Water–ethanol mixtures: strength, density and correct dilution

Introduction

Ethanol is the solvent diluted most often in the pharmacy and the pharmaceutical technology laboratory. Tinctures and extracts call for an extraction solvent of a given strength; skin and surface antisepsis uses 70 % alcohol; compounded prescriptions, solutions and granulation liquids need hydroalcoholic mixtures of various strengths. What is on the shelf, however, is usually 96 % or absolute alcohol. The task is always the same: to prepare a given amount of mixture at a given strength from a stock of known strength.

The familiar "C₁V₁ = C₂V₂" of aqueous solutions is misleading here. Even when the stock volume is right, adding water to make up the missing volume makes the mixture contract: the target volume is not reached and the strength ends up slightly above the target. A correct calculation treats mass, not volume, as the conserved quantity and takes the densities, including their temperature dependence, from an internationally agreed equation. This guide covers the concepts first, then the OIML equation and tables, and finally the numerical methods used by the tool and the dilution calculation itself.

Definitions

OIML R 22 defines the "alcoholic strength" of a water–ethanol mixture in two ways (OIML, 1975):

• Alcoholic strength by volume (% v/v, % vol, q): the volume of alcohol at 20 °C contained in the mixture divided by the total volume of the mixture at 20 °C.
• Alcoholic strength by mass (% m/m, % w/w, p): the mass of alcohol contained in the mixture divided by the total mass of the mixture.

Strength by volume depends on temperature, which is why the reference temperature is fixed explicitly at 20 °C. Strength by mass does not: warming or cooling a mixture leaves the masses unchanged. The two are related through the densities at 20 °C:

φ = w · ρ(w, 20 °C) / ρ_E(20 °C)

Here w is the mass fraction, φ the strength by volume (as a fraction), ρ(w, 20 °C) the density of the mixture and ρ_E(20 °C) that of pure ethanol at 20 °C (789.24 kg/m³). The denominator in the definition is the volume of the mixture, not the sum of the volumes of water and alcohol before mixing. This is not a mere subtlety: mixing 50 mL of alcohol with 50 mL of water does not give 50 % v/v but something slightly stronger, because the denominator has shrunk. Meija (2009) uses the "vodka challenge" attributed to Mendeleev to show why volume concentration and the volume fraction before mixing give different numbers.

Volume contraction

When water and ethanol are mixed, the volume change on mixing (the excess volume, V^E) is negative across the whole composition range: the mixture is smaller than the sum of the volumes its components occupied separately. The molecular reasons are explained in the next section. The excess volume has been measured in detail at several temperatures and by different methods (Benson & Kiyohara, 1980; Marsh & Richards, 1980; Grolier & Wilhelm, 1981). Computed from the OIML equation, the molar excess volume at 20 °C reaches its minimum of about −1.1 cm³/mol near an ethanol mole fraction of 0.4.

A more intuitive measure for the pharmacist is the percentage of the mixed volumes that "disappears". This relative contraction is largest at about 46–47 % alcohol by mass, a composition corresponding roughly to 1 mol of ethanol to 3 mol of water, which was the starting point of Mendeleev's hydrate interpretations. There, about 3.5 % of the mixed volumes is lost. A practical example: mixing 50 mL of absolute alcohol with 50 mL of water at 20 °C yields about 96.5 mL of mixture.

The contraction also changes somewhat with temperature (Marsh & Richards, 1980), so a correct calculation relies not on a single "contraction factor" but on a density equation that depends on both composition and temperature.

Why does the volume shrink?

There is no single cause of the contraction. Several complementary molecular effects act together, and their relative weight is still being studied. Experimental and theoretical work converges on three of them.

1. The open network of water and hydrophobic hydration. In liquid water each molecule forms a roughly tetrahedral hydrogen-bond network with its neighbours. This network is open and full of voids relative to the size of the molecules, which is also why ice is lighter than water. When a little ethanol is added, its non-polar ethyl group is accommodated within the network and the surrounding water molecules rearrange around it (hydrophobic hydration). As a result, ethanol in dilute solution occupies clearly less than its molar volume in the pure liquid (≈58 cm³/mol): at an ethanol mole fraction of 0.038 the partial molar excess volume of ethanol is about −5.2 cm³/mol (Marsh & Richards, 1980). Simulations explain this drop in the partial molar volume of ethanol by the hydration shells of the ethyl groups beginning to touch one another (Tan et al., 2015). As a rough analogy, when sand is poured into a bucket of marbles, part of the sand fills the gaps and the total volume stays below the sum of the two.

