$ of liquid out of thin air using this conversion algorithm.

Is this due to the negative excess volume of the ethanol/water mixture upon mixing? That's my current guess, but for the life of me I'm having a hard time figuring out the math here. It seems like the change in volume should be much smaller: $$10\%wt$$ water in ethanol is about $$10\%mol$$ and there's about $$4.5mol$$ of water in the mixture, so based on this dataset it seems like there should only be a difference of about $$3mL$$. Even with all the rounding I just did, it seems like this can't be the only error source.

Is there something wrong with the logic of my algorithm here? Can anyone point me to the reason I'm seeing this numerical error whenever I run this calculation?

• Hmmmm. I see 0.81942 in a different reference for 90 wt.% ethanol. Jan 7, 2019 at 23:23
• @JonCuster Ahh, is it hard copy or could you link it? Jan 7, 2019 at 23:25
• wissen.science-and-fun.de/chemistry/chemistry/density-tables/… for online, and 0.8194 in my old CRC... Jan 7, 2019 at 23:27
• @JonCuster Interesting! I usually trust both CRC and Perry's quite a lot, I wonder why the disparity. I'm at the gym, I'll look into it when I get home, or maybe when I get back to work tomorrow Jan 7, 2019 at 23:33
• My CRC has 0.8180 as the 'relative density at 20C, kg/l' while 0.8194 is the 'specific gravity at 20C'. Jan 7, 2019 at 23:43