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In
science Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of ...

science
and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

engineering
, the parts-per notation is a set of pseudo-units to describe small values of miscellaneous
dimensionless quantities In dimensional analysis In engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline ...
, e.g.
mole fraction In chemistry Chemistry is the scientific discipline involved with Chemical element, elements and chemical compound, compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they underg ...
or mass fraction. Since these
fractions A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, ...
are quantity-per-quantity measures, they are pure numbers with no associated
units of measurement A unit of measurement is a definite magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematic ...
. Commonly used are parts-per-million (ppm, ), parts-per-billion (ppb, ), parts-per-trillion (ppt, ) and parts-per-quadrillion (ppq, ). This notation is not part of the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ...
(SI) system and its meaning is ambiguous.


Overview

Parts-per notation is often used describing dilute solutions in
chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in the real world. T ...

chemistry
, for instance, the relative abundance of dissolved minerals or pollutants in
water Water (chemical formula H2O) is an Inorganic compound, inorganic, transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known li ...

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 The gram (alternative spelling: gramme; SI unit symbol: g) is a metric system The metric system is a that succeeded the decimalised system based on the introduced in France in the 1790s. The historical development of these systems culm ...
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 Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

engineering
to express the value of various proportional phenomena. For instance, a special metal alloy might expand 1.2 
micrometers The micrometre (American and British English spelling differences#-re, -er, international spelling as used by the International Bureau of Weights and Measures; SI symbol: μm) or micrometer (American and British English spelling differences# ...
per
meter The metre ( Commonwealth spelling) or meter (American spelling Despite the various English dialects spoken from country to country and within different regions of the same country, there are only slight regional variations in English ...
of length for every
degree Celsius The degree Celsius is a unit of temperature on the Celsius scale, a temperature scale Scale of temperature is a methodology of calibrating the physical quantity temperature in metrology. Empirical scales measure temperature in relation to conven ...

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 Uncertainty refers to epistemic Epistemology (; ) is the branch of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, ...

uncertainty
in measurements. For instance, the accuracy of land-survey distance measurements when using a
laser rangefinder and elevation The elevation of a geographic location (geography), location is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational eq ...
might be 1 millimeter per kilometer of distance; this could be expressed as "
Accuracy In a set of measurements, accuracy is closeness of the measurements to a specific value, while precision is the closeness of the measurements to each other. ''Accuracy'' has two definitions: # More commonly, it is a description of ''systematic er ...

Accuracy
 = 1 ppm." 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 nano = 2 × 10−9 = 2 ppb = 2 × ), so the
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
s are pure-number
coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s with positive values less than or equal to 1. When parts-per notations, including the
percent In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

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").BIPM
5.3.7 ''Stating values of dimensionless quantities, or quantities of dimension one
'].
Parts-per notations may be expressed in terms of any unit of the same measure. For instance, the
coefficient of thermal expansion Thermal expansion is the tendency of matter to change its shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask ...
of a certain
brass Brass is an alloy An alloy is an admixture of metal A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appea ...

brass
alloy, ''α'' = 18.7 ppm/°C, 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. Similarly, a
metering pump A metering pump moves a precise volume of liquid in a specified time period providing an accurate volumetric flow rate In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, it ...
that injects a trace chemical into the main process line at the proportional flow rate '' Qp'' = 125 ppm, is doing so at a rate that may be expressed in a variety of volumetric units, including 125 μL/L, 125 μ gal/gal, 125 cm3/ m3, etc.


In nuclear magnetic resonance (NMR) spectroscopy

In
nuclear magnetic resonance spectroscopy Nuclear magnetic resonance spectroscopy, most commonly known as NMR spectroscopy or magnetic resonance spectroscopy (MRS), is a spectroscopic Spectroscopy is the study of the interaction between matter and Electromagnetism, electromagnetic ra ...
(NMR),
chemical shift In nuclear magnetic resonance Nuclear magnetic resonance (NMR) is a physical phenomenon A phenomenon (; plural phenomena) is an observable In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗm ...
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 The hertz (symbol: Hz) is the derived unit of frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and an ...
. 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

