For those in search of a few minutes of algebra story problems, a holiday question:

I have just bought a half gallon (64 ounces) of non-alcoholic eggnog. I also happened to have some Virgin Island rum that is 40% alcohol. How much rum do I have to add to the eggnog to make it 16.666. . .% alcohol? Show your work.

. . . and yes, I just finished solving the problem myself.

__________________
Keeper of the Jabberwock

OK, here's how I would solve it:

0.40x = y/6
y=x+64

where y is the final amount of liquid, and x is the amount of rum added.

Simplifying eqn 1 yields:
y=2.4x

Now we can solve it from the equation:
2.4x = x+64
2.4x - x = 64
1.4x = 64
x=64/1.4
x=45.71oz of rum.

When we plug it back in to check, we get that the amount of alcohol in the rum is:

45.71*0.40 = 18.285oz

And the total volume of the final liquid is:

45.71+64=109.71

So, the final percentage alcohol is:

(18.285/109.71)*100 = 16.66...%

__________________
WANTED:

Schroedinger's Cat

Dead And Alive

Very good. That's the answer that I got.

__________________
Keeper of the Jabberwock

I'm going to be a stinker and make this complicated.

One cup of alcohol and one cup of water do not add to 2 cups total volume of mixture. It's less because of the way the molecules interact -- IIRC, the water sort of gets in the gaps of the alcohol. Believe it or not, the structure of alcohol and water mixtures is a very complex thing, and still not completely understood. They never mix completely to form a 100% "perfect" homogenous mixture at the micro level. And that has to do with entropy in some complex fashion. There will be always be regions with differing alcohol content.

And offhand I'm not completely sure of the meaning of the statement that a mixture is a certain percentage of alcohol by volume. Does that mean the ratio of the volume of the pure alcohol if it were separate to the volume of the mixture, or does that mean the volume of alcohol at the average partial density of the mixture to the total?

I think it's the former, but I'm not sure.

But this greatly complicates the notion of the volume ratio of a mixture. What you really should mean is simply the mass ratios.

-Richard

Here's a table of density vs. alcohol concentration (by mass) in
g/cc at STP, 20C and 1atm. 0% is pure water and 100% is pure ethanol. Now you have enough information to really solve the problem, allowing linear interpolation between these points.


0% 0.9982
10% 0.9819
20% 0.9687
30% 0.9539
40% 0.9352
46% 0.9227
50% 0.9139
60% 0.8911
70% 0.8676
80% 0.8436
90% 0.8180
100% 0.7893

Note that a 20% mixture is 97% as dense than pure water, but 100% alcohol is 93% the density of the 80% mixture. Or put another way, the 50% mixture is around 3% denser than the average of the two pure substances.

Add a little alcohol to pure water and the volume increases more than if you add a little water to pure alcohol.

This behavior is why a fellow and I got into a heated argument once where I insisted we measure something before we mixed it He was buying some liquid nitrogen fertilizer off of me, which I was doing as a favor. He had an applicator with a 300 gal tank and added about 15 - 25 gal of other chemicals to the tank, and wanted to say he purchased 300 - x gallons total. Uh-uh. Get your nitrogen, then add the other stuff.

He couldn't understand that, and got mad about it. He was a horse's hind quarters generally anyway. And this kind of stuff is why any place selling something commerically does it by weight -- you buy a load, you weigh it right there. But I don't have a set of big scales for that, of course.



-Richard

What you say is true. Unfortunately, every bottle of wine and spirits I have seen say x% by volume--a term I take to mean v/v rather than w/v.

But as your chart shows, the point at which the rate at which the density changes considerably doesn't happen until the 40% point, which means that I can probably ignore the small differences caused by the physical chemistry aspect of the mix (I truly hated that class in college). This gets really interesting when mixing a dry material with water vs mixing with oils (vegetable oils in the case of my job). If I have to mix something with water to a certain volume, I can initially start mixing by assuming the dry material has a density half of that of water. If I mix with oils, I have to assume it has the density of water.

