Parts-per notation: Difference between revisions

{{Redirect|Parts per billion|the film|Parts per Billion{{!}}''Parts per Billion''}}

[[File:Fluorescein (2).jpg|thumb|upright|[[Fluorescein]] aqueous solutions, diluted from 1 to 10,000 to 1 parts-per-million]] in intervals of 10 fold dilution.

==Overview Applications ==

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 solutionssolution]]s, 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&nbsp;[[micrometre|micrometers]] per [[metre|meter]] of length for every [[Celsius|degree Celsius]] and this would be expressed as {{nobr|"''[[coefficient of thermal expansion|{{mvar|α}}]]''&nbsp; {{=&nbsp;}} 1.2&nbsp; ppm/°C".}} Parts-per notation is also employed to denote the change, stability, or [[standard deviation|uncertainty]] in measurements. For instance, the accuracy of land-survey distance measurements when using a [[laser rangefinder]] might be 1&nbsp;millimeter per kilometer of distance; this could be expressed as "[[Accuracy and precision|Accuracy]]&nbsp;=&nbsp;1&nbsp;ppm."<ref>{{efn|
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:&nbsp; "Accuracy: {{nobr|±(1&nbsp; mm&nbsp; +&nbsp; 1&nbsp; ppm)".}} Consequently, a distance measurement of only a few meters would still have an accuracy of ±1&nbsp;mm in this example.</ref>

Parts-per notations are all dimensionless quantities: in mathematical expressions, the units of measurement always cancel. In fractions like "2&nbsp;nanometers per meter" {{nobr|(2&nbsp; n <s>m</s> / <s>m</s>&nbsp; {{=}} 2&nbsp; nano&nbsp; {{=}} 2&nbsp;×&nbsp;2×{{10^{^|−9}&nbsp;}} {{=}} 2&nbsp; ppb&nbsp; {{=}} 2&nbsp; ×&nbsp; {{val|0.000,000,001}}),}} so the [[quotient]]s are pure-number [[coefficient]]s with positive values less than or equal to&nbsp;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&nbsp;ppb" would generally be interpreted as "two parts in a billion parts").<ref name="BIPM">{{cite web |title=Stating values of dimensionless quantities, or quantities of dimension one |at=§&nbsp;5.3.7 |website=BIPM: [|url=http://www.bipm.org/en/publications/si-brochure/section5-3-7.html 5.3.7 ''Stating values of dimensionless quantities, or quantities of dimension one]''].}}</ref>

Parts-per notations may be expressed in terms of any unit of the same measure. For instance, the [[coefficient of thermal expansion|expansion coefficient]] of a certainsome [[brass]] alloy, ''{{nobr|[[coefficient of thermal expansion|{{mvar|α''&nbsp;}}]] {{=&nbsp;}} 18.7&nbsp; ppm/°C,}} may be expressed as 18.7&nbsp;([[micrometre|µmμm]]/[[meter|m]])/°C, or as 18.7&nbsp;(µμ [[inch|in]]/[[inch|in]])/°C; the numeric value representing a relative proportion does not change with the adoption of a different unit of length.<ref>{{efn|
In the particular case of coefficient of thermal expansion, the change to inches (one of the [[United States customary units|U.S. customary units]]) is typically also accompanied by a change to [[Fahrenheit|degrees Fahrenheit]]. Since a Fahrenheit-sized interval of temperature is only 5/{{sfrac| 5 |9}} that of a Celsius-sized interval, the value is typically expressed as {{nobr|10.4&nbsp; (µμ [[inch|in]]/[[inch|in]])/°F}} rather than {{nobr|18.7&nbsp; (µμ [[inch|in]]/[[inch|in]])/°C.</ref> }}
}}
Similarly, a [[metering pump]] that injects a trace chemical into the main process line at the proportional flow rate ''{{nobr|[[volumetric flow rate|{{mvar|Q}}]]<{{sub>|p</sub>''&nbsp;}} {{=&nbsp;125&nbsp;}} 12 ppm,}} is doing so at a rate that may be expressed in a variety of volumetric units, including {{nobr|125&nbsp;µ μ[[litre|L]]/[[litre|L]],}} {{nobr|125&nbsp;µ μ [[Gallon|gal]] / [[Gallon|gal]],}} 125&nbsp;cm³/[[cubic metre|m³]], etc.

