Parts-per notation

Encyclopedia

In science

and engineering

, the

, e.g. mole fraction or mass fraction

. Since these fractions

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

. Commonly used are

, for instance, the relative abundance of dissolved minerals or pollutants in water. The unit “1 ppm” can be used for a mass fraction

if a water-borne pollutant is present at one-millionth of a gram

per gram of sample solution.

Similarly, parts-per notation is used also in physics

and engineering

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

per meter

of length for every degree Celsius

and this would be expressed as “

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

= 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 n~~m~~/~~m~~ = 2 nano = 2 × 10

s are pure-number coefficient

s with positive values less than 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”).

Parts-per notations may be expressed in terms of any unit of the same measure. For instance, the coefficient of thermal expansion of a certain brass

alloy,

/m)/°C, or as 18.7 (µin

/in)/°C; the numeric value representing a relative proportion does not change with the adoption of a different unit of measure. Similarly, a metering pump

that injects a trace chemical into the main process line at the proportional flow rate

/gal, 125 cm

, etc.

Ppm is also a measure of the chemical shift

in nuclear magnetic resonance spectroscopy and represents difference of the frequency in parts per million from the reference frequency (signal).

(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

(SI). Note, that although “percent

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

, take the position that

,

starting with a “billion

” have different values in different countries, the BIPM suggests avoiding the use of “ppb” and “ppt” to prevent misunderstanding. In the English language

, named numbers

have a consistent meaning only up to “million

”. Starting with “billion”, there are

, and “billion” can mean either 10

or 10

(NIST) takes the stringent position, stating that

or a mole fraction. Since it is usually not stated which quantity is used, it is better to write the unit as kg/kg, or mol/mol (even though they are all dimensionless). 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. The usage 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 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, electrochemists

often use volume/volume, while chemical engineers

may use mass/mass as well as volume/volume. Many academic papers of otherwise excellent level fail to specify their usage of the part-per notation. The difference between expressing concentrations as mass/mass or volume/volume is quite significant when dealing with gases and it is very important to specify which is being used.

Note that the notations in the “SI units” column above are all dimensionless quantities

; that is, the units of measurement cancel in expressions like “1 nm/m” (1 n~~m~~/~~m~~ = 1 nano = 1 × 10

s are pure-number coefficient

s with values less than 1.

(IUPAP) in 1999 proposed the adoption of the special name “uno” (symbol: U) to represent the number 1 in dimensionless quantities. This symbol is not to be confused with the always-italicized symbol for the variable ‘uncertainty’ (symbol:

es to express the values of dimensionless quantities which are much less—or even

Common parts-per notations in terms of the uno are given in the table below.

In 2004, a report to the International Committee for Weights and Measures

(known also by its French-language initials CIPM) stated that response to the proposal of the uno

and it appears unlikely it will ever become an officially sanctioned way to express low-value (high-ratio) dimensionless quantities. The proposal was instructive, however, as to the perceived shortcomings of the current options for denoting dimensionless quantities.

concentration of 15 pCi

/L

" are not dimensionless quantities and may not be expressed using any form of parts-per notation, such as "15 ppt". Other examples of measures that are not dimensionless quantities are as follows:

Note however, that it is not uncommon to express aqueous concentrations—particularly in drinking-water reports intended for the general public—using parts-per notation (2.1 ppm, 0.8 ppb, etc.) and further, for those reports to state that the notations denote milligrams per liter

or micrograms per liter. Whereas "2.1 mg/L" is technically not a dimensionless quantity on the face of it, it is well understood in scientific circles that one liter of water has a mass of one kilogram and that "2.1 mg/kg" (2.1 ppm) is the true measure. The goal in all technical writing

(including drinking-water reports for the general public) is to clearly communicate to the intended audience with minimal confusion. Drinking water is intuitively a volumetric quantity in the public’s mind so measures of contamination expressed on a per-liter basis are considered to be easier to grasp. Still, it is technically possible, for example, to "dissolve" more than one liter of a very hydrophilic chemical in 1 liter of water; parts-per notation would be confusing when describing its solubility in water (greater than a million parts per million), so one would simply state the volume (or mass) that will dissolve into a liter, instead.