2. Water–ethanol hydrogen bonds. The hydroxyl group of ethanol hydrogen-bonds with water. X-ray absorption spectroscopy has shown that the hydrogen bonding originating from the ethanol OH group increases as water is added (Lam et al., 2016). Molecular dynamics simulations with polarisable models likewise found that the water–water bonds lost in the first hydration shell of ethanol are more than compensated in the second shell, so that up to an ethanol mole fraction of about 0.5 the mixture contains more hydrogen bonds in total than the pure liquids (Noskov et al., 2005). More bonds bring the molecules closer together on average. That mixing releases heat, i.e. that the enthalpy of mixing is negative, is consistent with this picture (Boyne & Williamson, 1967).

3. Incomplete mixing at the molecular level and close packing. Water and ethanol mix in all proportions, but they are not randomly distributed on the molecular scale. In concentrated alcohol–water solutions, neutron diffraction has shown water existing as small hydrogen-bonded strings and clusters in a "fluid" of close-packed alkyl groups (Dixit et al., 2002); a similar partial segregation, with both components forming their own percolating networks, has been observed in ethanol–water mixtures (Lenton et al., 2018). Simulations suggest that the ethyl groups align in parallel through hydrophobic interaction and pack better, contributing to the negative partial excess volume of ethanol (Banerjee et al., 2012).

An older explanation assumed that water builds ordered, ice-like cages ("icebergs" or clathrate-like structures) around the alkyl group (Frank & Evans, 1945). The idea stayed in textbooks for a long time; however, neutron diffraction studies found no direct evidence for such structures (Dixit et al., 2002), and simulations with polarisable models saw no extra hydrogen bonding right next to the alkyl group (Zhong & Patel, 2009). The current view is that the contraction is the combined outcome of hydrogen bonding, hydrophobic hydration and partial clustering. Their shares change with composition: hydrophobic hydration dominates in the water-rich region, hydrogen bonding and clustering in the intermediate and alcohol-rich regions.

The OIML R 22 equation

International Recommendation No. 22 "Alcoholometry" of the International Organization of Legal Metrology (OIML) was approved at the 4th International Conference of Legal Metrology in October 1972; its first edition appeared in 1973. The "International Alcoholometric Tables" based on it were published in 1975 by the International Bureau of Legal Metrology (BIML) (OIML, 1975). The tables rest on a single equation giving the density of a water–ethanol mixture as a function of mass fraction and temperature:

ρ(w, t) = A₁ + Σ_k=2¹² A_k w^k−1 + Σ_k=1⁶ B_k (t − 20)^k + Σ_i=1⁵ Σ_k=1^m_i C_i,k w^k (t − 20)ⁱ

ρ is the density in kg/m³, w the mass fraction (0–1) and t the Celsius temperature (IPTS-68 scale). The first sum gives the density at 20 °C as a function of composition; A₁ = 998.20123 kg/m³ is the density of water at 20 °C. The second sum describes how pure water changes with temperature and the third the interaction of composition and temperature; the upper limits of the inner sums are m = (11, 10, 9, 4, 2). The equation is valid from −20 °C to +40 °C. At 20 °C it gives 789.24 kg/m³ for w = 1 (pure ethanol; the value in the OIML errata) and 998.20 kg/m³ for w = 0 (water).

The coefficients were fitted by least squares to five data sets: Kawasaki, Minowa and Inamatsu at the National Research Laboratory of Metrology (NRLM), Tokyo; Jaulmes and Brun Tep at the Faculty of Pharmacy, Montpellier; Osborne, McKelvy and Bearce at the National Bureau of Standards (NBS), Washington (Osborne et al., 1913); Mendeleev's measurements as reconsidered by Georgievskii; and Wagenbreth at the PTB in Germany. A degree of 12 was chosen for the composition polynomial because the standard deviation no longer improved at higher degrees (OIML, 1975).

The subject is still being studied: Khattab et al. (2012) measured density, viscosity and surface tension between 293 and 323 K, and Danahy et al. (2018) examined in detail how to compute composition from measured density and temperature. In legal metrology, however, the reference remains the OIML equation.