{| class="wikitable" style="font-size:85%;line-height:0.9;text-align:center;margin-left:1ex;float:right;"

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!per<br />1&thinsp;000mille<br />(‰)

!per<br />10&thinsp;,000<br />(‱)

!per<br />100&thinsp;,000<br />(pcm)

|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}}

|1 ,000||100||10||1||0.1||0.0001

|10 ,000||1 ,000||100||10||1||0.001

*''One part per [[100 (number)|hundred]]'' is generally represented by the [[percent sign]] (%) and denotes [[percentage|one part [[percentage|per 100 ({{10^|2}}) parts]], one part in 10², and a value of 1&nbsp;×&nbsp;{{10⁻²^|-2}}. This is equivalent to about fifteenfourteen minutes out of one day.

*''One part per [[1000 (number)|thousand]]'' should generally be spelled out in full and '''not''' as "ppt" (which is usually understood to represent "parts per [[trillion (short scale)|trillion]]"). It may also be denoted by the [[per mille|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 1000 parts1,000 one part in ({{10^{^|3}}}) parts, and a value of 1&nbsp;×&nbsp;{{10⁻³^|-3}}. This is equivalent to about one and a halfninety minutesseconds 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 ({{val10^|100004}}) parts, one part in 10⁴, and a value of 1&nbsp;×&nbsp;{{10⁻⁴^|-4}}. This is equivalent to about nine seconds out of one day. <br/>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).<!--NOTE TO EDITORS:

--> 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>{{citeCite web |url=https://corporatefinanceinstitute.com/resources/knowledge/financefixed-income/basis-points-bps-definition/|title=What are Basis Points (BPS)?|website=Corporate Finance Institute}}</ref>

* ''One part per [[100,000|hundred thousand]]'', ''[[per cent mille]]'' ('''pcm''') or ''milli-percent'' denotes one part per {{val|100000}} parts100, one part in 10⁵, {{sfrac|{{val|100000}}}} × 100% = 0.001%000 (or 1% = {{val10^|10005}} pcm) parts, and a value of 1&nbsp;×&nbsp;{{10⁻⁵^|-5}}. It is commonly used in [[epidemiology]] for mortality, crime and disease prevalence rates, and nuclear reactor engineering as a unit of reactivity. In [[time measurement]] it is equivalent to about 5 minutes out of a year; in [[distance measurement]], it is equivalent to 1&nbsp;cm of error per km of distance traversed.

* ''One part per [[1,000,000|million]]'' ('''ppm''') denotes one part per {{val|1000000}} parts1, one part in 10⁶000, {{sfrac|{{val|1000000}}}} × 100% = 0.0001%000 (or 1% = {{val10^|100006}} ppm) parts, and a value of 1&nbsp;×&nbsp;{{10⁻⁶^|-6}}. It is equivalent to about 32 seconds out of a year or 1&nbsp;mm of error per km of distance traversed. In [[mining]], it is also equivalent to one [[gram]] per [[metric ton]], expressed as g/t.

* ''One part per [[1000000000 (number)1,000,000,000|billion]]'' ('''ppb''') denotes one part per {{val|1000000000}} parts1, one part in 10⁹000, {{sfrac|{{val|1000000000}}}} × 100% = {{val|0.0000001}}%000,000 (or 1% = {{val10^|100000009}} ppb) parts, and a value of 1&nbsp;×&nbsp;{{10⁻⁹^|-9}}. This is equivalent to about three seconds out of a [[century]].

* ''One part per [[Orders of magnitude (numbers)#1012|trillion]]'' ('''ppt''') denotes one part per 1,000,000,000,000 ({{val10^|100000000000012}}) parts, one part in 10¹², and a value of 1&nbsp;×&nbsp;{{10⁻¹²^|-12}}. This is equivalent to about thirty seconds out of every million years.

