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Parts-per notation

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(Redirected fromParts per trillion)

Set of units to describe small values

"Parts per billion" redirects here. For the film, seeParts per Billion.

Fluorescein aqueous solutions, diluted from10000 to 1 part per million in intervals of ten-fold dilution. At10000 ppm the solution is a deep red colour. As the concentration decreases, the colour becomes greenish-orange, then a vibrant yellowish-green, with the final 1 ppm sample a very pale yellowish-green.

Inscience andengineering, theparts-per notation is a set of pseudo-units to describe the small values of miscellaneousdimensionless quantities, e.g.mole fraction ormass fraction.

Since thesefractions are quantity-per-quantity measures, they are pure numbers with no associatedunits of measurement. Commonly used are

parts-per-million – ppm,10⁻⁶
parts-per-billion – ppb,10⁻⁹
parts-per-trillion – ppt,10⁻¹²
parts-per-quadrillion – ppq,10⁻¹⁵

This notation is not part of theInternational System of Units (the SI system) and its meaning is ambiguous. In chemistry, ambiguity arises because parts-per notation is able to be used to refer to amole fraction and amass fraction, which are unequal depending on themolar mass of the substance.

Applications

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Parts-per notation is often used describing dilute solutions inchemistry, for instance, the relative abundance of dissolved minerals or pollutants inwater. The quantity "1 ppm" can be used for a mass fraction if a water-borne pollutant is present at one-millionth of agram per gram of sample solution. When working withaqueous 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 inphysics andengineering to express the value of various proportional phenomena. For instance, a special metal alloy might expand 1.2 micrometers permeter of length for everydegree Celsius and this would be expressed as"α = 1.2 ppm/°C". Parts-per notation is also employed to denote the change, stability, oruncertainty in measurements. For instance, the accuracy of land-survey distance measurements when using alaser rangefinder might be 1 millimeter per kilometer of distance; this could be expressed as "Accuracy = 1 ppm."^[a]

Parts-per notations are all dimensionless quantities: in mathematical expressions, the units of measurement always cancel. In fractions like "2 nanometers per meter"(2 nm/m = 2 × 10⁻⁹ = 2 ppb = 2 ×0.000000001), so thequotients are pure-numbercoefficients with positive values less than or equal to 1. When parts-per notations, including thepercent 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").^[1]

Parts-per notations may be expressed in terms of any unit of the same measure. For instance, theexpansion coefficient of somebrass 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 length.^[b]Similarly, ametering pump that injects a trace chemical into the main process line at the proportional flow rateQ_p = 12 ppm, is doing so at a rate that may be expressed in a variety of volumetric units, including125 μL/L,125 μgal/gal, 125 cm³/m³, etc.

Innuclear 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 inMHz. 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 expressions

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1 of → = ⭨ of ↓	per cent (%)	per mille (‰)	per myriad (‱)	per cent mille (pcm)	per million (ppm)	per billion (ppb)
%	1	0.1	0.01	0.001	0.0001	10⁻⁷
‰	10	1	0.1	0.01	0.001	10⁻⁶
‱	100	10	1	0.1	0.01	10⁻⁵
pcm	1,000	100	10	1	0.1	0.0001
ppm	10,000	1,000	100	10	1	0.001
ppb	10⁷	10⁶	10⁵	10,000	1,000	1

Visualisation of 1%, 1‰, 1‱, 1 pcm and 1 ppm as fractions of the large block(larger version)

One part perhundred is generally represented by thepercent sign (%) and denotesone part per 100 (10²) parts, and a value of 10⁻². This is equivalent to about fourteen minutes out of one day.

One part perthousand should generally be spelled out in full andnot as "ppt" (which is usually understood to represent "parts pertrillion"). It may also be denoted by thepermille 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 (10³) parts, and a value of 10⁻³. This is equivalent to about ninety seconds out of one day.
One part perten thousand is denoted by thepermyriad sign (‱). Although rarely used in science (ppm is typically used instead), one permyriad has an unambiguous value of one part per 10,000 (10⁴) parts, and a value of 10⁻⁴. This is equivalent to about nine seconds out of one day.
In contrast, infinance, thebasis 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).^[2]
One part perhundred thousand,per cent mille (pcm) ormilli-percent denotes one part per 100,000 (10⁵) parts, and a value of 10⁻⁵. It is commonly used inepidemiology for mortality, crime and disease prevalence rates, and nuclear reactor engineering as a unit of reactivity. Intime measurement it is equivalent to about 5 minutes out of a year; indistance measurement, it is equivalent to 1 cm of error per km of distance traversed.

One part permillion (ppm) denotes one part per 1,000,000 (10⁶) parts, and a value of 10⁻⁶. It is equivalent to about 32 seconds out of a year or 1 mm of error per km of distance traversed. Inmining, it is also equivalent to onegram permetric ton, expressed as g/t.

One part perbillion (ppb) denotes one part per 1,000,000,000 (10⁹) parts, and a value of 10⁻⁹. This is equivalent to about three seconds out of acentury.