__________________
Keeper of the Jabberwock

Yes, alcohol content is always expressed as a v/v ratio, rather than by weight, and as I mentioned I'm not certain exactly what it means. I think it is the volume of the alcohol, if it were removed to the total volume of the mixture.

Note the way that would work is the v/v fraction of water and alcohol would sum to over 100%. But I'm not exactly sure if that's the way the v/v ratio is define.

I think v/v is just traditional. Proof, at least in the US is defined as twice the v/v percentage. 100 proof = 50% v/v. Proof used to be some more complicated scale, but I forget what it was.

-Richard

Speaking of mixtures and density reminds me of a time when my little mind was blown, thinking the laws of physics weren't working. I was filling up a 500 gal spray tank with water. The tank has sight gauges on the side, little clear plastic hoses to allow you to see the level. Well, the level in the gauge was still well below the top but the tank started overflowing, running right out the top.

I stood there and looked at that cross eyed for a few seconds, not believing my eyes. Water was not seeking its own level. Then it hit me. I'd left a little bit of the above mentioned nitrogen solution in the tank, which weighs about 10.8 lbs per gallon. The pure water just came in on top, didn't mix, and pushed the denser solution up the sight gauge.

-Richard

Volume of alcohol if it was pure to volume of mixture.

They use that because it makes for a bigger number, so sells better.

__________________
And the "driving on the freeway on a scooter" analogy still holds true because the pilots are sitting in 7 to 30 ton aircraft o' doom and you are running around them in your very own Meatbody, Mark I. Beep, beep.
Trying to make sense of computers, The Error Log.

bold mine

Not sure if that was the reason, maybe because it was easier to measure. You measure the alcohol volume before and than the total volume after it is mixed and there you have it. No high precision scale needed. I also think it was used in chemistry first.

BTW: I remeber that on American Whisky bottles it was always 80 proof. Which is the same as 40 % by volume. Obviously it is a factor of 2. Does anyone know were this comes from?

__________________
Andre

"They did not know it was impossible, so they did it!"
Mark Twain

Andre,

You don't usually have pure alcohol to start with. You make "likker" by first fermenting a mixture, letting some little buggers make the stuff. The little buggers usually kill their fool selves by stewing in their own product.

Now, you increase the concentration by distilling, basically boiling off the alcohol. But the way it works is, unless you're very careful, you still get a good bit of water in the output, just with a higher concentration of ethanol.

At all times in that process you have a mixture of alcohol and water plus some other stuff. The notion of the volume of the pure alcohol if it were completely separated is something that can only be calculated, never measured directly.

The notion of "proof" is an old one. The word originally came from "proving" the strength of a batch of liquor to the sastification of the customer. In the old days, sailors, soon to be drunken ones, used to soak gunpowder with the solution. If the gunpowder would still burn, the liquor was "proven". If it failed to light, there was too much water.

That point turns out to be just a tad under 50% ethanol by *weight*, 49 point something, which works out to around 57% by volume.

That was defined as "100 degrees proof", and with zero being pure water, a density scale was developed. There pure alchohol would've been around 170 proof, IIRC.

In the US, proof was later defined to be simply twice the alcohol by volume percentage. For lower concentrations, the original proof was very close to that, and so they just defined it to be so. However, note that by weight the original 100 proof was close to 50%.

But 80 proof whiskey in the US, 40% by volume, is IIRC 70 proof by the original scale.

By volume does indeed result in large numbers both percentage and proof the way it is currently defined.

Inflating numbers which are a common measure of how strong or "good" some product is a long tradition. The trick is not to outright lie, just play some game with definitions so you can emblazoon some large number on your product.

-Richard

Thanks for the explanation of proof. I always have been curious since I saw that first time (when I visited the U.S in April I noticed it is not in use any more. I know it from times when I hung out a lot with U.S. soldiers here around my home town, which is 20 years or more ago)

for the alcohol: I know how the stuff is usually made. What I think is that the by volume scale was set up in the lab. In Chemistry labs you usually have disttilled water and distilled "pure" Ethanol. So in this way it would be easy determin the persentage of the mixture.

I know a good Bourbon is not made that way.

__________________
Andre

"They did not know it was impossible, so they did it!"
Mark Twain