When adding water to a spirit or blending two spirits with significantly different alcohol levels, a well-known phenomenon occurs among spirit producers : the final volume is lower than the sum of the initial volumes. As a result, when adjusting the alcohol level, it is necessary to consider the percentage of contraction that will occur, thereby minimizing production steps. This article summarizes the physical origin of this phenomenon and provides a step-by-step guide for calculating the contraction factor.

**Origin of total volume contraction water + ethanol**

The observed volume contraction when adding water to a brandy is the result of highly complex reorganization phenomena within the space occupied by water and ethanol molecules. Over the past decade, numerous studies have been conducted using new physical measurement tools *(1) (2)* to better understand these phenomena. Based on the findings of these studies, the origin of the contraction can be summarized as follows :

Water molecules are linked together by electrostatic bonds called “hydrogen bonds”.

Upon contact with ethanol, these bonds will break in favor of stronger bonds with the ethanol molecules. For spirits, this reorganization leads to a contraction in the total volume of the mixture.

Other compounds inherent in spirits, such as organic acids, could play a role in strengthening these bonds, as suggested by studies on Sochu published by Japanese researchers *(3)*.

How to calculate the contraction rate ?

The calculation of the final contraction rate uses the principle of conservation of masses.

Initial mass of water + Initial mass of alcohol = Final Mass (Water + Alcohol)

By knowing the volumes of the two assembled products and the alcohol level of the starting product, it is possible to determine each of the masses, the final mass, to calculate the final volume and its alcohol level.

For this, it is necessary to use the official alcoholometer tables published by the International Organization of Legal Metrology in its Recommendation R22 *(4)*:

– Table I : for the correspondence between the % by mass and the volumic mass.

– Table II : for the correspondence between the volumic mass and the % vol. of alcohol.

**Principle and calculation steps**

For a given volume of eau-de-vie *(Vev)* to which is added a volume of water *(Vw), *the objective is to determine the final volume *(Vf)* to deduce the % contraction :

% contraction = [Vf – (Vev + Vw)] / (Vev + Vw)

To achieve this objective, the calculation steps are as follows :

**Obtaining the volumic masses of eau-de-vie***(MVev)***and of water***(Mw) :*with the Table II.**Calculations of the masses of eau-de-vie***(Mev)***and of water***(Mw)*, based on the volumic masses and volumes.**Calculation of the total mass of the blending***(Mf)*(law of masses conservation).**Obtaining % by mass of ethanol contained in eau-de-vie***(%Metev) :*with the Table I and knowing its density.**Calculation of Ethanol mass in the blending***(Metf) :*based on ethanol mass and % by mass.**Calculation of Ethanol % by mass in the blending***(%Metf) :*based on initial ethanol mass and final blending mass.**Obtaining the volumic mass of the blending***(MVf) :*with Table II, based on the % in mass of ethanol in the blending.**Final volume calculation***(Vf) :*based on the total mass of the blending and it’s volumic mass.**Calculation of the contraction factor***(Fc) :**(Vw+Vev)*and the contracted volume*(Vf)*, divided by the volume which would be obtained without contraction*(Vw+Vev)*.

Example of calculation from a practical case

Calculations based on 1l of eau-de-vie at 80% vol. of alcohol and adding 1l of water.

All volumes and alcohol levels taken into account in these calculations are at 20°C.

1. **Obtaining the volumic masses of each of the assembled products** :

– From the starting alcohol level of 80% vol., Table II – OIML – R22 returns a density of = 859.27 g/l *(MVev)*.

– Water volumic mass is done in this table at 998.2 g/l *(MVw)*.

2. **Calculations of the masses of water and eau-de-vie which are assembled **: The assembly being 1l of each product, so taking into account the densities of each constituent, the mass of starting eau-de-vie is 859.27 g (Mev) to which is added 998.2 g of water (ME).

3. **Calculation of the total mass of the blending** : eau-de-vie mass at 859.27 g + water mass at 998.2 g, gives a final total mass : 859.27 + 998.2 = 1857.5 g *(Mf)*.

4. ** Obtainning the % by mass of ethanol contained in eau-de-vie:** For a density and a % in alcohol known , Table I – OIML – R22 returns a % by mass of ethanol. However, for a density of 859.27 g/l, as this value has no direct correspondence with given mass %, it is necessary to make a rule of 3 between the 2 densities proposed by Table I and the 2%. masses of ethanol returned by this table. The result is 73.48% (%Metev)

5. **Calculation of the mass of Ethanol in the blending** : Knowing the eau-de-vie mass and the % by mass of ethanol in the eau-de-vie, calculation of the mass of ethanol in the blending = Eau-de-vie mass (Mev) * % by mass of ethanol (% Metev) : 859.3 * 0.7348 = 631.4 g of ethanol *(Metf)* which is completely found in the final volume.