When reporting air-borne rather than water-borne densities, a slightly different convention is used since air is approximately 1000 times less dense than water. In water, 1 µg/m

Science

Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

and engineering

Engineering

Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, the

**parts-per notation**is a set of pseudo units to describe small values of miscellaneous dimensionless quantitiesDimensionless quantity

In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and...

, e.g. mole fraction or mass fraction

Mass fraction

In aerospace engineering, the propellant mass fraction is a measure of a vehicle's performance, determined as the portion of the vehicle's mass which does not reach the destination...

. Since these fractions

Fraction (mathematics)

A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

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

Units of measurement

A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention and/or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of...

. Commonly used are

**ppm**(parts-per-million),**ppb**(parts-per-billion),**ppt**(parts-per-trillion) and**ppq**(parts-per-quadrillion).## Overview

Parts-per notation is often used describing dilute solutions in chemistryChemistry

Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

, for instance, the relative abundance of dissolved minerals or pollutants in water. The unit “1 ppm” can be used for a mass fraction

Mass fraction

In aerospace engineering, the propellant mass fraction is a measure of a vehicle's performance, determined as the portion of the vehicle's mass which does not reach the destination...

if a water-borne pollutant is present at one-millionth of a gram

Gram

The gram is a metric system unit of mass....

per gram of sample solution.

Similarly, parts-per notation is used also in physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

and engineering

Engineering

Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

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

Micrometre

A micrometer , is by definition 1×10-6 of a meter .In plain English, it means one-millionth of a meter . Its unit symbol in the International System of Units is μm...

per meter

Metre

The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

of length for every degree Celsius

Celsius

Celsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...

and this would be expressed as “

*α*= 1.2 ppm/°C.” Parts-per notation is also employed to denote the change, stability, or uncertaintyStandard deviation

Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

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

In the fields of science, engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which...

= 1 ppm.”

Parts-per notations are all dimensionless quantities

Dimensionless quantity

In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and...

: in mathematical expressions, the units of measurement always cancel. In fractions like “2 nanometers per meter” (2 n

^{−9}= 2 ppb = 2 × 0.000000001) so the quotientQuotient

In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...

s are pure-number coefficient

Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

s with positive values less than 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”).

Parts-per notations may be expressed in terms of any unit of the same measure. For instance, the coefficient of thermal expansion of a certain brass

Brass

Brass is an alloy of copper and zinc; the proportions of zinc and copper can be varied to create a range of brasses with varying properties.In comparison, bronze is principally an alloy of copper and tin...

alloy,

*α*= 18.7 ppm/°C, may be expressed as 18.7 (µmMicrometre

A micrometer , is by definition 1×10-6 of a meter .In plain English, it means one-millionth of a meter . Its unit symbol in the International System of Units is μm...

/m)/°C, or as 18.7 (µin

Inch

An inch is the name of a unit of length in a number of different systems, including Imperial units, and United States customary units. There are 36 inches in a yard and 12 inches in a foot...

/in)/°C; the numeric value representing a relative proportion does not change with the adoption of a different unit of measure. Similarly, a metering pump

Metering pump

A metering pump is a pump used to pump liquids at adjustable flow rates which are precise when averaged over time. Delivery of fluids in precise adjustable flow rates is sometimes called metering...

that injects a trace chemical into the main process line at the proportional flow rate

*Q*

= 125 ppm, is doing so at a rate that may be expressed in a variety of volumetric units, including 125 µL/L, 125 µgalVolumetric flow rate

The volumetric flow rate in fluid dynamics and hydrometry, is the volume of fluid which passes through a given surface per unit time...

_{p}Gallon

The gallon is a measure of volume. Historically it has had many different definitions, but there are three definitions in current use: the imperial gallon which is used in the United Kingdom and semi-officially within Canada, the United States liquid gallon and the lesser used United States dry...

/gal, 125 cm

^{3}/m^{3}Cubic metre

The cubic metre is the SI derived unit of volume. It is the volume of a cube with edges one metre in length. An alternative name, which allowed a different usage with metric prefixes, was the stère...

, etc.