The OIML tables

OIML R 22 contains a series of tables derived from the equation. The tool's three tabs correspond to them:

So that the tables derived by interpolation would not spoil the two printed decimals, OIML required the calculations to be carried out with 16 significant digits and the interpolation to be made from an intermediate table at 0.1 % by mass intervals. Tables VIII–X, which correct glass alcoholometer and hydrometer readings for the expansion of the glass, are not part of this tool.

The tool neither reads values from the tables nor interpolates: every result is computed directly from the equation. Where the equation has to be inverted (IVb, Va/Vb, VI/VII), the numerical methods described below are used.

The inverse solution: from strength by volume to strength by mass

Going from strength by mass to strength by volume is easy: once w is known, ρ(w, 20 °C) is computed and substituted into the φ relation above. The reverse requires solving an equation:

w · ρ(w, 20 °C) = φ · ρ_E(20 °C)

Since ρ(w, 20 °C) is a polynomial of degree 11 in w, the left-hand side is a polynomial of degree 12. By the Abel–Ruffini theorem there is no general formula expressing the roots of polynomial equations of degree five or higher in terms of the coefficients using arithmetic operations and radicals. OIML therefore performed the conversion by interpolation between tables; the tool uses a numerical method.

Fixed-point iteration. The equation is written as w = g(w) = φ · ρ_E(20) / ρ(w, 20) and w_n+1 = g(w_n) is repeated starting from w₀ = φ. The method converges if |g′(w)| < 1; here |g′(w)| = w · |dρ/dw| / ρ stays below 0.4 over the whole range (≈0.33 around 96 % v/v). Each step therefore at least halves the error, and the iteration always converges. It stops when two successive values differ by less than 10⁻¹² in mass fraction.

Strength from density: bisection. Finding the strength by mass from a measured density means solving ρ(w, t) = ρ_measured. It was checked at 0.01 % steps that, at every temperature from −20 to 40 °C, density decreases steadily as w increases; the root is therefore unique. The tool searches for it on [0, 1] by bisection: if the density at the midpoint of the interval is higher than the measured one, the root lies in the right half, otherwise in the left half. After about 40 halvings the interval is narrower than 10⁻¹². The w found is converted to strength by volume with the 20 °C relation.

Dilution calculation

In a dilution the conserved quantity is the mass of ethanol. With m_stock the mass of stock, m_final the mass of the mixture to be prepared and w_stock, w_final the strengths by mass:

m_stock · w_stock = m_final · w_final     m_water = m_final − m_stock     m_final = V_final · ρ_final(t)

When the target is given by volume, the strengths are first converted to strengths by mass and the required volume is turned into a mass using the density of the mixture at the preparation temperature. Once the masses of stock and water are known, their volumes are computed from their own densities at the same temperature. The mass balance holds at any temperature. For a dilution by volume prepared at 20 °C the stock volume equals the one given by simple proportion (V · φ_target / φ_stock), because the volume of ethanol at 20 °C is conserved by definition; the water required, however, is more than simple proportion suggests, because of contraction.

Why simple proportion is not enough. To make 100 mL of 70 % alcohol from 96 %, if 27.08 mL of water is added to 72.92 mL of stock by simple proportion, the mixture comes to about 97.5 mL and its strength is about 71.8 % instead of 70 %. This is why pharmacists often follow a "prepare 10 % extra" rule: for 50 mL, for example, the calculation is done for 55 mL and the surplus discarded. That closes the volume gap but does not correct the strength. The correct amount of water (here 29.74 mL) brings both the volume and the strength to target, and nothing extra needs to be prepared.

When the target is given by mass (% w/w, amount in g), the calculation is a plain mass balance; density and temperature are used only to compute the volumes shown for information.

Temperature

Strength by volume is defined with respect to volumes at 20 °C; 70 % v/v alcohol is 70 % once brought to 20 °C, whatever the temperature at which it was prepared. Volumes measured with glassware, however, are volumes at the temperature of measurement. The tool therefore computes the stock, water and mixture volumes from densities at the preparation temperature. For example, making 100 mL of 70 % alcohol from 96 % at 25 °C takes 72.96 mL (58.60 g) of stock and 29.63 mL (29.54 g) of water, with a final mass of 88.14 g. The difference from the 20 °C values is small, but because concentrated alcohol expands markedly more than water it grows as the temperature moves away from 20 °C.