6. **Calculation of % by mass of ethanol in the blending** = masse d’éthanol (Met)/ masse totale (Mf) : 631.4 / 1857.5 g = 33.99 *% (%Metf)*

7. **Obtaining the volumic mass of the blending** : Knowing the % by mass of ethanol in the blending, ltheTable I – OIML – R22, returns a volumic mass. However, for a % by mass of ethanol of 33.99%, as the step of the table is 1 in 1, it is necessary to make a rule of 3 between the 2 % by mass proposed by Table I and the 2 density masses returned by this table. The result is 946.78 g/l *(MVf)*.

8. **Calculation of the final volume **: Knowing the total mass (1857.5 g) and volumic mass of the blending (946.78 g/l), the final volume is total mass /volumic mass : 1857.5 / 946.78 = 1962 ml, so 1.962 l *(Vf)* instead of the expected 2l.

**9. Calculation of the contraction factor (Fc) **obtained by difference with the expected volume

**: 2 l – 1.962 l = 0.038 l, i.e. 1.9 % of 2 liters**

Other examples of calculations, reportedin the graph below, show that the % contraction is not a linear phenomenon. It would be too simple !

The addition of water can even lead to inverse phenomena of expansion of the mixture, as shown in the graph for a hydroalcoholic mixture at 20% vol. alcohol diluted by 2.

**Alcohol level adjustment calculations with Labox**

To optimize the alcohol level adjustment calculations given the complexity of these calculations and the risk of error, the use of the Boxette “Remontage et Réduction” is recommanded.

This boxette offers different options:

– **Reduction of alcohol level **: Calculation of the quantity of water or a product with a lower alcohol level, to obtain a given alcohol level.

– **Raise the alcohol level** : Calculation of the quantity of brandy necessary to “raise” the alcohol level, if it turns out that it is too low.

These calculations are made either from an initial volume to be rectified, or for a final volume to prepare.

**The calculated quantities are given in volume and mass.**

*The calculations with Labox take into account the contraction phenomena inherent to these operations and refer to the International Alcoholometer Tables published by the O.I.M.L (4), which have been computer generated by LABOX applications (5), from the notebook loads recommended by the O.I.M.L. and imposed by French regulations (6).*

In-depth study of contraction phenomena

The alcoholometer tables are empirical. They arise from experiments carried out over decades by different researchers from different nations, with increasingly precise measurement tools *(7)*.

These tables can be used to characterize contraction phenomena.

When adding water to a hydroalcoholic mixture, everything happens as if:**Only the volume occupied by the water molecules, which by electrostatically binding to the ethanol molecules, decreased.****Which would suggest that the ethanol molecules would retain the % volume they occupied before dilution.**

To demonstrate this, let’s take the previous example again, looking at it from a new angle: Or **1l of eau-de-vie at 80% vol.**, but this time, topped up to **2l with water**.

Due to the contraction phenomenon, the added water will be a little more than 1l. **Consequences :**

– **If the volume of alcohol contracts as much as the water**, the % of alcohol must be lower than that calculated by dilution (initial alcohol volume / final volume), i.e. lower than 40% vol.

– **If the volume of alcohol occupies the same space**, the final alcohol % must correspond exactly to the initial alcohol volume / final volume, or exactly 40% vol.

**Principle and calculation steps**

The objective is to determine the final alcohol content of an eau-de-vie, after adding water up to a given volume.

To achieve this goal, the steps are as follows :

**Calculation of the water volume added***(Vw+)*, to reach the final volume*(Vf)*, taking into account the contraction factor.**Calculation of the total mass of water added***(Mw)*, knowing the water volume and it’s volumic mass.**Calculation of eau-de-vie mass before dilution***(Mev)*, knowing it’s volume*(Vev)*and it’s volumic mass*(MVev)*.**Calculation of the total mass after dilution***(Mf)**(law of conservation of masses).***Calculation of % by mass of ethanol in the eau-de-vie after dilution***(% Metf) :*Ethanol mass*(Met)*/ Total mass*(Mf)***Obtaining the volumic mass of the eau-de-vie after dilution***(MVf)*, with the % by mass of ethanol Table I returns the volumic mass.**Obtaining the % vol. of alcohol in the eau-de-vie after dilution***(A.S.V..f) :*with the volumic mass the Table II returns the % vol. of alcohol.