Ppm is also a measure of the chemical shift

Chemical shift

In nuclear magnetic resonance spectroscopy, the chemical shift is the resonant frequency of a nucleus relative to a standard. Often the position and number of chemical shifts are diagnostic of the structure of a molecule...

in nuclear magnetic resonance spectroscopy and represents difference of the frequency in parts per million from the reference frequency (signal).

## Parts-per expressions

*One part per hundred*is generally represented by the percent (%) symbol and denotes one part per 100 parts, one part in 10100 (number)100 is the natural number following 99 and preceding 101.-In mathematics:One hundred is the square of 10...^{2}, and a value of 1 × 10^{−2}. This is equivalent to one drop of water diluted into 5 milliliters (one spoon-full), or about fifteen minutes out of one day.

*One part per thousand*should generally spelled out in full and**not**as “ppt” (which is usually understood to represent “parts per trillion”). It may also be denoted by the permillePermilleA per mil or per mille is a tenth of a percent or one part per thousand. It is written with the sign ‰ , which looks like a percent sign with an extra zero at the end...

(‰) symbol. Note however, that specific disciplines such as the analysis of ocean water salt concentration and educational exercises occasionally use the “ppt” abbreviation. “One part per thousand” denotes one part per 1000 parts, one part in 10^{3}, and a value of 1 × 10^{−3}. This is equivalent to one drop of water diluted into 50 milliliters (ten spoon-fulls), or about one and a half minutes out of one day.

*One part per ten thousand*is denoted by the permyriad (‱) symbol. It is used almost exclusively in financeFinance"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...

, where it is known as the basis pointBasis pointA basis point is a unit equal to 1/100 of a percentage point or one part per ten thousand...

and is typically used to denote fractional changes in percentages. For instance, a change in an interest rate from 5.15% to 5.35% would be denoted as a change of 20 basis points or 20 ‱. Although rarely used in science (ppm is typically used instead), one permyriad has an unambiguous value of one part per 10,000 parts, one part in 10^{4}, and a value of 1 × 10^{−4}. This is equivalent to one drop of water diluted into half a literLitér- External links :*...

, or about nine seconds out of one day.

*One part per million*(MillionOne 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 , from mille, "thousand", plus the augmentative suffix -one.In scientific notation, it is written as or just 106...**ppm**) denotes one part per 1,000,000 parts, one part in 10^{6}, 1/1,000,000 * 100% = 0.0001% (or 1% = 10,000 ppm), and a value of 1 × 10^{−6}. This is equivalent to one drop of water diluted into 50 liters (roughly the fuel tank capacity of a compact car), or about 32 seconds out of a year.

*One part per billion*(1000000000 (number)1,000,000,000 is the natural number following 999,999,999 and preceding 1,000,000,001.In scientific notation, it is written as 109....**ppb**) denotes one part per 1,000,000,000 parts, one part in 10^{9}, 1/1,000,000,000 * 100% = 0.0000001% (or 1% = 10,000,000 ppb) and a value of 1 × 10^{−9}. This is equivalent to one drop of water diluted into 250 chemical drums (50 m^{3}), or about three seconds out of a centuryCenturyA century is one hundred consecutive years. Centuries are numbered ordinally in English and many other languages .-Start and end in the Gregorian Calendar:...

.

*One part per trillion*(**ppt**) denotes one part per 1,000,000,000,000 parts, one part in 10^{12}, and a value of 1 × 10^{−12}. This is equivalent to one drop of water diluted into 20 Olympic-size swimming pools (50,000 m^{3}), or about three seconds out of every hundred thousand years.

*One part per quadrillion*(**ppq**) denotes one part per 1,000,000,000,000,000 parts, one part in 10^{15}, and a value of 1 × 10^{−15}. This is equivalent to 1 drop of water diluted into a cube of water measuring approximately 368 meters on a side (fifty million cubic meters, which is a cube about as tall as the Empire State BuildingEmpire State BuildingThe Empire State Building is a 102-story landmark skyscraper and American cultural icon in New York City at the intersection of Fifth Avenue and West 34th Street. It has a roof height of 1,250 feet , and with its antenna spire included, it stands a total of 1,454 ft high. Its name is derived...

