# Concentration Determination by Means of Density Measurement

Task: To simply determine the concentration of a substance in a solution
Solution: If there is a relation between the concentration of the substance and the density of the solution, the concentration can be determined by measuring the density of the solution.

## General aspects

The density of liquids changes if ingredients change: E.g. a soft drink containing sugar will have a higher density than a diet soft drink. So, sugar not only increases a liquid’s density, it can also be inversely measured by knowing the liquid’s density in a given liquid type. Using this correlation provides useful information in the production of many beverages.

Note: This example is only valid for liquids where the volumes are additive.

Figure 2 Mixing two samples with known densities A and B gives a mixture with a density that lies between value A and B.

### Binary and quasi-binary mixtures

Two-component mixtures are called binary mixtures. The density of the mixture is a function of its composition. Thus, the density value of a binary mixture can be used to calculate its composition with the aid of concentration tables.

Typical two-component mixtures are e.g. alcohol/water solutions, sugar/water solutions, salt/water solutions , and acids or bases dissolved in water.

Concentration determination is also possible with so-called quasi-binary mixtures. These are mixtures containing two major components and some additional ingredients which are present in very small concentrations compared to the two main components. Due to their small impact on the bulk density these additional ingredients can be ignored.

Concentration determination is also possible if a mixture contains several components and only one of them varies while all other ingredients are constant.

## Density measurement using a digital density meter

Countless analytical instruments prevail in today’s laboratories for quality and production control. However, not all of them are as easy, fast, and significant as a digital density meter.

Modern density meters require very little sample, do not change the sample’s composition, are not operator-dependent, reduce chemical consumption, and perform at the utmost precision.

The sample is introduced into a U-shaped tube that is electronically excited to oscillate at its characteristic frequency. The characteristic frequency changes depending on the density of the filled sample. Via a precise measurement of the characteristic frequency the density of the sample is determined. Due to the high temperature dependency of density, the measuring cell has to be accurately temperature controlled.

Figure 5: Oscillating U-tube principle (U-tube filled with air or water)

Modern high-precision density meters additionally provide a viscosity correction and a reference oscillator to enable accurate results over a large range of densities, temperatures, and viscosities.
The U-tube method is used to measure the true density of fluids.

### True density

Density (“True Density”) is defined as mass per volume. As mass is independent of external conditions, such as buoyancy in air or gravity, it corresponds to weight in vacuo. With the oscillating U-tube principle the true density of liquids or gases is measured. The unit of the true density is kg/m³ or g/cm³.

### Temperature dependence

Please note that the density of liquids and gases is highly temperature-dependent, as the volume of a substance changes with temperature.

In terms of a density measurement this means that precise temperature control is crucial to determine precise density values. However, the impact of temperature on density depends on the liquid’s characteristics and can vary extensively.

Example: An accurate density value in the 4th digit requires the measuring temperature to be held constant within ±0.2 °C for pure water, whereas a water-ethanol solution (30 % v/v) requires temperature stability within ±0.09 °C.

### Conversion of density into concentration units

The density of a mixture can be converted into a concentration unit using a suitable concentration table, or more generally, a conversion formula. Nearly all binary solutions can be characterized by using density-related functions and concentration tables . These tables may either come from literature or individual experimental data.

The concentration of the solutions or mixtures is often expressed in terms of the solution’s percentage composition on weight or volume basis.

QuantityUnit
Ethanol TablesAOAC 60 °F, HM C&E, IUPAC, KAEMPF, OIML, OIML-ITS-90, Proof 60 °F, Canadian Excise Alcohol Table, OIML-Shusei-Do % v/v (Vol.-%), % w/w
(% m/m, Gew.-%), °Proof
Extract/Sugar TablesBaumé, Concentration Sugar, Mass Concentration Sugar°Baumé, °Brix, °Plato,
g/L, kg/m³
Acid/Base TablesHCl, HNO₃, H₃PO₄, NaOH, H₂SO₄% w/w (% m/m, Gew.-%), mol/L, N
API FunctionsAPI Gravity 15 °C / 20 °C / 29.5 °C /
60 °F, API Density 15 °C / 20 °C /
29.5 °C / 60 °F, API Input Quantity, API Product Group, API Specific Gravity
15 °C / 20 °C / 29.5 °C / 60 °F
g/cm³, kg/m³, lb/gal, g/mL

Table 1: Preferred units for different quantities in several industries

The theoretical definition for concentrations is % weight/weight or % mass/mass, abbreviated % w/w or % m/m and meaning gram of the component per 100 grams of the solution.

Example:

10 % w/w sodium chloride means that 100 grams of solution contain 10 grams of sodium chloride (NaCl) or that there are 10 grams of NaCl in 100 grams of the solution. Temperature can be ignored here, since temperature does not have any influence on weight or mass.

While most industries use % weight/weight, other industry branches might report the concentration in % volume/volume, abbreviated % v/v, a temperature-dependent unit. Please note that due to thermal expansion of the volume the evaluation temperature has to be defined.

Example:

In the pharmaceutical industry e.g. a disinfection solution contains 70 % ethanol (v/v). This means that there are 700 mL of ethanol in 1 L solution.

The alcohol content on a bottle of beer is therefore typically reported in % v/v. In the United States the spirits industry uses °Proof, which is % v/v of alcohol multiplied by 2, at 60 °F.