Calculation from a practical case

**1l of hydroalcoholic mixture at 80% vol., topped up to 2l.**

*All volumes and alcohol levels taken into account in these calculations are at 20°C*

For these calculations, let’s take the information obtained previously:

- Volumic mass of water
*(MVw)*= 998.20 g/l - Volumic mass of eau-de-vie at 80% vol.
*(MVev)*= 859.3 g/l. - Mass of eau-de-vie
*(Mev)*= 859.27 g - % by mass of ethanol in eau-de-vie
*(%Metev) =*73.48 %. - Contraction factor at 1.9%
*(Fc)*to be applied to a volume of water equivalent to the starting volume.

**1. Calculation of the water volume added ***(Vw+)* : When 1l of water is added to 1l of a product at 80% vol. of alcohol, the contraction factor is 1.9%. The final volume is therefore 1,962 l. To reach 2l, you must supplement with 0.038 l of water, to which you must also apply the contraction rate which will be slightly lower than 1.9%. In fact, it decreases as water is added. 1.9% of 0.038 l = 0.0007. The total quantity of water added to reach 2 l is 1.038+0.0007 = 1.0387 l (Vw+).

2. **Calculation of the total mass of water added ***(Mw)* : The volumic mass of water being 998.2 g/l, the water mass is 1.0387 l * 0.9982 g/l i.e. 1036.83 g.

3. **Calculation of mass of eau-de-vie ***(Mev) *: Volume of 1l at 859.27 g/l i.e. 859.27 g of eau-de-vie.

4. **Calculation of total mass after dilution*** (Mf) *: Knowing the eau-de-vie mass (Mev) and the water mass, the final mass is : Eau-de-vie mass (Mev) 859.27 g + Water mass added (Mw) 1036.83 g = 1896.10 g (Mf)

5. **Calculation of % by mass of the ethanol in the eau-de-vie after dilution*** (%Metf)*** : **Knowing the ethanol mass and the total final mass calculation of % by mass after dilution is : = 63.14 / 1896.1 = 33.30%

6. **Obtaining the volumic mass of the eau-de-vie after dilution*** (MVf) :* With the % by mass, the Table I returns the correspondance with the volumic mass, i.e.** 948.04 g/l**

7. **Obtaining the % vol. in alcohol in the eau-de-vie after dilution ***(A.S.V.f) :* With the volumic mass the Table II returns the % vol. of alcohol, i.e. **40.01 % vol..**

In this practical case, if the contraction factor applied to both ethanol and water: the % alcohol volume of the final product would be reduced by the same amount, i.e. by 40 – (40 *1.9% ) = 40 – 0.76%, or 39.2% vol.. This alcohol content is much lower than that actually obtained of 40.0% vol..

These calculations and my own dilution experiments, carried out in the laboratory, seem to demonstrate that during a dilution with water, everything happens as if the contraction was mainly due to the contraction of the volume occupied by the water molecules around ethanol molecules. Given recent published work on this phenomenon *(2)(3)*, this is not what is actually happening.

**Physical reality of the phenomenon**

The volume occupied by the ethanol molecules changes with dilution. It also depends on other factors such as temperature, % water, or the presence of other compounds. These are very complex molecular rearrangements that take place. During dilution, the distances between the different molecules change.

However, whether by calculation using alcoholometer tables or by experimentation, everything goes well as if only the % volume of ethanol was contracting.

**Explanation of this contradiction**

Only Table I, which relates a % by mass of ethanol to the density, is empirical. The specifications described in the O.I.M.L. Recommendation. R22 *(4)* allows this table to be reconstructed. The other alcoholometer tables, including Table II which links the volumic mass to a % volume of ethanol or Table VIIIb (white pages of the Alcoholometer Guide) which allows a A.S.V. measurement to be converted at 20°C carried out with an alcoholometer, arise from calculations which involve certain physical constants, including the volumic mass of pure ethanol and the volumic mass of water.

However, it is not taken into account that for the establishment of these tables, the volumes actually occupied respectively by water and ethanol change, not only with the temperature, but also as a function of the % of ethanol and % of water. Which means that when these molecules are mixed, their respective volumic mass varies, depending on the proportions of each of these molecules.

**Therefore, the “Volume Title” as it is called in the O.I.M.L. Recommendation. R22 (4) and in the alcoholic tables which result from it, does not really correspond to a percentage of liter of ethanol in the mixture, but to a definition convention of this parameter which is the alcoholic strength by volume (A.S.V. % vol .).**

In-depth study of contraction phenomena

The alcoholometer tables are empirical. They arise from experiments carried out over decades by different researchers from different nations, with increasingly precise measurement tools *(7)*.

