Parts-per notation

One part per trillion (1 ppt) is a proportion equivalent to one-twentieth of a drop of water diluted into a two-meter-deep, Olympic-size swimming pool.

Parts-per notation is used, especially in science and engineering, to denote relative proportions in measured quantities; particularly in low-value (high-ratio) proportions at the parts-per-million (ppm) 10^–6, parts-per-billion (ppb) 10^–9, and parts-per-trillion (ppt) 10^–12 level. Since parts-per notations are quantity-per-quantity measures, they are known as dimensionless quantities; that is, they are pure numbers with no associated units of measurement. In regular prose, parts-per notations generally take the literal “parts per” meaning of a comparative ratio. However, in mathematical expressions, parts-per notations function as coefficients with values less than 1.

Overview

Parts-per notation is often used in the measure of dilutions (concentrations) in chemistry; for instance, for measuring the relative abundance of dissolved minerals or pollutants in water. The expression “1 ppm” means a given property exists at a relative proportion of one part per million parts examined, as would occur if a water-borne pollutant was present at a concentration of one-millionth of a gram per gram of sample solution.

Similarly, parts-per notation is used also in physics and engineering to express the value of various proportional phenomena. For instance, a special metal alloy might expand 1.2 micrometers per meter of length for every degree Celsius and this would be expressed as “α = 1.2 ppm/°C.” Parts-per notation is also employed to denote the change, stability, or uncertainty in measurements. For instance, the accuracy of land-survey distance measurements when using a laser rangefinder might be 1 millimeter per kilometer of distance; this could be expressed as “Accuracy = 1 ppm.”

The above parts-per notations are all dimensionless quantities; that is, in mathematical expressions, the units of measurement always cancel in fractions like “2 nanometers per meter” (2 nm/m = 2 nano = 2 × 10^–9 = 2 ppb = 2 × 0.000000001) so the quotients are pure-number coefficients with positive values less than 1. When parts-per notations, including the percent symbol (%), are used in regular prose (as opposed to mathematical expressions), they are still pure-number dimensionless quantities, however, they generally take the literal “parts per” meaning of a comparative ratio (e.g., “2 ppb” would generally be interpreted as “two parts in a billion parts”).

Note that although the International Bureau of Weights and Measures (an international standards organization known also by its French-language initials BIPM) recognizes the use of parts-per notation, it is not formally part of the International System of Units (SI). Consequently, according to IUPAP, “a continued source of annoyance to unit purists has been the continued use of percent, ppm, ppb, and ppt.” Also, because the named numbers starting with a “billion” have different values in different countries, the BIPM suggests avoiding the use of “ppb” and “ppt” to prevent misunderstanding. Nevertheless, parts-per notation, particularly the expression “ppm”, remains widely used in technical disciplines because of its convenience in denoting dimensionless quantities. See Alternatives to parts-per notation, below.

Parts-per expressions

• One part per hundred is generally represented by the percent (%) symbol and denotes one part per 100 parts, one part in 10², and a value of 1 × 10^–2. This is equivalent to one drop of water diluted into 5 milliliters (one spoon-full) or about fifteen minutes out of one day.

• One part per thousand is generally spelled out in full and not as “ppt” (which is usually understood to represent “parts per trillion”). It may also be denoted by the permille (‰) symbol. Note however, that specific disciplines such as the analysis of ocean water salt concentration and educational exercises occasionally use the “ppt” abbreviation. “One part per thousand” denotes one part per 1000 parts, one part in 10³, and a value of 1 × 10^–3. This is equivalent to one drop of water diluted into 50 milliliters (ten spoon-fulls), or about one and a half minutes out of one day.

• One part per ten thousand is denoted by the permyriad (‱) symbol. It is used almost exclusively in finance, where it is known as the basis point and is typically used to denote fractional changes in percentages. For instance, a change in an interest rate from 5.15% to 5.35% would be denoted as a change of 20 basis points or 20 ‱. Although rarely used in science (ppm is typically used instead), one permyriad has an unambiguous value of one part per 10,000 parts, one part in 10⁴, and a value of 1 × 10^–4. This is equivalent to one drop of water diluted into half a liter, or about nine second out of one day.

