Parts-per notation

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In
science Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earli ...
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 specializ ...
, the parts-per notation is a set of pseudo-units to describe small values of miscellaneous
dimensionless quantities A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate di ...
, e.g.
mole fraction In chemistry, the mole fraction or molar fraction (''xi'' or ) is defined as unit of the amount of substance, amount of a constituent (expressed in mole (unit), moles), ''ni'', divided by the total amount of all constituents in a mixture (also expr ...
or mass fraction. Since these fractions are quantity-per-quantity measures, they are pure numbers with no associated
units of measurement A unit of measurement is a definite magnitude (mathematics), magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can ...
. 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 wid ...
(SI) system and its meaning is ambiguous.

# Overview

Parts-per notation is often used describing dilute solutions in
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, properties ...
, for instance, the relative abundance of dissolved minerals or pollutants in
water Water (chemical formula ) 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 living ...
. 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 (originally gramme; SI unit symbol g) is a unit of mass in the International System of Units (SI) equal to one one thousandth of a kilogram. Originally defined as of 1795 as "the absolute weight of a volume Volume is a measure ...
per gram of sample solution. When working with
aqueous solution An aqueous solution is a Solution (chemistry), solution in which the solvent is water. It is mostly shown in chemical equations by appending (aq) to the relevant chemical formula. For example, a solution of table salt, or sodium chloride (NaCl), ...
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 Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
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 specializ ...
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 (British English, British spelling) or meter (American English, American spelling; American and British English spelling differences#-re, -er, see spelling differences) (from the French unit , from the Greek language, Greek noun , "m ...
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 Uncertainty refers to Epistemology, epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially ...
in measurements. For instance, the accuracy of land-survey distance measurements when using a
laser rangefinder A laser rangefinder, also known as a laser telemeter, is a Rangefinding telemeter, rangefinder that uses a laser beam to determine the distance to an object. The most common form of laser rangefinder operates on the time of flight principle by s ...
might be 1 millimeter per kilometer of distance; this could be expressed as "
Accuracy Accuracy and precision are two measures of ''observational error''. ''Accuracy'' is how close a given set of measurements (observations or readings) are to their ''true value'', while ''precision'' is how close the measurements are to each other ...
= 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, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division (mathematics), division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to a ...
s are pure-number
coefficient In mathematics, a coefficient is a multiplicative factor in some Summand, term of a polynomial, a series (mathematics), series, or an expression (mathematics), expression; it is usually a number, but may be any expression (including variables su ...
s with positive values less than or equal to 1. When parts-per notations, including the
percent In mathematics, a percentage (from la, per centum, "by a hundred") is a number or ratio expressed as a fraction (mathematics), fraction of 100. It is often Denotation, denoted using the percent sign, "%", although the abbreviations "pct.", "p ...
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, area, volume, and density in response to a change in temperature, usually not including phase transitions. Temperature is a monotonic function of the average molecular kinetic ...
of a certain
brass Brass is an alloy An alloy is a mixture of chemical elements of which at least one is a metal. Unlike chemical compounds with metallic bases, an alloy will retain all the properties of a metal in the resulting material, such as electrical ...
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. Delivery of fluids in precise adjustable flow rates is sometimes called metering. The term "metering pump" is based on the ap ...
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 Spectroscopy, spectroscopic technique to observe local magnetic fields around Atomic nucleus, atomic nuclei. The sampl ...
(NMR),
chemical shift In nuclear magnetic resonance (NMR) spectroscopy, the chemical shift is the resonance, resonant frequency of an atomic nucleus relative to a standard in a magnetic field. Often the position and number of chemical shifts are diagnostic of the str ...
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 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 (sometimes per cent sign in British English) is the symbol used to indicate a percentage, a number or ratio as a fraction (mathematics), fraction of 100. Related signs include the permille (per thousand) sign and the Basis p ...
(%) and denotes one part per 100 () parts, and a value of . This is equivalent to about fourteen minutes out of one day. *''One part per
thousand 1000 or one thousand is the natural number following 999 (number), 999 and preceding 1001 (number), 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period decimal mark#Digit grouping, separating ...
'' should generally be spelled out in full and not as "ppt" (which is usually understood to represent "parts per
trillion ''Trillion'' is a number with two distinct definitions: *1,000,000,000,000, i.e. one million 1,000,000, million, or (ten to the twelfth Exponentiation, power), as defined on the long and short scales, short scale. This is now the meaning in bot ...
"). 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 () parts, and a value of . This is equivalent to about ninety seconds out of one day. *''One part per
ten thousand The Ten Thousand ( grc, οἱ Μύριοι, ''oi Myrioi'') were a force of mercenary units, mainly Ancient Greece, Greeks, employed by Cyrus the Younger to attempt to wrest the throne of the Achaemenid Empire, Persian Empire from his brother, Ar ...
'' is denoted by the
