Calculating ABV when back sweetening

I am fermenting 2 gallons of apple juice + sugar @ .058 gravity

I will let this go until finish so, conservatively we can say it will go down to .005

Now if this was the finished product I could go us an ABV calc and be done but...

I am going to back sweeten with more juice before I put into bottles and yes, I know they can grenade if not pasteurized when the right carbonation has been reached.

So, if I put the two gallons of finished cider into the bottling bucket and add .5 to 1 gallon of un-fermented juice @.050 gravity, how do I figure out the ABV?

2 gallons at 1.005
+ 1 gallon at 1.050
Will be approximately 3 gallons at 1.018

The math:
(5 + 50) / 3 = 18.333

Check it with your hydrometer after mixing, I believe it will be close. Of course after bottle conditioning for maybe a week, that will probably drop to the 1.012 - 1.015 range

This will give you about a 6% ABV

That doesn't seem quite right to me...the math adds up...but...

If you go with the ABV of the orriginal 2 gallons, would be around 7% (6.96% actually) Basic formula (without an online calculator) is OG - FG = W : W x 131.25 = approx ABV

If you mix it with another gallon of straight juice, you've watered it down 50% (in spite of adding extra sugar), or in other words, reduced the ABV by 1/3. If you take 6.96 and subtract 33.33333% you get a rough ABV of 4.6%.

We don't usually calculate any extra ABV added during carbonation because it's fairly little...you might be able to guess it around 4.8%

I'm going to pass this question on to some Math teachers at the High School I work at, and see what they come up with...this intreagues me.

I was thinking of using total final volume's gravity (2@ 58 1@50) then use final gravity reading in the bottling bucket. But I can't think how to correctly add gravity. My guess would be and divide by three.
I guess this is similar to Mike's way.

If you had an OG of 1.058, and fermented down to 1.005, then added a gallon at 1.050, your math would only be correct if you then again fermented whatever you had back down to 1.005. but you're not fermenting it back down to 1.005, you're backsweetening and then bottling right away, and you'd have to take the extra gallon into consideration...the gravity has nothing to do with it, it's the ABV you have in the 2 gallons is the only alcohol you have, no matter how much sugar you then add to it (whatever the gravity is) you're not adding any more alcohol, just sugary liquid, which means you're backsweetening but also watering down.

If you want to know you're final ABV, the best way would be to calculate as you have... ([email protected] + 1@ 1.050) for your OG, but then take an FG out of a bottle you open after you've carbed and pasturized (be sure you shake the sample flat so you don't have carbonation latching to the bobber and altering the reading). Otherwise if you're guestamating you should take the 2 gal abv and subtract 1/3 of that to get what you'll have just by adding the juice as I mentioned in the above post.

If you want to figure out what the addition of 1 gallon at 1.050 is to the OG, just divide 50/3 and add that to the OG. It would be an addition of .017, making your OG 1.074. You'll find that, but if you then just again add the .016 to the FG, you get the same ABV calculation, 6.9%, and you can't water down 6.9% by 1 gallon and still have 6.9%....it won't work because you never fermented that extra gallon.

Thanks for your comments they helped me come up with a theory:

1) ABV needs two measurements (numbers) OG and FG.

2) If we add in the unfermented juice to the OG calc we should have .0553 as a OG: (.058+.058+.050)/3=.0553

3) I can then get a reading with my hydrometer in the bottling bucket with the added juice. It would be just like when you make beer and 25% of the sugar is unfermentable. Those sugars still affect your gravity. Here the added juice will still affect the gravity in the same way. Mike's idea for getting the FG should be close depending on what the final amount of conversion is. I am new to cider and haven't got a feel for what yeast will do with apple juice yet so I was conservative with my estimate. So, if we do say it will be at .005 FG then adding the juice should get us to .02 FG. (.05+.005+.005)/3=.02

4) Plugging those numbers into an ABV calc gives me an estimated ABV of 5.1% . This is a good number since that is right around the ABV I want an right around the sweetness I want.

I like it...never heard back from my math teachers but what you did looks like it should be pretty close as an estamite.

Go to the chemistry teachers first. Calculating strength of mixed solutions should be second nature in their classrooms.

jkoegel said:

Go to the chemistry teachers first. Calculating strength of mixed solutions should be second nature in their classrooms.

Click to expand...

yeah I thought about that. But I'm thinking my math approach should be pretty close.

That being said if anyone does talk to a chemist I sure would like to know what they say.

Urchen's got the math right. Calculate your abv based on the OG and FG of the 2 gallons that you fermented. Then take that number (approx. 7%) times 2 (two initial gallons) and divide by 3 (3 gallons in final product). For the gravity of the overall final product, just toss your hydrometer back in to the final mixture.

I'm fairly new to the homebrew world, but diluting alcohol like that is a linear function and I'm confident in that math. What I don't know is how blending back juices affects the SG. I don't know if the removal of the sugar affects the SG at a linear rate. If it does, you could take the same approach from an SG perspective and calculate like this (FG*2 + OG*1)/3 = new SG. (that's assuming you blending back in a gallon of the same stuff you have prior to fermentation). If that number holds true (and you can test it by tossing your hydrometer in final product to see if they match), then you could also use the differences between the OG and the (FFG - final final gravity) to determine abv. I stress again though, that this would only hold true if the gravity to sugar relationship is linear and so far, I haven't found it. Everyone just gives you a link to the homebrew app and tells you to use it.

hbrookie said:

Urchen's got the math right. Calculate your abv based on the OG and FG of the 2 gallons that you fermented. Then take that number (approx. 7%) times 2 (two initial gallons) and divide by 3 (3 gallons in final product). For the gravity of the overall final product, just toss your hydrometer back in to the final mixture.

I'm fairly new to the homebrew world, but diluting alcohol like that is a linear function and I'm confident in that math. What I don't know is how blending back juices affects the SG. I don't know if the removal of the sugar affects the SG at a linear rate. If it does, you could take the same approach from an SG perspective and calculate like this (FG*2 + OG*1)/3 = new SG. (that's assuming you blending back in a gallon of the same stuff you have prior to fermentation). If that number holds true (and you can test it by tossing your hydrometer in final product to see if they match), then you could also use the differences between the OG and the (FFG - final final gravity) to determine abv. I stress again though, that this would only hold true if the gravity to sugar relationship is linear and so far, I haven't found it. Everyone just gives you a link to the homebrew app and tells you to use it.

Click to expand...

See http://en.wikipedia.org/wiki/Density for a verification of your math. Since specific gravity is a ratio of a solution's density to a fixed reference density, it acts the same as density when blending solutions:
SG_new = ∑ₐ SGₐ*Vₐ/V
Where SGₐ is the specific gravity of volume Vₐ, V is the total mixed volume, and a ranges over the number of distinct solutions.

To recreate the above formula from the wiki's formula ρ = ∑ₐ ρₐ*Vₐ/V, divide by the density of the fixed reference solution (in our case, water) to change the ρ's to SG's.