• One part per million (ppm) denotes one part per 1,000,000 parts, one part in 10⁶, and a value of 1 × 10^–6. This is equivalent to one drop of water diluted into 50 liters (roughly the fuel tank capacity of a compact car), or about thirty seconds out of a year.

• One part per billion (ppb) denotes one part per 1,000,000,000 parts, one part in 10⁹, and a value of 1 × 10^–9. This is equivalent to 1 drop of water diluted into 250 chemical drums (50 m³), or about three seconds out of a century.

• One part per trillion (ppt) denotes one part per 1,000,000,000,000 parts, one part in 10¹², and a value of 1 × 10^–12. This is equivalent to 1 drop of water diluted into 20, two-meter-deep Olympic-size swimming pools (50,000 m³), or about three seconds out of every hundred thousand years.

• One part per quadrillion (ppq) denotes one part per 1,000,000,000,000,000 parts, one part in 10¹⁵, and a value of 1 × 10^–15. This is equivalent to 1 drop of water diluted into a cube of water measuring approximately 368 meters on a side (fifty million cubic meters, which is a cube about as tall as the roof of the Empire State Building), or two and a half minutes out of the age of the Earth (4.5 billion years). Although relatively uncommon in analytic chemistry, measurements at the ppq level are performed.

Alternatives to parts-per notation

SI-compliant expressions

In the English language, named numbers have a consistent meaning only up to “million”. Starting with “billion”, there are two numbering conventions: the “long” and “short” scales, and “billion” can mean either 10⁹ or 10¹².

For most of the 19th and 20th centuries, the United Kingdom uniformly used the long scale, while the United States of America used the short scale, so that the two systems were often referred to as “British” and “American” usage respectively. Today, the UK uses the short scale exclusively in official and mass media usage and, although some long-scale usage still continues, the terms “British” and “American” no longer reflect usage (see also Long and short scales). However, the long scale is dominant in many non-English-speaking areas, including continental Europe and Spanish-speaking countries in Latin America (see also Names of large numbers).

Although the BIPM recognizes the use of “parts per million” (ppm) to represent dimensionless quantities, it cautions that due to the above-mentioned language differences and also because “ppt” occasionally means “parts per thousand,” both “ppb” and “ppt” should be avoided to prevent misunderstanding. Clearly, this admonition would also apply to “parts per quadrillion” (ppq) for the same language-based reason. The U.S. National Institute of Standards and Technology (NIST) takes a more stringent position, stating that “the language-dependent terms ‘part per million,’ ‘part per billion,’ and ‘part per trillion’…are not acceptable for use with the SI to express the values of quantities.” Note however, that the NIST’s stated premise for its position is only partially true; “million” has only one meaning in all languages. Note too, that although “percent” (%) is not formally part of the SI, both the BIPM and the ISO, take the position that “in mathematical expressions, the internationally recognized symbol % (percent) may be used with the SI to represent the number 0.01” for dimensionless quantities.

Because parts-per notation generally has a well-understood meaning in modern, English-speaking scientific circles, and because its use simplifies the expression of dimensionless quantities, parts-per notation remains widely used in technical disciplines today. Expressions that the BIPM does not explicitly recognize as being suitable for denoting dimensionless quantities with the SI are shown in underlined green text text in the chart below.