's 102 stories), or two and a half minutes out of the age of the EarthAge of the EarthThe age of the Earth is 4.54 billion years This age is based on evidence from radiometric age dating of meteorite material and is consistent with the ages of the oldest-known terrestrial and lunar samples...

(4.5 billion years). Although relatively uncommon in analytic chemistry, measurements at the ppq level*are*performed.

## Criticism

Although the International Bureau of Weights and MeasuresInternational Bureau of Weights and Measures

The International Bureau of Weights and Measures , is an international standards organisation, one of three such organisations established to maintain the International System of Units under the terms of the Metre Convention...

(an international standards organization known also by its French

France

The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...

-language initials BIPM) recognizes the use of parts-per notation, it is not formally part of the International System of Units

International System of Units

The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

(SI). Note, that although “percent

Percentage

In mathematics, a percentage is a way of expressing a number as a fraction of 100 . It is often denoted using the percent sign, “%”, or the abbreviation “pct”. For example, 45% is equal to 45/100, or 0.45.Percentages are used to express how large/small one quantity is, relative to another quantity...

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

International Organization for Standardization

The International Organization for Standardization , widely known as ISO, is an international standard-setting body composed of representatives from various national standards organizations. Founded on February 23, 1947, the organization promulgates worldwide proprietary, industrial and commercial...

, 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 IUPAPInternational Union of Pure and Applied Physics

The International Union of Pure and Applied Physics is an international non-governmental organization devoted to the advancement of physics...

,

*“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:### Long and short scales

Because the named numbersNames of large numbers

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 numbers which are found in many English dictionaries and thus have a special claim to being "real words"...

starting with a “billion

Long and short scales

The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale...

” have different values in different countries, the BIPM suggests avoiding the use of “ppb” and “ppt” to prevent misunderstanding. In the English language

English language

English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...

, named numbers

Names of large numbers

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 numbers which are found in many English dictionaries and thus have a special claim to being "real words"...

have a consistent meaning only up to “million

Million

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 , from mille, "thousand", plus the augmentative suffix -one.In scientific notation, it is written as or just 106...

”. Starting with “billion”, there are

*two*numbering conventions: the “long” and “short” scalesLong and short scales

The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale...

, and “billion” can mean either 10

^{9}1000000000 (number)

1,000,000,000 is the natural number following 999,999,999 and preceding 1,000,000,001.In scientific notation, it is written as 109....

or 10

^{12}. The U.S. National Institute of Standards and TechnologyNational Institute of Standards and Technology

The National Institute of Standards and Technology , known between 1901 and 1988 as the National Bureau of Standards , is a measurement standards laboratory, otherwise known as a National Metrological Institute , which is a non-regulatory agency of the United States Department of Commerce...

(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 guessed from the context.### Mass fraction vs mole fraction

Another problem of the parts-per notation is that it may refer to either a mass fractionMass fraction

In aerospace engineering, the propellant mass fraction is a measure of a vehicle's performance, determined as the portion of the vehicle's mass which does not reach the destination...

or a mole fraction. Since it is usually not stated which quantity is used, it is better to write the unit as kg/kg, or mol/mol (even though they are all dimensionless). 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. The usage 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 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, electrochemists

Electrochemistry

Electrochemistry is a branch of chemistry that studies chemical reactions which take place in a solution at the interface of an electron conductor and an ionic conductor , and which involve electron transfer between the electrode and the electrolyte or species in solution.If a chemical reaction is...

often use volume/volume, while chemical engineers

Chemical engineering

Chemical engineering is the branch of engineering that deals with physical science , and life sciences with mathematics and economics, to the process of converting raw materials or chemicals into more useful or valuable forms...

may use mass/mass as well as volume/volume. Many academic papers of otherwise excellent level fail to specify their usage of the part-per notation. The difference between expressing concentrations as mass/mass or volume/volume is quite significant when dealing with gases and it is very important to specify which is being used.