* ''One part per [[Orders of magnitude (numbers)#1015|quadrillion]]'' ('''ppq''') denotes one part per 1,000,000,000,000,000 ({{val10^|100000000000000015}}) parts, one part in 10¹⁵, and a value of 1&nbsp;×&nbsp;{{10⁻¹⁵^|-15}}. 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>

Paper thickness = 0.0030 inch = 0.0000762 m (0.003 × 25.4 mm/inch ÷ 10001,000 mm/m)

Although the [[International Bureau of Weights and Measures]] (an international standards organization known also by its [[France|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 "[[Percentage|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 [[International Union of Pure and Applied Physics|IUPAP]], ''"a continued source of annoyance to unit purists has been the continued use of percent, ppm, ppb, and ppt"''.<ref name="IUPAP">{{cite web |title=Report on Recent Committee Activities on Behalf of IUPAP to the 1999 IUPAP General Assembly |first=Brian W. |last=Petley |date=September 1998 |url=http://archive.iupap.org/commissions/interunion/iu14/ga-99.html |access-date=2017-08-15 |url-status=live |archive-url=https://web.archive.org/web/20170815113700/http://archive.iupap.org/commissions/interunion/iu14/ga-99.html |archive-date=2017-08-15}}</ref> Although [[#SI-compliant expressions|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.

Because the [[names of large numbers|named numbers]] starting with a "[[Long and short scales|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: ''[http://physics.nist.gov/Pubs/SP811/sec07.html#7.10.3 Rules and Style Conventions for Expressing Values of Quantities: 7.10.3 ppm, ppb, and ppt]''.</ref>

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>{{cite web |title=Units |series=Introduction to Greengreen Engineeringengineering |date=2012-08-23 |df=dmy-all |publisher=[[University of Virginia]] |url=https://pages.shanti.virginia.edu/CE_2050_F12/2012/08/23/units/}}</ref>

The usage of the parts-per notation is generally quite fixed insidewithin mosteach specific branchesbranch of science, but often in a way that is inconsistent with its usage in other branches, leading some researchers to draw the conclusionassume that their own usage (mass/mass, mol/mol, volume/volume, mass/volume, or others) is thecorrect onlyand correctthat oneother usages are incorrect. This, inassumption turn,sometimes leads them to not specify the details of their own usage in their publications, and others may therefore misinterpret their results. For example, [[electrochemistry|electrochemists]] often use volume/volume, while [[Chemical engineering|chemical engineers]] may use mass/mass as well as volume/volume, while [[chemist]]s, 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, Manymany academic publications of otherwise excellent level fail to specify their usageuse 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.{{Citation needed|date=October 2021}}

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 shownmarked inwith <span style="color:darkred;">red text'''!'''</span>.

|align="center"|&nbsp;&nbsp;&nbsp; 2%<ref>Compliance with the SI regarding the percent symbol (%) is limited in this chart. According to the BIPM’s [http://www.bipm.org/en/publications/si-brochure/section5-3-37.html BIPM's SI brochure: Subsection 5.3.3, ''Formatting the value of a quantity''], a space is always used to separate the unit symbol from the numeric value. Notable exceptions are the unit symbols for degree, minute, and second for plane angle, °, ′, and ″ (e.g. a latitude of {{nowrap|47°&thinsp;38′&thinsp;8.8″).}} However, according to [http://www.bipm.org/en/publications/si-brochure/section5-3-7.htmlsection 5.3.7 ''Stating values of dimensionless quantities, or quantities of dimension one]'', the exception does not apply to the "%" symbol; it states as follows: ''"When it [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 [[WP:%|contrary to Wikipedia’sWikipedia's ''Manual of Style'']], and is not observed here._&nbsp;</ref>

|align="center"|2 ‰ <span style="color:darkred;">2 ‰'''!'''</span>

|align="center"|2 ‱ <span style="color:darkred;">2 ‱'''!'''</span>

|align="center"|2 [[Micro-|µμ]]V/V

|align="center"|2 parts per billion <span style="color:darkred;">2 parts per billion'''!'''</span>

|align="center"|2 ppb <span style="color:darkred;">2 ppb'''!'''</span>

|align="center"|2 parts per trillion <span style="color:darkred;">2 parts per trillion'''!'''</span>

|align="center"|2 ppt <span style="color:darkred;">2 ppt'''!'''</span>

|align="center"|2&nbsp;µgμg/kg<!--NOTE TO EDITORS: SI convention is to use the base unit kilogram in formulas and equalities. However, whenever an SI prefix, like milli (m) or micro (m or µμ), is used with the unit of mass, the unit gram (symbol g) is used. This results in a mix of units, such as mg (a thousandth of a gram) over a kilogram (a thousand grams). Thus, one microgram divided by one kilogram is one ppb.-->