Masses do not depend on temperature. The most reliable approach is therefore to prepare by weighing: the tool gives the mass alongside the volume at every step. Adding the stock alcohol and the water by weight into a tared vessel removes both the reading error of the measuring vessel and the effect of temperature. The thermal expansion of the glassware itself is not taken into account.

Mixing releases heat (Boyne & Williamson, 1967). In the small volumes of a prescription the heat dissipates quickly and its effect on the measurement is negligible. In large volumes (for example when a stock is prepared at the litre scale), however, the mixture warms noticeably and cools slowly, and a warm mixture looks larger than it is. In that case read the final volume only after the mixture has returned to the preparation temperature. Preparing by weight avoids this problem.

Using the tool

The Alcohol Dilution tool has three tabs.

1. Dilution recipe. Enter the stock strength, the target strength, the amount to prepare and the preparation temperature (−20…40 °C), and choose % v/v (amount in mL) or % w/w (amount in g) as the basis. The target must be greater than 0 and lower than the stock, and the stock at most 100. The result is a step-by-step recipe: the volume and mass of stock to take, the volume and mass of water to add, the final mass, the volume lost on mixing and the water volume simple proportion would have suggested.
2. Density from strength. Enter a strength (% v/v or % w/w) and a temperature; the density of the mixture is given in kg/m³ and g/mL, together with the strength on the other basis (% v/v ↔ % w/w). This tab corresponds to OIML Tables I, II, IIIa, IIIb, IVa and IVb.
3. Strength from density. Enter a measured true density and the measurement temperature; the strength by mass (% w/w) and the strength by volume at 20 °C (% v/v) are given. At 20 °C the result corresponds to Tables Va/Vb, at other temperatures to Tables VI/VII of Annex I. The density must lie between the densities of pure ethanol and pure water at that temperature. Example: 885.00 kg/m³ at 20 °C → 62.63 % w/w, 70.23 % v/v; the same density measured at 25 °C → 60.83 % w/w, 68.54 % v/v.

The strength-from-density tab does not correct glass alcoholometer or hydrometer readings (OIML Tables VIII–X); the value entered must be a true density, obtained for example with an oscillating U-tube density meter or a pycnometer.

Comparison with the printed tables

The tool was compared with the printed tables of OIML R 22. The tables were read from scanned pages into a test fixture; reading (OCR) errors were removed without using the calculation code, relying only on the internal consistency of each table row. The fixture contains 6 889 cells from 8 tables; in 6 884 of them the code's result rounded to two decimals equals the printed value. In the remaining 5 cells the difference is 0.01, and every one of them lies exactly on a rounding boundary (third decimal 5).

The comparison is a unit test (PTCalc.Core.Tests/AlcoholometryTests.cs) and runs with the test suite. The fixture is generated by the tools/make_oiml_reference.py script; the OIML publication itself is not in the repository.

Limitations

• Water–ethanol only. The equation is for a binary mixture. In tinctures and extracts, dissolved solids raise the density, and the strength derived from density departs from the true alcohol content; for such samples the alcohol should first be separated by distillation, or a correction for the dry residue applied (Bernatonienė et al., 2003). Congeners such as methanol and fusel oils can also shift the strength by 0.2–0.4 points (Kolesnov et al., 2023).
• No hydrometer correction. The glass expansion correction needed for glass alcoholometer and hydrometer readings (Tables VIII–X) is not applied; the input must be a true density.
• Freezing at low temperatures. Low-strength mixtures freeze at low temperatures, which is why OIML Table I starts, at each temperature, from the minimum permissible strength. The equation still returns a number in that region, but it does not correspond to a liquid mixture.
• Pressure. The equation applies at normal atmospheric pressure.
• Injectable solutions. The tool dilutes the stock with water. For ethanol-containing injections the diluent should be 0.9 % NaCl rather than water, otherwise the hypotonic mixture may cause haemolysis (Serdons et al., 2008). Because the density of normal saline differs slightly from that of water, the calculation is then approximate.
• Practice note. 70 % alcohol used for ethanol-lock therapy has been reported to remain stable for 14 days in polypropylene syringes (Cober & Johnson, 2007); storage conditions should be assessed separately.