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Parts-per notation

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File:Fluorescein (2).jpg
Fluorescein aqueous solutions, diluted from Template:Val to 1Template:Nbsppart per million in intervals of ten-fold dilution. At Template:Val the solution is a deep red colour. As the concentration decreases the colour becomes orange, then a vibrant yellow, with the final 1Template:Nbspppm sample a very pale yellow.

In science and engineering, the parts-per notation is a set of pseudo-units to describe the small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction.

Since these fractions are quantity-per-quantity measures, they are pure numbers with no associated units of measurement. Commonly used are

This notation is not part of the International System of Units - SI system and its meaning is ambiguous.

ApplicationsEdit

Parts-per notation is often used describing dilute solutions in chemistry, for instance, the relative abundance of dissolved minerals or pollutants in water. The quantity "1 ppm" can be used for a mass fraction if a water-borne pollutant is present at one-millionth of a gram per gram of sample solution. When working with aqueous solutions, it is common to assume that the density of water is 1.00 g/mL. Therefore, it is common to equate 1 kilogram of water with 1 L of water. Consequently, 1 ppm corresponds to 1 mg/L and 1 ppb corresponds to 1 μg/L.

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 Template:Nobr 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."Template:Efn

Parts-per notations are all dimensionless quantities: in mathematical expressions, the units of measurement always cancel. In fractions like "2 nanometers per meter" Template:Nobr so the quotients are pure-number coefficients with positive values less than or equal to 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").<ref name=BIPM>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Parts-per notations may be expressed in terms of any unit of the same measure. For instance, the expansion coefficient of some brass alloy, Template:Nobr may be expressed as 18.7 (μm/m)/°C, or as 18.7 (μ in/in)/°C; the numeric value representing a relative proportion does not change with the adoption of a different unit of length.Template:Efn Similarly, a metering pump that injects a trace chemical into the main process line at the proportional flow rate Template:Nobr is doing so at a rate that may be expressed in a variety of volumetric units, including Template:Nobr Template:Nobr 125 cm3/m3, etc.

In nuclear magnetic resonance spectroscopy (NMR), chemical shift is usually expressed in ppm. It represents the difference of a measured frequency in parts per million from the reference frequency. The reference frequency depends on the instrument's magnetic field and the element being measured. It is usually expressed in MHz. Typical chemical shifts are rarely more than a few hundred Hz from the reference frequency, so chemical shifts are conveniently expressed in ppm (Hz/MHz). Parts-per notation gives a dimensionless quantity that does not depend on the instrument's field strength.

Parts-per expressionsEdit

1 of →
= ⭨
of ↓   		per
cent
(%) 		per
mille
(‰) 		per
10,000
(‱) 		per
100,000
(pcm) 		per
million
(ppm) 		per
billion
(ppb)
% 1 0.1 0.01 0.001 0.0001 Template:10^
‰ 10 1 0.1 0.01 0.001 Template:10^
‱ 100 10 1 0.1 0.01 Template:10^
pcm 1,000 100 10 1 0.1 0.0001
ppm 10,000 1,000 100 10 1 0.001
ppb Template:10^ Template:10^ Template:10^ 10,000 1,000 1

Template:Visualisation parts per.svg

  • One part per hundred is generally represented by the percent sign (%) and denotes [[percentage|one part per 100 (Template:10^) parts]], and a value of Template:10^. This is equivalent to about fourteen minutes out of one day.

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  • One part per thousand should generally be 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 sign (‰). Note however, that specific disciplines such as oceanography, as well as educational exercises, do use the "ppt" abbreviation. "One part per thousand" denotes one part per 1,000 (Template:10^) parts, and a value of Template:10^. This is equivalent to about ninety seconds out of one day.
  • One part per ten thousand is denoted by the permyriad sign (‱). Although rarely used in science (ppm is typically used instead), one permyriad has an unambiguous value of one part per 10,000 (Template:10^) parts, and a value of Template:10^. This is equivalent to about nine seconds out of one day.
    In contrast, in finance, the basis point is typically used to denote changes in or differences between percentage interest rates (although it can also be used in other cases where it is desirable to express quantities in hundredths of a percent). For instance, a change in an interest rate from 5.15% per annum to 5.35% per annum could be denoted as a change of 20 basis points (per annum). As with interest rates, the words "per annum" (or "per year") are often omitted. In that case, the basis point is a quantity with a dimension of (time−1).<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

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  • One part per million (ppm) denotes one part per 1,000,000 (Template:10^) parts, and a value of Template:10^. It is equivalent to about 32 seconds out of a year or 1 mm of error per km of distance traversed. In mining, it is also equivalent to one gram per metric ton, expressed as g/t.

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  • One part per trillion (ppt) denotes one part per 1,000,000,000,000 (Template:10^) parts, and a value of Template:10^. This is equivalent to about thirty seconds out of every million years.

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  • One part per quadrillion (ppq) denotes one part per 1,000,000,000,000,000 (Template:10^) parts, and a value of Template:10^. This is equivalent to about two and a half minutes out of the age of the Earth (4.5 billion years). Although relatively uncommon in analytical chemistry, measurements at the ppq level are sometimes performed.<ref>

Measurements of dioxin are routinely made at the sub-ppq level. The U.S. Environmental Protection Agency (EPA) currently sets a hard limit of 30 ppq for dioxin in drinking water but once recommended a voluntary limit of 0.013 ppq. Also, radioactive contaminants in drinking water, which are quantified by measuring their radiation, are often reported in terms of ppq; 0.013 ppq is equivalent to the thickness of a sheet of paper versus a journey of Template:Val trips around the world. </ref>

CriticismEdit

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).<ref name="BIPM" /> Note that although "percent" (%) is not formally part of the SI, both the BIPM and the International Organization for Standardization (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.<ref name="BIPM" /><ref>Quantities and units. Part 0: General principles, ISO 31-0:1992.</ref> According to IUPAP, "a continued source of annoyance to unit purists has been the continued use of percent, ppm, ppb, and ppt".<ref name="IUPAP">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Although SI-compliant expressions should be used as an alternative, the parts-per notation remains nevertheless widely used in technical disciplines. The main problems with the parts-per notation are set out below.

