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Wednesday, March 13, 2019

The value of k *\delta_e

In a previous post, I showed that
  \begin{equation}ABV \approx ( \delta_i - \delta_f  ) ( k * \delta_e )^{-1} \end{equation}   
Where \delta_e is the density of ethanol
\begin{equation} \delta_e = 0.7892 \text{ g/cm}^3 \end{equation}
And k is the proportionality constant between the weights of CO_2 and \text{ethanol} that are produced during the fermentation process.
W_{CO_2} = k*W_e
We can derive the value of k by having a look at the fermentation reaction:
C_6H_{12}O_{6} \rightarrow 2 C_2H_5OH + 2 CO_2
In other words, for each molecule of carbon dioxide released in fermentation, one ethanol molecule is produced. Since the molar mass of ethanol is 46.07 g/mol, compared to 44.01 g/mol for carbon dioxide; we have 
\begin{equation}k =  \frac{W_{CO_2}}{W_e}= \frac{44.01 \text{g/mol}}{ 46.07 \text{g/mol}} \approx  0.9553 \end{equation}
And now it is time to plug the numbers...


Using the values of (2) & (3) into (1), we finally get

  ABV \approx ( \delta_i - \delta_f  ) ( 0.9553*0.7892\text{ g/cm}^3)^{-1} 
\Rightarrow ABV \approx ( \delta_i - \delta_f  ) *1.326\text{ cm}^3/ \text{g} 
In case the hydrometer readings for \delta_i and \delta_f are given in kg/m^3 instead of g/cm ^3, we need to divide by 1000; and if we want the Alcohol by Volume to be expressed as a percentage, we have to multiply by 100. In such case,

  \boxed{  \%ABV \approx ( \delta_i - \delta_f  ) *0.1326  \text{  [m}^3/  \text{kg]}   }

Saturday, March 2, 2019

Alcohol content formula... Where does it come from?

One of the most used equations by homebrewers is the formula to calculate the alcohol content by volume (ABV).  Hereby I explain how to derive it.


During the fermentation process, glucose is converted into ethanol and carbon dioxide. Since carbon dioxide escapes from the fermentation bucket, the relation between the initial and final weights is given by:
W_i - W_f = W_{CO_2}
We know that ethanol and carbon dioxide are always produced in the same proportion. This means that there is a fixed ratio among their weights:  W_{CO_2} = k*W_e , with k a constant. Thus,
W_i - W_f = k*W_{e}
By expressing the weights in terms of volumes and densities (W=V*\delta), we get:
V_i * \delta_i - V_f  * \delta_f = k*V_e * \delta_e
And here comes the trick... Let us assume that V_i \approx V_f. Thus,
V_f *( \delta_i - \delta_f ) \approx  k*V_e * \delta_e \Rightarrow \boxed{( \delta_i - \delta_f  ) ( k \delta_e )^{-1} \approx  \frac{V_e}{ V_f}  = ABV}

In the following posts, I will show that it is safe to assume that V_i \approx V_f , and determine the value of k \delta_e.

Calculating alcohol content in your homebrew beer

The method most employed by homebrewers involves the use of a hydrometer.
WortHydrometer
The process is very simple:
  1. Using a hydrometer, measure the initial density of the must before starting fermentation: \delta_i
  2. Then, after the fermentation has finished, take a second density reading: \delta_f
  3. Now subtract both readings and multiply by a given factor:
    (\delta_i - \delta_f)* f= ABV   
The value obtained is the ABV (alcohol by volume) often expressed as a percentage.

This is simple to calculate, but not many homebrewers know where this formula comes from.

In a later post, I will explain how to derive this equation.