|align="center"|2 parts per billion <span style="color:darkred;">2 parts per billion'''!'''</span>

|align="center"|2 ppb <span style="color:darkred;">2 ppb'''!'''</span>

|align="center"|2 parts per trillion <span style="color:darkred;">2 parts per trillion'''!'''</span>

|align="center"|2 ppt <span style="color:darkred;">2 ppt'''!'''</span>

|align="center"|2 parts per quadrillion <span style="color:darkred;">2 parts per quadrillion'''!'''</span>

|align="center"|2 ppq <span style="color:darkred;">2 ppq'''!'''</span>

|align="center"|5.2 µLμL/L

|align="center"|5.24 µmolμmol/mol

|align="center"|5.24 parts per billion <span style="color:darkred;">5.24 parts per billion'''!'''</span>

|align="center"|5.24 ppb <span style="color:darkred;">5.24 ppb'''!'''</span>

|align="center"|5.24 parts per trillion <span style="color:darkred;">5.24 parts per trillion'''!'''</span>

|align="center"|5.24 ppt <span style="color:darkred;">5.24 ppt'''!'''</span>

|align="center"|1 (µAμA/A)/[[minute|min]]

|align="center"|5 parts per billion <span style="color:darkred;">5 parts per billion'''!'''</span>

|align="center"|5 ppb <span style="color:darkred;">5 ppb'''!'''</span>

|align="center"|9 µgμg/kg

|align="center"|9 parts per billion <span style="color:darkred;">9 parts per billion'''!'''</span>

|align="center"|9 ppb <span style="color:darkred;">9 ppb'''!'''</span>

|align="center"|1 part per billion <span style="color:darkred;">1 part per billion'''!'''</span>

|align="center"|1 ppb <span style="color:darkred;">1 ppb'''!'''</span>

|align="center"|1 µmμm/m

|align="center"|0.3 (µHzμHz/Hz)/°C

|align="center"|0.35 part per billion <span style="color:darkred;">0.35 part per billion'''!'''</span>

|align="center"|0.35 ppb <span style="color:darkred;">0.35 ppb'''!'''</span>

Note that the notations in the "SI units" column above are allfor the most part [[Dimensionless quantity|dimensionless quantities]]; that is, the units of measurement factor out in expressions like "1&nbsp;nm/m" (1&nbsp;n<s>m</s>/<s>m</s>&nbsp;= 1&nbsp;nano&nbsp;= 1&nbsp;×&nbsp;10⁻⁹) so the [[quotientratio]]s are pure-number [[coefficient]]s with values less than 1.

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" /> ThisIn symbol2004, isa notreport to bethe confused[[International withCommittee thefor always-italicizedWeights symboland forMeasures]] (CIPM) stated that the variableresponse "uncertainty"to (symbol:the ''U'').proposal Thisof unitthe nameuno "unohad been almost entirely negative", and itsthe symbolprincipal couldproponent be"recommended useddropping inthe combinationidea".<ref>{{cite withweb |title=Report of the [[SI16th prefix]]esmeeting (13–14 May 2004) to expressthe International Committee for Weights and Measures, of the valuesInternational Bureau of dimensionlessWeights quantitiesand thatMeasures are|author=Consultative muchCommittee less—orfor evenUnits ''greater''—than|date=13–14 oneMay 2004 |url=http://www.bipm.org/utils/common/pdf/CCU16.pdf |url-status=dead |archive-url=https://web.archive.org/web/20140310054340/http://www.bipm.org/utils/common/pdf/CCU16.pdf |archive-date=2014-03-10}}</ref>Certain mathematicalTo functionsdate, canthe produceuno proportionalhas quantitiesnot withbeen valuesadopted greaterby thanany 1[[standards organization]]._&nbsp;</ref>

* [https://www.nist.gov/ National Institute of Standards and Technology] (NIST): [https://www.nist.gov/ Home page]

* [https://www.bipm.org/en/ International Bureau of Weights and Measures] (BIPM): [http://www.bipm.org/en/home/ Home page]