Good to know: There are several advantages of using density for concentration determination. For one, there is a very high reproducibility and repeatability in concentration over a very wide range. Whether 0 % or 100 % alcohol is contained in a binary solution, a single instrument can determine its concentration. If hydrometers are used, an entire family of them would be needed, from 5 to 7 individual hydrometers, since each of them covers a much smaller range.

In the soft drink industry the sugar concentration is often reported in °Brix or °Plato, degree saccharose or degree extract in the product.

In the Petroleum Industry products are often characterized according to their “API Gravity”. The API (short for American Petroleum Institute) came up with their own gravity term to easily characterize heavy or light petroleum products compared to water: If the API gravity (at 60 °F) of a petroleum product is greater than 10, it is “light” and floats on water, if it is less than 10, it is “heavier” than water and sinks.

## A non-linear relation

Sometimes a concentration table for a specific substance does not cover the whole concentration range of 0 % w/w to 100 % w/w. One reason might be that the solute is already present in an amount at which no more can be dissolved (“saturation”). Another reason might be that the density of the solution does not show a linear relation to the concentration value. This is the case with sulfuric acid: When determining sulfuric acid in water density measurement gives precise concentration readings in a range that covers a concentration range from 0 % to 87 %. But if the concentration of sulfuric acid is higher (87 % to 100 %), the density increases no more with increasing amounts of dissolved acid.

## Accuracy of the measurement result

The accuracy of the measuring result depends not only on the density accuracy of the measuring instrument but also on the quality of the reference data (i.e. density to concentration table) and the calculation behind the conversion function (e.g. a table or a polynomial function). The more accurate your density measurement, the more accurate your concentration determination.

Example: How to calculate accuracy
A sodium chloride (NaCl) solution is measured with an instrument with an accuracy of 0.000005 g/cm³. Two concentrations of the solution (0.5 % w/w and 1.5 % w/w NaCl) are considered as they span the expected sample concentration range. The densities of both solutions are measured and the difference ∆ between the densities is calculated.

The difference ∆ of 1.0 % w/w corresponds to a density change of 0.0071 g/cm³, thus the accuracy of the NaCl solution is: (1.0 x 0.000005)/0.00710 = 0.007 % w/w. This is the average accuracy for the overall range as defined above, and only true assuming the relation of density to concentration is linear.

You can achieve more accurate results with your user table when there are many obtained data pairs (density value and corresponding concentration) with high accuracy.
Due to the non-linear behavior of real solutions the accuracy is only valid in a certain concentration range.

## Other applications of density measurement

Material characterization: In certain industries, e.g. in the petroleum industry, the density of a product is a very important characterization property. Norms like the ASTM product specifications define a certain density for every product, including the density of fuels, lubricating oils, and crude oils. However, through precise density measurement , not only the characterization of materials is possible, but also a quality check, as density is a common quality parameter.

Academic research: For academic research studies the special physical behavior of a product such as the density of gasoline is measured at different pressures and temperatures to evaluate its performance in an engine.

Quality control: Density is a traditional quality control parameter for beverages , fuels (including biodiesel and bioethanol ), liquid or viscous food samples (e.g. dairy products , malt), cosmetics, fertilizers, chemicals, and many more.

Filling volume control: Density measurement is a very important tool to help determine the volume of a product. In some cases it is easier to determine the density and weight of a product (toothpaste, mustard, creams) and calculate the filled volume.

There are several advantages if density is used for concentration determination: for example, it covers an extremely wide field of applications. As a matter of fact, a chemical company can use the same density meter for several acids, for caustics , for the quality control of incoming raw material as well as final products, and for process control in the production line. Further, the instruments are easy to use and easy to adjust.

The results are highly reproducible and show good repeatability over a wide range. Other advantages of a density meter are that generally no dilution or sample preparation is needed. Sample recovery is therefore possible as the sample remains unchanged during the measurement. There are no manual calculations needed and the method allows a quick measurement that is completely user-independent.

In contrast to alternative methods for concentration determination like titration, photometry, or gas chromatography, the handling of chemicals is reduced to a minimum; only 1 mL to 2 mL of sample have to be filled into the measuring cell, e.g. with a syringe or an automatic filling unit.

### Alternative methods

Titration: A common disadvantage of titration is that it requires a skilled operator with chemical know-how. A limited concentration range is covered, so the sample has to be diluted accordingly. Further, a special setup depending on each individual sample is needed. Another drawback of titration is the use of aggressive chemicals and the need of costly consumables. These points make this method very time-consuming and not user-friendly.

Pycnometer: A pycnometer is an affordable measuring device that gives highly accurate measurement results. However, to achieve highly accurate results a very skilled operator is needed as well as a very precise balance. Another drawback is the temperature control of the sample of which a large amount (up to 100 mL) is needed. All in all this method is quite time-consuming and it is only possible to determine the apparent density of the sample, so the buoyancy of air has to be compensated.

Hydrometer: Hydrometers are affordable and provide a very fast concentration measurement. However, the scale can easily be misread and a temperature correction is necessary. As is the case with pycnometers, a large sample amount is needed and the hydrometer can break easily.

Hydrostatic balance: Hydrostatic balances are used in special facilities, like the National Bureau of Standards, because they are incredibly precise and reliable. However, a hydrostatic balance is a huge, complex apparatus with a concrete foundation that needs to be insulated and extensively temperature-controlled. This “instrument” is very expensive and not suitable for the standard industry. Measurements are immensely time-consuming and might take 1 to 2 days.