NOTATIONS FOR DIMENSIONLESS QUANTITIES
Measure	SI units	Named parts-per ratio	Parts-per abbreviation or symbol	Value in scientific notation
A strain of…	2 cm/m	2 parts per hundred	2%	2 × 10–2
A sensitivity of…	2 mV/V	2 parts per thousand	2 ‰	2 × 10–3
A sensitivity of…	0.2 mV/V	2 parts per ten thousand	2 ‱	2 × 10–4
A sensitivity of…	2 µV/V	2 parts per million	2 ppm	2 × 10–6
A sensitivity of…	2 nV/V	2 parts per billion	2 ppb	2 × 10–9
A sensitivity of…	2 pV/V	2 parts per trillion	2 ppt	2 × 10–12
A mass concentration of…	2 mg/kg	2 parts per million	2 ppm	2 × 10–6
A mass concentration of…	2 µg/kg	2 parts per billion	2 ppb	2 × 10–9
A mass concentration of…	2 ng/kg	2 parts per trillion	2 ppt	2 × 10–12
A mass concentration of…	2 pg/kg	2 parts per quadrillion	2 ppq	2 × 10–15
A volume concentration of…	5.2 µL/L	5.2 parts per million	5.2 ppm	5.2 × 10–6
A molar concentration of…	5.24 µmol/mol	5.24 parts per million	5.24 ppm	5.24 × 10–6
A stability of…	1 (µA/A)/min.	1 part per million per min.	1 ppm/min.	1 × 10–6/min.
A change of…	5 nΩ/Ω	5 parts per billion	5 ppb	5 × 10–9
An uncertainty of…	9 µg/kg	9 parts per billion	9 ppb	9 × 10–9
A shift of…	1 nm/m	1 part per billion	1 ppb	1 × 10–9
A strain of…	1 µm/m	1 part per million	1 ppm	1 × 10–6
A temperature coefficient of…	0.3 (µHz/Hz)/°C	0.3 part per million per °C	0.3 ppm/°C	0.3 × 10–6/°C
A frequency change of…	0.35 × 10–9 ƒ	0.35 part per billion	0.35 ppb	0.35 × 10–9

Note that the notations in the “SI units” column above are all dimensionless quantities; that is, the units of measurement cancel in expressions like “1 nm/m” (1 nm/m = 1 nano = 1 × 10^–9) so the quotients are pure-number coefficients with values less than 1.

Uno

Because of the cumbersome nature of expressing certain dimensionless quantities per SI guidelines, the International Union of Pure and Applied Physics (IUPAP) in 1999 proposed the adoption of the special name “uno” (symbol: U) to represent the number 1 in dimensionless quantities. This symbol is not to be confused with the always-italicized symbol for the variable ‘uncertainty’ (symbol: U). This unit name uno and its symbol could be used in combination with the SI prefixes to express the values of dimensionless quantities which are much less—or even greater—than one.

Common parts-per notations in terms of the uno are given in the table below.

IUPAP’s “uno” proposal

Coefficient	Parts-per example	Uno equiv.	Symbol form	Value of quantity
10–2	2%	2 centiuno	2 cU	2 × 10–2
10–3	2 ‰	2 milliuno	2 mU	2 × 10–3
10–6	2 ppm	2 microuno	2 µU	2 × 10–6
10–9	2 ppb	2 nanouno	2 nU	2 × 10–9
10–12	2 ppt	2 picouno	2 pU	2 × 10–12

In 2004, a report to the International Committee for Weights and Measures (known also by its French-language initials CIPM) stated that response to the proposal of the uno “had been almost entirely negative” and the principal proponent “recommended dropping the idea.” To date, the uno has not been adopted by any standards organization and it appears unlikely it will ever become an officially sanctioned way to express low-value (high-ratio) dimensionless quantities. The proposal was instructive, however, as to the perceived shortcomings of the current options for denoting dimensionless quantities.

Improper applications of parts-per notation

Parts-per notation may properly be used only to express true dimensionless quantities; that is, the units of measurement must cancel in expressions like “1 mg/kg” so that the quotients are pure numbers with values less than 1. Mixed-unit quantities such as “a radon concentration of 15 pCi/L” are not dimensionless quantities and may not be expressed using any form of parts-per notation, such as “15 ppt”. Other examples of measures that are not dimensionless quantities are as follows:

• Particulate matter in the air: 50 µg/m³  but not:  50 ppb. Also see Air measurements, below.
• A stepper motor/gear system that produces a motion of 1 µm/pulse  but not:  1 ppm
• Mercury vapor concentration in air: 0.6 ng/L  but not:  0.6 ppt

