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While proofing down some potato vodka, there was a big difference in what I calculated versus what the hydrometer was reading. It should have ended up at 80 proof, but the hydrometer was reading 92 proof. Adjusting for temperature (85°) according to the chart that came with the hydrometer, I subtracted .4 to end up with roughly 91.6 proof. Baffled, I checked the HoochWare app to see what it said the true proof at the temperature is, which said 80 proof. It gave me some confidence that I didn't do my math wrong, but left me confused about temperature adjustment. TLDR: what is the best way to adjust for true proof when the temperature isn't reading 60°?

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The "official" correction is TTB Table 1.  This indicates that an apparent proof of 92 at 85F is equal to a true proof of 81.1.

The correction calculator in AlcoDens indicates a true proof of 81.13

My opinion is that the difference between 85F and 60F is rather large to make this correction accurately and it would be better to allow the batch to cool before you make your final gauging.  Although the expansion of alcohol-water mixtures is very well known and can be calculated very accurately, the unknown factor is the exact expansion factor of the glass used for the hydrometer.  AlcoDens and the Table 4 values are based on typical high quality hydrometers, but this is the one factor that is not known for certain when you are doing the correction.  Fortunately the expansion of the glass has a much smaller effect on the reading than the expansion of the spirit, but the smaller the temperature difference that you are correcting for, the better your accuracy will be.

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Also if the temperature of your spirit being measured is a long way from the air temperature it can be hard to get a steady reading.

eg If spirit is at 85f and air at 50f the spirit is cooling, density is increasing, hydrometer is glass is shrinking, hydrometer is rising, glass thermometer is shrinking and spirit inside the thermometer is also shrinking. Some components are shrinking faster than others so it really is impossible to get a true reading until spirit temperature is same as air temperature.

Also you referred to Hoochware app. I have never used it but because of the name I suspect it may not be as accurate as TTB requires. Do some comparisons with TTB Table 1, downloadable from TTB website  https://www.ttb.gov/foia/Gauging_Manual_Tables/Table_1.pdf

Also is your hydrometer a "Hooch" type as well?  Over what range does the one you used measure?

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Actually hoochware uses the ttb tables to do the conversion.  It can be downloaded as an app for the conversion part or used within the tracking software.

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