science Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...

and

engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...

, the parts-per notation is a set of pseudo-units to describe the small values of miscellaneous dimensionless quantities, e.g.

mole fraction In chemistry, the mole fraction or molar fraction, also called mole proportion or molar proportion, is a quantity defined as the ratio between the amount of a constituent substance, ''ni'' (expressed in unit of moles, symbol mol), and the to ...

or mass fraction. Since these

fractions A fraction (from , "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, thre ...

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

units of measurement A unit of measurement, or unit of measure, 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 qua ...

. Commonly used are * parts-per-million - ppm, * parts-per-billion - ppb, * parts-per-trillion - ppt, * parts-per-quadrillion - ppq, This notation is not part of the

International System of Units The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official s ...

- SI system and its meaning is ambiguous.

Applications

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 physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...

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

water Water is an inorganic compound with the chemical formula . It is a transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance. It is the main constituent of Earth's hydrosphere and the fluids of all known liv ...

. 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 thousandth of a kilogram. Originally defined in 1795 as "the absolute Mass versus weight, weight of a volume ...

per gram of sample solution. When working with

aqueous solution An aqueous solution is a 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, also known as sodium chloride (NaCl), in water ...

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 scientific study of matter, its Elementary particle, 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 whi ...

and

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

meter The metre (or meter in US spelling; symbol: m) is the base unit of length in the International System of Units (SI). Since 2019, the metre has been defined as the length of the path travelled by light in vacuum during a time interval of of ...

of length for every degree Celsius and this would be expressed as Parts-per notation is also employed to denote the change, stability, or

uncertainty Uncertainty or incertitude refers to situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown, and is particularly relevant for decision ...

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 or laser distance meter, is a 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 ...

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''. ''Precision'' is how close the measurements are to each other. The ...

= 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" so the

quotient In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...

s are pure-number

coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...

s with positive values less than or equal to 1. When parts-per notations, including the

percent In mathematics, a percentage () is a number or ratio expressed as a fraction of 100. It is often denoted using the ''percent sign'' (%), although the abbreviations ''pct.'', ''pct'', and sometimes ''pc'' are also used. A percentage is a dime ...

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"). Parts-per notations may be expressed in terms of any unit of the same measure. For instance, the expansion coefficient of some

brass Brass is an alloy of copper and zinc, in proportions which can be varied to achieve different colours and mechanical, electrical, acoustic and chemical properties, but copper typically has the larger proportion, generally copper and zinc. I ...

alloy, 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 is doing so at a rate that may be expressed in a variety of volumetric units, including 125 cm³/ m³, etc. 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 based on re-orientation of Atomic nucleus, atomic nuclei with non-zero nuclear sp ...

(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 of ...

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), often described as being equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose formal expression in terms of SI base u ...

. 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 mathematics 100 is the square of 10 (number), 10 (in scientific notation it is written as 102). The standar ...

'' 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'' 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 The phrase per mille () indicates parts per thousand. The associated symbol is , similar to a per cent sign but with an extra zero in the divisor. Major dictionaries do not agree on the spelling, giving other options of per mil, per mill, p ...

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 (, ''hoi 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 Battle of Cunaxa and bac ...

'' 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. Changes of interest rates are often stated in basis points. For example, if an existing interest rate of 10 percent is increased ...

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 refers to monetary resources and to the study and Academic discipline, discipline of money, currency, assets and Liability (financial accounting), liabilities. As a subject of study, is a field of Business administration, Business Admin ...

, the

basis point A basis point (often abbreviated as bp, often pronounced as "bip" or "beep") is one hundredth of 1 percentage point. Changes of interest rates are often stated in basis points. For example, if an existing interest rate of 10 percent is increased ...

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 Risk factor (epidemiology), determinants of health and disease conditions in a defined population, and application of this knowledge to prevent dise ...

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 1,000,000 (one million), 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 ...

'' (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. In

mining Mining is the Resource extraction, extraction of valuable geological materials and minerals from the surface of the Earth. Mining is required to obtain most materials that cannot be grown through agriculture, agricultural processes, or feasib ...

, it is also equivalent to one

per

metric ton The tonne ( or ; symbol: t) is a unit of mass equal to 1,000 kilograms. It is a non-SI unit accepted for use with SI. It is also referred to as a metric ton in the United States to distinguish it from the non-metric units of the sh ...

, expressed as g/t. * ''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 now the most common sense of the word in all varieties of ...

'' (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 or 10 decades. 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. ...

. * ''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 Depending on context (e.g. language, culture, region), some large numbers have names that allow for describing large quantities in a textual form; not mathematical. For very large values, the text is generally shorter than a decimal numeric repres ...

'' (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 (astrophysics), accretion, or Internal structure of Earth, core formation, or of the material from which Earth formed. This dating ...

(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 (, BIPM) is an List of intergovernmental organizations, intergovernmental organisation, through which its 64 member-states act on measurement standards in areas including chemistry, ionising radi ...

(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 "

" (%) is not formally part of the SI, both the BIPM and the

International Organization for Standardization The International Organization for Standardization (ISO ; ; ) is an independent, non-governmental, international standard development organization composed of representatives from the national standards organizations of member countries. M ...

(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, "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 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,

volume fraction In chemistry and fluid mechanics, the volume fraction \varphi_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: :\varphi_i = \frac . Being dimensionless quantit ...

. Since it is usually not stated which quantity is used, it is better to write the units out, such as kg/kg, mol/mol or m³/m³, even though they are all dimensionless. 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 in air (Molar mass of CFC-11 / Mean molar mass of air = 137.368 / 28.97 = 4.74). For volume fraction, the suffix "V" or "v" is sometimes appended to the parts-per notation (e.g. ppmV, ppbv, pptv). However, ppbv and pptv are usually used to mean mole fractions"volume fraction" would literally mean what volume of a pure substance is included in a given volume of a mixture, and this is rarely used except in the case of

alcohol by volume Alcohol by volume (abbreviated as alc/vol or ABV) is a common measure of the amount of Alcohol (drug), alcohol contained in a given alcoholic beverage. It is defined as the volume the ethanol in the liquid would take if separated from the rest ...

. 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, mass/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, while

chemist A chemist (from Greek ''chēm(ía)'' alchemy; replacing ''chymist'' from Medieval Latin ''alchemist'') is a graduated scientist trained in the study of chemistry, or an officially enrolled student in the field. Chemists study the composition of ...

s, the field of occupational safety and the field of

permissible exposure limit The permissible exposure limit (PEL or OSHA PEL) is a legal limit in the United States for exposure of an employee to a chemical substance or physical agents such as high level noise. Permissible exposure limits were established by the Occupational ...

(e.g. permitted gas exposure limit in

air An atmosphere () is a layer of gases that envelop an astronomical object, held in place by the gravity of the object. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A stellar atmosph ...

) may use mass/volume. Unfortunatelly, many academic publications of otherwise excellent level fail to specify their use 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.

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

ratio In mathematics, a ratio () shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...

s are pure-number

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 ...

(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 (abbreviated CGPM from the ) is the supreme authority of the International Bureau of Weights and Measures (BIPM), the intergovernmental organization established in 1875 under the terms of the Metre C ...

(CIPM) stated that the 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 ...

Footnotes

References

External links

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