Arithmetic

[spot]

Someone might like to check my assumptions. I'm sure everyone here has their way, I've had to work my way out on paper.

I was brought up on the British meaning of proof alcohol, effectively the ABV (alcohol by volume) times 1.75 - other than that there's no difference, they are proportionate. 40%abv = 70°proof, 100%abv = 175°proof. So what I'm thinking applies whether ABV or proof are being discussed, but I'll use the modern measure, ABV.

ABV is what a hygrometer or alcohol concentration meter measures. A hygrometer shows ABV at the present temperature. There are correction tables to bring it to a nominal 20°C. Which makes ABV plain odd to choose as a standard to show on a bottle of spirits. The ABV marked on a bottle in a supermarket is only true at 20°C. Why the standard isn't alcohol by weight I have no idea.

So, some physical details. A litre of water weighs 1000g at 4°C. The same volume, 1 litre, only contains 998.23g of water at 20°C.

40%abv at 20°C weighs 951g and contains 300g ethanol and 651g water.

Just to make that clearer, a litre bottle of 40%abv on a supermarket shelf containing 951g of fluid, if in practice it genuinely contains that at 20°C which I doubt, will have 300g of alcohol in it and 651g of water and flavouring. Withdrawn erroneous text, see next post: If you chill it down to 4°C in a fridge then the bottle will actually measure on a hygrometer, and actually contain, 33.5%abv. It's exactly the same weight of alcohol in the litre bottle - 300g - but that alcohol now occupies less volume compared to the water. That's what the hygrometer tables say anyway.

This makes mixing what comes out of a still down to "40%abv" a bit tricky, especially since nobody's going to do it with both the still product and the water at 20°C.

I reckon the easy way is to do it on the kitchen digital scales.

You can take the still product and measure the abv content at its present temperature and weigh it, and let's say it's 3085g at 91%abv at 24°C.

91%abv at 24°C, according to the alcohol meter tables, is 92%abv at 20°C and 0.814g of 92%abv is 1ml at 20°C so 92%abv is 90.3%abw at 20°C.

Therefore the product of the still contains 3085g * 0.903 = 2786g of alcohol.

So to get 300g into each litre, or 951g, of final product at 20°C I need to end up with 2786g * 951g / 300g = 8830g of 40%abv at 20°C (which is 9.29 litres).

That needs 8830g - 2786g = 6044g of water, and the still product currently contains 299g of water.

Which means I can pour 5745g of water into what I got out of the still and know that if it were adjusted to 20°C it would be 9.29 litres at 40%abv, and the first time I try this I'll damn well check it's right too by measuring it.

If anyone does it different they might, please, elaborate.

[Anavrin]

Hi Spot

If a supermarket bottle is at a different temp other than 20°c, it doesn't contain any more or less alcohol, it's just your alcometer reads wrong and needs to be corrected for temperature.

There is also something that happens when alcohol is diluted with water, as well as producing some heat, "it shrinks" 700ml of water and 300ml of alcohol will never add up to 1000ml because of this shirinkage, it's something that comercial distillers have to account for but for our small amounts, plus the fact that we won't get in trouble if it's not exactly 40% I don't concern my self with it.

All I do is divide my quantity alcohol by the ABV I want, then multiply by the AVB it is, the difference between the starting quantity and the calculated quantity is how much water to add.

For example if I have 1.5 litres at 94% and I want to make it 40%

1.5/40 = 0.0375
0.0375*94 = 3.523
3.523-1.5 = 2.025 litres of water to be added

[spot]

duplicate post, mod please delete

[spot]

Yes, on reflection the uncorrected reading on the alcometer is wrong at a non-standard temperature. The ABV ratio will shift too, but not to the wrong value shown on the scale. But the ABV ratio does vary with temperature, whereas the ABW ratio doesn't.

I'll adjust my post to indicate the error, thank you.

[Dave]

Actually ABV is exactly half that of proof. 200% proof is exactly 100% ABV.

[spot]

Actually ABV is exactly half that of proof. 200% proof is exactly 100% ABV.

That, I agree, is the US definition.

[Easydrinker]

Struggling to see how this thread helps in the making.
Seems to be aimed at regulations?

Robert.

[spot]

Not at all, Robert. I'm just very puzzled about how I can bottle a 40%abv product after distillation. My guess was that it's easier by juggling the weights rather than the volumes but those who have experience may easily show me something simpler.

I'd not actually do the sums by hand, I'd write a computer script once I know what to do.

[Mash]

I'm just very puzzled about how I can bottle a 40%abv product after distillation.

Buy a refractometer.

[Dave]

For sake of argument, say you have 900ml of 60%abv product, and you want to dilute it down to 40%abv. You would add 300ml of water (or whatever) and end up with 1200ml of 40%abv....simples! That's a nice easily worked out quantity, but the principle carries through.

[spot]

For sake of argument, say you have 900ml of 60%abv product, and you want to dilute it down to 40%abv. You would add 300ml of water (or whatever) and end up with 1200ml of 40%abv....simples! That's a nice easily worked out quantity, but the principle carries through.

I'd have thought you'd add roughly half as much again, 450ml, and end up with 1350ml of 40%abv, if the 900ml and the 450ml were both at 20°C, but as Anavrin mentions there's different volumes due to shrinkage so the final volume of that mixing isn't actually going to be exactly 1350ml. I think the result would be approximately 40%abv but I'm sure it's not exact and I don't know how much it would be out by, that's why I started the thread to see if anyone knows what sums are correct.

[Dave]

Argh!....I'm confused now! But I'm sure that's what I did.....you may be right...in fact I think you ARE right....yes, cos I had to add a little more.. ..I need a rest!

Edit:- You're absolutely right Spot, here's a useful calculator

http://homedistiller.org/distill/dilute/calc