Note however, that it is not uncommon to express aqueous concentrations—particularly in drinking-water reports intended for the general public—using parts-per notation (2.1 ppm, 0.8 ppb, etc.) and further, for those reports to state that the notations denote milligrams per liter or micrograms per liter. Whereas “2.1 mg/L” is technically not a dimensionless quantity on the face of it, it is well understood in scientific circles that one liter of water has a mass of one kilogram and that “2.1 mg/kg” (2.1 ppm) is the true measure. The goal in all technical writing (including drinking-water reports for the general public) is to clearly communicate to the intended audience with minimal confusion. Drinking water is intuitively a volumetric quantity in the public’s mind so measures of contamination expressed on a per-liter basis are considered to be easier to grasp. Still, it is technically possible, for example, to "dissolve" more than one liter of a very hydrophilic chemical in 1 liter of water; parts-per notation would be confusing when describing its solubility in water (greater than a million parts per million), so one would simply state the volume (or mass) that will dissolve into a liter, instead.

When reporting air-borne rather than water-borne densities, a slightly different convention is used since air is approximately 1000 times less dense than water. In water, 1 µg/m³ is roughly equivalent to parts-per-trillion whereas in air, it is roughly equivalent to parts-per-billion. Note also, that in the case of air, this convention is much less accurate. Whereas one liter of water is almost exactly 1 kg; one cubic meter of air is often taken as 1.143 kg—close enough for many practical uses.

Convertibility to other units of measurement

Parts-per notations may be expressed in terms of any unit of the same measure. For instance, the coefficient of thermal expansion of a certain brass alloy, α = 18.7 ppm/°C, may be expressed as 18.7 (µm/m)/°C, or as 18.7 (µin/in)/°C; the numeric value representing a relative proportion does not change with the adoption of a different unit of measure. Similarly, a metering pump that injects a trace chemical into the main process line at the proportional flow rate Q_p = 125 ppm, is doing so at a rate that may be expressed in a variety of volumetric units, including 125 µL/L, 125 µgal/gal, 125 µ(m³)/m³, etc.

Air measurements

Often, parts-per notation is the simplest and best measure. But other times, it is less suitable or even impossible. This sometimes results in agencies such as the World Health Organization or US EPA presenting entire tables of air concentrations that use the weight/volume method, when parts-per notation might have more readily worked for many (but not all) substances present. While many contaminants can be easily described using parts-per notation, a few of them present a much more complex picture.

A case in point is when measuring gasoline (a.k.a. petrol). It's a very complex mixture which might have dozens or hundreds of types of molecules present, in air. It would require sophisticated, extremely expensive analysis to enumerate all of a given sample's molecules and give it an accurate parts-per measurement - and that measurement would probably not be comparable to any other gasoline parts-per concentration, because the petroleum mixture of the second sample would be different. (Batches of petroleum product differ considerably; as an extreme example, gasoline fractions have a far higher parts-per count - many more molecules per molecule of air - compared to a heavier petroleum product like diesel, for a given weight per volume measure at STP.) Therefore, a mass per volume concentration unit (such as mg/m³) is almost always used for petroleum vapors, as well as for certain other complex mixtures.

If one is dealing with a single known diluent (contaminant), it is easy to convert from parts-per notation to mass per volume concentration, using the substance's molecular weight in the following types of equations:

           ppm × MW                     mg/m3 × 24.45
mg/m3  =  ----------    or     ppm  =  ---------------
             24.45                           MW

For example, 1 ppm of toluene (MW 92.15) is equivalent to a 3.8 microg/m³ concentration. The constant 24.45 is the number of liters in a mole of gas at 25°C and 1 atmosphere. For different pressures and temperatures, this value needs to be recomputed.

See also

• International Bureau of Weights and Measures (BIPM)
• International Committee for Weights and Measures (CIPM)
• International Electrotechnical Commission (IEC)
• International Organization for Standardization (|SO)
• International System of Units (SI)
• Long and short scales
• Mole fraction

• Names of large numbers
• National Institute of Standards and Technology (NIST)
• Percentage (%)
• Permille (‰)
• Permyriad (‱) (Basis point)
• SI prefixes

External links

• National Institute of Standards and Technology (NIST): Home page
• International Bureau of Weights and Measures (BIPM): Home page
• Concentration Calculator and Concentration Converter