These tables can be used to characterize contraction phenomena.

When adding water to a hydroalcoholic mixture, everything happens as if:**Only the volume occupied by the water molecules, which by electrostatically binding to the ethanol molecules, decreased.****Which would suggest that the ethanol molecules would retain the % volume they occupied before dilution.**

To demonstrate this, let’s take the previous example again, looking at it from a new angle: Or **1l of eau-de-vie at 80% vol.**, but this time, topped up to **2l with water**.

Due to the contraction phenomenon, the added water will be a little more than 1l. **Consequences :**

– **If the volume of alcohol contracts as much as the water**, the % of alcohol must be lower than that calculated by dilution (initial alcohol volume / final volume), i.e. lower than 40% vol.

– **If the volume of alcohol occupies the same space**, the final alcohol % must correspond exactly to the initial alcohol volume / final volume, or exactly 40% vol.

**Principle and calculation steps**

The objective is to determine the final alcohol content of an eau-de-vie, after adding water up to a given volume.

To achieve this goal, the steps are as follows :

**Calculation of the water volume added***(Vw+)*, to reach the final volume*(Vf)*, taking into account the contraction factor.**Calculation of the total mass of water added***(Mw)*, knowing the water volume and it’s volumic mass.**Calculation of eau-de-vie mass before dilution***(Mev)*, knowing it’s volume*(Vev)*and it’s volumic mass*(MVev)*.**Calculation of the total mass after dilution***(Mf)**(law of conservation of masses).***Calculation of % by mass of ethanol in the eau-de-vie after dilution***(% Metf) :*Ethanol mass*(Met)*/ Total mass*(Mf)***Obtaining the volumic mass of the eau-de-vie after dilution***(MVf)*, with the % by mass of ethanol Table I returns the volumic mass.**Obtaining the % vol. of alcohol in the eau-de-vie after dilution***(A.S.V..f) :*with the volumic mass the Table II returns the % vol. of alcohol.

Calculation from a practical case

**1l of hydroalcoholic mixture at 80% vol., topped up to 2l.**

*All volumes and alcohol levels taken into account in these calculations are at 20°C*

For these calculations, let’s take the information obtained previously:

- Volumic mass of water
*(MVw)*= 998.20 g/l - Volumic mass of eau-de-vie at 80% vol.
*(MVev)*= 859.3 g/l. - Mass of eau-de-vie
*(Mev)*= 859.27 g - % by mass of ethanol in eau-de-vie
*(%Metev) =*73.48 %. - Contraction factor at 1.9%
*(Fc)*to be applied to a volume of water equivalent to the starting volume.

**1. Calculation of the water volume added ***(Vw+)* : When 1l of water is added to 1l of a product at 80% vol. of alcohol, the contraction factor is 1.9%. The final volume is therefore 1,962 l. To reach 2l, you must supplement with 0.038 l of water, to which you must also apply the contraction rate which will be slightly lower than 1.9%. In fact, it decreases as water is added. 1.9% of 0.038 l = 0.0007. The total quantity of water added to reach 2 l is 1.038+0.0007 = 1.0387 l (Vw+).

2. **Calculation of the total mass of water added ***(Mw)* : The volumic mass of water being 998.2 g/l, the water mass is 1.0387 l * 0.9982 g/l i.e. 1036.83 g.

3. **Calculation of mass of eau-de-vie ***(Mev) *: Volume of 1l at 859.27 g/l i.e. 859.27 g of eau-de-vie.

4. **Calculation of total mass after dilution*** (Mf) *: Knowing the eau-de-vie mass (Mev) and the water mass, the final mass is Eau-de-vie mass (Mev) 859.27 g + Water mass added (Mw) 1036.83 g = 1896.10 g (Mf)

5. **Calculation of % by mass of the ethanol in the eau-de-vie after dilution*** (%Metf)*** : **Knowing the ethanol mass and the total final mass calculation of % by mass after dilution is 63.14 / 1896.1 = 33.30%

6. **Obtaining the volumic mass of the eau-de-vie after dilution*** (MVf) :* With the % by mass, the Table I returns the correspondance with the volumic mass, i.e.** 948.04 g/l**

7. **Obtaining the % vol. in alcohol in the eau-de-vie after dilution ***(A.S.V.f) :* With the volumic mass the Table II returns the % vol. of alcohol, i.e. **40.01 % vol..**

In this practical case, if the contraction factor applied to both ethanol and water : the % alcohol volume of the final product would be reduced by the same amount, i.e. by 40 – (40 *1.9% ) = 40 – 0.76%, or 39.2% vol.. This alcohol content will be much lower than that actually obtained of 40.0% vol..