*''One part per
hundred 100 or one hundred (Roman numerals, Roman numeral: C) is the natural number following 99 (number), 99 and preceding 101 (number), 101. In medieval contexts, it may be described as the short hundred or five 20 (number), score in order to different ...
'' is generally represented by the
percent sign The percent sign (or per cent sign in British English British English (BrE) is the standard dialect of the English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon Engl ...
(%) and denotes
one part per 100 () parts
one part per 100 () parts
, and a value of . This is equivalent to about fifteen minutes out of one day. *''One part per
thousand 1000 or one thousand is the natural number In mathematics, the natural numbers are those used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). ...
'' should generally be spelled out in full and not as "ppt" (which is usually understood to represent "parts per
trillion A trillion is a number with two distinct definitions: *1,000,000,000,000 This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantity and probability, probabilities. Each number is given a n ...
"). It may also be denoted by the
permille Per mille (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the ...
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 () parts, and a value of . This is equivalent to about one and a half minutes out of one day. *''One part per
ten thousand The Ten Thousand ( grc, οἱ Μύριοι, ''oi Myrioi'') were a force of mercenary A mercenary, sometimes known as a soldier of fortune, is an individual who takes part in military conflict for personal profit, is otherwise an outsider to ...
'' is denoted by the
permyriad A basis point (often abbreviated as bp, often pronounced as "bip" or "beep") is (a difference of) one hundredth of a percent or equivalently one percent of one percent or one ten thousandth. A very rarely used term, permyriad, means parts per te ...
sign (‱). Although rarely used in science (ppm is typically used instead), one permyriad has an unambiguous value of one part per 10,000 () parts, and a value of . This is equivalent to about nine seconds out of one day.
In contrast, in
finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in the left corn ...

finance
, the
basis point A basis point (often abbreviated as bp, often pronounced as "bip" or "beep") is (a difference of) one hundredth of a percent or equivalently one percent of one percent or one ten thousandth. A very rarely used term, permyriad A basis point (o ...
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). * ''One part per hundred thousand'', ''
per cent mille A per cent mille or pcm is one one-thousandth of a percent. It can be thought of as a "milli-percent". It is commonly used in epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, ...
'' (pcm) or ''milli-percent'' denotes one part per 100,000 () parts, and a value of . It is commonly used in
epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined populations. It is a cornerstone of public health, and shapes policy decisions ...
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 cm of error per km of distance traversed. * ''One part per
million One million (1,000,000), or one thousand thousand, is the natural number In mathematics, the natural numbers are those used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' l ...

million
'' (ppm) denotes one part per 1,000,000 () parts, and a value of . It is equivalent to about 32 seconds out of a year or 1 mm of error per km of distance traversed. * ''One part per
billion A billion is a number with two distinct definitions: *1,000,000,000 1,000,000,000 (one billion A billion is a number with two distinct definitions: *1,000,000,000 1,000,000,000 (one billion, short scale; one thousand million or milliar ...
'' (ppb) denotes one part per 1,000,000,000 () parts, and a value of . This is equivalent to about three seconds out of a
century A century is a period of 100 years. Centuries are numbered names of numbers in English#Ordinal numbers, ordinally in English and many other languages. The word ''century'' comes from the Latin ''centum'', meaning ''one hundred''. ''Century'' is s ...

century
. * ''One part per
trillion A trillion is a number with two distinct definitions: *1,000,000,000,000 This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantity and probability, probabilities. Each number is given a n ...
'' (ppt) denotes one part per 1,000,000,000,000 () parts, and a value of . This is equivalent to about thirty seconds out of every million years. * ''One part per
quadrillion This article lists and discusses the usage and derivation of names of large numbers, together with their possible extensions. The following table lists those names of large number Numbers that are significantly larger than those typically used ...
'' (ppq) denotes one part per 1,000,000,000,000,000 () parts, and a value of . This is equivalent to about two and a half minutes out of the
age of the Earth The age of Earth is estimated to be 4.54 ± 0.05 billion A billion is a number with two distinct definitions: *1,000,000,000 1,000,000,000 (one billion A billion is a number with two distinct definitions: *1,000,000,000 1,000,000,00 ...
(4.5 billion years). Although relatively uncommon in analytical chemistry, measurements at the ppq level are sometimes performed.


Criticism

Although the
International Bureau of Weights and Measures The International Bureau of Weights and Measures (french: Bureau international des poids et mesures, BIPM) is an intergovernmental organisation An intergovernmental organization (IGO) is an organization composed primarily of sovereign state ...
(an international standards organization known also by its
French
French
-language initials BIPM) recognizes the use of parts-per notation, it is not formally part of the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ...
(SI). Note that although "
percent In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

percent
" (%) is not formally part of the SI, both the BIPM and the
International Organization for Standardization The International Organization for Standardization (ISO ) is an international standard An international standard is a technical standard A technical standard is an established norm Norm, the Norm or NORM may refer to: In academic discipline ...
(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. According to
IUPAP The International Union of Pure and Applied Physics (IUPAP ) is an international non-governmental organization A non-governmental organization, or simply an NGO, is an organization that is, generally, formed independent from government. The ...
, ''"a continued source of annoyance to unit purists has been the continued use of percent, ppm, ppb, and ppt"''. 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 scales

Because the named numbers starting with a "
billion A billion is a number with two distinct definitions: *1,000,000,000 1,000,000,000 (one billion A billion is a number with two distinct definitions: *1,000,000,000 1,000,000,000 (one billion, short scale; one thousand million or milliar ...
" 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 The National Institute of Standards and Technology (NIST) is a physical sciences Physical science is a branch of natural science that studies abiotic component, non-living systems, in contrast to life science. It in turn has many branches, e ...
(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"''.


