To determine the amount of alcohol in the beer they are making, brewers compare the specific gravity of the beer before it starts fermenting to its specific gravity when it is finished fermenting.
The specific gravity is a measure of the density of a liquid relative to water. The density of water is 1 kilogram per liter, so if the specific gravity of a liquid is 1.06, one liter of that liquid will weigh 1.06 kg.

The liquid that will form the beer is called a wort (pronounced wert). Its specific gravity is always higher than water because it contains a lot of dissolved sugars. Yeast added to the wort will convert some of these dissolved sugars into ethyl alcohol. When the beer is finished, the specific gravity is always less than when it started, because some of the sugars have been converted into alcohol, which is less dense than water (0.79 kg/L).

Glucose (C₆H₁₂O₆) is the main sugar that will be converted to alcohol. Many reactions take place inside the yeast that ultimately convert each glucose molecule into two molecules of ethyl alcohol (CH₃CH₂OH) and two molecules of carbon dioxide (CO₂).

C₆H₁₂O₆ => 2(CH₃CH₂OH) + 2(CO₂)

If you check the periodic table, you can figure out the molecular weights of these two molecules. The molecular weight of ethyl alcohol is 46.0688 and the molecular weight of carbon dioxide is 44.0098. You will need these numbers to calculate the alcohol content of the beer.

During the fermentation process most of the carbon dioxide that forms from the reaction bubbles out of the solution and leaves the fermentation vessel by way of a vent. You can say that all of it leaves, because the amount the remains in the beer is very small compared to the amount that leaves.

If you look at the equation for the reaction you see that each glucose is split into two ethyl alcohol molecules and two carbon dioxide molecules. That means for each carbon dioxide molecule that leaves the fermentation vessel, one ethyl alcohol molecule must be formed inside the vessel. If you look back at the molecular weights you can say that for each 44.0098 grams of CO₂ that leaves the vessel 46.0688 grams of ethyl alcohol are formed. Put another way, for each gram of CO₂ that bubbles off, about 1.05 grams of ethyl alcohol are produced.

You can compare the starting specific gravity to the final specific gravity. If the starting gravity of the wort is 1.06, and after fermentation, the gravity is 1.02. Subtracting the second from the first gives us the weight of CO₂ that left the vessel. That is equal to 0.04 kg/L. Then you multiply by 1.05 to get the weight of the alcohol in the container. That is 0.042 kg/L. Now that you know both the mass of the solution (1.02 kg/L) and the mass of the alcohol (0.042 kg/L) you can calculate the percentage of alcohol by mass by dividing the two. This gives 0.042 / 1.02, which equals 0.041, or 4.1 percent.

It is important to note that the percentage of alcohol by mass is higher than the percentage of alcohol by volume because an equal mass of alcohol occupies more volume than water would. So to convert from percent alcohol by mass to percent alcohol by volume you just divide by the density of alcohol. In this case you get 4.1/0.79 or 5.2 percent alcohol by volume.

In Colorado the beer sold in grocery stores has to be low alcohol "3.2" beer. It can contain no more than 3.2 percent alcohol by mass. But the beer sold in liquor stores is labeled by volume, and most beer is about 5 percent alcohol by volume. But when you convert from weight to volume you find that the "3.2" beer is really 4 percent alcohol by volume. Still a pretty big difference, but maybe not quite as big as you might think.

More on the Hydrometer

A hydrometer is an instrument used for determining the specific gravity of liquids. It is usually made of glass and consists of a cylindrical stem and a bulb weighted with mercury or shot to make it float upright. The liquid is poured into a tall jar, and the hydrometer is gently lowered into the liquid until it floats freely.
The point where the surface of the liquid touches the stem of the hydrometer is noted. Hydrometers usually contain a paper scale inside the stem, so that the specific gravity (or density) can be read directly in grams per cubic centimeter.
In light liquids like kerosene, gasoline, and alcohol, the hydrometer must sink deeper to displace its weight of liquid than in heavy liquids like brine, milk, and acids. In fact, it is usual to have two separate instruments, one for heavy liquids, on which the mark 1.000 for water is near the top, and one for light liquids, on which the mark 1.000 is near the bottom of the stem.
The function of the hydrometer is based on Archimedes principle that a solid suspended in a liquid will be buoyed up by a force equal to the weight of the liquid displaced. Thus, the lower the density of the substance, the lower the hydrometer will sink.

Commercial uses
Because the commercial value of many liquids, such as sugar solutions, sulfuric acid, alcohol, and wine, depends directly on the specific gravity, hydrometers are used extensively.
Perhaps the best-known form of hydrometer is the kind used in testing milk, called a lactometer. The specific gravity of cow's milk varies from 1.027 to 1.035. Since only the last two digits are important, the scale of a lactometer runs from 20 to 40, which means from 1.020 to 1.040. The specific gravity of milk does not give us a conclusive test as to its worth. In addition to water (which is about 87%), milk contains some substances that are heavier than water, such as albumen, sugar, and salt, and others that are lighter than water, such as butter fat. So, in addition to the specific gravity, one needs to determine the amount of fat, and, if possible, of other solids in the milk, to know its richness. Of course, the very important question as to the cleanliness of milk must be left to the bacteriologist.
An alcoholometer is a hydrometer which is used for determining the alcoholic strength of liquids.
A saccharometer is a hydrometer used for determining the amount of sugar in a solution. It is primarily used by brewers and winemakers.
A thermohydrometer is a hydrometer that has a thermometer enclosed in the float section
 

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