These calculations and my own dilution experiments, carried out in the laboratory, seem to demonstrate that during a dilution with water, everything happens as if the contraction was mainly due to the contraction of the volume occupied by the water molecules around ethanol molecules. Given recent published work on this phenomenon *(2)(3)*, this is not what is actually happening.

**Physical reality of the phenomenon**

The volume occupied by the ethanol molecules changes with dilution. It also depends on other factors such as temperature, % water, or the presence of other compounds. These are very complex molecular rearrangements that take place. During dilution, the distances between the different molecules change.

However, whether by calculation using alcoholometer tables or by experimentation, everything goes well as if only the % volume of ethanol was contracting. **Where does this contradiction between reality and calculations come from ?**

**Explanation of this contradiction**

Only Table I, which relates a % by mass of ethanol to the density, is empirical. The specifications described in the O.I.M.L. Recommendation. R22 *(4)* allows this table to be reconstructed. The other alcoholometer tables, including Table II which links the volumic mass to a % volume of ethanol or Table VIIIb (white pages of the Alcoholometer Guide) which allows a A.S.V. measurement to be converted at 20°C carried out with an alcoholometer, arise from calculations which involve certain physical constants, including the volumic mass of pure ethanol and the volumic mass of water.

However, it is not taken into account that for the establishment of these tables, the volumes actually occupied respectively by water and ethanol change, not only with the temperature, but also as a function of the % of ethanol and % of water. Which means that when these molecules are mixed, their respective volumic mass varies, depending on the proportions of each of these molecules.

Therefore, the “Volume Title” as it is called in the O.I.M.L. Recommendation. R22 *(4)* and in the alcoholic tables which result from it, does not really correspond to a percentage of liter of ethanol in the mixture, but to a definition convention of this parameter which is the alcoholic strength by volume (A.S.V. % vol .).

**Conclusions**

The work presented in this document shows the complexity of the calculations to take into account the contraction phenomena inherent in blends, or during “Reduction” or “Raise” operations of the alcohol content of spirits.

They highlight that the definition of the % volume of alcohol is conventional and does not represent a physical reality of the occupancy rate of the volume of ethanol. **To calculate ethanol balances, it would be preferable to measure a % by mass of ethanol.**

In the field of analytical applications, using alcohol volume percentage as a measurement parameter has various advantages. This is the case, for example, when it is necessary to dilute a sample to remain within the range of the distillation apparatus. However, this practice raises questions about the accuracy of the results.

To address these concerns, I have dedicated an article to this blog. This aims to provide answers to questions asked to me by users of distillation devices “Alcohol measurement – Part 3 : Distillation with dilution“.

**Evelyne CHANSON – Consultant in quality control of Wines and Spirits of EC Consulting**

*Features of the Temperature and Concentration Dependences of the Contraction of Aqueous Solutions of Ethanol – V.Ya. Gotsul’skii, N.P Malomuzh, V.E. Chechko – Mechnikov National University, ODESSA, J. Phys. Chem. A., 2013, 87, 10, 1638-1644.**“Physical Properties Hydrogen Bonding Network of Water-Ethanol Mixtures from Molecular Dynalics Simulations” – A. Ghoufi (*), F. Artzner (*), P. Malfreyt (**), – J. Phys. Chem. B 2016, 120, 793-802. (*) Institut de physique de Rennes, (**) Institut de chimie de Clermont Ferrand.**“Hydrogen Bonding in Alcoholic Bevarges (Distilled Spirits) and Water-Ethanol Mixtures – A. Nose, T. Hamasaki, R. Kato,**K. Uehara, T. Ueda – J. Agric. Food Chem. 2005, 53, 7074-7081.**International Alcoholometric tables**published published by l’OIML Recommendation R022 -f75 .**Traceability and computerization of alcoholometric tables – E. CHANSON – OIML Bulletin Volume LVI – N°3 – July 2015- p7.**Décret 79-200 du 05/03/79 (on french public administration regulations regarding alcoholometers, hydrometers for alcohol and alcoholometric tables). Although this decree has been repealed, the new decrees refer to it with regard to the specifications to be taken into account to generate the alcoholometer tables.**The bibliography of studies from which the O.I.M.L. established the specifications from which the Alcoholometer Tables result is published in the introduction to its document ” International Alcoholometric tables“.*

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