## SI-compliant expressions

SI-compliant units that can be used as alternatives are shown in the chart below. Expressions that the BIPM does not explicitly recognize as being suitable for denoting dimensionless quantities with the SI are shown in__underlined green text__.NOTATIONS FOR DIMENSIONLESS QUANTITIES | ||||
---|---|---|---|---|

Measure |
SIunits |
Namedparts-per ratio |
Parts-perabbreviation or symbol |
Value inscientific notation |

A strain Strain (materials science) In continuum mechanics, the infinitesimal strain theory, sometimes called small deformation theory, small displacement theory, or small displacement-gradient theory, deals with infinitesimal deformations of a continuum body... of… |
2 c |
2 parts per hundred | 2% | 2 × 10^{−2} |

A sensitivity of… | 2 mV/V | 2 parts per thousand | 2 ‰ |
2 × 10^{−3} |

A sensitivity of… | 0.2 mV/V | 2 parts per ten thousand | 2 ‱ |
2 × 10^{−4} |

A sensitivity of… | 2 µV/V | 2 parts per million | 2 ppm | 2 × 10^{−6} |

A sensitivity of… | 2 nV/V | 2 parts per billion |
2 ppb |
2 × 10^{−9} |

A sensitivity of… | 2 pV/V | 2 parts per trillion |
2 ppt |
2 × 10^{−12} |

A mass fraction of… | 2 mg/kg | 2 parts per million | 2 ppm | 2 × 10^{−6} |

A mass fraction of… | 2 µg/kg | 2 parts per billion |
2 ppb |
2 × 10^{−9} |

A mass fraction of… | 2 ng/kg | 2 parts per trillion |
2 ppt |
2 × 10^{−12} |

A mass fraction of… | 2 pg/kg | 2 parts per quadrillion |
2 ppq |
2 × 10^{−15} |

A volume fraction of… | 5.2 µL/L | 5.2 parts per million | 5.2 ppm | 5.2 × 10^{−6} |

A mole fraction of… | 5.24 µmol/mol | 5.24 parts per million | 5.24 ppm | 5.24 × 10^{−6} |

A mole fraction of… | 5.24 nmol/mol | 5.24 parts per billion |
5.24 ppb |
5.24 × 10^{−9} |

A mole fraction of… | 5.24 pmol/mol | 5.24 parts per trillion |
5.24 ppt |
5.24 × 10^{−12} |

A stability of… | 1 (µA/A)/min. | 1 part per million per min. | 1 ppm/min. | 1 × 10^{−6}/min. |

A change of… | 5 nΩ/Ω | 5 parts per billion |
5 ppb |
5 × 10^{−9} |

An uncertainty of… | 9 µg/kg | 9 parts per billion |
9 ppb |
9 × 10^{−9} |

A shift of… | 1 nm/m | 1 part per billion |
1 ppb |
1 × 10^{−9} |

A strain of… | 1 µm/m | 1 part per million | 1 ppm | 1 × 10^{−6} |

A temperature coefficient Temperature coefficient The temperature coefficient is the relative change of a physical property when the temperature is changed by 1 K.In the following formula, let R be the physical property to be measured and T be the temperature at which the property is measured. T0 is the reference temperature, and ΔT is the... of… |
0.3 (µHz/Hz)/°C | 0.3 part per million per °C | 0.3 ppm/°C | 0.3 × 10^{−6}/°C |

A frequency change of… | 0.35 × 10^{−9} ƒ |
0.35 part per billion |
0.35 ppb |
0.35 × 10^{−9} |

Note that the notations in the “SI units” column above are all dimensionless quantities

Dimensionless quantity

In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and...

; that is, the units of measurement cancel in expressions like “1 nm/m” (1 n

^{−9}) so the quotientQuotient

In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...

s are pure-number coefficient

Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

s with values less than 1.

## Uno

Because of the cumbersome nature of expressing certain dimensionless quantities per SI guidelines, the International Union of Pure and Applied PhysicsInternational Union of Pure and Applied Physics

The International Union of Pure and Applied Physics is an international non-governmental organization devoted to the advancement of physics...

