This question was put to me recently, and I realised that as many people have never had to weigh anything accurately before, a little guidance might be helpful.
1) Buy a laboratory balance
You can pay a fortune for these, but can also make do with an inexpensive machine. The model I use is the PCE-BS 3000, which can weigh up to 3000g to a resolution of 0.1g. This model is mostly intended for school use, and can be bought for around £107 inc. VAT.
Most magnums weigh less than 3000g. You might be able to find a cheaper balance with a 2000g weight limit if you only wanted to weigh bottles, moreover as 99% of bottles weigh less than 1500g, that limit would not greatly cramp your style.
2) Find a good place to put it
A balance needs to be on a solid surface, free of vibration. It does not have to be perfectly flat, as balances have knurled feet that can be adjusted to get the balance level. There is usually a little round spirit level bubble to help you adjust it. Make sure that the bubble is dead centre, and all feet are in contact with your bench. As this takes a minute or so, it is helpful if your balance is somewhere where it does not have to be put away.
3) Make a reference weight
There is a little bit of drift in the reading on a balance, due to temperature/atmospheric conditions. In the case of mine it is about +/- 0.2g.
You can either choose to live with this, or make a reference weight. My reference weight is a sealed port bottle loaded with small glass marbles to a weight of 1346g - which is very close to the average weight of a full bottle. Be careful not to use anything in your reference weight that might contain moisture and dry off - I would advise against using sand, as even kiln dried sand still has some residual moisture.
I record the weight of the reference weight before each weighing session and use a spreadsheet to create an adjusted weight. An alternative approach would be to create a reference weight that is slightly lighter than your lightest bottle - about 1200g for bottles - place the reference weight on the balance, hit the tare button and then record the weight of your bottles, over and above the reference weight.
4) Using the balance
Turn the balance on and give it a few seconds to settle before using it. Before putting a bottle on the balance, hit the tare button to make sure that zero really means zero, and put the bottle on gently. (Balances do not like rough handling) If the last digit flits back and forth between two numbers, lift the bottle, hit the tare button again, and see which one it settles for.
Keep a spreadsheet record, not just of the weight, but also the date you weighed it. Make sure the date on your spreadsheet can be read as a serial number.
5) What results should you be hoping for?
Weight loss of less than 0.1g per annum should be considered excellent. 0.1g to 0.15g is normal for most sound older bottles. A loss of over 0.2g p.a. demands prioritisation for consumption, and anything over double that demands urgent attention, either for consumption or preservation.
Bottles that have been kept vertical for a while can sometimes show an alarming weight loss in their first year after being laid down. If you note a high weight loss, but no aroma from a recently bought bottle, make a note to weigh it again a year hence.
I recommend keeping a paper record as you weigh, which you then transfer to a spreadsheet. By keeping the paper records, you can guard against any computer calamity.
Anything to do with Port.
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