science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence for ...

and

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 of engineering encompasses a broad rang ...

, 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 to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...

, e.g.

mole fraction In chemistry, the mole fraction or molar fraction (''xi'' or ) is defined as unit of the amount of a constituent (expressed in moles), ''ni'', divided by the total amount of all constituents in a mixture (also expressed in moles), ''n''tot. This ex ...

or mass fraction. Since these

fractions A fraction (from la, fractus, "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 ...

are quantity-per-quantity measures, they are pure numbers with no associated

units of measurement A unit of measurement is a definite 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 be expressed as a multi ...

. 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 pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...

(SI) system and its meaning is ambiguous.

Overview

Parts-per notation is often used describing dilute solutions in

chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...

, for instance, the relative abundance of dissolved minerals or pollutants in

water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as a ...

. 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 Physical unit, 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 of pure wate ...

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 fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

and

to express the value of various proportional phenomena. For instance, a special metal alloy might expand 1.2

micrometers The micrometre ( international spelling as used by the International Bureau of Weights and Measures; SI symbol: μm) or micrometer (American spelling), also commonly known as a micron, is a unit of length in the International System of Unit ...

per

meter The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...

of length for every

degree Celsius Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

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 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 observable ...

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 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 oth ...

= 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 of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

s 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").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 kineti ...

of a certain

brass Brass is an alloy of copper (Cu) and zinc (Zn), in proportions which can be varied to achieve different mechanical, electrical, and chemical properties. It is a substitutional alloy: atoms of the two constituents may replace each other wit ...

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 '' Q_p'' = 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

nuclear magnetic resonance spectroscopy Nuclear magnetic resonance spectroscopy, most commonly known as NMR spectroscopy or magnetic resonance spectroscopy (MRS), is a spectroscopic technique to observe local magnetic fields around atomic nuclei. The sample is placed in a magnetic fie ...

(NMR),

chemical shift In nuclear magnetic resonance (NMR) spectroscopy, the chemical shift is the 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 structure o ...

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 unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that one he ...

. 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 numeral: C) is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the short hundred or five score in order to differentiate the English and Germanic use of "hundred" to de ...

'' 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 of 100. Related signs include the permille (per thousand) sign and the permyriad (per ten thousand) ...

(%) 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 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000. A group of one thousand th ...

'' 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 million, or (ten to the twelfth power), as defined on the short scale. This is now the meaning in both American and British English. * 1,000,000,000,0 ...

"). It may also be denoted by the

permille Per mille (from 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 percent sig ...

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 Greeks, employed by Cyrus the Younger to attempt to wrest the throne of the Persian Empire from his brother, Artaxerxes II. Their march to the Bat ...

'' 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'' means one hundredth of 1 percent. Changes of interest rates are often stated in basis points. If ...

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, distribution, and consumption of money, assets, goods and services (the discipline of fina ...

, 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 rates, mort ...

'' (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 determinants of health and disease conditions in a defined population. It is a cornerstone of public health, and shapes policy decisions and evidenc ...

for mortality, crime and disease prevalence rates, and nuclear reactor engineering as a unit of reactivity. In

time measurement Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to c ...

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 range can be measured. The most commonly used approaches are the rulers, followed by transit-time methods and the in ...

, 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 following 999,999 and preceding 1,000,001. The word is derived from the early Italian ''millione'' (''milione'' in modern Italian), from ''mille'', "thousand", plus the au ...

'' (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, i. ...

'' (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 ordinally in English and many other languages. The word ''century'' comes from the Latin ''centum'', meaning ''one hundred''. ''Century'' is sometimes abbreviated as c. A centennial or ...

. * ''One part per

'' (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 Two naming scales for large numbers have been used in English and other European languages since the early modern era: the long and short scales. Most English variants use the short scale today, but the long scale remains dominant in many non-E ...

'' (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 (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, through which its 59 member-states act together on measurement standards in four areas: chemistry ...

(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

(SI). Note that although " 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 development organization composed of representatives from the national standards organizations of member countries. Membership requirements are given in Ar ...

(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 "

" 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 physical sci ...

(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,

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, its unit is ...

. 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 (CFC). It is a colorless, faintly ethereal, and sweetish-smelling liquid that boils around room temperature. CFC-11 is a Class 1 ozone-depleting substance ...

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 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 elements by physical separation methods, i.e., wi ...

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

; that is, the units of measurement factor out in expressions like "1 nm/m" (1 nm/m = 1 nano = 1 × 10⁻⁹) so the

s are pure-number coefficients 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 ...

(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.

References

External links

* {{Commonscatinline * National Institute of Standards and Technology (NIST)
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* International Bureau of Weights and Measures (BIPM)
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