(IUPAP) in 1999 proposed the adoption of the special name “uno” (symbol: U) to represent the number 1 in dimensionless quantities. This symbol is not to be confused with the always-italicized symbol for the variable ‘uncertainty’ (symbol:

*U).*This unit name uno and its symbol could be used in combination with the SI prefixSI prefix

The International System of Units specifies a set of unit prefixes known as SI prefixes or metric prefixes. An SI prefix is a name that precedes a basic unit of measure to indicate a decadic multiple or fraction of the unit. Each prefix has a unique symbol that is prepended to the unit symbol...

es to express the values of dimensionless quantities which are much less—or even

*greater—*than one.Common parts-per notations in terms of the uno are given in the table below.

Coefficient | Parts-per example | Uno equiv. | Symbol form | Value of quantity |
---|---|---|---|---|

10^{−2} |
2% | 2 centiuno | 2 cU | 2 × 10^{−2} |

10^{−3} |
2 ‰ | 2 milliuno | 2 mU | 2 × 10^{−3} |

10^{−6} |
2 ppm | 2 microuno | 2 µU | 2 × 10^{−6} |

10^{−9} |
2 ppb | 2 nanouno | 2 nU | 2 × 10^{−9} |

10^{−12} |
2 ppt | 2 picouno | 2 pU | 2 × 10^{−12} |

In 2004, a report to the International Committee for Weights and Measures

International Committee for Weights and Measures

The Interglobal Committee for Weights and Measures is the English name of the Comité international des poids et mesures . It consists of eighteen persons from Member States of the Metre Convention...

(known also by its French-language initials 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 organizationStandards organization

A standards organization, standards body, standards developing organization , or standards setting organization is any organization whose primary activities are developing, coordinating, promulgating, revising, amending, reissuing, interpreting, or otherwise producing technical standards that are...

and it appears unlikely it will ever become an officially sanctioned way to express low-value (high-ratio) dimensionless quantities. The proposal was instructive, however, as to the perceived shortcomings of the current options for denoting dimensionless quantities.

## Improper applications of parts-per notation

Parts-per notation may properly be used only to express true dimensionless quantities; that is, the units of measurement*must*cancel in expressions like "1 mg/kg" so that the quotients are pure numbers with values less than 1. Mixed-unit quantities such as "a radonRadon

Radon is a chemical element with symbol Rn and atomic number 86. It is a radioactive, colorless, odorless, tasteless noble gas, occurring naturally as the decay product of uranium or thorium. Its most stable isotope, 222Rn, has a half-life of 3.8 days...

concentration of 15 pCi

Curie

The curie is a unit of radioactivity, defined asThis is roughly the activity of 1 gram of the radium isotope 226Ra, a substance studied by the pioneers of radiology, Marie and Pierre Curie, for whom the unit was named. In addition to the curie, activity can be measured using an SI derived unit,...

/L

Litre

pic|200px|right|thumb|One litre is equivalent to this cubeEach side is 10 cm1 litre water = 1 kilogram water The litre is a metric system unit of volume equal to 1 cubic decimetre , to 1,000 cubic centimetres , and to 1/1,000 cubic metre...

" are not dimensionless quantities and may not be expressed using any form of parts-per notation, such as "15 ppt". Other examples of measures that are not dimensionless quantities are as follows:

- Particulate matter in the air: 50 µg/m
^{3}; not 50 ppb. Also see air measurements, below. - A stepper motorStepper motorA stepper motor is a brushless, electric motor that can divide a full rotation into a large number of steps. The motor's position can be controlled precisely without any feedback mechanism , as long as the motor is carefully sized to the application...

/gear system that produces a motion of 1 µm/pulse; not 1 ppm - Mercury vapor concentration in air: 0.6 ng/LLitrepic|200px|right|thumb|One litre is equivalent to this cubeEach side is 10 cm1 litre water = 1 kilogram water The litre is a metric system unit of volume equal to 1 cubic decimetre , to 1,000 cubic centimetres , and to 1/1,000 cubic metre...