permyriad A basis point (often abbreviated as bp, often pronounced as "bip" or "beep") is one hundredth of 1 percentage point. The related term ''#Permyriad, permyriad'' means one hundredth of 1 percent. Changes of interest rates are often stated in basis ...
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 the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of Production (economics), production, Distribution (economics), distribution, and Consumption (economics) ...
, the
basis point A basis point (often abbreviated as bp, often pronounced as "bip" or "beep") is one hundredth of 1 percentage point. The related term '' permyriad'' means one hundredth of 1 percent. Changes of interest rates are often stated in basis points. If ...
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, and in nuclear reactor engineering as a unit of reactivity. Epidemiology Statistics of Crime statistics, c ...
'' (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 a defined population. It is a cornerstone of public health, and shapes policy decision ...
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 Length measurement, distance measurement, or range measurement (ranging) refers to the many ways in which length, distance, or slant range, range can be measurement, measured. The most commonly used approaches are the rulers, followed by transit ...
, 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 numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' larg ...
'' (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 Billion is a word for a large number, and it has two distinct definitions: * 1,000,000,000, i.e. one thousand million, or (ten to the ninth power), as defined on the short scale. This is its only current meaning in English. * 1,000,000,000,000, ...
'' (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 ...
. * ''One part per
trillion ''Trillion'' is a number with two distinct definitions: *1,000,000,000,000, i.e. one million 1,000,000, million, or (ten to the twelfth Exponentiation, power), as defined on the long and short scales, short scale. This is now the meaning in bot ...
'' (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'' (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 years This age may represent the age of Earth's accretion, or core formation, or of the material from which Earth formed. This dating is based on evidence from radiometric age-dating of ...
(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 List of intergovernmental organizations, intergovernmental organisation, through which its 59 member-states act together on measur ...
(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 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 wid ...
(SI). Note that although "
percent In mathematics, a percentage (from la, per centum, "by a hundred") is a number or ratio expressed as a fraction (mathematics), fraction of 100. It is often Denotation, denoted using the percent sign, "%", although the abbreviations "pct.", "p ...
" (%) 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 development organization composed of representatives from the national standards organizations of member countries. Membership requirements are given in Art ...
(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 whose mission is to assist in the worldwide development of physics, to foster international cooperation in physics, and to help in 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 Billion is a word for a large number, and it has two distinct definitions: * 1,000,000,000, i.e. one thousand million, or (ten to the ninth power), as defined on the short scale. This is its only current meaning in English. * 1,000,000,000,000, ...
" 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 an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into Outline of p ...
(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, the mole fraction or molar fraction (''xi'' or ) is defined as unit of the amount of substance, amount of a constituent (expressed in mole (unit), moles), ''ni'', divided by the total amount of all constituents in a mixture (also expr ...
or
volume fraction In chemistry and fluid mechanics, the volume fraction φ''i'' is defined as the volume of a constituent ''V'i'' divided by the volume of all constituents of the mixture ''V'' prior to mixing: :\phi_i = \frac Being dimensionless quantity, dim ...
. 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 hydrocarbons that contain carbon Carbon () is a ch ...
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 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, 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. 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 A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate di ...
; 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, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division (mathematics), division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to a ...
s are pure-number
coefficient In mathematics, a coefficient is a multiplicative factor in some Summand, term of a polynomial, a series (mathematics), series, or an expression (mathematics), expression; it is usually a number, but may be any expression (including variables su ...
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 (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 (BIPM), the intergovernmental organization established i ...
(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.

*
International Electrotechnical Commission The International Electrotechnical Commission (IEC; in French: ''Commission électrotechnique internationale'') is an international standards organization that prepares and publishes international standards for all electrical, electronic an ...
(IEC) * Milligram per cent *
Percentage In mathematics, a percentage (from la, per centum, "by a hundred") is a number or ratio expressed as a fraction (mathematics), fraction of 100. It is often Denotation, denoted using the percent sign, "%", although the abbreviations "pct.", "p ...
(%) 1 part in 100 *
Per mille Per mille (from New Latin, Latin , "in each thousand") is an expression that means parts per thousand. Other recognised spellings include per mil, per mill, permil, permill, or permille. The associated sign is written , which looks like a ...
(‰) 1 part in 1,000 *
Permyriad A basis point (often abbreviated as bp, often pronounced as "bip" or "beep") is one hundredth of 1 percentage point. The related term ''#Permyriad, permyriad'' means one hundredth of 1 percent. Changes of interest rates are often stated in basis ...
(‱) 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, and in nuclear reactor engineering as a unit of reactivity. Epidemiology Statistics of Crime statistics, c ...
(pcm) 1 part in 100,000 *
Per-unit system In the power systems, 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-uni ...