Long and short scalesEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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. The U.S. National Institute of Standards and Technology (NIST) takes the stringent position, stating that "the language-dependent terms [...] are not acceptable for use with the SI to express the values of quantities".<ref>NIST: Rules and Style Conventions for Expressing Values of Quantities: 7.10.3 ppm, ppb, and ppt.</ref>

Thousand vs. trillionEdit

Although "ppt" usually means "parts per trillion", it occasionally means "parts per thousand". Unless the meaning of "ppt" is defined explicitly, it has to be determined from the context.Template:Citation needed

Mass fraction vs. mole fraction vs. volume fractionEdit

Another problem of the parts-per notation is that it may refer to mass fraction, mole fraction or volume fraction. Since it is usually not stated which quantity is used, it is better to write the units out, such as kg/kg, mol/mol or m3/m3, even though they are all dimensionless.<ref>Template:Cite journal</ref> The difference is quite significant when dealing with gases, and it is very important to specify which quantity is being used. For example, the conversion factor between a mass fraction of 1 ppb and a mole fraction of 1 ppb is about 4.7 for the greenhouse gas CFC-11 in air (Molar mass of CFC-11 / Mean molar mass of air = 137.368 / 28.97 = 4.74). For volume fraction, the suffix "V" or "v" is sometimes appended to the parts-per notation (e.g. ppmV, ppbv, pptv).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite book</ref> However, ppbv and pptv are usually used to mean mole fractionsTemplate:Dash"volume fraction" would literally mean what volume of a pure substance is included in a given volume of a mixture, and this is rarely used except in the case of alcohol by volume.

To distinguish the mass fraction from volume fraction or mole fraction, the letter "w" (standing for "weight") is sometimes added to the abbreviation (e.g. ppmw, ppbw).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

The usage of the parts-per notation is generally quite fixed within each specific branch of science, but often in a way that is inconsistent with its usage in other branches, leading some researchers to assume that their own usage (mass/mass, mol/mol, volume/volume, mass/volume, or others) is correct and that other usages are incorrect. This assumption sometimes leads them to not specify the details of their own usage in their publications, and others may therefore misinterpret their results. For example, electrochemists often use volume/volume, while chemical engineers may use mass/mass as well as volume/volume, while chemists, the field of occupational safety and the field of permissible exposure limit (e.g. permitted gas exposure limit in air) may use mass/volume. Unfortunatelly, many academic publications of otherwise excellent level fail to specify their use of the parts-per notation, which irritates some readers, especially those who are not experts in the particular fields in those publications, because parts-per-notation, without specifying what it stands for, can mean anything.Template:Citation needed

SI-compliant expressionsEdit

SI-compliant units that can be used as alternatives are shown in the chart below. Expressions that the BIPM explicitly does not recognize as being suitable for denoting dimensionless quantities with the SI are marked with !.

Notations for dimensionless quantities
Measure SI
units 		Named
parts-per ratio
(short scale) 		Parts-per
abbreviation
or symbol 		Value in
scientific
notation
A strain of... 2 cm/m 2 parts per hundred     2%<ref>According to BIPM's SI brochure, section 5.3.7, "When [the percent symbol] is used, a space separates the number and the symbol %." This practice has not been well adopted with regard to the % symbol, is contrary to Wikipedia's Manual of Style, and is not observed here.</ref> 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 fraction of... 2 mg/kg 2 parts per million 2 ppm 2 × 10−6
A mass fraction of... 2 μg/kg 2 parts per billion ! 2 ppb ! 2 × 10−9
A mass fraction of... 2 ng/kg 2 parts per trillion ! 2 ppt ! 2 × 10−12
A mass fraction of... 2 pg/kg 2 parts per quadrillion ! 2 ppq ! 2 × 10−15
A volume fraction of... 5.2 μL/L 5.2 parts per million 5.2 ppm 5.2 × 10−6
A mole fraction of... 5.24 μmol/mol 5.24 parts per million 5.24 ppm 5.24 × 10−6
A mole fraction of... 5.24 nmol/mol 5.24 parts per billion ! 5.24 ppb ! 5.24 × 10−9
A mole fraction of... 5.24 pmol/mol 5.24 parts per trillion ! 5.24 ppt ! 5.24 × 10−12
A stability of... 1 (μA/A)/min 1 part per million per minute 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 for the most part dimensionless quantities; that is, the units of measurement factor out in expressions like "1 nm/m" (1 nm/m =1 × 10−9) so the ratios are pure-number coefficients with values less than 1.

Uno (proposed dimensionless unit)Template:AnchorEdit

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.<ref name="IUPAP" /> In 2004, a report to the International Committee for Weights and Measures (CIPM) stated that the response to the proposal of the uno "had been almost entirely negative", and the principal proponent "recommended dropping the idea".<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> To date, the uno has not been adopted by any standards organization.

FootnotesEdit

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See alsoEdit

ReferencesEdit

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External linksEdit

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