; not 0.6 ppt

Note however, that it is not uncommon to express aqueous concentrations—particularly in drinking-water reports intended for the general public—using parts-per notation (2.1 ppm, 0.8 ppb, etc.) and further, for those reports to state that the notations denote milligrams per liter

Litre

pic|200px|right|thumb|One litre is equivalent to this cubeEach side is 10 cm1 litre water = 1 kilogram water The litre is a metric system unit of volume equal to 1 cubic decimetre , to 1,000 cubic centimetres , and to 1/1,000 cubic metre...

or micrograms per liter. Whereas "2.1 mg/L" is technically not a dimensionless quantity on the face of it, it is well understood in scientific circles that one liter of water has a mass of one kilogram and that "2.1 mg/kg" (2.1 ppm) is the true measure. The goal in all technical writing

Technical writing

Technical writing, a form of technical communication, is a style of writing used in fields as diverse as computer hardware and software, engineering, chemistry, the aerospace industry, robotics, finance, consumer electronics, and biotechnology....

(including drinking-water reports for the general public) is to clearly communicate to the intended audience with minimal confusion. Drinking water is intuitively a volumetric quantity in the public’s mind so measures of contamination expressed on a per-liter basis are considered to be easier to grasp. Still, it is technically possible, for example, to "dissolve" more than one liter of a very hydrophilic chemical in 1 liter of water; parts-per notation would be confusing when describing its solubility in water (greater than a million parts per million), so one would simply state the volume (or mass) that will dissolve into a liter, instead.

When reporting air-borne rather than water-borne densities, a slightly different convention is used since air is approximately 1000 times less dense than water. In water, 1 µg/m

^{3}is roughly equivalent to parts-per-trillion whereas in air, it is roughly equivalent to parts-per-billion. Note also, that in the case of air, this convention is much less accurate. Whereas one liter of water is almost exactly 1 kg, one cubic meter of air is often taken as 1.143 kg—much less accurate, but still close enough for many practical uses.## See also

- International Bureau of Weights and MeasuresInternational Bureau of Weights and MeasuresThe International Bureau of Weights and Measures , is an international standards organisation, one of three such organisations established to maintain the International System of Units under the terms of the Metre Convention...

(BIPM) - International Committee for Weights and MeasuresInternational Committee for Weights and MeasuresThe Interglobal Committee for Weights and Measures is the English name of the Comité international des poids et mesures . It consists of eighteen persons from Member States of the Metre Convention...

(CIPM) - International Electrotechnical CommissionInternational Electrotechnical CommissionThe International Electrotechnical Commission is a non-profit, non-governmental international standards organization that prepares and publishes International Standards for all electrical, electronic and related technologies – collectively known as "electrotechnology"...

(IEC) - International Organization for StandardizationInternational Organization for StandardizationThe International Organization for Standardization , widely known as ISO, is an international standard-setting body composed of representatives from various national standards organizations. Founded on February 23, 1947, the organization promulgates worldwide proprietary, industrial and commercial...

(ISO) - International System of UnitsInternational System of UnitsThe International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

(SI) - Long and short scalesLong and short scalesThe long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale...
- Mass fractionMass fraction
- Mole fraction

- Names of large numbersNames of large numbersThis 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 numbers which are found in many English dictionaries and thus have a special claim to being "real words"...
- National Institute of Standards and TechnologyNational Institute of Standards and TechnologyThe National Institute of Standards and Technology , known between 1901 and 1988 as the National Bureau of Standards , is a measurement standards laboratory, otherwise known as a National Metrological Institute , which is a non-regulatory agency of the United States Department of Commerce...

(NIST) - PercentagePercentageIn mathematics, a percentage is a way of expressing a number as a fraction of 100 . It is often denoted using the percent sign, “%”, or the abbreviation “pct”. For example, 45% is equal to 45/100, or 0.45.Percentages are used to express how large/small one quantity is, relative to another quantity...

(%) - PermillePermilleA per mil or per mille is a tenth of a percent or one part per thousand. It is written with the sign ‰ , which looks like a percent sign with an extra zero at the end...

(‰) - PermyriadBasis pointA basis point is a unit equal to 1/100 of a percentage point or one part per ten thousand...

(‱) (Basis point) - SI prefixSI prefixThe International System of Units specifies a set of unit prefixes known as SI prefixes or metric prefixes. An SI prefix is a name that precedes a basic unit of measure to indicate a decadic multiple or fraction of the unit. Each prefix has a unique symbol that is prepended to the unit symbol...

es

## External links

- National Institute of Standards and Technology (NIST): Home page
- International Bureau of Weights and Measures (BIPM): Home page
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