Thousand vs. trillion

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.


Mass fraction vs. mole fraction vs. volume fraction

Another problem of the parts-per notation is that it may refer to mass fraction,
mole fraction In chemistry Chemistry is the scientific discipline involved with Chemical element, elements and chemical compound, compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they underg ...
or
volume fractionIn chemistry Chemistry is the scientific discipline involved with Chemical element, elements and chemical compound, compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they undergo ...
. Since it is usually not stated which quantity is used, it is better to write the unit as kg/kg, mol/mol or m3/m3 (even though they are all dimensionless). 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 Trichlorofluoromethane, also called freon-11, CFC-11, or R-11, is a chlorofluorocarbon Chlorofluorocarbons (CFCs) and hydrochlorofluorocarbons (HCFCs) are fully or partly halogenated Alkane, paraffin hydrocarbons that contain only carbon (C ...
in air. For volume fraction, the suffix "V" or "v" is sometimes appended to the parts-per notation (e.g. ppmV, ppbv, pptv). Unfortunately, ppbv and pptv are also often used for mole fractions (which is identical to volume fraction only for ideal gases). 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). The usage of the parts-per notation is generally quite fixed inside most specific branches of science, leading some researchers to draw the conclusion that their own usage (mass/mass, mol/mol, volume/volume, or others) is the only correct one. This, in turn, leads them to not specify their usage in their publications, and others may therefore misinterpret their results. For example, often use volume/volume, while
chemical engineers A chemical substance is a form of matter having constant chemical composition and characteristic properties. Some references add that chemical substance cannot be separated into its constituent Chemical element, elements by physical separation me ...
may use mass/mass as well as volume/volume. Many academic publications of otherwise excellent level fail to specify their usage of the parts-per notation.


SI-compliant expressions

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 !. Note that the notations in the "SI units" column above are all
dimensionless quantities In dimensional analysis In engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline ...
; that is, the units of measurement factor out in expressions like "1 nm/m" (1 nm/m = 1 nano = 1 × 10−9) so the
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
s are pure-number
coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s with values less than 1.


Uno (proposed dimensionless unit)

Because of the cumbersome nature of expressing certain dimensionless quantities per SI guidelines, the
International Union of Pure and Applied Physics The International Union of Pure and Applied Physics (IUPAP ) is an international non-governmental organization whose mission is to assist in the worldwide development of physics, to foster international cooperation in physics, and to help in the a ...
(IUPAP) in 1999 proposed the adoption of the special name "uno" (symbol: U) to represent the number 1 in dimensionless quantities. In 2004, a report to the
International Committee for Weights and Measures The General Conference on Weights and Measures (GCWM; french: Conférence Générale des Poids et Mesures, CGPM) is the supreme authority of the International Bureau of Weights and Measures The International Bureau of Weights and Measures ( ...
(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 A standards organization, standards body, standards developing organization (SDO), or standards setting organization (SSO) is an organization whose primary function is developing, coordinating, promulgating, revising, amending, reissuing, interpr ...
, and it appears unlikely that it will ever become an officially sanctioned way to express low-value (high-ratio) dimensionless quantities.


See also

*
International Electrotechnical Commission The International Electrotechnical Commission (IEC; in French: ''Commission électrotechnique internationale'') is an international standards organization A standards organization, standards body, standards developing organization (SDO), or s ...
(IEC) * Milligram per cent *
Percentage In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

Percentage
(%) 1 part in 100 *
Per mille Per mille (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of ...
(‰) 1 part in 1,000 *
Permyriad A basis point (often abbreviated as bp, often pronounced as "bip" or "beep") is (a difference of) one hundredth of a percent or equivalently one percent of one percent or one ten thousandth. A very rarely used term, permyriad, means parts per te ...
(‱) 1 part in 10,000 *
Per cent mille A per cent mille or pcm is one one-thousandth of a percent. It can be thought of as a "milli-percent". It is commonly used in epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, ...
(pcm) 1 part in 100,000 *
Per-unit systemIn the power systems analysis field of electrical engineering, a per-unit system is the expression of system quantities as fractions of a defined base unit quantity. Calculations are simplified because quantities expressed as per-unit do not chang ...


References


External links

* {{Commonscatinline * National Institute of Standards and Technology (NIST)
Home page
* International Bureau of Weights and Measures (BIPM)
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Analytical chemistry Chemical nomenclature Dimensionless numbers Environmental chemistry Mathematical terminology Measurement Metrics Physical constants Units of measurement