One part pertrillion (ppt) denotes one part per 1,000,000,000,000 (10¹²) parts, and a value of 10⁻¹². This is equivalent to about thirty seconds out of every million years.

One part perquadrillion (ppq) denotes one part per 1,000,000,000,000,000 (10¹⁵) parts, and a value of 10⁻¹⁵. This is equivalent to about two and a half minutes out of theage of the Earth (4.5 billion years). Although relatively uncommon in analytical chemistry, measurements at the ppq level are sometimes performed.^[3]

Criticism

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Although theInternational Bureau of Weights and Measures (an international standards organization known also by itsFrench-language initials BIPM) recognizes the use of parts-per notation, it is not formally part of theInternational System of Units (SI).^[1] Note that although "percent" (%) is not formally part of the SI, both the BIPM and theInternational 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.^[1]^[4] According toIUPAP, "a continued source of annoyance to unit purists has been the continued use of percent, ppm, ppb, and ppt".^[5]^[6] AlthoughSI-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 scales

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Main article:Long and short scales

Because thenamed 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".^[7]

Thousand vs. trillion

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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.^[6]

Mass fraction vs. mole fraction vs. volume fraction

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Another problem of the parts-per notation is that it may refer tomass fraction,mole fraction orvolume 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.^[6] 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 gasCFC-11 in air (Molar mass of CFC-11 / Mean molar mass of air = 137.368 / 28.97 = 4.74). For volume fraction, the suffixV orv is sometimes appended to the parts-per notation (e.g. ppmV, ppbv, pptv).^[8] To distinguish the mass fraction from volume fraction or mole fraction, the letterm (for mass, but is ambiguous with mole fraction) orw (standing for weight) is sometimes added to the abbreviation (e.g. ppmw, ppbw).^[6]

SI-compliant expressions

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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
Astrain of...	2cm/m	2 parts per hundred	2%^[9]	2 × 10⁻²
A sensitivity of...	2mV/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...	2μV/V	2 parts per million	2 ppm	2 × 10⁻⁶
A sensitivity of...	2nV/V	2 parts per billion!	2 ppb!	2 × 10⁻⁹
A sensitivity of...	2pV/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
Atemperature 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 partdimensionless quantities; that is, the units of measurement factor out in expressions like "1 nm/m" (1 nm/m =1 × 10⁻⁹) so theratios are pure-numbercoefficients with values less than 1.

Uno (proposed dimensionless unit)

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Because of the cumbersome nature of expressing certain dimensionless quantities per SI guidelines, theInternational 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.^[5] In 2004, a report to theInternational 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".^[10] To date, the uno has not been adopted by anystandards organization.

Footnotes

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^This is a simplified explanation. Laser rangefinders typically have a measurement "granularity" of one to ten millimeters; thus, the complete specification for distance measurement accuracy might read as follows: "Accuracy±(1 mm + 1 ppm)". Consequently, a distance measurement of only a few meters would still have an accuracy of ±1 mm in this example.
^In the particular case of coefficient of thermal expansion, the change to inches (one of theU.S. customary units) is typically also accompanied by a change todegrees Fahrenheit. Since a Fahrenheit-sized interval of temperature is only⁠ 5 /9⁠ that of a Celsius-sized interval, the value is typically expressed as10.4 (μin/in)/°F rather than18.7 (μin/in)/°C.

References

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^^a ^b ^c"Stating values of dimensionless quantities, or quantities of dimension one".BIPM. § 5.3.7.
^"Basis Points (BPS)".Corporate Finance Institute.
^Measurements ofdioxin are routinely made at thesub-ppq level. TheU.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 of146000 trips around the world.
^Quantities and units. Part 0:General principles, ISO 31-0:1992.
^^a ^bPetley, Brian W. (September 1998)."Report on Recent Committee Activities on Behalf of IUPAP to the 1999 IUPAP General Assembly".Archived from the original on 2017-08-15. Retrieved2017-08-15.
^^a ^b ^c ^dSchwartz, S.E.; Warneck, P. (1995)."Units for use in atmospheric chemistry (IUPAC Recommendations 1995)"(PDF).Pure and Applied Chemistry.67 (8–9):1377–1406.doi:10.1351/pac199567081377.S2CID 7029702.
^NIST:Rules and Style Conventions for Expressing Values of Quantities: 7.10.3 ppm, ppb, and ppt.
^Beychok, Milton R. (2005). "Air dispersion modeling conversions and formulas".Fundamentals of Stack Gas Dispersion (4th ed.). Milton R. Beychok.ISBN 0964458802.
^According toBIPM'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, iscontrary to Wikipedia'sManual of Style, and is not observed here.
^Consultative Committee for Units (13–14 May 2004)."Report of the 16th meeting (13–14 May 2004) to the International Committee for Weights and Measures, of the International Bureau of Weights and Measures"(PDF). Archived fromthe original(PDF) on 2014-03-10.